ST/SOA/Series A/42

Manuals on methods of estimating population

MANUALIV

Methods of EstimatingBasic Demographic Measuresfrom Incomplete Data

UNITED NATIONS

Department of Economic and Social Affairs

POPULATION STUDIES, No. 42

Manuals on methods of estimating population

MANUAL IV

Methods of EstimatingBasic Demographic Measuresfrom Incomplete Data

UNITED NATIONSNew York, 1967

NOTE

Symbols of United Nations documents are composed of capital letters combinedwith figures. Mention of such a symbol indicates a reference to a United Nationsdocument.

ST/SOA/Series A/42

UNITED NATIONS PUBLICATION

Sales No.: 67. XIII. 2

Price: $U. S. 2.00(or equivalent in other currencies)

FOREWORD

The United Nations manuals on demographic methodology, as a means ofdisseminating international experience, have a history extending over more than adecade.As a firststage,the Population Divisionof the United Nations prepared a seriesof manuals on methods of derivingpopulation estimates.Manual I dealt with the meth­ods of estimating total population for current dates. Manual II described the proce­dures for appraising the quality of basic demographic data. Manual III presentedmethods of calculating future population estimates by sex and age.

While those manuals have found widespread use, the Population Commissionexpressed the need for additional technical manuals dealing with methods of esti­mating fundamental demographic variables, particularly in countries where thenecessary statistics are incomplete. A technical study was prepared entitled TheConcept of Stable Population-Application to the Study of Populations of Countrieswith Incomplete Demographic Statistics.t As a further step in assisting technicians indevelopingcountries, the Population Commission recommended at its twelfth sessionthat "at the earliest possible date a manual should be prepared on methods of esti­mating fundamental demographic measures from incomplete data 2 ". Owing to thelong-standing interest and international experience of the Office of PopulationResearch, Princeton University, in the development and application of methods ofestimating fertility and mortality from defectivedata, that office was asked to under­take the preparation of the Manual for the United Nations. The present study, whichis the fourth in this series of manuals, is the result.

This Manual was written by Professor Ansley J. Coale, Director, Office of Popu­lation Research, Princeton University, and Professor Paul Demeny of the same office,with valuable assistance from Mr. R. D. Esten, Mrs. Erna Harm, and Mr. S. B.Mukherjee. In recognizing the important contribution of the two authors, the UnitedNations also wishes to mention the help rendered by Princeton University, whosefacilities, including those of the electronic computer, were utilized in the work. Thetables in annexes I and II are taken from Regional Model Life Tables and StablePopulations, (Princeton University Press, copyright 1966). The Office of PopulationResearch has also provided the graphs included in the Manual.

".1 Population Studies, No. 39, United Nations publication, Sales No.: 6S.XIII.3.2 Official Records of the Economic and Social Council, thirty-fifth Session, Supplement No.2,

para. 48.

iii

Chapter

INTRODUCTION .

CONTENTS

Pages

1

Part One. Methods ofestimation

I. METHODS OF ESTIMATION BASED ON RECORDS OF POPULATION GROWTH ANDDISTRIBUTION BY AGE . . . . . . . . . . . . . . . . . • • . . •. 7

A. Estimation of mortality from census survival rates and the consequentestimation of birth and death rates . . . . . . . . . . . . . .. 71. Model life tables. . . . . . . . . . . . . . . . . . . . .. 72. Selection of a model life table consistent with census survival rates 8

B. Estimation of fertility and mortality by stable population analysis whenfertility and mortality have been constant . . . . . . . . . . . 12

1. Model stable populations . . . . . . . . . . . . . . . . .. 142. Selecting a model stable population on the basis of an accurately

recorded age distribution . . . . . . . . . . . . 153. Characteristic forms of age-mis-statement. . . . . 17

(a) Female age distributions with large distortions . 19(b) Female age distributions with smaller distortions 21(c) Distortions in male and female age distributions 21

4. Selecting a model stable population on the basis of a distorted agedistribution . . . . . . . . . . . . . . . . . . . . . . . . 22

5. Assigning the characteristics of a model stable population . . . . 23C. Adjustment of estimates based on model stable populations for the

effectsof recent decreases in mortality . . . . . . . . . . . . . . 25D. Concluding remarks on estimates adjusted for the effects of recent

declines in mortality. . . . . . . . . . . . . . . . . . . . . . 28E. The estimation of fertility from the age distribution recorded in one

census. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

II. METHODS OF ESTIMATION BASEDON RESPONSESTO QUESTIONS ABOUT FERTILITYAND MORTALITY • . . . . . . . . . . . . . . . . • . . . . . .. 31

A. Estimation of fertility from reports on childbearing in the past . . . . 31B. Estimates of mortality based on proportions surviving among chil­

dren ever born . . . . . . . . . . . . . . . . . . . . . . . . 34

III. ESTIMATES OF FERTILITY AND MORTALITY BASED ON REPORTED AGE DISTRI­BUTIONS AND REPORTED CHILD SURVIVAL. . . • . . • . . . . . . . . 37

A. Estimation of birth and death rates from childhood survival rates and asingle enumeration by reverse projection . . . . . . . . . . . . . 37

B. Estimation of birth and death rates from child survival rates and asingle enumeration by model stable populations. . . . . . . . . . 38

C. Estimation of birth and death rates from child survival rates and agedistribution in a population enumerated several times . . . . . . . 391. Estimation of birth and death rates in a non-stable population . . 392. Estimation of birth and death rates in a stable population enumer­

ated more than once . . . . . . . . . . . . . . . . . . . . 39D. Adjustment of estimates of fertility derived from child survival rates

and the age distribution when mortality has been declining 40

IV. ACCURACY OF ESTIMATION • . . . • • . . . . • . . . . . • . . . • 41

A. Differencesbetween assumed and actual conditions . . . . . . . . 411. Errors arising from differences between the actual age pattern of

mortality and that embodied in the model life tables . . . . . . 41

v

t,

Chapter Pages

(a) Effects of assumed age patterns of mortality on estimates derivedfrom census survival rates . . . . . . . . . . . . . . . . 41

(b) Effects of assumed age patterns of mortality on estimates derivedfrom stable populations chosen on the basis of C (x) and r. . . 42

(c) Effects of assumed age patterns of mortality on estimates derivedfrom reported child survival in combination with records of theage distribution originating from one census, or from two ormore censuses . . . . . . . . . . . . . . . . . . . . . 44

2. Errors caused by non-stability of a population assumed to be stable 46B. Errors caused by faulty data . . . . . . . . . . . . . . . . . . 48

1. Differential rates of omission in consecutive censuses . . . . . . 482. Age-misreporting in censuses or surveys. . . . . . . . . . . . 49

(a) Age-mis-statement and mortality estimation by census survival 49(b) Age-mis-statement and stable population analysis . . . . . . 50(c) Age-mis-statement and the estimation of fertility and mortality

from special questions on past experience . . 50C. Suggestions for best estimation . . . . . . . . . . 51

V. DATA USEFUL FOR ESTIMATES OFFERTILITY AND MORTALITY 53

A. Data on age . . . . . . . . . . . 53B. Data on children ever born 53C. Data on the age structure of fertility 54

Part Two. Examples ofEstimation

VI. EXAMPLES OF ESTIMATES BASED ON RECORDS OF POPULATION GROWTH ANDDISTRIBUTION l3Y AGE . . . . . . . . . . . . . . . . . . . .. 57

A. Estimation of mortality and of the birth rate from census survival rates 57B. Estimation of fertility and mortality by stable population analysis 61

1. England and Wales, 1871 . . . . . . . . . . . . . . . . . . 612. India, 1911 . . . . . . . . . . . . . . . . . . . . . . . . 653. Brazil, 1950. . . . . . . . . . . . . . . . . . . . . . . . 67

C. Estimation of fertility and mortality by stable population analysis whenthe population is quasi-stable. 681. India, 1911 . . . . . . . . . . . . . . . . . . . . . . . . 682. Mexico, 1960 . . . . . . . . . . . . . . . . . . . . . . . 70

VII. EXAMPLES OF ESTIMATES BASED ON QUESTIONS ABOUT FERTILITY AND MOR-TALITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A. Estimation of fertility from reports on childbearing in censuses orsurveys . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

B. Estimation of mortality from reported numbers of children ever born,and children surviving . . . . . . . . . . . . . . . . . . . . . 74

VIII. EXAMPLES OF ESTIMATION BASED ON CHILD SURVIVAL AND AGE DISTRIBUTIONS 76

A. Estimation of fertility and mortality from data in a single census thatrecords the age composition of the population, the number of childrenever born and the number surviving. . . . . . . . . . . . . . . 76

B. Estimation of fertility and mortality from data on age distribution, theintercensal rate ofincrease, and survival rates of children ever born 77

.Annex Part Three. A1UIexes

I. Model life tables 81

II. Model stable populations . 95

III. Tables for adjusting stable estimates for the effects of declining mortality 119

IV. Tables for estimating cumulated fertility from age-specific fertility rates 124

V. Tables for estimating mortality from child survivorship rates 125

VI. Anote on interpolation . . .. 126

vi

INTRODUCTION

There is a serious gap between the quantitative infor­mation about populations that is essential for manypurposes and the amount and quality of data actuallyavailable. Among the kinds of fundamental informationneeded about populations are age and sex compositionand fertility and mortality rates. Data of this sort areneeded in assessing problems and formulating plans forhealth, education, employment, social services and manyother critical functions of government and private organi­zations. In countries with fully developed statisticalsystems data on the state of the population are obtainedfrom censuses at decennial intervals or less, often supple­mented by large-scale sample surveys; and informationabout births, deaths, marriages and divorces is usuallyobtained from the continuous registration of these eventsas they occur. When a national system of data collectionhas been operating for many years in a country wherealmost all persons of school age and beyond are literate,the demographic information needed for administrativepurposes and for social and economic planning is usuallyadequate; but in less developed countries the registrationof vital events is often nearly non-existent or at best onlyfractional in its coverage, and information about thepopulation contained in censusesor demographic surveysis often unavoidably deficient particularly with regardto the reported ages of the population.

The disadvantages of not possessing adequate demo­graphic data in developing countries are becoming evermore acute. Governments in these countries are expand­ing their efforts to promote social and economic develop­ment, but they cannot formulate practical plans andprogrammes without data on the current size and compo­sition of the population, and on the rate of increase ofthe total population and of various important subgroups,such as children of school age and persons in the ages ofprincipal economic activity. Specifically, planning agenciesare finding some form of population projections analmost indispensable tool for their work. Moreoverthe importance of population data is accentuated becauseof the rapid changes in population that are taking place.The precipitous fall in the death rate that many lessdeveloped countries have experienced, combined with acontinued high level of fertility, has produced a rapid andaccelerating increase in numbers. One consequence hasbeen that in many countries each census has tended toprovide a considerable surprise in revealing the unexpec­tedly great increase in the population since the precedingcensus. In other words, the assumption that populationremains more or less as it was in the most recent censusis no 'Ionger serviceable. Another result of the rapidacceleration of population growth in some developingcountries has been the formulation of policies intendedto affect the rate of increase itself. Obviously such policiescan be designed and implemented intelligently only on

1

the basis of reasonably complete and accurate currentstatistics.

The only satisfactory method of recording the demo­graphic data needed in the administration of a moderncountry is a sequence of carefully designed and accuratecensuses, supplemented by frequent sample surveys tocollect social and economic data required at short inter­vals, and by the prompt and complete registration ofbirths, deaths, marriages, and dissolutions of marriage.Individual records are needed for many purposes-bythe courts in adjudicating inheritance, by administrators ofhealth programmes, by individuals in establishing proofof citizenship, age or nativity etc. But the establishmentof complete registration of vital statistics requires anextensive administrative apparatus and a thorough re­education of the public, and, even if given high priority,cannot be achieved in less than one or two decades inthe less developed countries. In the meantime, it isessential to make use of data already available to deter­mine at least approximately the principal demographiccharacteristics.

In some countries there has been an effort to establishregistration of births and deaths in a sample list of villagesand urban areas, and in others attempts to collect vitalstatistics by periodic surveys.Such methods of continuingmeasurement are a promising interim substitute forcomplete registration while the latter is being designedandinstituted. But there is still a third source of estimates ofpopulation parameters: data on age and on growthusually contained in or derivable from population censusesor broad-purpose surveys, plus information on recentfertility and mortality that is sometimes included. Thesubject of this Manual is estimation from this last kindof information.

The original plans for this Manual, when work on the firstdraft began, were for a non-technical, do-it-yourself setof instructions for methods of estimation that have alreadybeen widelyemployed and discussed,but are not accessibleto the majority of demographers and statisticians becausedetailed instructions for their use have not been published.The original purpose was to show in a simple non­technical manner how to use stable population analysis,and how to modify this analysis when mortality hadrecently been declining rather than remaining constant.There was already a substantial body of published materialon stable populations, a United Nations manual expound­ing stable population theory and a recent book tabulatingmodel stable populations; but this material containsrather complicated mathematical discussion, and is notdesigned to guide the estimator to the most efficientmethods of extracting information from data of variouskinds. There was therefore an apparent need for amanual that would make it possible for a demographer-

statistician with only a moderate levelof training, perhapsworking in isolation in a provincial capital of a lessdeveloped country, to derive the maximum of reliableinformation from data in a census or demographic survey.

The process of writing the manual, however, disclosedthe advisability of extending its scope to include anumber of different methods of analysis and estimation,and since some of these were developed after the workhad started, and are therefore not described anywhereelse, it was essentialto provide a fuller and more technicalexposition at some points than was originally intended.It is still hoped that all the methods can be employed bypersons with no more than a one-year special course indemography. To minimize the difficulty of actual esti­mation, there are assembled in part two a set of examplesin which actual calculations of the most important formsof estimation are fully worked out. These examplescontain references to the discussion in part one. A personinterested in obtaining figures for a specific populationcan follow the procedures in the appropriate sections ofpart two, and read perhaps only a few pertinent pagesfrom part one. But there are some wholly or partly newpoints in part one-the use of census survival ratios to

choose a model life table, the analysis of typical patternsof age misreporting in different populations in under­developed countries, the estimation of total fertility fromthe average parity of women 20 to 24 and 25 to 29, andthe discussion of the susceptibility of the various methodsto different sorts of error-that it is hoped will be worth­while reading for professional demographers.

There is one respect, at least, in which the examplesfall short of the procedure that we would recommendfor the construction of optimum estimates. A thorough­going examination of primary sources, of the nature ofthe individual censusesand surveys (wording of questions,evidence of completeness of coverage, and the like)should be part of the effort to arrive at the best possibleapproximation of population parameters. Since thepurpose of the examples is to illustrate the mechanics ofestimation, data have been taken from the DemographicYearbook of the United Nations, except where otherwisenoted, without examination of census procedures, inter­view schedules, and the like.

The following table is provided to help the estimator tofind the examples and the discussion relevant to hispurposes.

GUIDE TO USE OF MANUAL: METHODS OF ESTIMATION SUITABLE FOil VAIlIOUS KINDS OF DATA

I Two or more censuses, withage distributions

MetluHl01est/_tlolt Paramete,.est/_ed

Selection of model life table 0eo, 0es, b, dduplicating census survival bypopulation projection

DescriptloltolmetluHl

Chapter IA

Discussiolt Examples0/ pm:islolt

Chapters Chapter VIAIVA,IVB

2 Two or more censuses with Selection of model stable popu-age distributions, plus evidence lation from ogive of ageof constant fertility and distribution and intercensalmortality rate of increase

3 Proportions married by age, Use of "standard" age scheduleevidence of small proportion of of marital fertilitybirths to non-married womenand little use of birth control

4 Average number of children Use of regression equationever born to women 20-24 and25-29 (P2 and Pa)

°eo, °es, b, d, andadjusted agedistribution

m-themeanage of the ferti­lity schedule

Chapter IB

Chapter IB

Chapter IB

ChaptersIVA,IVB

Chapter VIB

Chapter VIB

5 Data listed in (2), plus thoselisted (3) or (4)

Determination of ORR in model Gross reproduc-stable population tion and total

fertility

Chapter IB Chapter VIB

6 Two or more censuses with age Selection of model stable popu- b, d, °eo, GRR,distributions, and evidence of lation, estimation of rate and TFconstant fertility and declining duration of mortality decline,mortality (accelerating popu- adjustment of parameters inlation growth, or changing age model stable populationcomposition of deaths)

7 Single census with age distribu- Selection of model stable popu- b, GRR, TF,don, rough guess of r or 0eo lation

8 Children ever born and births Adjustment of reported fertility GRR, TF, blast year, by age of woman rates to match reported parity

of younger women

2

Chapter IC

Chapter ID

Chapter IIA

ChapterIVB

ChapterID

ChapterIVB

Chapter VIC

Chapter VIlA

GUIDE TO USE OF MANUAL (continued)

NaturtJ 01data Method 01estll1UJtlon PIR_"" Description Dlscussloll ExomplesestlmatM ofmethod 01prtlClslon

9 Children ever born tabulated Regression ofTF/Pa on Pa/P2 TF Chapter IIA Chapterby age of woman (Pl, P2, ... , IIA

P7)

10 Children ever born, and Selection of adjustment factors lqO, 2qO, aqo, liqo, Chapter liB Chapters, Chapter VIIBsurviving children, by age of converting proportion dead loqO, 15qO, 20QO IIB,IVBwoman among children ever born to

nQO

11 Age distribution, children ever Reverse projection based on b, d, r, °eo ChapterIHA Chapterborn and children surviving, model life table selected from IVAby age of woman estimated /2

12 Same as (11), plus evidence of Selection of model stable popu- b, d, r, 0eo Chapter nIB Chapters Chapterconstant fertility and mortality lation from ogive of age adjusted age IVA,IVB VIllA

distribution and estimated /2 distribution

13 Two censuses with age distribu- Life table up to 5 from child b, d, r, Deli, 0eo Chapter IIIC Chaptertion, children ever born and survival, above 5 by matching IVAsurviving children, by age of census survivalwoman

14 Same as (13), plus evidence of Fertility estimated from model b, d, r, GRR TF Chapter IllC Chapters Chapterconstant fertility and mortality stable population matching IVA,IVB VIllB

C(x) and /2, death rate as b-r

15 Same as (14) Mortality under age 5 estimated 0eo, 0eli Chapter IllC Chapters Chapterfrom child survival, above IVA,IVB VIllBage 5 from C(x) and r

16 Two or more censuses with age Same as (14), adjusted for effect b, d, GRR, TFdistributions, children ever born of mortality declineand surviving children, by ageof woman, plus evidence ofcenstant fertility and decliningmortality

3

Chapter IllD

Part One

METHODS OF ESTIMATION

Chapter I

METHODS OF ESTIMATION BASED ON RECORDS OF POPULATIONGROWm AND DISTRIBUTION BY AGE

There are many populations that have no usable directrecords of births or deaths, but have been enumeratedin one or more censuses in which the age and sex of eachperson was recorded. In this chapter methods of esti­mating fertility and mortality from such census data aredescribed. The underlying rationale of such methods isthat the growth and age structure of a population aredetermined by the mortality, fertility, and externalmigration to which it has been subject, and consequentlythe possibility exists of estimating those forces fromrecords of the evolving size and structure of the popu­lation.

The description of estimation in this chapter proceedsfrom methods applicable to any closed population (orto any for which records of gains and losses by migrationexist) to methods applicable to closed populations withspecial histories-of essentially unchanging fertility andmortality, and of unchanging fertility combined withdeclining mortality.

A. ESTIMATION OF MORTALITY FROM CENSUS SURVIVAL

RATES AND THE CONSEQUENT ESTIMATION OF BIRTH ANDDEATH RATES

Suppose that a closed population is enumerated in twocensuses at an interval of exactly ten years, and that eachcensus contains tabulations of males and females by age,in five-year intervals. Each cohort enumerated in thefirst census is counted again ten years later, and it is asimple matter to calculate the apparent fraction of eachcohort surviving the decade. Thus the ratio of persons20 to 24 in the later census to those 10 to 14 in the earlieris equivalent to SL20!sLIO in a life table representingthe mortality risks of the intercensal decade. A sequenceof life table values can be based on the sequence ofcalculated census survival ratios, and by well-testedactuarial procedures, a life table can be constructed forages above five-provided that the two censuses achievedaccurate coverage of the population, and that ages wereaccurately recorded. However, this procedure does notyield estimates of survival rates in infancy and childhoodunless the number of births during the intercensal decadehas been recorded. Therefore, when, as is usual, adequaterecords of births are lacking, a complete life table can bebased on census survival rates only by estimating infantand child mortality indirectly-for example, by assuminga typical relationship between mortality rates under agefive and rates for persons over five.

7

There is a substantial literature on the construction oflife tables from census survival rates, and the method hasbeen applied to data from India, Egypt, Brazil and othercountries in Latin America. 1 The demographers andactuaries using census survival rates have been forcedto adjust the original census age distributions beforecalculating survival, or to adjust the rates after calcu­lation, because of the usual effects of age-misreportingcombined with differential omission by age. Unadjustedsurvival rates that are over one or that are absurdlylow are common. The methods of adjusting the agedistributions (or the raw survival rates) to remove theeffects of age-mis-reporting are essentially arbitrary,and when the reported age distributions are seriouslydistorted, the age pattern of mortality embodied in theestimated life table contains a strong component of thesmoothing procedure used as well as of the actual ageschedule of mortality.

Once a life table-however approximate it may be inform above age five, and in level of childhood mortality­has been constructed from census survival rates, it canbe combined with additional data from the two censusesto provide estimates of birth and death rates during theintercensal interval. A good approximation to the deathrate (on the assumption that the censuses are accurateand the life table valid) can be obtained by calculatingthe average of the two age distributions and applyingthe me values from the life table. The birth rate can thenbe estimated by adding the average annual rate of increaseto the estimated death rate.

1. Model life tables

The two problematic aspects of constructing a lifetable from survival rates are the estimation of infant andchild mortality, and the determination of the age pattern

1 For general methodological discussion, practical applicationand for further references to earlier writings, see Clyde V. Kiser,"The Demographic Position of Egypt", The Milbank MemorialFund Quarteriy, vol. XXII, No.4 (October 1944), pp. 383 to 408;Giorgio Mortara, Methods of Using Census Statistics for theCalculation of Life Tables and Other Demographic Measures (withApplication to the PopulationofBrazil), see United Nations publica­tion, Sales No.: 50.XIII.3; Kingsley Davis, The Population ofIndia and Pakistan (princeton, Princeton University Press, 1951)pages. 238 to 242; Hugh H. Wolfenden, Population Statistics andTheir Compilation (University of Chicago Press, 1954), pp. 115to 117, and Jorge Somoza, "Trends of Mortality and Expectationof Life in Latin America", The Milbank Memorial Fund Quarterly,vol. XLIII, No.4 (October 1965), part. 2, pp. 219 to 233.

of mortality above childhood from data distorted byage-misreporting. A convenient solution to these diffi­culties is provided by model life tables once suitablemodels have been calculated and published. An abridgedset of such tables is reproduced in annex 1. The tables arebrieflydescribed in this section, and their use with censussurvival rates outlined in the following section.

It has often been observed that the mortality risksexperienced by different age-and-sex-defined segmentsof a population are interrelated: i.e., if death rates arerelatively high among (for example) middle-aged womenin a given population, the normal expectation is thatinfant mortality is also relatively high. There is a greatdeal of statistical evidencein support of this commonsensere1ationship-a relationship expressing the fact that whenhealth conditions are especially good or especially poorfor one group in the population, conditions tend to begood or poor for other groups as well.

The result of a tendency for death rates experienced bydifferent groups to be uniquely related would be thatdeath rates for all age-sex groups but one could beprecisely estimated from knowledge of the mortality ofthat one group. It would then be possible to constructa set of life tables that stated the proportions survivingfrom birth to each age under mortality.conditions rangingfrom the highest to the lowest death rates observed inhuman populations, and a life table appropriate to agivenpopulation could then be chosen (with interpolation,if necessary) from this set of model tables.

Of course the mortality experienced by different popu­lations is not in fact so perfectly uniform. Although thereis a strong general tendency for relatively high rates tooccur among all segments of a population if they occurin any, there are populations with especially high (orlow) rates at certain ages, for one or both sexes. Variousapproaches have been tried in efforts to express in analy­tical or tabular form the variety of frequently observedsex and age patterns of mortality. 2 The most widely usedmodel tables are those previously published by the UnitedNations, and these might have been (but were not)employed in this Manual; instead, a set of model lifetables rather closelyresemblingthe earlier United Nationstables were employed-a set based on a large body ofaccurately recorded national mortality experience con­forming closely to a single pattern of death rates by age.This group of model tables is one of four calculated atthe Office of Population Research, Princeton University.3

The selection of this set of model tables-extracts froma more extended set appear in annex I-for use in thisManual was based primarily on convenience. The modellife tables reproduced in annex I are accompanied inannex II by a set of model stable populations that includea wide range of useful parameters, and it is the existenceof these auxiliary tables, already calculated on an elec­tronic computer, that dictated the use of this set of model

2 see Age and Sex Patterns of Mortality, Model Life Tablesfor .Uuder-Developed Countries (United Nations publication,Sales No.: 55.XIII.9); Methods for Population Projections bySex and Age, Manual III United Nations publication, Sales No.:56.XIII.3) ;and other tabulations cited in the work listedin foot-note 3.

3 A.J. Coale and Paul Demeny, Regional Model Life Tablesand Stable Populations (Princeton, Princeton University Press, 1966).

life tables rather than the earlier United Nations tables.In most instances estimates based on these model lifetables are little different from those that would be obtainedfrom the earlier United Nations life tables; however,the absence of associated model stable populations meansthat estimation from the earlier United Nations life tableswould be much more laborious.

This is not the place for an extended description ofalternative possible forms of model life tables, nor even ofthe four families of which one was chosen for use in thisManual. Three of the sets summarize mortality patternscharacteristic of regions of Europe, and the fourth­the one partially reproduced in annex I, the so-called" West" family-expresses an age pattern of mortalitycommon to twenty-one countries (Australia, Canada,Israel, Japan, New Zealand, South Africa, Taiwan, theUnited States and thirteen in western Europe). The agespecific mortality rates in this set of model tables arematched quite closely at the appropriate mortality levelby the published life tables of these twenty-one ratherdiverse populations. The 125 life tables for each sexfrom which these model tables were calculated wereselected because they showed no systematic tendency todeviate from a preliminary set of model tables designedto express median recorded world experience. In contrast,each of the other families of model tables was based onregional patterns of consistent and persistent deviationfrom average world age patterns of mortality. Forexample, one regional set is based on life tables fromcontiguous populations in central Europe with a persistenttendency towards unusually high infant mortality ateach level of adult mortality.

The model lifetables in annex I can be logicallyemployedto construct an approximate schedule of mortality fora population with an unknown age pattern of mortalityprovided there is no specific evidenceof an unusual pattern.Of course the absence of specific evidence of an unusualmortality pattern does not imply that the" usual" patternin fact prevails. It is highly probable that, if accuraterecords of mortality existed for all populations, patternsof deviation would be found different from and moreextreme than in the four regional families. Nevertheless,the existence of high intercorrelations among mortalityrates at different ages, and the existence of patterns towhich many mortality schedules closely conform providethe soundest empirical basis of estimation available atpresent.

2. Selection of a model life table consistent with censussurvival rates

If two consecutive censuses of a closed population wereperfectly accurate, and if the mortality scheduleexpressingaverage experience during the interval conformed exactlyto the model life tables in annex I, the appropriate tablecould readily be located by comparing the values ofsLx + 101sLx in the model tables with the correspondingsurvival rates calculated from the two censuses. Underthese hypothetical circumstances all of the survival rateswould fall between the values for the same pair of modeltables, and an interveningmodel table could be constructedby interpolation. These circumstances are, for instance,

8

rather closely approximated by the censuses of Koreabefore the Second World War as indicated by the com­parison of female survival rates from 1925 to 1935 inthat country with survival rates found in model lifetables shown in figure I.

In most censuses of populations without vital statistics(so that mortality must be estimated) the recorded survivalrates are highly erratic in the mortality level they indicatebecause of the effect ofage-misreporting. As an illustrationof such situations figure I also shows female survival ratescalculated from Indian and Turkish censuses 4. One way

4 Note the striking similarity in pattern in the sequence ofapparentsurvival rates for Turkey and India, doubtless reflecting a basicsimilarity in the pattern of age-mis-statement. See section B.3 below,

of estimating the level of mortality from a sequence oferratic rates based on erroneous age distribution wouldbe to determine the level implied by the survival rate ofeach cohort, and then to take some sort of average of thelevels so indicated. Specifically, the mortality levels inthe model tables indicated by the census survival ratesof persons 0-4, 5-9, .. " 40-44 in the earlier census couldbe determined, and ranked from highest to lowest indi­cated level, and the median selected as the final estimate.(It is prudent to avoid survival values for persons abovesixty or seventy because of the prevalence of systematicage-misreporting among older persons, and the possibilityof special age patterns of mortality.) In fact, as figure Iindicates, the individual survival values are often soaffected by age-misreporting that many of the survival

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.4

5 10 15 20 25 30 35 40 45 50 55 60 AGE

Figure I. Census survivorship rates of females from age x to x+5 at time t to age x+ 10 to x+ 15at time t +10, according to censuses of India, Korea and Turkey, and according to selected ..West"

model life tables

9

rates are outside the limits of the model tables (whichextend from an expectation of life at twenty to one ofseventy-five years), and little confidence could be attachedto the median of such a wildly erratic sequence. It is betterto take advantage of the dampening effect that cumulationhas on age-misreporting, and to try to determine the levelof mortality from the proportions surviving from theentire earlier population to age ten and over in the latercensus, the proportion five and over that survives toage fifteen and over, and so on. Survival rates of the lattertype (calculated from the same census data from whichthe cohort survival rates plotted in figure I were obtained)are shown in figure II. Naturally, unlike simple cohortsurvival rates, such rates cannot be directly expressedin terms of mortality levels since their value is significantly

_PORTIONSURVIVING

.9

influenced not only by mortality but also by the agedistribution of the population in question. The compu­tational procedure that permits the translation of theserates into mortality levels requires that the initial popu­lation be projected to the later date by applying thesurvival rates of model life tables at different levels ofmortality, e.g., levels 3, 5, 7 and 9. Each projection yieldsa ratio of the surviving population over ten to the initialpopulation, of the surviving population over fifteen tothe initial population over age five, etc. By comparingthe recorded survival ratios with those obtained by projec­tions with alternative model tables, one obtains a seriesof estimates of the level of mortality--a series consistentwith the recorded survival of the whole population, thepopulation five and over, ten and over, etc., rather than

.8

.7

.6

.5

.4

.3

.2

.1

o

.......................,KOREA.1925-1935

...........~

5 10 15 20 25 30 35 40 45 50 55 60 65 AGE

Figure II. Census survivorship rates of females from age x and over at time t to age x +10 and overat time t+10 according to censuses of India, Korea and Turkey

10

with the recorded survival of individual cohorts. 5 Thesequence of mortality levels obtained in this manner ismuch less affected by age-misreporting than the seriesbased on the recorded survival of individual cohorts.

The reduced effectof age-misreporting is seen in a com­parison of figure III and figure I. Determining the levelof mortality by examining the survival rates of largesegments of the age distribution minimizes the distortingeffect of age-misreporting because the survival ratesare not affected by age misreporting within the groupswhose survival is calculated. For example, the survivalratio of the population over twenty in the later censusto the population over ten in the earlier is distorted byage-mis-statements that transfer persons across twentyin the later census and across age ten in the earlier, but isunaffected by all other forms of age-mis-statement.6

This method of estimating the level of mortality pro­duces a series of alternative figures that are confined to anarrow range when age-misreporting is mild, but thatvary extensively when age-misreporting is extreme. Insection B of this chapter stable populations methods areused to show that certain populations (including manyin tropical Africa, some in northern Africa and the

a A worked-out example of this method is given in chapter VI.8 Age-misreporting can, however, affect the number of survivors

projected with a given life table, and hence influences the levelof mortality that matches the reported number surviving. Specifically,overstatement of age for persons past middle age reduces theprojected number of survivors, and leads to the selection of a modellife table with too low mortality rates, or too high an expectationof life at birth. This effect is important only in the projection ofthe population that is older than forty and over in the earliercensus.

Near East, plus India, Indonesia and Pakistan) havecensus age distributions by five-year intervals that arequite substantially distorted by age-misreporting, in apattern that has many common features. In contrast,censuses in the Philippines and Latin America have five­year age distributions that are much less distorted, andcensuses in parts of Asia, including China (Taiwan),the Republic of Korea and Thailand have five-year agedistributions that appear distorted only to a minorextent by age-misreporting.

The large distortions in the African-Indian-Indonesian­Pakistani censuses mean that the application of themethod described above of determiningthe levelof mortal­ity from census survival rates produces a series of esti­mates with a rather wide range, and with a characteristicsequence of ups and downs. The sequence is consistentwith the characteristics of age-misreporting (overestima­tion of the age of late adolescent girls and young women,for example) discerned by stable population analysis.This analysis suggests that certain survival rates areoverstated and others understated for these populations,and that the level of mortality estimated from such ratesis biased. Even when ratios with predictable biases arediscarded, the remaining censussurvivalrates may indicatelevelsof mortality that would lead to estimates of a deathrate differing by several points. The range of mortalitylevels consistent with census survival rates in LatinAmerica is typically much smaller and in censuses littledisturbed by age-misreporting-such as in Korea from1925 to 1940-the mortality levels indicated by differentsurvival ratios are confined within a narrow range. Themost satisfactory rule of thumb appears to be the selectionof the median level of mortality indicated by the propor-

21

MORTALlnr------------------:-----------...LEVEL ~.

70

II

9

65

45

40

Figure III. Mortality levels in "West" model life tables implied by female survival rates from age xand over at time 1 to age x+l0 and over at time 1+10 according to censuses of India, Korea and

Turkey

11

77"

tion surviving among the first nine groups-i.e., allpersons, persons five and over, ten and over, ... , fortyand over.

This procedure permits the selection of a model lifetable consistent with the proportions recorded as survivingfrom one census to the next. It is possible to use the sameprocedure to select a table from other families of modellife tables embodying other age patterns of mortality.It is interesting to note that experimental comparisonsof life tables chosen in this way show that the expectationof life at ages five, ten and fifteen in model tables selectedfrom the four families differs very little but that theexpectation of life at birth varies widely among the lifetables selected. Also, the death rates estimated by applyingthe selected model table to an estimated mid-decadepopulation differ widely from one family of model tablesto another, while the estimated death rates for thepopulation over age five are very nearly the same. Thereason for these similarities and differences is that thecensus survival method establishes essentially themortality of the non-child population-the populationover age five-and that therefore the life table selectedfrom any family of model tables must have the over-allnon-child mortality indicated by census survival.

Consider the mortality indicated by a comparisonof the population over ten with the whole population tenyears earlier. The difference between these populationsis precisely the deaths that occurred during the decadeto the persons alive at the time of the earlier census.At the midpoint of the decade, this population" at risk"is the population five and over, so that the survival ratefor the whole population is very closely linked to theaverage death rate for the population past age five. Allmodel life tables that give a projected population ten andover equal to that recorded at the end of the decade mustconnote about the same death rate for the population overfive. Since the expectation of life at age five is the recipro­cal of the death rate over age five in the stationary popu­lation, it is not surprising that the different model lifetables have nearly equal °es s.

In other words, the use ofcensus survival rates to choosea model life table at a fitting level of mortality permits theestimation ofthe death rate over age five with some confi­dence. However, the death rate for the whole population isstrongly affected in populations of high fertility andmoderate to high mortality by death rates amonginfants and young children. The death rate obtained byapplying the age specific mortality rates in the selectedmodel table to the estimated mid-period population isvalid only if the relation of infant and child mortalityto mortality above age five in the family of model tablesmatches the relation in the population in question. Andthis relationship is far from the same in the four familiesof model life tables." Thus census survival rates (if derivedfrom accurate enumerations not excessively distorted byage-misreporting) establish the level of mortality forpersons five and over, but leave uncertain (unless the agepattern of mortality is known) the level of child mortality,and thus the expectation of life at birth and the over-alldeath rate. Since this procedure yields an estimate of

7 For a detailed discussion of this point see chapter IV.

the birth rate by adding the intercensal rate of increaseto the estimated death rate, the uncertainty surroundingthe level of infant and child mortality affects the estimatedbirth rate precisely as it affects the estimated death rate.

B. ESTIMATION OF FERTILITY AND MORTALITY BY STABLE

POPULATION ANALYSIS WHEN FERTILITY AND MOR­

TALITY HAVE BEEN CONSTANT

Populations subject to approximately constant mortalityand fertility schedules come to have the age compositioncharacteristic of Alfred J. Lotka's stable populations.

This Manual is not the place for a summary of theextensive literature on stable populations ;" instead ittries only to show when and how useful estimates offertility and mortality can be based on stable populationtheory. The question of when estimates can be based onstable analysis is easily answered in principle: wheneverfertility has been subject to no more than low-amplitudeand short duration variations during the previous five orsix decades, and mortality has changed only slightly andand gradually during the past generation. The approxi­mate constancy of fertility is a very common, if notuniversal, feature of populations that are mainly agricul­tural, and low in .literacy and income, except whenfertility has been affected by wars, revolutions, majorepidemics or other such episodes. The absence of majortrends in mortality has also been a common characteristicof less developed areas until the past few decades whenvery rapid declines in death rates have been frequent.In this section the use of stable population techniques isdiscussed in those instances where it is clearly appropriate,namely, when there have been no major trends or fluc­tuations in fertility, and no sustained important changesin mortality.

Stable population analysis has also been applied bydemographers to populations whose mortality has beendeclining, although it has been demonstrated that theresultant estimates are biased. In section C of this chaptera method of adjustment is described for altering theestimates derived from stable analysis to compensatefor the effect of a history of recent decreases in mortality.

A stable population is generated by the continuation ofa fixed schedule of fertility and a fixed schedule of mor­tality; it is characterized by an unchanging proportionateage distribution, and a constant annual rate of increase.In populations essentially closed to migration wherethere is no evidence of spreading use of deliberate birthcontrol or of changing patterns of nuptiality, and noreason to believe that mortality is changing rapidly,confirmation of the conjecture that the population may bestable can be sought in comparisons of the recordedage distributions in successive censuses, and of successiveintercensal rates of increase. If the census age distributionsshow marked differences (e.g., such as are seen in thecensuses of Turkey from 1935 to 1960), it is probable

8 The fundamental work on the subject is Alfred J. Lotka'sTheorte des associations biologiques, deuxleme partie (Paris, Her­mann et Cie., 1939). For a comprehensive treatment see the UnitedNations Study entitled The Concept ofa Stable Population. Applica­tion to the Study of Populations of Countries with Incomplete Demo­graphic Statistics (Sales No.: 65.XIII.3).

12

that fertility has not been constant, in the cited examplebecause the series of wars and other disturbances expe­rienced by the Turkish population undoubtedly causedmajor fluctuations in fertility, and in mortality as well,at least for males. When the rate of increase rises markedly,it is likely that mortality is falling. But essentially constantage distributions and rates of increase observed in a seriesof censuses can be considered justification for consideringthe population stable.

The age distribution of a stable population is describedby a well-known formula of Lotka's:

c(a) = be-,a p(a) (1)

where c(a) is the proportion of the population at age a, bis the birth rate of the stable population, e is the base ofnatural or Naperian logarithms, r is the annual rate ofincrease, and p (a) is the proportion surviving from birthto age a according to the prevalent mortality risks.The proportion p(a) is an alternative expression for thesurvivor function (lz/lo) in the life table, and all tbat isneeded to fix the proportion at every age in a stablepopulation is the life table expressing the constant

mortality conditions and the average annual rate ofincrease. The birth rate b is determined by the fact thatthe sum of the proportions at all ages must be equal toone.

In England and Wales in 1881 the age composition ofthe female population was much the same as it had beenin the two preceding censuses, and the estimated inter­censal rate of natural increase had been nearly constant.To be sure, the population was not closed, primarilybecause of a flow of emigrants to America, but between1871 and 1881 the rate of average annual loss for femaleswas only .00070, so that the effect on the age distributionwas minor. The theory of stable populations leads usto expect, then, that the age composition of the femalepopulation of England and ~ Wales in 1881 is closelyapproximated by a synthetic age distribution calculatedaccording to formula (1), using the life table for 1871-1881to provide p(a), and equating r to the average annualintercensal rate of natural increase in that decade.Figure IV provides a comparison of this calculated agedistribution with that recorded in the censuses of 1881.and gives a demonstration by example that theoretically

cI!)'----------------------------------------.

.14

.13

.12.

.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

.01

5

•• •••~NGLAND AND WALES, FEMALES,I881.......

••• STABLE POPULATION WITH r •.01414,• BASEDON OFFICIAL LIFE TABLE FOR 1871-1881

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85AGE

Figure IV. Female age distribution in England and Wales, by five-year intervals, as recorded in the census of 1881 and as approximatedby a stable population constructed on the basis of the intercensal (1871-1881) rate of natural increase (r = .01414) and the official English

life table for the same period, both for females

13

derived stable age distributions fit actual populationsvery closely.

This example does not show the potential usefulness ofstable populations as a way of estimating demographicvariables, because when data exist to calculate an accuratelife table, as in England and Wales in the 1870s, indirectestimation is not required. Suppose, therefore, that deathregistration had been non-existent or very incomplete.Couldthe theory of stablepopulations and the near stabilityof the population of England and Walesbeusedto estimatemortality and fertility? A positive answer is providedthrough the device of model stable populations based onmodel life tables.

1. Model stable populations

As is indicated by equation (1) above, the age distri­bution of a stable population is jointly determined by themortality schedule (or life table) and the annual rate ofincrease. In the preceding section there is a descriptionof model life tables which can be constructed to embodytypical age patterns of mortality at different mortality

c(~)

.14

levels. A set of such model tables embodying what canbe considered "normal" or " typical" world mortalityexperience is reproduced in abridged form in annex I.Corresponding to each model life table is a set of possiblestable populations at rates of increase correspondingto the highest and lowest levels of fertility that mightaccompany the life table. Annex II presents such a set ofstable populations for each model life table, with rates ofincrease ranging from - .010 to .050.

This set of model stable populations includes a range ofage distributions bracketing all those likely to be found inactual populations that (a) are themselves approximatelystable (i.e., have had approximately constant fertility andmortality), and (b) experience mortality where the agepattern conforms more or less closely to that embodiedin the set of model life tables. If an actual populationappears, in the light of observed characteristics, tobelong to the family of model stable populations, it ispossible to locate the model stable population matchingthe actual one, and to estimate various demographicparameters of the actual population by attributing to itthe parameters of the model stable population.

.13

.12

.11

.10

.09

.08

.07

.06

.05

.04

.03

.02

.01

o 5 \0 15

•••• ENGLAND AND WALES, FEMALES,.188\...........

·WEST" STABLE POPULAT ION, C(20l =.4514'.'"

20 25 :30 :35 40 45 50 55 60 65 70 75 80 85AGE

Figure V. Female age distribution in England and Wales, 1881, by five-year intervals, as recorded in census and as approximated by"West" female stable population constructed with the same proportion under age twenty as recorded in 1881 and with the intercensal

(1871-1881) rate of natural increase of the female population (r = .01414)

14

In the comparison of a calculated stable age distri­bution and a recorded age distribution represented infigure IV, the life table and the rate of increase wereknown, and the stable age distribution was calculated.When stable population analysis is used as a method ofestimation, the process is partially reversed. The agedistribution is known, as is some other parameter suchas the rate of increase, and this knowledge is used tolocate the appropriate stable population among thefamily of stable populations, and the birth rate, deathrate, expectation of life at birth and other characteristicstabulated for the model population are ascribed to theactual. (The details of the method are illustrated in theexamples given in part two, especially in the examplewhere a model stable population is fitted to the femalepopulation of England and Wales enumerated in the 1871census.) Figure V shows a comparison of the 1881 agedistribution with a model stable with a growth rate equalto the estimated average yearly rate of natural increasebetween 1871 and 1881 and the same proportion underage twenty as recorded in the census. It is essential to notethat this is not the same stable population shown infigure IV. Both incorporate the same rate of increase,but one incorporates the recorded female life table of thedecade and does not involve the 1881 age distributionin any direct way, so that its virtual identity with thisdistribution can be taken as confirming the stable natureof the actual population. The other calculated stablepopulation, in contrast, is not based at all on the recordedEnglish life table, but on the recorded rate of increase,the presumed stability of the population, and the assump­tion that the age pattern of mortality conforms to thisfamily of model life tables. The model stable populationselected in this manner has an expectation of life at birthwithin .3 of a year of the recorded value, and a birth rateapparently closer to the actual figure than the registeredbirth rate, because of a slight under-registration of birthsin England and Wales during the decade in question.

2. Selecting a model stable population on the basis of anaccurately recorded age distribution

Characteristics ofan actual population can be estimatedby locating the model stable population that best fitscertain recorded or calculated features of the populationin question, and then assigning the characteristics ofthe model stable to the actual population. Relevantfeatures of the actual population that are usually recordedinclude its age distribution and the average rate of naturalincrease between two censuses estimated by adjustingthe average annual intercensal rate of increase for anyknown difference in completeness of coverage, or for anyknown minor rates of net gain or loss through migration9 •

The problem of selecting the appropriate model stablepopulation usually reduces, then, to finding a stablepopulation with a given annual rate of increase and

most closely resembling the recorded population in agecomposition.!0

How can one judge whether a stable age distribution"closely resembles" a recorded age distribution? Ifthe recorded age distribution is generated by genuinelyconstant fertility and mortality, if the coverage of thecensuses is complete, if the population is not substantiallyaffected by migration, and if ages are accurately reported,various alternative features of the recorded age distri­bution would serve to locate essentially the same modelstable population. The proportion under age ten, orfive to fourteen years of age, or over age fifty would leadto the selection of much the same stable distribution. Forexample, the population of England and Wales in 1881fits the model stable population remarkably well. Butage-misreporting in the 1881 census was limited, and theage pattern of English mortality conforms quite closelyto the so-called "West" family of model life tables.

In many of the less developed countries the age distri­bution reported in censuses or demographic surveys isaffected by gross misreporting of age. The distribution bysingle years of age is very often conspicuously distortedby "age heaping" - by a tendency for many persons toreport a preferred nearby number (one ending in zero or5, for example) rather than the correct age. Indeed itappears likely that often many more ages are misreportedthan given correctly. A reported number of persons atage sixty greater than ages sixty-one to sixty-nine com­bined is not unusual.

On a priori grounds, it is clear that the effects ofage­misreporting on the cumulative age distribution (theso-called ogive) are less than on the proportion in a par­ticular five-year interval. In fact the proportion underage x (which will be designated C (x» is affected only bythose age-mis-statements that cause a net transfer ofpersons across age x. The proportion reported as underage thirty is not altered if children under age five arereported as five, six or seven, or if persons in their latefifties or early sixties are reported as sixty years old.

One of the advantages of the cumulative age distri­bution is the simple general relation that exists amongthe ogives of model stable populations with the samerate of increase: stable populations with higher fertility(and hence higher mortality at the same rate of increase)have greater cumulative proportions to every age thanlower fertility stable populations (see figure VI). In otherwords, the ogives of stable populations with the same rateof increase do not cross, and each model stable populationis thus completely determined by knowledge of the rateof increase and C (x) for any value of x. Consequently,one stable population is identified by the intercensal rateof increase and C (5), another by the rate of increase andC (10) etc. If the reported population in fact has a stableform, and if age reporting is accurate, the series of stablepopulations identified with C(5), C(lO), ... , C(65), willexhibit little variation, as would be evident in a verytight cluster of ogives of the stable populations thus

10 If the proportion of children surviving to age two is known(from the methods described in chapter II) the problem is one of

9 If gains and losses because of migration are substantial, the selecting a model stable population with the given 12 and mostuse of stable population analysis becomes questionable. closely resembling the recorded population in age composition.

15

selected. Deviations from constant fertility (and to a lessextent from constant mortality) in the recent past wouldcause the stable populations identified with cumulativeproportions under some ages to differ from those iden­tified with the proportions under other ages. The set ofmodel stable ogivesconsistent with C(5), C(lO), ... , C(65)would then be spread out rather than tightly clustered.The same effect of diverse rather than consistent esti­mation of the appropriate model stable population wouldalso result from age-misreporting of a sort that causedlarge net transfers of persons across ages divisible by 5(cf. figureVII).

Suppose a population has been subject to approximatelystable conditions and that its cumulative age distributioniscompared with the ogivesof stable populations that areconsistent with C(5), C(lO), ... , C(40) and the intercensal

rate of increase. 11 There will be a highest and a lowestcumulative stable distribution, setting upper and lowerlimits to the choice of a model stable population. If thehighest stable ogive is accepted as approximating the trueage distribution, it follows that C (x) at all other agesfrom zero to forty is too low, and that therefore theremust have been a net transfer of persons by age overstate­ment at all ages except that where the ogive agrees withthe highest stable. Conversely, acceptance of the lowestogive implies net understatement of age across all agesdivisible by five except one.

11 It is wise to avoid comparisons at the older ages becausedifferences in age-patterns of mortality have an increasing effecton the cumulative age distribution above age forty or fifty. andbecause of the prevalence of systematic age-misreporting at olderages.

f·

.9

.8

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. 6

. 5

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o

.",.....;:::-;.~.;. ............:.~--: ....»:»: ....,. ", .-". ,' .....,,',' ..,

/'/ ..../',' ....

£•.060/',' ./'/. , .. ', " ..',,', ..'".,' ....

. i' ..'", .'/',' ....b'~OJ.·/ .,/- , .. , .'

II .'. II ..'./ .'I, :'., .

I, .:" .I, ...

il ...." .

I, .../, ...., :

I' ...., ..I, ...I' ...tI.....,.

I,.:., .It ....,.

It .:I':.,:

IIIit/./:

IF./:".:.,.It;','

l5 \0 15 20 25 30 35 40 45 50 55 60 65 70 75

AGE

Figure VI. Ogives of age distributions (proportions up to age x) in stable populations ("West"females) with a growth rate of .02 and with birth rates (b) as indicated

16

These relationships are sometimes useful in detecting"patterns" of age-mis-statement, and are so employedin the next section.

were the "West" model tables of annex II, with0eo = 40 years. 12

2. The calculation of

3. Characteristic forms of age-mis-statement

We have compared a large number of recorded agedistributions with stable ogives as a way of uncoveringtypical patterns of deviation from stable populations incertain categories of censuses or surveys. The comparisonwas twofold:

1. The calculation of C (x) - Co (x), where C (x) is thecumulative age distribution of the given (male orfemale) population, and C8(X) is the middle (median)stable population of those with ogives that agree withC(lO), C(15), ... , C(4O). The set of stables employed

c(0-4) c(40-44)...,-----,£:,(0-4) £:,(40-44)

when c (0-4) is the proportion aged 0-4 in the givenpopulation, and C8 (0-4) is the proportion aged 0 to 4in the median stable population defined above.

If the given age distribution conformed exactly to the

12 The reader may naturally suspect that the employment ofogives of alternative model stable populations with a given lifetable is very different from using ogives with a given rate of increaseIn fact, the comparisons obtained by holding 0eo constant at fortyyears are virtually indistinguishable from those that would beobtained from a fixed value of r (see figure IX.)

AGE

DIA, FEMA LES,I911

10 15 20 25 30 35 40

,,,. ,. ,.. ,... ,

.. I.. ,... ,. ,STABLE. ·WEST" FEMALES,",)'r"·00725, CuO)' .2~."" ~

.:".,/STABLE:WEST" FEMALES

I r •.00725. C(20)' .455I:?"'"

II

/I

I,,I

Ir,

5o

.6

.7

.5

.1

.2

.3

.4

Figure VII. Ogive of the age distribution ofthe female population of India as reported by the censusof 1911 and ogives of the age distributions in "West" female stable populations with a growth ratesame as the female intercensal (1901-1911) rate of increase in India (r = .00725) and with the highest

and lowest ogives consistent with values of C(5), C(lO), ... , C(40) in the census population

17

\0 15 20 25 30 35 40AGE

5

.8

C( ,INDIAC ! ,STABLE

1.3

10 15 20 25 30 35 40AGE

5

1.2 1.2

\1.\ 1.1

,\1\1

1.0 1.0\

.9 .9

ctll,IN01Ac(!)~STAILE

1.3

CtIC),IN01A-C tIC ),STAILE- -.04C(~ ),INOIA -CC !),STABLE

.04

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...... '.. .,..- ., ••• ... / <,

',',••MEDIAN•••,.··' ""-

\'. . ", -. ....\ .. ' ,\ ,~

HIGH \,..../

.02

-.02

-.04

. .. .···,.MEDIAN•••··'. .

, ". ". ".'

o 5 10I , I I I

15 20 25 30 35 40AGE

o 5 \0 15 20 25 30 35 40AGE

Figure VIII. Comparisons of the reported female age distributionby five-year age groups-<:(x)-and its ogiv~C(x)-as reportedin the 1911 census of India with corresponding values in threemodel stable populations defined by the Indian female growthrate for 1901·1911 and by agreement with C(lO). C(20) and C(15)in the census population resulting in the highest. the lowest andthe median ogive respectively among those corresponding to C(5).

C(lO)..... C(40)

Figure IX. Comparisons of the reported female age distributionby five-year age groups-<:(x)-and its ogive -C(x)-as reportedin the 1911 census of India with corresponding values in threemodel stable populations defined by an expectation of life at birthof forty years and by agreement with C(35). C(20) and C(15) inthe census population resulting in the highest. the lowest and themedian ogive respectively among those corresponding to C(5).

C(lO)..... C(40)

18

"""1'1"&'" n T 7! n

model stable, C(x)-C,(x) would be zero at each age, and

c(x-y)

c.(x-y)

would be one in each age interval. A positive value ofC(x)-C,(x) implies age understatement that shiftedpersons across age x, and a

c(x-y)

c.(x- y)

greater than one implies that age-misreporting hasinflated the reported number of persons. in the given ageinterval-both implications following if C,(x) is acceptedas a valid estimate of the true age distribution. Butfigure VIII shows that the sequential pattern of upsand downs of C(x)-C,(x) and

c(x-y)

c.(x-y)

is maintained whether the comparison stable is thehighest, the lowest or an intermediate ogive. It is evidentin figure IX that the apparently arbitrary choice of stableogives with °eo = 40 years has no important influenceon the sequence.

The purpose of the comparisons is to bring to lightcommon patterns of deviation from the stable form whenapproximate conformity to a model stable populationmight be expected. Over 150 censuses or surveys ofpopulations of each sex thought to have a history ofapproximately constant fertility were analyzed in this way.

The analysis reveals the existence of certain generalpatterns of deviation from approximately appropriatemodel stable populations-patterns in each instanceshared by censuses and surveys of several different popu­lations. One pattern is clearly the product primarilyof age-misreporting, and others appear to result in largepart from past variations in fertility and mortality-theresult of departures from the prerequisites of a genuinelystable population, in other words.

Censuses and surveyswith extreme age heaping evidentin the single-year age distributions could be expectedto have ogives and distributions by five-year intervals thatdo not conform closely to model stable populations. Butin fact some censuses and surveys obviously subject topoor age-reporting (by single years) deviate only slightly,and some quite strongly from the expected form ofogives and five-year age distributions. Figure X (upperpanel shows

c(x- y)

c.(x- y)

for females in the Philippines (1960), Colombia (1951),Venezuela (1950) and Ecuador (1950) on the one hand,and in India (1911), Morocco (1960), Ghana (1960) andIndonesia (1961) on the other. All are populations inwhich age heaping is extensive. The lower panel of thefigure compares C(x)-C,(x) for the same censuses. Themost conspicuous contrast in the upper panel of figure Xis between the large deviations from the stable, with two ormore consecutive age intervals deviating in the same sense

in one set of censuses, and the more modest deviations,usually with a saw-tooth quality (positive deviationsfollowed by negative ones) in the other set of censuses.This contrast is manifested in a U-shaped sequence oflarge differences between the ogive of the reported distri­bution and the model stable in one group of censuses,and a more or less alternating pattern of small differencesin the other group. The implication of these contrastingpatterns is obvious: in the first group of censuses, thereis a systematic form of unidirectional age-misreportingover a broad range that distorts the reported age distri­butions even as an ogive; in the second group, althoughage-misreporting causes pronounced age heaping by singleyears, the distribution by five-year intervals tends to alter­nate excessesand deficits, and the cumulative distributionis not much distorted.

(a) Female age distributions with large distortions

An examination of twenty-nine female age distributionsof the sort affected by large-scale misreporting shows thefollowing commo~ characteristics in the pattern of thecumulative age distribution, as revealed by C(x)-C,(x):

1. The cumulative age distribution rises (relative to thestable) from age 5 to 10;13

2. It falls from age 10 to 1513and from 15to 20; 143. It rises from 25 to 30,13and from 30 to 35.15

The proportion in five-year intervals shows the followingcharacteristics, relative to the stable:

1. The proportion 5-9 is above the stable; 132. The proportions 10_1413 and 15-19 14 are below the

stable;3. The proportions 25-2913 and 30-3415 are above the

stable.

In other words, the age distributions have a surplus at5-9, and a deficit in the adolescent age intervals (10-14and 15-19) followed by a surplus in the central ages ofchild-bearing (25-34). Censuses and surveys in all of thecountries of tropical Africa, in India, Indonesia, Morocco,and Pakistan show this pattern. It is repeated in all of thecensuses of India before partition (except the census of1931, where the published age distribution was smoothed)as well as in both Pakistan and India in 1961. The under­lying pattern of age-misreporting can be detected in thecensuses of most Near Eastern and North Africancountries. It is not evident in Latin American countries,Ceylon, Taiwan (China), Malaya, the Philippines, theRepublic of Korea, or Thailand.

Why should African censuses and surveys show thesame kinds of misreporting of age as censuses of India,Indonesia and Pakistan ? A plausible explanation of thesimilar patterns is that in these enumerations the ageentered on the interview schedule was often an estimatemade by the interviewer, rather than the transcription of anumber supplied by the respondent. In other words thecommon form of distorted age distributions is caused bythe common biases in the estimation of women's ages

13 No exception in twenty-nine censuses or surveys.1. Only one exception in twenty-nine cases.15 Two exceptions in twenty-nine cases.

19

by another person. Unfortunately, this explanation of thesimilar patterns of distorted age distributions does not byitself indicate what the true age distribution is. It suggeststhat the age distributions affectedare distorted in a similarway, but to determine where there is net overestimationof age and where net underestimation it is necessary tofind by some other means which of the alternative modelstable populations does in fact resemble the actual agedistributions.

There are a few instances where there is an independent

basis for selecting the appropriate stable population, andby considering these it is possible to obtain an insight intothe typical age-mis-statements prevailing in theseinstances-and presumably in other censuses and surveyswith the same pattern of distortion in the age distribution.In the sample census of the Congo of 1955-1957, the agesof all but a minority of young children were verified bythe interviewer through the examination of birth certi­ficates or of entries in the mother's identity booklets,and the stable population consistent with the proportion

_ COLOMBIA,I951___ ECUA DOR,I950

...... PHILIPPINES,I960

._••• VENEZUELA,I950

.8

.7

.6

'( I I I I I I I0 5 10 15 20 25 30 35 40

AGE

C~l-C.(!l.06

.04

.02

0

_ COLOMBIA,I951___ ECUADOR"I950

...... PHILIPPINES,I960'.'_' VENEZUELA,1950

'C I I I I I I I ,0 5 10· 15 20 25 30 3!l 40

AGE,-~

1.2

~1.3 ~\ll

1.2 .-.~

1.1

1.0

;;

.9 i,;

.8 I •'" l ! _ INDIA,1911'. ~ ,1: . ___ INDONESIA,1961, I , ....h. GHANA,I960

~' ..f I , ._._. MOROCCO,1960

\ i.6 \J

'( I I I I I I I I

0 5 10 15 eo 25 30 3!l 40AGE

Cll!l-C.ll!l.06

1\i \

.04 i \,; 1\ \, \

.02 l·"\\ \"' ~\ \I ••~.. \

0\

-D2

-.04

'C I I I I I I I I0 5 10 15 20 25 30 35 40

AGE

Figure X. Comparisons of the female age distribution by five-year age groups-c(x)-and itsogive-C(x)-as reported in various censuses. with corresponding values in stable populationsmedian among those defined by an 0eoof forty years and by agreement with C(5). C(IO)..... C(40)in the census populations. The comparisons illustrate two typical patterns of age-misreporting;the" African-South Asian" pattern (left-hand diagrams) and the" Latin American" pattern (right-

hand diagrams)

20

...

of children can be accepted as a valid fit. In several otherAfrican territories fertility and mortality could be esti­mated on the basis of retrospective data on children everborn, children surviving, and births and deaths in theyear before the census, and a stable population chosento conform to these fertility and mortality estimates.The stable populations selected in this way in someinstances agreed with the reported cumulated distributionup to age ten, and in other instances had a lower propor­tion under age ten but a larger proportion under age 15than the census or survey. If these examples are represen­tative, the distorted age distributions of Africa, India,Indonesia and Pakistan are the result of the followingtypical errors in estimating the ages of females:

1. A tendencyto overestimatethe age of young children,contributing to the typical excess proportion at ages 5-9,and the relative deficitat 0-4.

2. A tendency to overestimate the age of girls 10-14who have passed puberty, especially if they are married,combined sometimes, but not universally, with a tendencyto underestimate the ages of girls 10-14 who have not

.reached puberty, causing a net transfer downwards acrossage ten, and contributing to the peak at ages 5-9.

3. A tendency toward overestimation like that affectingsome of the 10 to 14 year olds, for females 15-19, 20-24,and 25-29, causing net upward transfers across ages 15,20, 25, and 30, and causing deficits at 10-14 and 15-19,and excessive proportions at 25-29 and 30-34. Thisoverestimation of the age of young women may be causedby an unconscious upward bias associated with marriageand child-bearing, or from a mechanical assumption thatwomen were married at some alleged conventional ageat marriage and have then experienced an allegedlytypical passage of time between marriage and first birth,and in each subsequent interbirth interval.

(b) Female age distributions with smaller distortions

When, on the other hand, the recorded age is suppliedby the respondent, the age distribution is naturally lessdistorted- first of all because when most respondentssupply a plausible figureto the enumerators, the maximumerror is generally below the level of the ridiculous, andthe averageerror is thus diminished. That is, when respon­dents are usually prepared to supply an age on request,and then the figure given is acceptable in the sense ofbeing only rarely absurd (e.g, seven for a grown manwith a beard), it is plausible that members of the popu­lation know their approximate age, and that the broadoutline of the age distribution is approximately correct.Rough knowledge of age on the part of most personsdoes not requirea high levelof literacybut merelya culturein which numerical age has importance. Apparently ageswere accurately known in Sweden in the middle of theeighteenth century, for example. Distortions in the cumu­lative age distributions are relatively minor in the Philip­pines and Latin America---even in Honduras, where the

. proportion of persons over fifteen recorded as illiterateis nearly as high as in Indonesia.

Some of the features of the less pronounced age-mis­reporting in the populations where apparently ages weresupplied by the respondents are just what would be

expected from respondent's errors. For example, theremay be a reluctance to pass certain milestones, such asage thirty or age forty. The relatively inflated age groupsof 25-29 and 35-39 found in several of the censuses ofLatin America and the Philippines perhaps include somewomen who have really passed thirty and forty, respec­tively. This possibility is reinforced by the relative deficitfound at ages 30-34 and 40-44.

(c) Distortions in male and female age distributions

Estimations of population parameters can be based onthe age distribution of either sex. The stable populationmethods of estimation described in this section can beapplied either to males or females, as can the censussurvival techniques outlined earlier. The ratio of malebirths to female births is confined to the limits of 1.04to 1 to 1.07 to 1 in almost all populations where birthregistration is essentially complete, the consistent excep­tions being in populations of African origin, where theratio varies from 1.02to 1.04.Becauseof the approximateconstancy of the sex ratio at birth, it is possible to checkthe consistency of estimates derived from male agedistributions on the one hand, and female on the other:the estimated male births should exceed the female byabout 5 or 6 per cent in non-African populations, andby 2 or 3 per cent in African populations.

The approximate constancy of the sexratio at birth alsomakes it possible to base the estimates for both sexes onthe analysis of the age distributions of only one. Forexample, if female births are estimated by stable popu­lation techniques, male births can be estimated as 6 percent more than the female. This estimate divided by therecorded male population gives an estimated male birthrate. The male birth rate less the intercensal rate ofincrease of the male population gives the male deathrate.

The results of analysing age-misreporting in femalepopulations with approximately stable age distributionsby comparisons with model stable populations havealready been described. The corresponding male popula­tions have been examined in precisely the same manner.In general, age distortions among male populations showsimilarities analogous to those found among females:one pattern of major distortions in both ogives and five­year distributions is found to characterize surveys inmost of Africa and southern Asia while another patternof substantial age heaping, but relatively minor distor­tions in ogivesand five-year distributions, is characteristicof Latin America and the Philippines.However,among themale populations in Africa-South Asia, the similaritiesin distortion from country to country are less for malesthan for females, and the distortions themselves appearlarger on the average. In the Latin American populations,the distortions of the female age distributions are almostalways larger than in the male; moreover there is usuallya slight systematic bias found among the female agedistributions (ogives that yield continuously increasingestimates of fertility as age increases from ten to forty)not found in the males.

A summary comparison of the effectof distortions in agedistribution on estimation is shown in figure XI. Birth

21

MALESbma -b

.020

.018

• •.016 •

""

.014"

".012

".010

" ".008

..

" AFRICAN-SOUTH ASIAN PATTERN

• LATIN AMERICAN PATTERN

.018FEMALES

bmax-bmin

Figure XI. Ranges ofstable population estimates of the female birth rate (b) versus those of the malebirth rate in various censuses. Ranges are the difference between the highest and the lowest birthrate in stable populations having an 0eo of forty years and agreeing with C(5), C(lO), ... , C(40) inthe census populations. Estimates shown were obtained from two groups of censuses, each character-

ized by a typical pattern of age-misreporting

rates were estimated (assuming °eo = 40) on the basisof ogives to age 5, 10, 15, ... ,40 for males and females,employing "West" model stable populations. Thesecalculationswerelimited to age distributions not obviouslyaffected by rapid changes in fertility or mortality. Therange (highest estimate minus lowest estimate) for malesand females in the African - southern Asian and LatinAmerican censuses so selected were then plotted. Notethat the range is much less for both sexes in the LatinAmerican group; that in a large proportion of LatinAmerican censuses the male age distribution leads to asmaller range than the female; but that in the African­southern Asian populations, the opposite is seen-a largerrange of estimates when based on male age distributions.

The smaller gross displacements in female than inmale age distributions in the African - southern Asiancensuses is consistent with the hypothesis that in thesesurveys age is often estimated by someone other thanthe person in question. The estimation of female ages isassistedbyclues-bodilychangesassociatedwithmenarche,number and age of children, and a fairly well-definedupper limit to the ages of childbearing-that do not

exist, or are lessevident, for males. The smaller distortionsaffecting male age distributions in Latin America arealso consistent with the surmise that age in these popula­tions is usually supplied by the respondent. The moreextensive education and greater worldliness of the malewould lead to a better average knowledge of age.

4. Selecting a model stable population on the basis of adistorted age distribution

The observations on age-misreporting in the precedingsection imply that the pattern of distortion must be takeninto account when determining what model stable popu­lation best fits the population in question. As a preliminarystep, the female age distribution may be compared witha model stable population with the same proportion underage thirty five (and 0eo = 40) to see if it has the character­istics (African-southern Asian)listed under sub-section (a)above. If the age distribution does have these character­istics it is likely that the female age distribution is a bettersource of estimated population characteristics than isthe male. The choice of a female stable population is

22

...

made and the consequent estimate of the birth rate isobtained as follows:

(1) The female model stable populations with a rate ofincrease equal to that of the female population andwith the same C(5), C(10), ... , C(40) are selected andthe birth rate of each is recorded.

(2) The sequence of estimated birth rates will typicallyhave a peak at age ten, fall at ages fifteen and twenty,and rise to a second peak at thirty-five or forty, if theage distribution is of the African - southern Asian type.On the basis of the analysis of this pattern given earlier,the birth rate estimate based on C(lO) may be about right,or may be too large, because there is sometimes, but notalways, understatement of the ages of enough 10-14year­olds to inflate the proportion under ten. The birth rateestimated from C(15), C(20), C(25) and C(30) wouldbe too low because of the almost universal overstatementof age among young women in these populations. Theestimate based on C (35) is expected to be at about theright level, except for the possible effects of recent mor­tality or fertility trends.

(3) A minimum estimate of the birth rate in the femalepopulation is obtained from rand C(15).

(4) If evidence in favour of stability is convincing (i.e.,little change in intercensal growth rate or in age distri­bution for two or three decades), the estimate based onC (35) can be accepted, and with more confidence ifconfirmed by the figure derived from C(lO). If growthhas been accelerating, the estimate based on C (35) can beadjusted by methods describedin section C of this chapter.

(5) The birth rate in the male population is estimated as

sexratio at birth x female birth rate x fe~ale POP;I~tionma e popu ation

In the absence of reliable direct information on the sexratio at birth a value of 1.05 should be taken in popu­lations other than those of tropical Africa. In Africanpopulations a multiplier of 1.03should be employed.

Now suppose that the comparison of the female agedistribution with the model stable population with thesame C (35) shows that the recorded population has asubstantial deficit under age five relative to the stable,and a tendency for the proportions under age ten, fifteen,twenty and twenty-five also to fall somewhat short ofthis stable, but to a minor and decreasingextent, and theproportion in successive five-year age groups does notdepart by more than perhaps 5 to 10 per cent from thestable from age ten to age forty. It may then be assumedthat the age distribution is of the Latin American patternand the choice of a stable population is made as follows:

(1) Find the male model stable populations with a rateof increase equal to that of the male population, and withthe same proportions under age 5, 10, 15, ... ,40 as inthe recorded male population, and note the birth rate ineach. Perform the analogous calculations with the femalepopulation;

(2) The sequence of male birth rates will typically risefrom age five to ten, and then fluctuate mildly until ageforty or forty-five. The female birth rate will follow asimilar sequence except that there will be a tendency

toward mildly increasing estimated birth rates from ageten to forty or forty five. 16

(3) Select the model stable population for males thatproduces the median value of the birth rate among thoseagreeing with C(5), C(10), ... , C(45). The birth rate ofthe female stable population can then be taken as

male birth rate male populationsex ratio at birth x female population

This estimate can be checked by comparing it with theaverage of the birth rates in the female model stablepopulations that agree with the recorded female popu­lations in growth rate and C (20), and in growth rate andC(25).

5. Assigningthe characteristics ofa modelstablepopulation

Once a model stable population has been selected as aclose approximation of the actual population, the charac­teristics of the stable population can be ascribed to thepopulation in question. At a later point, in chapter IV,we shall comment on the accuracy of correspondencebetween the model stable population and the actual withregard to mortality and fertility. At this place we wishonly to indicate what characteristics of the populationcan usually be attributed to the actual.

When circumstanceswarrant the assumption of approx­imate stability of the population (i.e., when there areindications of a past history of approximately constantfertility and mortality), the age distribution of the stablepopulation can be attributed to the actual population.This attribution is useful in those populations with largesystematic errors in age reporting. For example, anappropriately chosen stable population undoubtedly hasan age distribution closer to reality than the reported agedistribution for the population of India in 1911. In factthe stable population age distribution can be used toprovide a base population for population projectionsthat will provide a more valid basis for estimating thefuture evolution of the population of schoolgoing ageor of the ages of labour force participation or the likethan would be obtained by basing a projection on theoften highly distorted age distribution recorded in acensus or survey. The principal purpose of the stablepopulation analysis described in this Manual is, however,to provide estimates of fertility and mortality.

As will be seen later, the estimates of fertility derivedfrom stable analysis are ordinarily more trustworthy thanestimates of mortality. Strictly speaking, information onage composition and growth which permit the choice of amatching model stable population provide an estimateof the crude birth rate as a measure of fertility and do notenable us to estimate total fertility (the number of children

16 Frequent exceptions are found in the Latin American censusestaken since 1950 because of the effect on the age distributions ofdeclining mortality, combined in some instances with a slight risein fertility. The effect is to produce estimates of the birth rate amongmales that fall from age ten to forty or forty-five, and amongfemales that are about constant, or declining less sharply than themale estimates. The normal sequence of approximate constancyfor the males and slightly rising estimates for the females is restoredby the adjustments described in section C of this chapter.

23

Age Index of marital fertility rate

Q m(15-19) = proportion of married females at ages 15-19.

TABLE 1. STANDARD AGE PA'ITIlRN OF FEMALE MARITAL FERTILITYRATES

born per woman passing through the fertile part of life)or the gross reproduction rate (the number of daughtersborn per woman in the same span). The reason is that twopopulations with the same fertility in the sense of the samebirth rate and with the same agecomposition can have quitedifferent numbers of children per woman passing throughthe childbearing span, depending upon whether, on theaverage, births occur relatively early or relatively latein the span. When fertility is as high as it is in almost allof the populations for which stable analysis is appropriate,a low mean age of child bearing permits women with asmaller total fertility to produce the same birth I ate asachieved by a population in which women of higher totalfertility produce their children somewhat later in life.This relationship-i.e., that low fertility and early child­bearing produces the same birth rate as higher fertilityand late child-bearing-is represented in our tabulationof model stable populations by the presentation of fourgross reproduction rates with each population; each grossreproduction rate being associated with a particular meanage of the fertility schedule.1 7 This multiple tabulation ofgross reproduction rates with each stable population ofcourse implies that the identification of a model stablepopulation as essentially identical with a given actualpopulation is not a sufficientbasis for estimation of totalfertility or the gross reproduction rate. In addition, theanalyst must have some knowledge of the mean age ofthe fertilityschedulecharacterizing the women in the popu­lation in question.

The mean age of the fertility schedule can be calculatedfrom tabulated responses to a question about births occurr­ing in the preceding year (with due allowance for the factthat women who report a birth during the preceding yearwould on the average have been six months younger atthe time of birth than at the time of the survey). In manypopulations no such direct evidence on the age patternof fertility is available. In such populations the mean ageof the schedule can sometimes be estimated by variousindirect approaches. One possibility is to estimate theage pattern of fertility by assuming that marital fertilityfollows a standard pattern. Table 1 shows a pattern ofmarital fertility-fertility rates expressed in terms of therate for age 20 to 24-that can serve this purpose.

The rates shown in table I for ages 20-49 are based onaverage experience of a number of populations in whichlittle or no birth control is practised. 18 The value for age15-19 is a rough approximation according to which if,in the absence of birth control, marital fertility for age20-24, i.e., 1 (20-24), and the proportion of marriedfemales at age 15-19, i.e., m(15-l9), are known,/(15-l9)may be estimated as 1.2/(20-24)-.7 [(20-24) m(15-l9).In a population in which it is known that the fertility ofnon-married women is a negligible factor in the over-allbirth rate, and where the census includes a tabulation ofmarital status by age, the age pattern of fertility can beapproximated by multiplying this standard maritalfertility schedule by the proportion married among thewomen in each age group and from this approximatedschedule the mean age of the fertility schedule calculated.

The method described in the previous paragraph is notapplicable in populations with high proportions of birthsoutside of marriage, or in which there is a variety ofsanctioned sexual unions, including for example wide­spread consensual unions.

When women have been asked about the number ofchildren ever born, and the responses have been tabulatedby age of woman, it is possible to estimate the mean ageof the fertility schedule from the relation of averageparity at age 20-24 to average parity at ages 25-29. If theearlier average parity is called P2 and the later Pa, theratio P3/P2 (in a population not practising birth control)depends primarily on the ages at which women begintheir child-bearing, a high value of P3/P2 indicates a latestart, and a low value an early start. It may be assumedthat the decline of fertility with age in populations notpractising birth control follows a fairly common pattern,so that the mean age of the fertility schedule is determinedprimarily by the rising portions. The relationship betweenthe mean age of the fertility schedule and the value ofP3/P2 has been calculated on the basis of the recordedschedules of a number of populations in which there islittle practice of birth control. The mean age of eachschedule was calculated, and the values of P2 and P3implied by the schedules computed. The relation of iiito P3/P2 is very close, as is shown in figure XII, and iswell expressed by the following linear expression:

iii = 2.25 P3 + 23.95P2

When, due to the lack of required data, neither of theapproximations described above is applicable, the bestalternative is to assign to the population in question themean age of the fertility schedule of another populationpresumed to have similar factors affecting the age of thefertility schedule. Thus in a Latin American country apossible expedient is to utilize the mean age of child­bearing from a nearby neighbour, if the two countries inquestion are both known to have a roughly similarprevalence of consensual unions and of marriages per­formed by the State or the church. The reader's attentionis directed to the expedient used in examples worked outin part two of this Manual.

1.2-.7m(15-19)Q1.000.935.853.685.349.051

15-19 .20-24 .25-29 .30-34 .35-39 .40-44 .45-49 .

17 The arithmetical mean of the schedule itself, unweighted bythe age distribution.

18 They are adopted from Louis Henry, "Some Data on NaturalFertility", Eugenics Quarterly, vol. 8, No.2 (June 1961),pages 81-91.

24

34

33

32

3\

30

29

28

27

2.0 2.2 2.4 2.6 2.8 3.0 3.2 v.4 3.6 3.8 4.0 42. 4.4

Pa/P2

Figure XII. Mean age of the fertility schedule in populations not practising birth control versusratio of Ps (average parity at age 25-29) to P2 (average parity at age 20-24)

C. ADJUSTMENT OF ESTIMATES BASED ON MODEL STABLEPOPULATIONS FOR THE EFFECTS OF RECENT DECREASESIN MORTALITY

It goes without saying that many populations have agedistributions that do not conform at all closely to any ofthe model stable populations. Major fluctuations infertility create unusually large or unusually small birthcohorts that stand above or below the corresponding agegroup in any stable population during their lifetime.Persistent trends in fertility can create an age distributionfar different from any stable population. Because thepopulations of all highly industrialized countries of theworld have experienced sustained decreases in fertilityamounting in almost every instance to a 50 per centreduction of total fertility, and since most of thesepopulations have experienced substantial fluctuations infertility in addition to the downward trend, age distri­butions of industrialized populations cannot in generalbe matched by those of model stable populations. Eventsin the history of some of the less developed countries havealso caused irregularities in the age distribution thatwould find no match in one of the model stable popu­lations. Some populations have age compositions stronglyinfluenced by age-and-sex-selective migration. Suchmigration is typical of the urban populations ofdevelopingcountries and a non-stable form of age distribution istherefore to be found at certain census dates in suchpopulations as that of Singapore' and Hong Kong. Theprolonged mobilization of a large fraction of the youngadult male population into military service can have apronounced effect on fertility and create a lasting notchin the age distribution. Military casualties of course affectthe age distribution of the male population. Invasions andrevolutions can leave similar traces. Finally, major

epidemics such as the world-wide infleunza epidemic of1918 to 1920 cause a temporary reduction in fertilityand an excess of infant and child mortality that againproduce a small cohort evident in subsequent age distri­butions.

On the other hand, in developing countries where thepopulation is little affected by international migration,and in the absence of major catastrophes such as wars orgreat epidemics, fertility tends to remain at a fairly levelplateau. The only apparent exception is in areas whererelatively late marriage is the custom, such as in westernEurope before the systematic decline in marital fertilitybegan. In these populations, for example, in Tuscanyduring the nineteenth century, changes in total fertilityamounting to 20 or 25 per cent can occur caused by longintervals in which the average age of marriage wasincreased and other intervals in which it was reduced. Itappears possible that in Latin America changes in age atmarriage, and perhaps differential recourse at differenttimes to marriage on the one hand and to less stableconcensual unions on the other have raised and loweredthe average level of fertility.

The conclusion that emerges from these observations isthat there are populations in less developed countries thatcannot be analysed by stable population techniques, andothers in which the precision of the estimates may bedegraded, even though the general approach remainsuseful. Among the contemporary populations for whichthe method does not appear useful are those stronglyaffected by migration and those that have suffered asequence of wars or other major disturbances. Thereremains the majority of populations in the less developedcountries, where the assumption of a history of moreor less constant fertility is warranted, but where mortality

25

has followed a strong and sustained downward trend inrecent years.

The prevalence of rapidly falling death rates in the lessdeveloped countries is well known and need not bedescribed in detail in this Manual. Falling mortality hasfollowed as many different courses (if these are consideredin detail) as there are identifiable populations in thedeveloping countries. It is of course not possible todescribe how every imaginable decline in mortality wouldaffect the age composition of a population. Many demo­graphers have noticed that different mortality schedulesproduce only slightly different stable populations, andthat populations experiencing approximately constantfertility and changing mortality show only restrictedalterations in age composition. Finally, it has beennoted that the age composition produced at each momentof time during a prolonged period of declining mortalitybears a closer relationship to the stable implied by thecurrent fertility and mortality conditions than to the agedistribution of an earlier period when mortality washigh~r. ~uch observations have been used to justify theapplication of the methods of stable population analysisto populations experiencing approximately constantfertility and steadily declining mortality. Indeed aspecial designation of quasi-stable has been invented forthese populations. Coale and Demeny have analysedthe .w.ays in which populations that have experienceddeclining rather than constant mortality differ from thestable population that would have resulted had currentmortality and fertility conditions prevailed throughoutthe past. 19 They noted that in spite of the close visualresemblance between the age composition of a stablepopulation and of a so-called quasi-stable population anestimate of total fertility or the birth rate based on' theogive of the quasi-stable population and the current rateof increase can be in error by as much as 10 to 15 per cent.In this section a method is described by which the demo­graphic analyst can adjust estimates of population para­meters extracted from model stable populations to com­pensate for the distorting effects of a history of recentlydeclining mortality.

On the basis of previous analytical work it is knownthat th~ principal effect of declining mortality on ageC?~Posltl.o.n closely reseJ?ble~ the influence of steadilynsmg fe~Ility. Over a specifiedinterval of falling mortality,It IS possible to find a proportionate change in fertility­say a 7-per cent increase-that is equivalent to therecorded decline in mortality so far as the effect on agecomposition is concerned. It is possible, by an extensionof this idea, to determine what sequence of annuallyrising expectation of life in the "West" model life tablesgiven in annex I would be equivalent to an annual increasei~ fertility of one per cent. A series of population projec­tions were constructed along these lines in which the initialpopulation was a model stable population with a totalfertility of about 5, 6 and 7 respectively, and with various

18 Ansley J. Coale, "Estimates of Various Demographic Mea­sures Through the Quasi-Stable Age Distribution", in EmergingTechniques in Population Research, Milbank Memorial Fund,1963,pp. 175-193,and Paul Demeny, "Estimating Vital Rates for Popula­tions in the Process of Destabilization", Demography (Chicago),vol. 2, 1965, pp. 516-530.

levels of the initial expectation of life at birth. Thesepopulations were then projected for forty years withsteadily rising expectation oflife at birth (i.e., with steadilyfalling mortality) at a pace which in each year was equi­yalent to a one per cent increase in fertility in the principalinfluence on changing age distribution. These projectionswere performed on an electronic calculator. The pro­grammed computations included the calculation of theaverage rate of increase during each five-year period andalso during each ten-year "intercensal" interval, birthrates, death rates, age distributions in five year groupsat the end of each five-year time interval, and ogives ofthese distributions. Birth rates and total fertility rateswere then "estimated" at five-year intervals by choosinga model stable population with the same rate of increaseas the. "intercensal" rate of increase during either thepreceding five years or ten years in the populationprOjectIOn, an~ the value~ of C(5), C(lO) etc. in the pro­jected population. The difference between the estimatedrate calculated in this fashion and the average value ofthe "true" birth rate or of the "true" total fertilityduring .the intercensal period (calculated as part of theprojection programme) was then obtained. These differ­ences, expressed as a proportion of the true value of thebirth rate or of total fertility, turned out to be very nearlythe same whether the population projection was for af~rtility o.f5, 6 or 7 births per woman. Thus these propor­tionate differences can be used as the basis for adjustmentfactors to reconstruct the true birth rate from the birthrate ~stimated from the model stable population - if thedur~tIOn of the decline in mortality is known and if thefalling death. rates ar~ at a. J?ace equivalent to a one percent annual increase m fertility so far as age compositioneffectsare concerned. That is so since the calculations alsoshowed that. the adJustments needed, e.g., at fifteen years~fter mortality decbn~ ?~gan, were for practical purposes,independent ?f the initial level of mortality, i.e., theywer~ subst~ntIally the same whether the initial expectationof life at birth before the onset of declining mortality was,for example, twenty years or thirty years. Finally sets ofpopulation projections were programmed that'differed~It~ resI?ect t~ the speed of mortality decline, e.g., pro­jections m whI~h the decline in mortality was only halfas fast as specified above; in short, in which mortality,c~nges were equivalent in their principal age composi­tional effects to an annual increase in fertility of one-halfof one per cent. These projections showed that the differ­ences in the estimated and actual values of birth rates andtota~ fert~lity :were proportionate to the speed of mortalitydecltne, i.e., m the example just given they were almostexactly half as great as with the more rapid decline inmortality.

This Manual includes as a result of these calculations atabl~ of adj~stments (see annex III, table I1I.1) to beapplied to birth rates and to gross reproduction rates(or total fertility rates) derived from the model stablepopulations- adjustments that are appropriate for ogivesup to age 5, 10, ... , 40 and which assume different valuesat t = 5, 10, ... , 40, where t is the time after the declinein mortality begins. These adjustments should be applied~h~n t?e value of k equals .01, where k is a parameterindicating the rate of mortality change expressed interms of the equivalent proportionate annual increase in

26

fertility in so far as age:distribution effects are concerned.For values of k other than .00-i.e., in the general casewhen the age distribution effects of the changing mortalityare equivalent to an average annual fertility change thatis greater or smaller than one per cent per year-thetabulated adjustments are to be scaled up or down inthe same proportion as the actual value of k differs from.01. For instance if the value of k is .012 the appropriateadjustment factors are to be increased by 20 per cent.Annex table III.1 thus enables the analyst who has madea preliminary set of stable population estimates of thebirth rate and of the gross reproduction rate (from the agedistribution cumulated to age 5, 10 etc. in conjunctionwith the intercensal rate of increase) to adjust thesepreliminary estimates in order to correct the bias presentin the stable estimates due to the fact that contrary tothe assumptions underlying the stable estimates mortalityin fact has been declining. A quasi-stable estimate of thedeath rates is obtained by subtracting the intercensalgrowth rate from the adjusted birth rate. A quasi-stableestimate of the expectation of life at birth is finallyderivedby findingthe °eo of the stablepopulation characterized bythis death rate plus any of the other parameters (ORR, bor r) for which the quasi-stable estimates have previouslybeen calculated.

The application of the method described in the pre­ceding paragraph assumes that estimates of the durationand average pace of mortality decline-i.e., estimates ofthe parameters t and k-have already been obtained.Preferably such estimates should be based on informationconcerning the rate of population growth in the decadespreceding the census that is being analysed- informationthat is sufficient to locate approximately the time when thedeparture from the stable state has occurred and whichindicates the tempo at which destabilization has takenplace. Specifically, on the basis of the same calculationsthat were outlined above, it was found that k can be esti­mated as 17.8x lir/lit where fir is the absolute changein the rate of growth as compared to the original stablerate, and lit is the number of years that have elapsedwhile that change took place.20

The acceleration of population growth is not the onlyevidence from which the parameters k and t can be esti­mated. What is needed is any approximate indication ofthe duration and pace of recent mortality declines. In

20 This formula assumes that the acceleration of growth isattributable in its entirety to a change in mortality, i.e, that fertilityhas remained constant. It may be noted at this juncture that in aformal sense the procedure adjusting stable estimates for the effectsof changing mortality may be extended to the case where destabili­zation has been brought about by changing fertility, or a mixtureof changing fertility and mortality, rather than by changing mor­tality alone. Assume, for example, that fertility has been changingfor t years (following an original stable situation) while mortalityhas remained constant. The adjustment factors tabulated in table 111.1would still be applicable-with proper attention to sign, i.e.,multiplied by -1 in case of declining fertility-the value of k tobe used being simply the average annual change in fertility. Notethat the value of the multiplier connecting k and ti.rlti.t wouldthen be about t.wice as large as in the case of mortality change;approximately 36 instead of 17.8. In view of the fact that sustainedchanges in fertility are less regular and, under the conditionsnecessitating the application of quasi-stable techniques for estima­ting vital rates, are less common, no systematic discussion of thistopic is offered in this Manual.

chapter II a method of estimating child mortality fromspecial census data is described, and in chapter III thecombination of estimated child mortality and the agedistribution to selecta model stable population is outlined.At the end of chapter III it is noted that changes in theestimated level of child mortality from census to censuscan be used to determine approximate values of k and t 21.

Another basis for estimating k and t is the changing agecomposition of deaths. The proportion of deaths by agechanges when mortality declines and fertility remainsconstant. It is possible to obtain a very rough estimate of kand t from even an inexact record of the changing agecomposition of deaths. The reader should note that themethod outlined in.thenextparagraphs is clearlyimprecise,but that it is used only to indicate the magnitude of anadjustment for the effects of declining mortality, andthat an approximate adjustment usually produces anestimate superior to the unadjusted stable value.

The estimation procedure rests on the followingsupporting facts and relationships:

(1) When initially stable populations are projectedwith declining mortality but constant fertility, certainchanges in the age composition of deaths are very closelycorrelated with the associated change in expectation oflife at birth. The most usable index of the changingcompo­sition of deaths is perhaps the ratio of deaths to personsover sixty-five to deaths to persons over age five. Thisindex is as closelyrelated as any other to lioeo, and has themerit of omitting infant and child mortality, which couldhave an overidingand possiblymisleadingeffect, especiallywhen completeness of registration changes.

(2) The change in the proportion of deaths over sixty­five may be faithfully represented in registered deaths,even when these are so incomplete in coverage thatneither the level nor the trend of the crude death ratecan be derived directly.

(3) An approximate value of Ii°eo can be obtained fromannex table 111.2 as the difference between two stablepopulation estimates of °eo referring to different periodsof time, each based on a value of the index deaths65+/deaths 5+ and on a rough measure of fertility

(births of a given sex )22

persons 15-44of the same sex

(4) The value of k. t associated with a given lioeodepends rather strongly on the base level ofoeo, and anapproximate value of the latter is needed before k. tcan be estimated from table III.3. A crude but usableindication of the terminal level (which lessIi°eo gives thebase level) of °eo can again be obtained by the provisionalassumption of stability. It is recommended that theprovisional terminal level be taken as the average of the°eo 's in the stable populations associatedwith r and C (10),and rand C (15). The same procedure is to be used inobtaining the rough measure of fertility that is neededin estimating Ii°eo, as explained in point (3) above.

21 See chapter II, section B, chapter III, section C.2 and annextable I1I.4.

22 This ratio is one of the stable population parameters tabulatedin annex II.

27

It must be noted that the recommended procedure forestimating kt from the changing age composition ofdeaths (i.e., approximating aOeo and the levelof °eo fromthe provisional assumption of stability before usingtable 111.3) generates provisional data on mortality­namely the levelofoeo-that are useful only in determininga correction factor, and that must not be confused witha final estimate of that parameter.

40 years. in figure XIV the plot of C(x)-Cs(x) of severalLatin American populations with censuses in the early1960s is shown in comparison with the projected popu­lation where mortality has been declining for twentyyears. The imprint of the effect of declining mortalityon the age composition of these populations is clearlyevident.

D. CoNCLUDING REMARKS ON ESTIMATESADJUSTED FOR THE

EFFECTS OF RECENT DECLINES IN MORTALITY C (x) - C lx)- $-

.04Whether adequate direct or indirect information on past

changes in mortality are available or not it is to beexpected that with reasonably good reporting of ages sucha deviation from stability will be discernable from the agedistribution itself. The nature of the effect of a history ofdeclining mortality on the stable age distribution isindicated in figure XIII which shows C(x)-Cs(x) where

.02

01---..lj,L----~1IO:""""-----

, ...---.-.-.

Cl!)'_ MODEL QUASI STABLE, ~".Ol, 1-20___ VENEZUELA,I961

•••••• HONDURAS, 1961•__• PERU,I961

The age distributional effects of declining mortality arealso present in recent censuses of such populations asthat of India and Pakistan, but because of the nature ofthe distortions in the reported age distribution the effectsare not so easy to discover as in the Latin American cen­suses. However it is interesting to note that when stablepopulation analysis is applied to the Indian age distri­bution (females) in 1911 there emerges from the inter­censal rate of increase and the proportion under ageten andunder age thirty-five estimates of total fertility in approx­imate agreement-estimates of 6.5 children per womanin the former case and 6.3 in the latter case. When stablepopulation analysis is applied to the Indian census of1961 the total fertility estimated from the proportionunder age ten and the intercensal rate of increase is 6.6and from the intercensal rate of increase and the propor­tion under thirty-five is only 5.8. However, when theappropriate adjustments are made to allow for the influence

Figure XIV. Comparisons of the ogive-C(x)-of a modelquasi-stable population and of male populations as recorded inrecent Latin American censuses, with that of stable populations-Cs(x)-having the same proportion under age twenty and an

0eo of forty years

o 5 10 15 20 25 30 35 40AGE

Cl~): MODELQUASI STABLE (h.Il.O!)

t - 15== i- 20..... i· 25__._. t~ 30

··_··1=35

o

02

-.02

o 5 10 15 20 25 30 35 40AGE

C(x)-Clx)- s-

04

Figure XIII. Comparisons of the ogive-C(x)-of model quasi­stable populations (populations which were originally stable butwhich have experienced a decline of mortality for t years). withthat of stable populations-Cs(x)-having the same proportion

under age twenty and an 0eo of forty years

C (x) is the ogive of the projected population with mortal­ity declining for IS, 20, ... , 35 years and Cs(x) is that ofthe stable populations having the same C (20) and °eo of

28

TABLE 2. STABLE ESTIMATES OF THE FEMALE BIRTH RATE OF INDONESIADERIVED FROM THE 1961 FEMALE AGE DISTRIBUTION

10 IS 3S

°eo = 30 .................... .0511 .0492 .0502= 35 •• 0 ••••••••••••••••• .0534 .0460 .0472= 40 '0 •••••••••••••••••• .0505 .0434 .0450

r = .010 ................... .0811° .0559 .0565= .015 ................... .0615 .0491 .0503= .020 ................... .0594 .0435 .0462

OJ Extrapolated figure.

The estimates based on C(l5) are a series of minimumestimates, those based on C(1O) appear in this instanceto be a series of maximum estimates, and those basedon C(35), to be the best obtainable. While the interpre­tation of the Indonesian age distribution in the light ofoutside experience does drastically reduce the width ofthe interval within which the birth rate is likely to belocated, lacking further information it would be unwar­ranted to go beyond the cautious assertion that the birthrate in Indonesia is probably not lessthan 45 per thousandand not more than 56 per thousand.

The Indonesian population was enumerated in 1930ina census that did not record the distribution by chrono­logical age. Strictly speaking, then, Indonesia is not aproper example of a single census. For example, over-

enumerated population in fact departs from the stable agedistribution.

In the hypothetical example just described, the impli­cation that the demographer would know that theproportion under age 10 is .3 is unrealistic, since if apopulation has had its age distribution recorded only once,it is likely that the recorded distribution is seriouslydistorted by age-misreporting. A more realistic exampleis provided by considering the problem of estimating vitalrates for Indonesia on the basis of the age distributioninformation provided by the 1961 census of that country.As shown in figure XV combination of various indicesof this age distribution (proportions up to age 5, 10, ... etc.)with specified hypothetical growth rates or levels ofmortality imply widely varying birth rates. Unless someparticular indices of the age distribution can be acceptedas more reliably reported than others, thus leading to anarrower range of uncertainty, the information providedin figure XV would be of little value. Inspection of thisfigure as well as comparisons of the Indonesian agedistribution in cumulative form with the ogives of modelstable populations revealsthe pattern of age-mis-statementcharacteristic of censuses in India, Pakistan and manyAfrican populations. It is therefore appropriate to usethe rules of estimation devised for such age distributions.Table 2 highlights the figures that are relevant in applyingthese rules.

Birth rates based on assumed leoelsofmortality or rate of Increase,

and proportion rmder age:

Assumption aboutmortality or rate

of Increase

E. THE ESTIMATION OF FERTILITY FROM THE AGE DISTRI­

BUTION RECORDED IN ONE CENSUS

of declining mortality in the preceding forty years at apace estimated from the average rate of acceleration ofgrowth the resultant estimatesare 6.6and 6.4, respectively.

What is the minimum information that permits demo­graphers to form approximate estimates of fertility ormortality? In the earlier sections of this chapter methodsare described for basing estimates on a series of two ormore population censuses, and in the next chapter thereis an outline of techniques of estimation to extract themaximum of reliable inference from special questionsconsciously inserted in a census or survey to measurefertility and mortality. What can be learned from asinglecensus (so that no intercensal rate of increase can becalculated) that provides no special fertility or mortalitytabulations?

Age composition is more strongly affected by fertilitythan by mortality, so that with minimal information morereliable estimates can be made of the birth rate than ofthe death rate. For example,if a closedpopulation containsa very high proportion of young children, it must be apopulation that has recently experienced high fertility­an inferencethat is valid whether the expectation of life atbirth is high or low. However, because high infant andchild mortality diminish the fraction under age five or ten,a population with a large proportion of children wouldbe a moderately high fertility population if mortalitywere low, and a very high fertility population if mortalitywerehigh.

Consider a specific example: suppose the proportionunder ten years of age in a population was 30 per cent.If mortality were assumed to be that of the model lifetable with 0eo = 30 years, births in the precedingten yearscould be estimated as 30per cent of the current populationtimes a factor derived from the model life table expressingthe reciprocal of the proportion surviving, and the guessmay be hazarded that with such a low expectation of life,the population grew moderately-at perhaps 1.5 per centper year so that it was about 7.5 per cent larger than itwas at the midpoint of the preceding ten years. On theseassumptions the average birth rate during the decadepreceding the census would be 50 per thousand. On theother hand, if mortality were assumed to match that ofthe model life table with °eo = 40 years, and if a morerapid rate of population growth-say, 2 per cent annually-were viewed as a sensible guess, the resultant estimateof the birth rate would be 44.3. Often the conditions inwhich the population lives are known well enough, andthe mortality of other populations in similar circumstancesare well enough recorded to give some confidence that abroad estimated range of the level of mortality probablyencompasses the actual figure. The use of model stablepopulations can be used to translate a recorded proportionunder age five, ten or fifteen and an assumed level ofmortality into an estimated birth rate without elaboratecalculations. If the basis of estimation is confined to agesunder fifteen, the procedure is closely equivalent toreverse projection and the estimate is valid even if the

29

BIRTH.RATE

,07

.06

.05

.04

4 .

",.\1::\,: :\'i :1,. .\! h. :\: :1

u:\: \ I ••···r=.02: \ I. -: \ I·''. \_01 :-. :-. :......:

BIRTHRATE

.07

.06

/\.05 / ••••\..' \\

\\'.' r:,... '-' ....

.04 ••••••••••

,03 .03

,02 .02

,01 .01

AGE

0'-----''-----'---1.---'---'---'------10 20 30 40 50 60

AGE

o'----''---~_--'-_--L._ _'__-L-- _

Figure XV. Stable population estimates of the female birth rate in Indonesia derived from C(x) as reported by the census of 1961and fromhypothetical rates of population growth (left panel) or hypothetical levels of mortality (right panel)

looking differences in geographical coverage and com­pleteness of enumeration, one can calculate the intercensalrate of increase of the female population (for thirty-oneyears) as 1.63per cent per annum. This information wouldlead us to guess that the growth rate just before 1961 was2 per cent (or more), and that the estimate of 46.2 perthousand for the birth rate was therefore to be preferredto the 56.5 and 50.3associated with lower rates of increase.But if a rate of increase above the intercensal average isaccepted for the period just before the census, populationgrowth must have been accelerating, primarily, oneassumes, because of falling mortality. If mortality hadbeen falling for fifteen years at a rate causing an accele-

ration in the growth rate of about 5 per thousand eachdecade, the estimated female birth rate based on C(35)and r = .02 should be increased by 6 to 10 per cent-to50 per thousand, or a little higher. The male birth rate isprobably about 8 per cent higher than the female, becausethe number of males is about 2 per cent less, and thenumber ofmale births is normally about 5 per cent greater.As a result of the above arguments then, it is possible toconclude that the birth rate in Indonesia was probablyat least 2 or 3 points above 50 per thousand in the yearsbefore the census. A corresponding minimum estimatefor total fertility, assuming that the mean age of thefertility schedule is twenty-nine years, is 6.5.

30

Chapter II

METHODS OF ESTIMATION BASED ON RESPONSES TOQUESTIONS ABOUT FERTILITY AND MORTALITY

A. EsTIMATION OF FERTILITY FROM REPORTS ON CHILD­BEARING IN THE PAST l

In many censuses and surveys there appear data on thenumber of children women have ever borne, tabulated byage of woman. If fertility rates have been approximatelyconstant in the recent history of the population in question,if the reported fertilityhistories have not been substantiallyaffected by migration, and if differential mortality accord­ing to the prolificacy of the woman has not had an impor­tant effect on the survival of mothers, the average numberof children ever born to women past age forty-five or fiftyequals total fertility. MOle precisely, the average numberof children ever born per woman aged forty-five to forty­nine equals the total fertility of this cohort of women,which in turn is about the same as the total fertility of thepopulation at the time of the census or survey, providedfertility rates in the population have been approximatelyconstant. If the assumptions stated above are valid, it ispossible to estimate the age specific fertility rate for eachfive-year age interval within the child-bearing span by(for example)fitting a polynomial to the reported numberof children ever born by age of mother.

The uncritical use of the reported average number ofchildren ever born as a means of estimating the fertilityof a population is risky, however, because of a widespreadtendency for the number of children ever born to beunder-reported, especially for older women. In manycensuses or demographic surveys the average number ofchildren ever born increases too gradually with age,especially at ages above thirty or thirty-five, and indeeda common feature of many censuses is reported averagenumbers of children ever born that declinewith age aboveage forty-five 01 fifty. One can speculate about the causesof the apparent under-reporting of the average numberof children on the part of older women. Perhaps the mostimportant factor is that some women tend to omit childrenwho have grown up, or who have left home. A secondpossibility is the inability of some illiterate respondentsto report large numbers accurately. A third hypothesisis that older women tend to omit offspring who died,especially many years earlier, although there are manyinstances in which the proportion reported as dead riseswith the age of woman in a consistent fashion. Forwhatever reason, the reported average number of children

1 W. Brass. A.J. Coale, P. Demeny, D. Heisel, F. Lorimer,A. Romaniuk, and E. van de Walle, The Demography of TropicalAfrica (Princeton, Princeton University Press), (in press), chapter III.

31

ever born is very frequently a downward-biased estimateof the cumulative fertility experience of women overthirty or thirty-five, and the average parity of womenpast age forty-five or fifty, being typically understated,would usually provide an underestimate of total fertility.

On the other hand, younger women presumably reportthe number of children ever born to them with muchbetter accuracy. Such women are not asked to recallevents from the remote past or to count accurately to alarge number; a higher proportion of the children everborn have survived to the time of the interview and few,if any, have left the household. If age-misreporting doesnot cause excessive distortion, the sequence of averagenumbers of children ever born by age of woman duplicatesclosely the curve of cumulative age specific fertility rates,until an age is reached where the proportion of childrenever born that are omitted by the respondents becomessignificant. In other words, the early part of a curveshowing the rise in the average parity of women with ageshould resemble closely the cumulation by age of fertilityrates. On the other hand, towards the end of the child­bearing interval, as the average number of children everborn approaches total fertility, the tendency towardsomission or understatement causes reported parity tofall short of cumulated fertility rates. To provide a morevalid estimate, then, it would be useful to splice to therising curve of children ever born with age at the youngerages a curve which continued to rise with age in a wayreflecting more accurately the actual fertility rates aboveage thirty or so. If we could be sure of the approximaterelationship between fertility rates at different ages in thegiven population, we could determine what fertility ratesat the younger ages would account for the average numberof children ever born reported by the younger women,and then ascribe to the older women fertility ratesconsistent with these rates established at the younger agesof child-bearing. For example, if all populations in lessdeveloped countries had approximately the same agepattern of fertility, and differed only according to the sizeof a factor expressing the level at which this age patternof fertility operated, the level could be ascertained bylooking at the average number of children ever born byyounger women-say, at ages twenty to twenty-four andtwenty-five to twenty-nine. It would then be possible toprepare a rough estimate of the remainder of the fertilityschedule and thereby to estimate total fertility. However,no such universal age pattern of fertility exists, althoughit is probably true that many populations share a similarage pattern of fecundability, that is a similar age pattern

of the probability of conception among women livingregularly in sexual union, without practising birth control.

The age pattern of fecundability is not identical in allpopulations primarily because of differences in theincidence of secondary sterility, or in the average age atwhich childbearing ceases. But even if differences in theage pattern of fecundability could be ignored, it would stillnot be possible to ignore differences by age in exposure tothe risk of pregnancy. The major source of such differencesis in the age pattern of the establishment of regular sexualunions-through marriage or other socially sanctionedinstitutions. The average age at first marriage for femalesin populations not commonly practising contraceptionvaries from less than fifteen in India before the secondWorld War to over twenty-five in many parts of Europeduring the nineteenth and early twentieth centuries. Insocieties where consensual unions and formal marriagesare both common, the age of entry into the former istypically much younger than into the latter, and con..censual unions do not ordinarily involve as regular expo­sure to intercourse as marriage. There are similar thoughquantitatively less important differences in the dissolutionof sexual unions. The incidence of widowhood dependsupon the level of mortality, and the frequency of divorceupon law and custom. Societies also differ with regardto remarriage of the widowed and divorced.

To adjust the reported numbers of children ever bornfor a characteristic omission on the part of older womenit is therefore necessary to have reliable evidence of theparticular age pattern of fertility in the population inquestion. There are two potential sources of directinformation on child-bearing rates by age of mother. Thefirst is birth registration, and the second responses to asurveyor census question on births during the yearpreceding the survey, either tabulated by age of woman.Of course if either of these sources were known to beaccurate in the coverage of births the resultant data couldbe used directly to construct fertility measures. But evenwhen the fertility rates derived from these sources arenot accurate, they may be approximately correct in form.In other words there is the possibility that even if thenumber of births registered or reported understates oroverstates the true number, the degree of understatementor overstatement is not age selective. This possibility is aplausible one with regard to the events reported byrespondents for the preceding year, but is much less so forregistered births. Incomplete registration cannot ordi­narily be used to indicate the age pattern of fertilitybecause there is no reason for supposing that the popu­lation covered by registration is representative of thewhole population with respect to the fertility schedule.If birth registration is associated with literacy, for example,there may well be a tendency for the births occurring toyounger women to be more completely registered thanthose occurring to older women. Other likely sources ofbias include differential completeness of registration inregions that are not uniform with respect to the agepattern of fertility-in urban areas as compared to rural,for example.

The other potential source of information about the agepattern offertility is survey data on births occurring duringsome preceding period, typically a year, before a census or

demographic survey. The number of births reported inresponse to such questions has not proven accurate.Experience seems to indicate that the source of the inaccu­racy is not a systematic tendency for women to fail toreport births that have occurred or to exaggerate thenumber of these births but rather the difficulty thatrespondents have in identifying properly the length of theinterval for which births should be reported. In somesurveys there is a net tendency for women to report birthsthat occurred in a period that is less than a year and inother surveys to report on the average events that occurredduring more than the preceding year. The factors causingthis reference period error seem likely to depend on gene­ral cultural conditions, the circumstances of the particularsurvey, including the wording of questions, the instruc­tions to enumerators and the like. There seems no reasonto expect an association between errors in reference periodand the age of respondents. Moreover, the populationcovered by questions on births during the precedinginterval is the same as that covered by questions aboutchildren ever born so that inconsistencies caused bydifferences in coverage or in forms of age-misreporting areavoided-differences that may exist between a populationincluded in a register and a population covered by asurvey.

The necessary questions on births during the precedingyear on the one hand and on children ever born on theother, both tabulated by age of woman, have been includedin a number of demographic surveys in Africa. WilliamBrass has designed for use primarily in Africa a method offertility estimation that accepts as essentially correct thepattern of fertility rates by age indicated by the birthsreported as occurring during the preceding year, and thataccepts as an essentially correct indication of the levelof fertility the average number of children reported asever born by younger women. The method requires theestimation of the average value of cumulative fertility byage over the same age intervals (usually 15 to 19, 20 to24, 25 to 29 etc.) for which the average number of childrenever born is reported. It is then assumed that the source ofthe difference between the estimated average value ofcumulated fertility at the younger ages (such as 20 to 24or 25 to 29) and the average number of children ever bornat these ages is an erroneous perception of the referenceperiod by the respondents. The multiplier that would beneeded to bring cumulative fertility at the younger agesin line with the reported average number of children everborn is determined, and the reported numbers of births atall ages are multiplied by this factor.

The basic principle underlying the Brass method offertility estimation is simple enough, and the computationsare complicated only by the difficulty of estimating theaverage value of cumulated fertility for the same ageintervals for which average children ever born are given",If approximate age specific fertility rates based on reportedbirths by age of mother are tabulated only in five-year ageintervals, cumulative fertility can be calculated directlyonly at the boundaries of these age intervals. Thus fivetimes the age specific fertility rate of women 15 to 19 givesthe cumulative fertility to age 20; this value plus five

2 W. Brass, et al. loco cit.

32

times the fertility rate of women 20 to 24 gives the cumu­lative fertilityto age 25,and so on. It is necessarythereforeto estimate from cumulative fertility to ages 20, 25, 30etc.,what would be the average value of cumulative fertilityin the five-year age groups for which average parity (oraverage number of children ever born) is reported. If agespecific fertility rates were constant within each five-yearage interval the average value of cumulative fertilitywould be closely approximated by simple linear inter­polation to the midpoint of each age interval. In fact,the typical pattern of fertility rates (increasing as they dofrom the earliest age of child-bearing to a peak usually inthe late twenties) creates a curve of cumulative fertilitythat is not linear and that therefore requires a more com­plicated form of estimation than simple linear interpo­lation. Brasshas calculated variable interpolation factorsto be applied to the readily calculated values of cumulatedfertilityto the boundaries of the age intervals.The selectionof appropriate interpolating factors is determined by howrapidly the reported fertility rates increase from the firstto the second age group of women, since it is the steepnessof the rise in fertility that determines the curvature incumulative fertility with age over the early portion of thecumulative fertility function. Tables in annex IV presentthe interpolation factors-table IV.1 to be used when agespecific fertility rates are available for the five-year agegroups bounded by exact ages l4.S and 19.5, 19.5 and24.S etc.; 3 and table IV.2 to be used when such rates areavailable for the conventional five-year age groups. Inchapter VII there is a fully worked out example of theestimation of fertility by this method.

There are many censuses and surveys that have includeda question on children ever born, with the responsestabulated by age of woman, but that have not included aquestion on births in the precedingyear, so that there is nopossibility of using the method just described. As statedearlier, it is a mistake to assume that differentpopulationshave the same age pattern of fertility-even populationsnot employing contraception or abortion. However, theform of the fertility schedules in populations that employlittle birth control differs primarily in the way in whichfertility rises from the first ages of childbearing to theageswherefertilityis a maximum, and relatively much lessin the way fertility declines after the peak is reached.This greater relativevariability in the early part of fertilityschedules results from the fact that the rise of fertilitywith age is strongly affected by customs and institutionsgoverning the establishment of sexual unions-stronglyaffected, that is to say, by the age pattern of nuptiality insocieties where formal marriage is a principal determinantof cohabitation. In other words, when birth control is notwidelypractised, the shape of the early part of the fertilityschedule is dominated by the age pattern of entry intoregular sexual relations rather than by the age scheduleof fecundability; and in contrast the decline of fertilitywith age-when birth control is not practised-is generally

8 When age specific ferti1ity rates are based on births reportedin the preceding year, the mothers were approximately one half-yearyounger when the births occurred than at the time of the survey.Therefore births reported by women between exact ages 15 and20 serve as a basis for estimating fertility rates for women 14.5 to19.5.This slight displacement of age must be allowed for in calcula­ting cumulative fertility.

governed more by decliningfecundability than by customsand institutions. The decline of fecundability with age indifferent populations is likely to follow a roughly similarpattern while no such similarity is found with respect tothe age of entry into cohabitation. These considerationssuggest the hypothesis that the ratio of the average parityof women at the end of child-bearing to the averageparity of a younger group (say women 25-29) is closelyrelated to the relative parity of women early and late intheir twenties. The reasoning behind the hypothesis is asfollows: (a) if the average parity of women 25-29 is anunusually large multiple of the average parity of women20-24, child-bearingdoes not begin early in this populationsince the high ratio implies that fertility is unusually highat ages 22.S to 27.S compared to before 22.S. It followsthat in this population an unusually large fraction of totalfertility occurs in the later years of child-bearing, and theratio of final average parity to the average at ages 2Sto 29is therefore unusually large; (b) on the other hand, anunusually low ratio of parity at 2S-29 to parity at 20-24indicates that high rates of child-bearing began early,that an unusually small fraction of total fertility occursin the later years of child-bearing, and that the ratio offinal averageparity to the average at 2S-29 is unusually low.

Suppose that the average number of children ever born(average parity) to women lS-19 is designated PI, towomen 20-24P2, and so on, until P7designates the averageparity of women 4S-49, Suppose the average parity ofwomen reaching age S0-assumed to be the upper limit ofchild-bearing-is designated TF (for total fertility). Ourhypothesis is, then, that TF/P3 is closely related to P3/PS.The usefulness of this hypothetical relationship, should itprove valid, is that it provides a possible method ofestimating total fertility when older women under-reportthe number of children they have borne, and youngerwomen report parity accurately.

The hypothesis obviously cannot be tested by examiningthe relationships among average numbers of childrenever born reported by women over SO, 2S-29 and 20-24ina number of populations that do not practise birth control,because it is the very inaccuracy of these reports thatleads us to consider the expected relationship. The testemployed is based on the cumulation of age specificfertility rates in a number of apparently reliable fertilityschedules (based on virtually complete birth registration)to construct a set of necessarilyconsistent average parityschedules.Thus the average number of children ever bornwas calculated at ages 20-24,2S-29 and on reaching age SOin a group of women subject to each of the given fertilityschedules. By constructing the average parties in this way,we insured that there would be no distortion from faultyreporting, or changing fertility.The ratios ofTF/P3 and ofP3/P2 are plotted in figure XVI. The relationship isgratifyingly close, and is well represented by simpleequality of the two fractions, or by the formula TF= (P 3)2/P2·

This formula provides an estimate of total fertilityunder the following conditions:

(1) Fertility at ages lS-29 has been constant in therecent past;

(2) The age pattern of fertility conforms to the typicalage relationships found in populations practising little

33

•4.4

4.2

4.0

3.8•

3.6

3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0

•

•1.8

o 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4

P3/p'

Figure XVI. Ratio of TF (total fertility) to Pa (average parity at age 25-29) versus ratio of Ps to Pa(average parity at age 20-24) calculated from fertility schedules of populations not practising birth

control

birth control, implying (a) that the age pattern of decliningfecundability is typical; and (b) that widowhood, divorce,and other forms of dissolution of sexual unions do nothave an unusual age incidence from age thirty to forty-fivein the population in question.

The value of (P3)2IP2 can be compared with the averageparity reported by women over 50, and 45-49. Lowerparity reported by women over 50 than at 45-49 is anindication of likely omission of children ever born byolder women. If under these circumstances (P3)2IP2

exceeds P7, the estimate gains in credibility and is to bepreferred to the numbers supplied by the older women.Approximate equality of P7' the average parity reportedby women over 50, and (P3)2IP2 indicates that any of thethree figures is an acceptable estimate of total fertility.But if (P3)2IP2 is substantially lessthan P7' or if the averageparity reported by women over 50 is much greater thanP7' the wisest course is not to attempt an estimate ofcurrent fertility by manupulation of the data on childrenever born.

B. ESTIMATES OF MORTALITY BASED ON PROPORTIONSSURVIVING AMONG CHILDREN EVER BORN

In some of the censuses and demographic surveys inwhich women are asked to report the total number ofchildren ever born to them, there is an additional questionasking the number of surviving children. It has long beenrealized that the proportion surviving depends on thelevel of infant and child mortality, and census reports inwhich data on surviving children are included often con­tain comments that variations within the enumeratedpopulation in proportions surviving can be consideredan index of differential mortality. Recently WilliamBrass has greatly increased the usefulness of data of thissort by developing a method of translating proportionssurviving and proportions dead among the children everborn to women in different age groups into conventionalmeasures of mortality. His technique makes it possibleunder certain circumstances to estimate the proportionofchildren born who survive to age 1, 2, 3, 5, 10, 15, ... ,35

34

from the proportion reported as surviving among childrenever born to women 15 to 19, 20 to 24, 25 to 29, ... , 60to 64. Because a full account of this technique appearsin another publication4 there is no attempt in thisManual to explain the somewhat surprising correspon­dence between Ix values and data on child survival inwomen's reports of children ever born, nor to justifyin detail the adjustments described below. Our discussionis restricted to a brief outline of the conditions underwhich the technique may be applied, and of the stepsinvolved in constructing estimates.

The Brass method of estimating mortality enables theanalyst to construct the survival function of a life tableup to early adult ages. The conditions that would makethis computation accurate are:

(1) The age specific fertility schedule has been approxi­mately constant in the recent past (at least for the youngerwomen), and the approximate form of the schedule isknown;

(2) Infant and child mortality rates have been approxi­mately constant in recent years;

(3) There is no powerful association between age ofmother and infant mortality or between death rates ofmothers and of their children;

(4) Omission rates of dead children and of survivingchildren are about the same in the reported numbersever born;

(5) The age pattern of mortality among infants andchildren conforms approximately to the model life tables.

Under such ideal conditions it has been shown that theproportion of children dying before their first birthdayis not very different from the proportion dead among thoseever born to women 15 to 19, the proportion dying beforethe second birthday not very different from the proportiondead among the children ever born to women 20 to 24,before the third birthday to the proportion dead amongchildren ever born to women 25 to 29, before the fifthbirthday to the proportion dead among children ever bornto women 30 to 34, before the tenth equal to the propor­tion dead among children ever born to women 35 to 39 etc.These approximations are very close for a populationcharacterized neither by a very early nor by a very latestart in child-bearing. If fertility begins at a very earlyage the children ever born to women in each age groupwould be exposed to more prolonged risks of mortality,and, therefore, the proportion dead would tend to behigher for each age group of mothers than when fertilityhas a later start. Brass has constructed a set of adjustmentfactors that can be used to modify the estimates ofproportions dying defore age 1, 2, 3, 5 and so on, inaccordance with whether the starting point of fertilityin the population in question is early or late. The indexof early or late fertility is the ratio of the average numberof children ever born in the first two age groups of women-ptiP2' In making adjustment for estimates derived from

4 See the discussion in W. Brass, et al., op. cit. For an earlierexposition see W. Brass, "The Construction of Life Tables fromChild Survivorship Ratios", in Union internationale, InternationalPopulation Conference, New York, 1961, (London, 1963), vol. I,pages 294-301.

the proportion dead among the children ever born toolder women, the index of early or late fertility is themean age (iii) or the median age (iii') of the fertilityschedule. These adjustment factors are given in annex V;in table V.l (to be used when children ever born andsurviving are tabulated by the conventional five-year agegroups) and in table V.2 (to be used when the censustabulations are by ten year groups of women: 15-24,25-34 etc.). Examples of estimating mortality by thismethod are worked out in chapter VII.

Of course the conditions specified as ideal are seldom ifever completely fulfilled. Because of the sensitivity of theestimate of the proportion dying before age one topeculiarities or defects in the data the estimate of infantmortality directly derived by this technique does notjustify much confidence. On the other hand, estimates ofchild mortality up to older childhood ages based on thereports by older women of still-living children and ofchildren who have died are especially subject to reportingerrors and also to the effects of the possibly differentmortality levels in the more distant past. The estimatesthat appear to reflect the best chance of minimizing errorfrom various sources are those for the proportion dyingbefore age two and age three. Because of the prevalenceof falling death rates in many of the less developed coun­tries in recent years, it should be borne in mind that thelife table value estimated by these procedures representsthe average mortality experience over the preceding fouror five years in the determination of the proportion deadbefore the second birthday, and in the preceding six toeight years, in the estimate of proportion dead before thethird birthday.

The ideal conditions listed above were not fully realizedin the population of Hungary before the censuses of1930, 1949 1960; nor in the population of Canada justbefore the census of 1941. Mortality was changing, theage pattern of mortality in Hungary does not conformto the "West" pattern, and fertility was not constant.Questions in the censuses concerning children ever bornand surviving were limited to married women only.Nonetheless, when estimates are made by the Brasstechniques they come remarkably close to estimating theproportions dying before age two in the four or five yearspreceding each of the censuses, even though the rangeof mortality estimated is from a high of over 200 perthousand to a low of less than 70 per thousand, as shownin table 3. Columns 2 and 3 of table 3 were obtained bycalculating the average level of infant mortality in thegiven country and time for a four and five year periodbefore the given census and then using the relationshipbetween lqO and 2QO as shown in the nearest official lifetables to estimate the proportion dying before age two.Column 4 is derived from the reported proportionssurviving to women aged twenty to twenty-four, using themultipliers of annex table V.I, selected according to thereported PtiP2 ratios.

When the potential sources of bias are considered, it isevident that the estimation of child mortality by thisprocedure tends almost always to err on the low side,if the estimate is not accurate. The presumption of biasin this direction is based on a judgement that respondentsare much more likely to omit from a summary of their

35

From vital reglstratloll data

Average 0/4 Average of j Estimated/rom censusCensus years preced/llg years precedlllg survivorship

the census the census rates

TABLE 4. INFANT MORTALITY RATES (tqo) ASSOCIATED WITH aqoVALUES THAT WERE ESTIMATED FROM PROPORTIONS OF CHILDRENSURVIVING AS REPORTED IN VARIOUS CENSUSES AND AVERAGEINFANT MORTALITY RATES DURING THE FOUR YEARS PRECEDINGEACH CENSUS AS REGISTERED IN VITAL STATISTICS

6 Concerning the possibility of obtaining such information fromcensus reports on the proportion of orphans in a population, seehowever Louis Henry, .. Mesure indirecte de la mortalite des adultes",Population, 1960, No.3, pages 457-466.

experience to date children who have died than those whohave survived.

Estimates of infant mortality have been obtained in anumber of populations by first applying the methodhere described to detmine the proportion dying before thesecond birthday and then assuming that this proportionis related to infant mortality as in the "West" modellife tables. It is remarkable in view of the expected down­ward bias that these estimates exceed (often by a majorextent) the average level of infant mortality derived fromregistered births and deaths in the four years before thesurvey in the countries shown in table 4. This comparisonindicates the widespread prevalence of under-registrationof infant mortality.

The demonstrated accuracy of the Brass method ofestimating infant and child mortality in Hungary andCanada, in conjunction with the fact that it appears inmany of the less developed countries to provide an estimatemuch closer to actuality than can be obtained fromregistered data suggests that this method will prove apowerful and welcome addition to the techniques availableto demographic analysts. Unfortunately, the method doesnot provide information about adult mortality." Toestimate the expectation of life at birth or the crudedeath rate from these childhood survival rates, it isnecessary to make an assumption about the relationshipbetween death rates at different ages. It is a simplemechanical procedure to select a model life table havingthe same proportions surviving to age two as is indicatedby this technique of estimation. However, there is littlebasis for confidence that the relationship between mortalityrates at different ages in the "West" model tables holdsclosely in a population in Africa, Asia or Latin America .

.162

.131

.119

.050

.212

.115

.093

.186

.202

.114

.069

.095

.080

.129

.133

Estimates of lqOobtained from ,qOaccording to"West" mode/llfetable" based allcensusreports

.161

.119

.097

.030

.152

.067

.051

.114

.1280

.065

.049

.038

.023

.112

.117

Av#rageQ/'lqO IIIthe fort year, prece­dlllg the cellSUS,accordlllgto vitalregistrat1011 data

Hungary1930 ............ .204 .203 .2031949 ............ .138 .133 .1371960 ............ .063 .063 .067

Canada1941 ......... , .. .069 .071 .072

Barbados 1946British Guiana 1946Brunei 1960Cyprus 1960Egypt 1947Fiji Islands 1946Fiji Islands 1956North Borneo 1951Peru 1940Sarawak 1960Seychelles 1960Western Samoa 1956Western Samoa 1961Windward Islands .. 1946Yugoslavia 1953

TABLE 3. PROPORTIONS DYING BBFORE AGB 2 (VALUES OF 2QO) IN

SPBCIFIED PBRIODS AS DBRIVED FROM VITAL REGISTRATION DATA

FOR HUNGARY AND CANADA, AND AS ESTIMATED FROM PRO­PORTIONS OF CHILDREN SURVIVING REPORTED IN CBNSUSES OF

THE SAME COUNTRIES

Coulltry Year

a 1940 only.

36

T

Chapter III

ESTIMATES OF FERTILITY AND MORTALITY BASEDON REPORTED AGE DISTRIBUTIONS AND REPORTED CHILD SURVIVAL

In chapter I there are described methods of estimatingfertility and mortality that make use of the age distributionof a population recorded in one or more enumerations. Asecondenumeration providesvaluablesupplementaryinfor­mation in the form of the rate of increase and survivalrates. When there has been only one usable enumerationof a population, only very rough estimation is usuallypossible. However, in chapter II a technique of cal­culation is outlined that makes it possible to determineproportions survivingfrom birth to age, 2, 3, 5 and some­times to ages up to 30 or 35 from data supplied by wom­en about the history of the children they have borne.

Knowledge of childhood mortality is an extremelyuseful adjunct to knowing the age composition of apopulation, particularly for the estimation of fertility.In fact, responses about the survival of children ever bornare probably more useful in estimating recent fertilityfrom the age composition recorded in a census or surveythan is the existence of an earlier enumeration. Speci­fically, if C (x) is known, 12 is a more useful supplementarydatum than r, the rate of natural increase, in estimatingthe birth rate. On the other hand, knowledge of 12 (withor without the age distribution) gives only inferentialevidence about adult mortality (i.e., mortality above agefive). From data on childhood mortality, mortality aboveage five can be estimated only on the basis of assumedregular relations between mortality at different ages, andin different populations the relation of child mortalityto adult mortality varies substantially. For example, inthe four families of model life tables expressing averagemortality patterns in different "regions", expectations oflifewere calculated at age 5 ranging from forty-six years toover fifty-three years associated with the same proportion(about 75 per cent) surviving from birth to age two.

Two enumerations spaced five or ten years apart, onthe other hand, provide a good indication of the level ofadult mortality, but no direct evidence on childhoodmortality. Expectations of life at age five estimated frommodel stable populations with a given C (30) and a givenrate of increase vary only from 51.7 to 52.4 years whenbased on the different "regional" model tables; the esti­mated level of mortality above age five in this instance isessentially independent of variation in age pattern. Butwhen rand C (x) are known, childhood mortality mustbe approximated by assuming some kind of "normal"association between child and older age death rates. Itwill be seen in each of the techniques presented in thischapter that a solidly based figure for child mortalityis needed for any precision in an estimate of the birth ratederived from an enumeration of a population.

A. ESTIMATION OF BIRTH AND DEATH RATES FROM can.o­HOOD SURVIVAL RATES AND A SINGLE ENUMERATION BY

REVERSE PROJECTION

An accurate census that records the number of personsin each five-year age interval provides the basis forreconstructing recent birth and death rates, if migrationeither is known accurately or is negligible in magnitude,and if survival rates by age are known for recent periods.The method of estimation is simply to reversethe custom­ary procedures of population projection-to reconstructby reverse survival the births that brought into being thechildren recorded as under age five or ten, and to recon­struct the population among which the births occurredby reverse survival of persons over fiveor ten in the census.

The specific steps employed in reverse projection are todivide the population under five by sLo/5 x 10 from a lifetable representing the mortality of the preceding fiveyears to obtain an estimate of births, and to divide eachfive-year age group by the appropriate survival factorfrom the same life table to reconstruct age group by agegroup the population five years before. The sum of allsuch estimated age groups is the estimated total populationfive years earlier. The estimated average annual number ofbirths can then be divided by the average of the enumer­ated population at the end and the estimated populationat the beginning of the period to give an average birthrate during the preceding five years. The average annualrate of increase (l/510ge Pt/Pt-s) can be subtracted fromthe birth rate to estimate the death rate. Similarly, thepopulation 5-9 can be projected back to provide theestimated birth rate in the next earlier five years, etc.The estimated total population becomes subject to increas­ing uncertainty as the reverseprojection procedes,however,even if mortality is somehow accurately known: the oldestage group in the past population has no current survivors,and this segment of the past population must be estimatedby some assumption about the nature of the past agedistribution. For earlier and earlier dates, the portionof the population estimated in this way is larger and larger.

Given an accurate census, the crucial additionalelement for reverse projection is an appropriate life table.The Brass method of estimating child survival providesapproximate values of 12 (during the preceding four orfive years) and 13 (during the preceding six or eight).A model life table can be selected with the given 12 , andsurvival factors from this table employed for the reverseprojection. The value of sLo in the model table is veryclose to the correct one, if the data from which 12 was

37

estimated are accurate. Differences in age patterns ofmortality do not much affect the ratio of 12 to sLo• Onthe other hand survival rates above age five are estimatedby assuming that the age pattern of mortality conformsto the "West" familyofmodellife tables, and if one judgesby differences among life tables based on accurate data,the actual survival rates above five may diverge from the"West" family.However,differences in mortality above agefive in life tables with a given 12 would rarely produceestimates of over-all population 2.5 years earlier differingby more than one per cent, so that the error in the esti­mated birth rate from this source would rarely exceedhalf a point (e.g., an estimate of 50.5 per thousand in­stead of 50).

If the questionson children everborn and survivinghavebeen asked and recorded separately for males andfemales, separate estimates of child mortality can be madefor each sex and (if the internal consistency of the data isacceptable) the model life table selected for each sex canbe based on this separate evidence. However, the questionson children ever born are typically asked (or tabulated)only for both sexes combined. One is tempted to derive,from values of 12 and 13 for the two sexes together;:estimates for each sex on the assumption that the relationof female to male child mortality in the given populationis the same as in the population whose experience under­liesthe "West" model tables. The typical relation betweenmale and female mortality in these populations is thatmale and female life tables tend to be at about the samelevel; i.e., when 12 for females is .72765 (level7, °eo = 35.0),the typical 12 for males is .69537 (also level7, °eo = 32.48).It is easy to construct a table containing the combined l«values for x = I to 5 for both sexes (assuming a typicalsex ratio at birth of 105 males per 100 females) at each"level", and then to assume that male and female mor­tality is expressed by the life table for each sex at the"level" indicated by the value of 12 and 13 for the twosexes together. However, the evidence available on sexdifferences in mortality in the less developed countriesdoes not warrant the assumption that these differencesalwaysconform to the relations found in the experience­primarily from Europe, North America, and Oceania­underlying the model life tables. It is possible to find1

many examples of female mortality higher than malemortality and this resort to male and female model tablesat the same level may introduce a mortality differentialopposite to the actual one.

If the age distributions of both sexes are about equallyusable as a basis for estimating fertility, the uncertainty ofsex differences in mortality can be ignored, and the valuesof 12 or 13 estimated for the two sexes combined can beemployed as if it were a valid estimate for each sex sepa­rately. If in fact the sex differences in child mortality aresubstantial, estimates based on the common value of 12or 13 will overstate the birth rate and death rate for onesex, and understate the rates for the other, but provideunbiased figures for the whole population. In general,however, the reliability of estimation is greater when

1 Pravin M. Visaria, The Sex Ratio of the Population of India(unpublished doctoral dissertation, Princeton University, 1963).Available at University Microfilms, Ins., Ann Arbor, Michigan.

based on age data from one sex than when based on theother. It is then necessary to make some rough estimateof the 12 or 13 implied for males and females by 12 or 13for the two sexescombined. Often there is indirect evidenceindicating the direction and approximate extent of sexdifferences in mortality: the sex ratio of a closed popula­tion that has not experienced large sex selective militarydeaths indicates the sex incidence of mortality since thesex ratio at birth can often be closely estimated; and thesex ratio of mortality in registration areas, in samplesurveys, or even in neighbouring populations can betaken as relevant evidence.

Reverse projection cannot be recommended as agenerally satisfactory basis of estimating birth rates evenwhen calculation of child survival rates is possible becauseof the frequent unreliability of recorded age distributions.The tendency for the proportion under five to be under­reported in many censuses and surveys has often leddemographers to base estimates of the birth rate on thereverse projection of persons five to nine. However, theproportion of the population five to nine is frequentlyoverstated by a wide margin (because of understatementof the ages of adolescent girls, for example, and becausethere is a common tendency to overstate the age of somechildren under five), so that this procedure cannot beendorsed as always valid. In fact, if there is evidence ofsubstantial age-misreporting, it is not possible to makegood use of reverse projection unless some means isavailable for identifying a valid part of the reported agedistribution or of adjusting the reported figures.

B. ESTIMATION OF BIRTH AND DEATH RATES FROM CHILDSURVIVAL RATES AND A SINGLE ENUMERATION BY MODELSTABLE POPULATIONS

The use of model stable populations to estimate charac­teristics of a population requires the identification of astable population among the tabulated age distributionsthat shares some of the observed or inferred characteristicsof the recorded population.' In chapter I, the identifyingfeatures used to locate a model population were theintercensal rate of increase (assumed equal to the rate ofgrowth of the stable population), and the cumulativeage distribution or ogive up to some age that depends onthe apparent pattern of age-misreporting. Under theconditions considered in this section, a model stablepopulation is identified by the estimate of 12 , which bymeans of table 1.2 in annex I determines the level ofmortality, and by the ogive, which is used in a mannerwholly analogous to the procedures outlined in section Bof chapter I. For example, if the age distribution is of theIndian-Pakistani-Indonesian-African pattern, a minimumestimate of b can be obtained from C(lS), and a lessbiased estimate from the value associated with C(35).

a As noted earlier, the identification of a model stable populationdoes not determine unique value of the gross reproduction rate orof total fertility. To estimate these quantities, the mean age of thefertility schedule must first be estimated. For a discussion of thistopic, see section B.5 in chapter I.

38

Two features of estimation by model stable populationsselected by /2 and C (x) are worth noting: first, if an esti­mate of the birth rate is based on C (5), the result isessentially identical with the results obtained by reverseprojection, whether or not the population is stable; andsecond, estimates of the birth rate obtained from 12and C (x) are insensitive to differences in age pattern ofmortality, at least the differences found in the fourfamilies of model tables.

The birth rate in the model stable population with thesame /2 and C (5) as the observed population is identicalto that which would be obtained by applying reverseprojection to the children under five in the stable popu­lation to obtain births, and reverse projection to thewhole stable population to obtain an average number ofpersons (the denominator of the birth rate). The numberof births estimated for the actual and the stable popu1a­tions is identical, so that the only source of differencebetween the birth rate estimated by reverse projectionand that found in the model stable population is in thedenominator, which in each case is a number obtainedby applying reverse projection, with the same life table, topopulations with the same number of persons, and thesame number over and under five, but possibly differingin the internal age structure above age five. This pointloses relevance as the choice of a model stable population ismade dependent on C(lO), C(15) and ogives to higherages, because with possibly different internal age compo­sition in the population (e.g., under fifteen) that isimplicitly or explicitly projected back to birth, and withdifferences in the size of the reverse projected denominatorbecoming more pronounced as the time period of reverseprojection is extended, the virtual identity of the twoestimates is lost. This feature does imply, however, thatinterpolation in stable populations is a convenient way ofdetermining the birth rate implied by reverse projectionof children under one, under five, one to five, five to tenor under ten.

The insensitivity of birth rate estimates from 12 and C (x)to differences in age patterns of mortality is an importantadvantage of such estimates. The advantage lies in thefact that the true age pattern of mortality.is usually notknown, and if estimates based on alternative plausibleage patterns are widely different, the range of uncertaintyis great. This point is discussed in greater detail inchapter IV.

C. ESTIMATION OF BIRTH AND DEATH RATES FROM CHILDSURVIVAL RATES AND AGE DISTRIBUTION IN A POPU­LATION ENUMERATED SEVERAL TIMES

When a population has been enumerated more thanonce at an interval of about five or ten years, the methodsof estimation presented in chapter I can be applied, andat first thought the mortality estimates derived from dataon survival among children ever born might be consideredmerely as a verification of the mortality estimated on thebasis of C (x) and r, or C (x) and a model life table consis­tent with fractions surviving from one census to the next.However, the best use of such data is to accept theestimates they provide of mortality under age five, and

of any other population parameters dependent onmortality in this age range, and to accept the estimates ofmortality over age five that can be derived from ananalysis of the two censuses-either by survival analysis,or by accepting the mortality above age five consistentwith C (x) and r. Of course such general advice is con­tingent on the quality of the basic figures. If two censusesare unequal in completeness of coverage, or if interna­tional migration is substantial and inadequately recorded,it would be necessary to rely more on inferences thatcould be based solely on age distribution and estimatedchild survival. Similarly, if internal inconsistencies wereapparent in the reports of children ever born, the estimatesof child survival might not be usable.

1. Estimation of birth and death rates in a non-stablepopulation

Suppose a population is enumerated in censuses at thebeginning and end of a decade, and that the secondcensus includes data permitting the calculation of 12and 13 , The best estimate of the over-all life table isobtained by accepting °es (and the mz and qz values forage five and over) from the model life table chosen asbest fitting census survival rates, and by taking Is (and51110) from the model life table with the value of 12 or 13obtained by the Brass methods. The expectation of lifeat birth in this hybrid model table is easily calculated.The average death rate for the decade is then calculated.by applying the m, values in this hybrid life table to arough mid-decade age distribution obtained by averagingthe distributions at the beginning and end. The birth ratecan then be estimated as equal to the death rate plus theaverage annual rate of increase.

2. Estimation ofbirth and death rates in a stable populationenumerated more than once

If the popu1ation enumerated in two or more censuseshas closely similar age distributions in each census, themethods just described can be supplemented by thefollowing procedures: (a) estimate the birth rate byselecting a model stable population from 12 and C(x);and (b) estimate the death rate as the birth rate. less theintercensal rate of increase. A slightly more elaborateprocedure may be applied when an estimated life table,an adjusted age distribution, and other detailed parametersare sought. This procedure recognizes the superiorityof the Brass estimation procedures for determiningchildhood mortality, and of stable population techniquesusing C (x) and r to estimate mortality above age five. Theprocedure entails: (a) selecting a model stable populationfrom 12 and C (x), and accepting the proportion underfive and the child death rate in this population, and (b),selecting a model stable population from C (x) and r,and accepting the age specific death rates above five,and the age distribution within the span above five in thispopulation. The over-all death rate is then estimated as thesum of the death rate under five in (a) times the proportionunder five in stable popu1ation (a) plus the death rateover five in (b) times one minus the" proportion under 5in stable population (a).

39

D. ADJUSTMENT OF ESTIMATES OF FERTILITY DERIVED FROM

CIDLD SURVIVAL RATES AND THE AGE DISTRIBUTIONWHEN MORTALITY HAS BEEN DECLINING

If the population for which childhood survival can beestimated appears to have experienced declining mortalityduring the recent past, the stable estimates of the birthrate and of the gross reproduction rate (assuming that anestimate of the mean age of the fertility function has been

40

obtained previously) can be adjusted to allow for theeffects of declining mortality on the age distribution. Theprocedure is the same as that described in section C ofchapter I, using the adjustment factors from the appro­priate part (part (c) or (d)) of table IILl in annex III.If questions on child survival were asked in two consecu­tive censuses,it is also possible to estimate the parameter kby means of table IlIA.

Chapter IV

ACCURACY OF ESTIMATION

The extraction of approximate birth and death ratesfrom censuses of varying completeness, in which age,parity and other relevant data are inaccurately or incom­pletely reported, can scarcely be expected to producefigures of great precision. Moreover, the range of errorin the estimates about the true figure cannot in general bedetermined at all exactly. The purpose of this chapter is tocall attention to the imprecision of estimates based ondata of poor quality, even when they have been madewith the help of elaborate tables and adjustments; togive some rough impression of the magnitude of errorthat is routinely encountered; and to distinguish theforms of estimation most subject to large errors fromthose less susceptible.

The sources of error discussed are of two principalkinds: errors caused by a discrepancy between actual andassumed conditions, and inaccuracies in the basic data.Crucial assumptions that may be wrong are that the agepattern of mortality in a given population conforms to afamily of model life tables, and that the age compositionof a population has a form stable in the sense of Lotka.The kinds of inaccuracy that affect almost all forms ofestimation are omission of persons from censuses andsurveys and age-misreporting. The effects of these formsof imprecision are illustrated in this chapter principallyby synthetic estimates in which each source of error isassumed to operate in the absence of others.

A. DIFFERENCES BETWEEN ASSUMED AND ACTUAL CON­

DITIONS

1. Errors arising from differences between the actualage pattern ofmortality and that embodied in the modellife tables

There is no conclusive way of delineating the errorsthat might arise because populations with incompleterecords may have age patterns of mortality that differfrom those with full records. Even when there are recordeddata from which age schedules of mortality can be calcul­lated, it is often uncertain whether an extreme pattern ofmortality - e.g., unusually low infant mortality, given theprevalent death rates above age one - is genuine, or theproduct of unusual inaccuracy in the data rather than anunusual pattern of death rates.

In this section the errors that originate in age patternsof mortality different from the model life tables of annex Iare illustrated by examples in which estimates based onalternative families of model life tables are compared.

The four families are described in chapter I. The distinc­tive age patterns of mortality they embody are examplesof differences found among populations with especiallyaccurate data, and undoubtedly do not nearly exhaust thevariety of patterns to be found among all populations.Nevertheless, it can safely be assumed that forms of esti­mation that yield nearly identical figures in all fourfamilies are insensitive to age pattern differences, and areon that account preferable to forms of estimation thatyield divergent figures. The divergence itself can be takenas a minimum index of the uncertainty of estimationassociated with variations in age patterns; it is obviousthat variations at least this large do occur.

The effectson various estimates of different age patternsof mortality will be illustrated by utilizing the fourfamilies of model life tables to make calculations for apopulation that is assumed recorded without error. The"test" population is a "West" female stable populationwith an expectation of life at birth of forty years, a grossreproduction rate of 3.00, and mean age of fertility oftwenty-nine years. The various characteristics of thispopulation can be obtained from the appropriate table inthe model stable populations. To illustrate the importanceof the assumption that the population has a given agepattern of mortality, estimates are made with all four agepatterns. The variation among the four figures is thesignificant result: the fact that the estimate based on the" West" tables always agrees with the "true" figure is ofcourse a purely automatic consequence of how theexample is formulated.

(a) Effects of assumed age patterns of mortality onestimates derived from census survival rates

In section A of chapter I, the reader can find a descrip­tion of how to select a model life table consistent with thenumbers recorded in two censuses taken at a ten yearinterval: project the first population by life tables atvarious levels of mortality and by interpolation find thelevel that matches the recorded total over ten, over,fifteen etc. It is suggested that the median of the first ninelevels so indicated is a sensible choice. By summing theproducts of the age specific death rates from the lifetable so determined and of the average intercensalnumber of persons in the corresponding age groups oneobtains an estimate of the death rate. This death rateadded to the average annual rate of growth during theintercensal period yields an estimate of the birth rate.The procedure at no point makes use of the assumptionof stability.

41

TABLE 5. ExPECTATION OF LIFE AT AGES 0 AND 5IN VARIOUS FAMILIES OF MODEL LIFE TABLES PRODUCING

A PROJECTED POPULATION OVER AGE X MATCHING THE TEST POPULATION AT THE END OF A DECADE

Expectation of II/e at birth Expectation of II/e at age 5

AgBX '·West" "North" "East" ·'South" "West" "North" «East" ,.South "

10 ................. , .. 40.0 40.5 36.3 37.2 49.7 49.8 50.1 50.615 •••••••• 0 ••••••••••• 40.0 38.6 35.5 35.0 49.7 48.6 49.6 49.220 .................... 40.0 37.7 35.6 34.5 49.7 47.9 49.6 48.925 ••••••••••• 0.0 •••••• 40.0 37.5 35.9 34.3 49.7 47.8 49.8 48.830 .................... 40.0 37.6 36.2 34.0 49.7 47.9 50.0 48.635 .................... 40.0 37.7 36.4 33.8 49.7 47.9 50.1 48.540 .................... 40.0 37.7 36.8 33.8 49.7 47.9 50.4 48.545 .................... 40.0 37.5 37.4 34.1 49.7 47.8 50.7 48.750 .................... 40.0 37.3 38.1 34.7 49.7 47.7 51.1 49.055 .................... 40.0 37.3 39.1 35.6 49.7 47.7 51.7 49.760 .................... 40.0 37.6 40.3 37.0 49.7 47.9 52.3 50.565 .................... 40.0 38.1 41.6 38.6 49.7 48.2 53.1 51.570 ................. , .. 40.0 38.7 42.9 40.3 49.7 48.6 53.8 52.675 .................... 40.0 39.1 43.8 41.9 49.7 48.9 54.3 53.6

Median of first 9 ....... 40.0 37.7 36.3 34.3 49.7 47.9 50.1 48.8

TABLE 6. PARAMETERS ASCRIBED TO THE TEST POPULATION (WEST MODELSTABLE, 0eo = 4O,GRR = 3.00),BY SELECTING THE MEDIAN LEVEL MODEL LIFE TABLE FROM EACH FAMILY FROM AMONG TABLESMATCHING THE PROPORTIONS SURVIVING IN TWO CENSUSES

Est/mated by median model life table

Estimated parameter

°eo .°es ./2 ./s .Death rate .Birth rate .Death rate of population under age 5 .Death rate of population over age 5 .

"West"

40.049.7

.773

.725

.0234

.0445

.0720

.0137

"North"

37.747.9

.778

.704

.0252

.0463

.0769

.0149

"East"

36.350.1

.701

.654

.0276

.0487

.0976

.0137

"South"

34.348.8

.706

.629

.0290

.0501

.1019

.0146

Suppose that our hypothetical population (" West"model stable female population, °eo = 40 years, ORR= 3.00, iii = 29) were enumerated at the beginning andend of a decade, and projections made employing variouslevelsof the four sets of model life tables.

Table 5shows°eo's and °es's in the model life tables thatproduce the numbers over age 10, 15, 20 etc. in the" actual" population at the end of the decade. Table 6shows various parameters that would be ascribed to thetest population by assuming that the median life tableindicated in the last row in table 5 represented the popu­lation's mortality schedule.

The differences in age pattern among the four familiesproduces estimates of over-all mortality (estimated deathrate and expectation of life at birth) and mortality underage five (12 , 15' and death rate under five) that are muchmore divergent than the estimates of mortality in thepopulation 5 and over °es and death rate over five). Theuniformity diminishes among estimates based primarilyon survival at the older ages. As a consequence of the

differences in the estimated population death rates,arising primarily from differences in the estimated mor­tality under age five, the estimated birth rates differ bysome 5.6 points or 11 per cent of the largest estimate.Note that when there is no evidence of the age pattern ofmortality nor separate indications of child mortality,the assumption of the" West" pattern of mortality pro­duces low (or conservative) estimates of birth and deathrates.

(b) Effects of assumed age patterns of mortality onestimates derived from stable populations chosen onthe basis ofC (x) and r

If the test population conforming exactly to the Westmodel stable with °eo= 40 years and ORR = 3.00 wereenumerated twice, the intercensal rate of increase could becalculated, and model stable populations found matchingthe given population in the proportion under age x andin the rate of increase. Table 7 shows the expectation oflife at birth and at age five in model stable populations

42

TABLE 7. EXPECTATION OF LIFE AT AGES OANDS IN MODEL STABLE POPULATIONS BASED ON VARIOUS

FAMILIES OF MODEL LIFE TABLES THAT MATCH THE TEST POPULATION IN PROPORTION UNDER AGE XAND IN THE RATE OF INCIUlASIl

EJq¥ctatlon of life at birth Expectation of life at age 5

Age x "West" "North" "East" "South" "West" "North" "East" "South"

5 .0 ..•••••.••...••••• 40.0 42.2 36.8 39.5 49.7 50.9 50.4 52.110 .0 •••••••••••••••••• 40.0 41.3 36.1 37.9 49.7 50.3 50.0 51.115 .................... 40.0 40.4 35.7 36.9 49.7 49.7 49.7 50.520 .................... 40.0 39.6 35.5 36.3 49.7 49.2 49.6 50.125 .................... 40.0 39.2 35.4 35.8 49.7 48.9 49.6 49.730 '0' ••••••••••••••••• 40.0 38.9 35.5 35.3 49.7 48.7 49.6 49.435 .0' ••••••••••••••••• 40.0 38.7 35.5 35.0 49.7 48.6 49.6 49.240 .................... 40.0 38.5 35.6 34.7 49.7 48.5 49.7 49.045 .................... 40.0 38.4 35.8 34.5 49.7 48.4 49.8 48.950 •••••••••••••••••• 0. 40.0 38.2 36.1 34.5 49.7 48.3 50.0 48.955 .................... 40.0 38.1 36.6 34.6 49.7 48.2 50.3 49.060 .................... 40.0 38.0 37.3 35.1 49.7 48.2 50.7 49.365 .................... 40.0 38.0 38.2 35.9 49.7 48.2 51.1 49.8

Median of first 9 ....... 40.0 39.2 35.6 35.8 49.7 48.9 49.7 49.7

IIItTHRATI

.070

.060

"SOUTH"

.=-~-----------~~----.050

.040

.030

.020

.010

"EAST"~ ",.,........

.""".-... -NORTH" ._._._._._._._~-- ~._.-._.~._._._._.-

:.,;,::;;:,;,: W.£5i' 0 •••••••••0••

10 III 20 211 30 311 40 'Ill !$O 115 60 65AGE

Figure XVII. Birth rates in the test population derived by the stable population method fromG(x) and r assuming various patterns of mortality as the appropriate one

43

TABLE 8. PARAMETERS ASCRIBED TO THE TEST POPULATION BY SELECTING THE MODEL STABLE POPULA­

TION FROM BACH FAMILY WITH THE MEDIAN BIRTH RATE FROM AMONG THOSE WITH THE SAME rAND C(S), C(10), ... , C(4S)

Estimated from median model stable population

Estimated porQllUlter "West" "North" "East" "South"

°eo .................................... 40.0 39.2 35.6 35.8°e5 .................................... 49.7 48.9 49.7 49.712 .0 ••. · .•••••...••••••.•••••••••.••••• .773 .790 .694 .719/5 ..................................... .725 .720 .646 .646Death rate ............................. .0234 .0244 .0287 .0287Birth rate ••••••••••••••••••••••• 0 •••••• .0445 .0455 .0498 .0498ORR (fli = 29) ......................... 3.00 3.11 3.36 .342Death rate of population under age 5 ..... .0720 .0724 .1030 .0992Death rate of population over age 5 ....... .0137 .0144 .0136 .0138

(based on various families of model life tables) thatduplicate the rate of increase and the proportion underage 5, 10, 15 etc. in the test population. Figure XVIIshows the birth rate in these stable populations. Table 8shows various parameters that would be ascribed to thetest population by assuming that the model stable popu­lation with median fertility shown in figure XVII wasrepresentative of the test population.

The same features of variation in estimation are seenin these tables as in the preceding two. Again estimatesof mortality above age five are insensitive to differences inage pattern while mortality estimates below age five andother measures strongly affected by infant and earlychildhood mortality, such as the over-all death rate or theexpectation of life at birth, are markedly influenced.

(c) Effects of assumed age patterns of mortality onestimates derived from reported child survival incombination with records of the age distributionoriginating from one census, or from two or morecensuses

Assume that the test population has been enumeratedin a census that includes questions about the number ofchildren ever born to each woman, and the number stillalive at the time of the census, tabulated by sex of thechild and age of the woman. By methods described insection B of chapter II, proportions surviving to agetwo and age three (/2 and 13) can be estimated for females,and (as outlined in section B of chapter III) these valuescan be used to select a model life table, and with C(5),C(lO) etc., to select model stable populations at theindicated mortality level. Table 9 shows 15' °eo, and °esselected on the basis of 12 according to each of the"regional" patterns of mortality. Figure XVIII indicatesthe birth rate in the stable population based on theseregional patterns with the given 12 and the proportionunder age 5, 10, 15 etc., in the test population. Table 9also includes various parameters of the "median" modelstable populations selected on the basis of the fertilityshown in figure XVIII. Note that the patterns of estimatedbirth rates shown in figure XVIII are much more uniformthan in figure XVII, and that, therefore, the estimationof the birth rate from C(x) and 12 is much more nearly

independent of mortality pattern than when the estimatesare based on C (x) and r. On the other hand parametersshown in table 9 that measure mortality above age fiveare directly dependent on the assumed age pattern ofmortality, since the only observed quantities relate tochild mortality. Thus estimates of °es and the death rateof persons over five have a much larger range than intables 6 and 8.

Now assume that the test population had been enumer­ated in two censuses a decade apart, and that the secondcensus included the data needed to calculate 12 , It is nowapparent that 12 gives estimates of the death rate underage five that are less sensitive to age patterns of mortalitythan estimates based on age composition, and that on theother hand, estimates of the death rate over age five basedon age composition (either census survival or C(x) and r)are less sensitive to differences in mortality patterns. It istherefore recommended (cf. section C of chapter III)that parameters related to child mortality be estimatedfrom 12 and those related to adult mortality from the agecomposition; from census survival in the general case orfrom the intercensal growth rate and the (stable) agedistribution when stability may be assumed. To illustratethe advantages of this procedure in terms of increasedprecision, and in particular to illustrate the increasedinsensitivity of the resulting estimates to differences inmortality patterns, the following calculations were made:

(a) Death rates estimated from census survival (ages fiveand over), and from 12 (ages under five), and b estimated asd--r. A model life table was selected on the basis of bestagreement between the projected population and thepopulation recorded in the second census (the medianlevel in table 5), and the smx values in this model table wereapplied to the recorded (test) population above age fiveto obtain the death rate over five. Death rates under agefive (tmO and 4ml) from the life table selected on thebasis of 12 (cf. table 9) were applied to the test populationat these ages. The expectation of life at birth was calcu­lated as sLo+°es x 15110 where sLo and 15 were taken fromthe life table associated with 12 and °es was taken from thelife table obtained for the population over age five.

(b) Death rates estimated from stable population (agesover five), and from 12 (ages underfive), b estimatedas d +r.

44

I ••,HRATE

.0701-

.06(1-

.0501-

---:':....................................•..,..~ .~..~...•....;,;::..._._._._._._.!II'lI'I'.~_._.- ._._._.-

·WES. • ·ST•

.040f.-

.030 f.-

.020f.-

.010 -

o 10

I • •

I 5 20 25 30 35 40 45 50 55 60 65AGE

Figure XVIII. Estimated birth rates in the test population derived by the stable population methodfrom C(x) and la assuming various patterns of mortality as the appropriate one

TABLE 9. PARAMETERS ASCRIBED TO THE TEST POPULATION BY SELECTING THE MODEL STABLE POPULA­TION FROM EACH FAMILY WITH THE MEDIAN BIRTH RATE FROM AMONG THOSE WITH THE SAME 12AND C(5), C(10), ... , C(45)

Estimated from median model stable population

Estimated parameter

°eo .°e5 .15 ...........•.................•......•Death rate .Birth rate .Growth rate .GRR (m = 29) .Death rate of population under age 5 .Death rate of population over age 5 .

"West"

40.049.7

.725

.0234

.0445

.02II3.00

.0720

.0137

45

"North"

37.047.5

.697

.0265

.0465

.02013.18

.0797

.0154

"East"

44.054.4

.736

.0207

.0451

.02443.07

.0697

.0109

"South"

42.654.1

.716

.0221

.0461

.02403.19

.0730

.0113

The death rates for ages over fivewere obtained by select­ing the life table underlying the stable population deter­mined from C(x) and r (the median level in table 7);otherwise the same procedure was followed as in (0).1

(c) Birth rate estimated from stable population, deathrate as b-r. The birth rate was obtained from C(x)and 12 (table 9), and the death rate was calculated as thedifference between this birth rate and the reported inter­censal rate of increase.

Table 10showsthe population parameters calculated bythese methods. Method (a) is applicable whether thepopulation is stable or not, but (b) and (c)can be employedonly with stable populations. The most striking featureof this table is the small variability in the estimate of suchover-all characteristics as the birth and death rates andthe expectation of life at birth caused by differences inage pattern of mortality when mortality above age fiveis derivedfrom the age distribution and intercensal changeand mortality under age five from reported child survival.It is clear how valuable is the supplementary informationprovided by data on proportions survivingamong childrenever born.

2. Errors caused by non-stability of a population assumedto be stable

If the age distribution of a population conforms closelyto that of a stable population, estimation of manycharacteristics is greatly simplified, especiallythrough theuse of tabulated model stable populations. The methodsof selecting an appropriate model population are givenin chapters I and III; and in the preceding discussion inthis chapter the errors that may arise because of variationsin age patterns of mortality are explained. Another sourceof error is that the population in question may not in

1 In a less artificial example of the application of the stablepopulation method the life table death rates would be applied notto the reported age distribution but to the stable age distributionas explained in section C.2 of chapter III.

fact have the age distribution of a stable populationbecause of age-selective migration, or past variations infertility or mortality.

Any extended discussion of the deviations in stabilitythat can and do occur is beyond the scope of this Manual.Only a few comments about general principles and fre­quently encountered cases will be attempted.

Ideally, stable estimation should be employed only fora closed population with constant mortality during thepreceding 25-30years, and constant fertility for some twogenerations. A useful practical test is the absence ofsubstantial change in age composition and of intercensalrate of increase in three consecutive quinquennial ordecennial censuses. For example, examination of the agedistributions in Turkey from 1935 on reveals clearly(in spite of conspicuous distortions caused by age-mis­reporting) that fertility was greatly reduced during certainperiods since 1910: the evidence is a low point in the agedistribution that does not remain at the same age fromone census to the next as it would if age-misreporting werethe cause of the low point, but rather advances by fiveyears in each subsequent quinquennial census. In contrast,the Indian age distributions from 1891 to 1911, also irreg­ular, are much the same in form, indicating that theirregularities are caused by age-misreporting, and that theunderlying age composition was essentially constant (seefigure XIX). Stable analysis is appropriate for India in1911, but not for Turkey in the years shown.

Few populations for which estimation is necessary havebeen enumerated in an extended series of censuses ofcomparable quality, and it is often impossible to applythe suggested criterion of an essentially unchanging agedistribution and rate of increase. The assumption ofstability must often be made without much direct evidencein its support.

In general, stable methods of estimation should beattempted only in populations where there is no wide­spread use of birth control, since where the practice iscommon there are usually pronounced variations or

TABLE 10. ESTIMATED PARAMETERS OF THE TEST POPULATION CALCULATED BY VARIOUS METHODS BASED ON POPULATION AGE DISTRIBUTIONSFROM TWO CENSUSES AND FROM REPORTS ON CHILD SURVIVAL

Estimatedparameter

Pattern Death rate Death rateof under age over age

OeoMethad ofestimation mortality S 5 °e5 Birth rate Death rate

(a) Death rates from "West" .0720 .0137 40.0 49.7 .0445 .0234census survival (ages "North" .0791 .0149 37.3 47.9 .0466 .0255over 5) and from 12 "East" .0697 .0137 40.8 50.1 .0441 .0230(under 5); b = d-s-r "South" .0734 .0146 38.8 48.8 .0454 .0243

(b) Death rates from "West" .0720 .0137 40.0 49.7 .0445 .0234C(x) and r (ages over "North" .0791 .0142 38.0 48.9 .0460 .02495) and from 12 (under "East" .0697 .0140 40.5 49.7 .0443 .02325); b = d-rr "South" .0734 .0139 39.5 49.7 .0448 .0237

(c) Birth rate from C(x) "West" .0445 .0234and 12;d = br-r "North" .0465 .0254

"East" .0451 .0240"South" .0461 .0250

46

C(~l

It'16~

.14

.12

.10

.08

.06

.04

INDIA_1911___ 1901

....... 1891

AG E

I 8

.16

.14

.12

.10

.08

.06

.04

.02

o 5 10 15 20 25 30

TURKEY__ 1945___ 1940

........ 1935

I I

35 40 45 50 55 60 65AGE

Figure XIX. Distribution of the female population by age in five-year intervals as recorded in selected censuses in India and in Turkey

trends in fertility. Stable analysis should be avoided inpopulations in which migration has had a pronouncedinfluence on age composition-as is typical of the popu­lations of many cities in developing countries. Onereason for the emphasis on analysing female age distri­butions in this Manual is that female distributions are veryoften less affected by migration than are male. With someconspicuous exceptions (e.g., rural-urban migration inLatin America) female migration is usually less than male,and because women are usually accompanied by theirchildren, the effect on the age distribution is sometimesnegligible, even when migrants form a substantial fractionof the population. It has been shown for example that aconstant stream of female immigrants, constitutingannually 4 to 5 per cent of the receiving population,consisting of young adults and their children, does notproduce an age composition markedly different from whatwould exist in the absence of immigration.2

If mortality and fertility have fluctuated rather thanbeing constant, but without a regular trend, the stablepopulation that has the the same ogive up to age fifteen,twenty or thirty (and the same rate of increase) hasfertility and mortality close to the average during the

I See J.M. Boute, S.J. ,La demographie de la branche indo­paklstanaise d'Afrique, Etudes morales, sociales et juridiques,(Louvain, 1965). On this topic see also Leon Tabah and AlbertoCataldi, "Effets d'une immigration dans quelques populationsmod~les", Population, No.4, 1963, pages 683-696.

past few decades. What may produce poor estimates areeither major swings in fertility producing one or moreconsecutive small five-year cohorts, or a sustained trendin either fertility or mortality. A marked and continueddecline in mortality has occurred in many populations thatappear to have essentially constant fertility, and chapter Iincludes a section (section C) that shows how estimatesbased on stable populations can be adjusted for theeffects of a history of falling death rates. But theseadjustments can be made only if there are clues indicatingapproximately how long and how rapidly mortality hasfallen. When there is no reliable basis for detecting thedownward course of mortality, the adjustments givenin table I1I.l in annex III can be interpreted as indicationsof the errors that occur from stable population calcula­tions made in ignorance of a downtrend in mortality.For example, suppose that a stable population is chosenwith C (30) = .7, and ten-year intercensal r = .020,leading to an estimated b of .04912 and GRR29 of 3.312.But if in fact because of falling mortality populationgrowth has increased from 0.144 in the next earlierintercensal decade and about .009 in still earlier decadesthe approximate value of k would be .01, and the valueof t about 20 years. It can be seen, then, in table IlL1that the true value of the birth rate would be about 8.5per cent higher, and the true value of the gross reproduc­tion rate about 9.6 per cent higher than the estimatesarrived at on the basis of the false assumption of stability.In other words the correct estimates for the birth rate andthe GRR in this example are .0533 and 3.63, respectively.

47

TABLE 11. EXPECTATION OF LIFE AT BIRTH IN "WEST" MODEL LIFE

TABLES PRODUCING A PROJECTED POPULATION OVER AGE X THATMATCHES THE TEST POPULATION AT THE END OF A DECADE ASSUM­ING A 2 PER CENT RELATIVE OVERCOUNT OF THE TEST POPULATIONAT THE LATTER DATE

earlier discussion) that the population is the same "test"population used before-the" West" female model stable,with °eo = 40 years, and GRR2 9 = 3.00. Table 11 showsthe levels of mortality required to project the earlier

10 44.5 50 42.415 44.6 55 42.120 44.2 60 41.725 43.8 65 41.530 43.5 70 41.235 43.2 75 .. 41.040 43.0 80 40.945 .. 42.7

population to the later one at ages over to, 15, 20 etc.The assumed conditions imply that a mortality level mustbe chosen producing a proportion surviving that is 2 percent above the true numbers. It requires a bigger differencein mortality level to increase survival of the whole popu­lation by 2 per cent than of the population over fifty orsixty. Thus the biggest mistake in selecting a level ofmortality when there is a change in the completeness ofcoverage occurs, as is evident in table II, in levels basedpartly on survival of the population at younger ages.However, as will be seen later, age-mis-statements makethe projected number of persons derived from thepopulation over forty especiallyunreliable.

°00Age xAge"

Table III.I can also be used as an approximate indi­cation of the errors caused by the assumption of stabilitywhen fertilityhas recentlyfollowed a rising trend such thatthe GRR has risen by one per cent annually-beenmultiplied by (1+k) in each of the last t years, where k= 0.01.The reason that table III.1 indicates the effectof ahistory of rising fertility is that the age distribution isdisplaced from the stable in an almost identical mannerby a recent trend of falling mortality on the one hand, orof rising fertility on the other. If the sign of the adjustmentfactors in table 111.1 is reversed, the factors then indicatethe errors in stable estimates if in fact fertility has beenfalling by one per cent annually for t years.

It does not seem advisable to attempt to extend the useof table III.1 or to construct similar or more complicatedtables, to cover estimation for populations in which thevoluntary control of fertility is widespread. The appli­cation of stable analysis should probably be confined topopulations that apparently do not practise deliberatecontraception or abortion. But fertility may be subject toprolonged, if limited, upward and downward movementseven when conscious birth control is rare. In tropicalLatin America there appears to be little contraception orabortion except in some urban populations, but in manycountries fertility was reduced in the depressed 1930s,and rose during the 1940s and 1950s.3 The source of thevariation was changes in marital status reflecting some­what later marriage in the 1930s. In other populationsproportions married increasewhen mortality falls becauseof a reduced incidence of widowhood, and in still othersfecundity is changed by the spread or the conquest ofpathological sterility. Table III.1 shows the orders ofmagnitude of the errors in stable estimates introducedby such trends, if not allowed for explicitly.

B. ERRORS CAUSED BY FAULTY DATA

TABLE 12. STABLE POPULATION ESTIMATES OF THE BIRTH RATE INTHE TEST POPULATION DERIVED FROM CORRECTLY REPORTEDPROPORTIONS UP TO AGE X AND FROM THE OBSERVED INTERCENSALGROWTH RATE WHEN THE LATTER IS DISTORTED BY A 2 PER CENTRELATIVE OVERCOUNT IN THE SECOND OF TWO CENSUSES TAKENTEN YEARS APART

5 ......... , ...... .0429 40 ................. .043210 ................ .0425 45 . ................ .043415 ................ .0425 50 . ................ .043620 " .............. .0426 55 ..... '" ......... .043825 ................ .0428 60 ................. .044030 ................ .0429 65 ..... '" ......... .044235 ................ .0431 Test population ..... .0445

Now suppose that the estimate is made by stablepopulation methods. It has been assumed that the improve­ment in coverage has not affected the reported agedistribution, but the intercensal rate of increase is over­estimated by about 2 per thousand. Table 12 shows theestimates of b obtained from this erroneous rand C (5),C (to) etc. Note that the error in estimating b varieslittle from about ages five to forty and that its absolute

The discussion of mistaken estimation caused bymistakes in the basic data will be confined to two impor­tant forms of defective information: omission of personswho should have been included in a census or survey(or the opposite mistake of erroneous inclusion), andmisreporting of age. Distortions in the recorded agedistribution are caused by age selective omissions as wellas by age-mis-statement, but it is not generally possibleto determine which of these factors has caused a givenirregularity in age composition, and it will be implicitlyassumed that omissions affect primarily the total numberof persons enumerated, and that distorted age compo­sition is caused by misreporting of ages.

1. Differential rates of omission in consecutive censuses

Consider a population enumerated in two censuses adecade apart, and suppose the coverage of the secondcensus is 2 per cent more complete. What is the effect onestimates of mortality and fertility?

It is assumed (to maintain comparability with the

3 Cf. O. Andrew Collver, Birth Rates in Latin America, NewEstimates of Historical Trends and Fluctuations (Institute of Inter­national Studies, University of California, Berkeley, 1965).

Age x Birth rate Age x Birth rate

48

TABLE 13. EXPECTATION OF LIFE AT BIRTH AND VITAL RATES IN THE TEST POPULATION, AND ESTIMATES

OF THESE QUANTITIES BASED ON THE CENSUS SURVIVAL METHOD AND THE STABLE POPULATIONMETHOD. BIAS OF ESTIMATES REFLECTS AN ASSUMED RELATIVE OVERCOUNT IN THE SECOND OF TWOCENSUSES OF THE TEST POPULATION, TAKEN TEN YEARS APART

oeo Birth rate Death rate Growth rate

Test population ................ 40.0 .0445 .0234 .0211Census survival method ......... 43.5 .0434 .0203 .0231Stable population method ....... 44.3 .0429 .0198 .0231

magnitude is larger than when obtained with ogives upto higher ages.Once again, however,due to age-mis-state­ments at such ages, this observation is of no consequencein selecting a stable estimate of the birth rate from areported age distribution.

Table 13 summarizes the values of various parametersbased on the census survival method and on the stablepopulation method i.e., obtained by accepting themedian model life table and the median stable populationamong the first nine in tables 11 and 12, respectively. Inthe former case the m, values of the selected life table arecombined with an estimated mean age distribution toderive the (crude) death rate; the estimate of the birthrate is obtained by adding to this death rate the observedintercensal growth rate. In the case of the stable popu­lation method, the selection of the median stable popula­tion naturally implies the acceptance of all other stableparameters of that population. Note that with the censussurvival method a more complete enumeration in thesecond census causes an appearance of higher rates ofsurvival and hence causes the estimated death rate to betoo low; of course, the calculated intercensal rate ofincrease is too high, and the two errors are partiallycompensating when the birth rate is calculated. In thecase of the stable population method, an upward-biased rin combination with C (x) causes b to be underestimated­in the case of the median stable population this error isapproximately equal to the error in r. Since its sign isopposite, however, the estimate of d is about twice as farremoved from the true value as the estimate of b.

Variations in completeness of coverage tend to com­promise the adjustments for declining mortality. Forexample, if the middle of three decennial censuses isespecially incomplete, mortality in the earlier decade isoverstated and in the later decade understated, creatingthe impression of declining mortality when death rateswere actually constant, or exaggerating the extent of areal decline.

2. Age-misreporting in censuses or surveys

A large proportion of the ages recorded in most censusesand demographic surveys in less developed countries areinaccurate. In tabulation of the population by singleyears of age, peaks at ages ending in zero and five, and to aless extent at two and eight, are usual, with deficits at theother digits. Numbers reported at forty which are severaltimes bigger than those at forty-one are not uncommon,for example. In section B.3 of chapter I comparisons with

stable populations were used to demonstrate the existenceof typical distortion in the.ogives of age distribution, andin distributions by five-year age intervals, caused bycharacteristic forms of age-misreporting. The mostconspicuous systematic distortions are found in popula­tions in which apparently the ages of many persons areestimated by the interviewer rather than the respondent,and rules were suggested for selectingages at which ogivesare likely to be most reliable. Of course such rules cando no more than minimize errors that are unavoidablysubstantial, and the question remains concerning therange of unavoidable variation introduced into estimatesof birth and death rates by age-rnis-statements.

(a) Age-mis-statement and mortality estimation by censussurvival

The effects of age-mis-statement on the estimation ofmortality by finding the model life table that duplicatesobserved census survival values cannot be simply summa­rized. There are two principal ways in which age-mis­statement affects the level of mortality that gives aprojected population over x +10agreeingwith the enumer­ated population. First is the effect of age-mis-statementon the reported numbers over x+ 10 in the later censusrelative to the effect on those reported as over x in theearlier census. Both numbers may be inflated or bothdeflated without causing an error in the estimated level ofmortality, provided relative inflation or deflation is thesame. But if the increase in the proportion over thirtyin the later census by age overstatement is less than theincrease in the proportion over twenty in the earliercensus, the estimated survival rates will be too low,and estimated mortality too high. Secondly, the estimatedlevel of mortality is affected by age errors that exaggerateor understate the proportions at ages of high mortality.The typical exaggeration of the age of persons pastfifty or sixty means that too many old persons are reportedin the earlier census. This upward shift of age reduces theexpected number of survivors in a projection by anygiven life table, and therefore requires overstated survivalrates (too low mortality) to produce expected survivorsequal to the actual. Experiments show that in the high­fertility populations for which estimates are usuallyneeded, overstatement of age by the aged tangibly affectsonly the estimates of mortality based on projections of thepopulation forty-five and over. Because of the complexityof the effects of age-mis-statement on census survivalratios, no preferred ages are suggested for estimating thelevel of mortality. Selection on the basis of projection

49

of the population at ages above forty-five cannot betrusted, and the median of the first nine estimates is aneutral choice.

(b) Age-mis-statement and stable population analysis

The use of model stable populations to estimate variousparameters involves the selection of a stable age distri­bution matching the recorded age distribution in someway. In this Manual the recommended procedure is toselect a stable population with the same proportion undersome age-the selection of what age depending on thepattern of apparent distortion of the recorded age distri­bution. With the extreme distortions characteristic ofcertain Asian and African censuses, it is recommendedthat the stable ogive matching the given female populationat age thirty-five be used. For age distributions subjectto less distortion such as found in the Philippines andLatin America use of the male distribution was recom­mended, specifically the selection of the median ogiveamong those matching the census at 5, 10, ... ,45.

Whatever procedure is followed, it can only avoidextreme errors, and cannot insure that the proportionrecorded as under the age prescribed by the rule is exact.The value of C(x) in the model population selected isthus generally somewhat above or below the true valueof C(x), because age-mis-statement has caused a nettransfer of persons across age x. How much does an errorin C (x) affect the estimated birth rates, death rates, andother parameters ?

TABLE 14. VALUES OF Ab/AC(x) FOR THE TEST POPULATION WHENTHE BIRTH RATE IS ESTIMATED BY THE STABLE POPULATION METHODGIVEN C(x) AND r, AND C(x) AND /2

L1b/LlC(x) L1b/LlC(x)glvell C(x) glvell C(x)

Q/Idr aIId12Age x (a) (b) Col. alb

5 .................... .568 .304 1.86810 .................... .388 .189 2.05315 .................... .319 .155 2.05820 .................... .293 .146 2.00725 .................... .283 .145 1.95230 .......... , ......... .289 .153 1.88935 .................. ,. .305 .171 1.78440 .................... .336 .192 1.75045 .................... .379 .225 1.684

Table 14 shows the ratio of errors in the estimation of bto errors in the recorded value of C (x) (x = 5 to 45)when C(x) is combined with r on the one hand and 12on the other. These calculations apply to the same testpopulation ("West" model female, °eo= 40, GRR2 9= 3.00) employed before. The meaning of the entries inthis table is illustrated by this example: Suppose C (35)were recorded as .7627 rather than the correct figure of.7527, or that there were an error of .01 in C(35). If rwere known, the error in estimating b would be (.01)(.305) or .00305, and if 12 were known, the error inestimating b would be .00171. (Since the "true" birth rate

ofthe test population is 44.48 per thousand, the erroneousestimates would be 47.53 and 46.19, respectively.) Notethat knowledge of 12 yields more "robust" estimates ofthe birth rate than does knowledge of r-estimates lesssensitive to errors in the reported age distribution as wellas possible differences in age pattern of mortality. Esti­mates of d are in error to the same extent as is b when ris known. Because knowledge of 12 implies (within thecontext of a family of model life tables) that the level ofmortality is known, errors in C (x) would generallyproduce trivial errors in the estimate of d, and errors in theestimate of r essentially equal to those in estimating b.

In the earlier discussion of typical pattern of age-mis­reporting, the hypothesis was advanced that when agereporting is not subject to the gross distortions seen inthe censuses of Africa, India, Pakistan and Indonesia,populations have an approximate knowledge of age,accounting for the relatively orderly form of the ogive,and even of the five-year age distribution, even thoughage heaping is extensive. This hypothesis suggests thatgross overstatement or understatement of age is not thenorm and that age-mis-statement is usually the result ofsmall errors, with a preference for round numbers. InLatin America and other areas where there appears to beknowledge of the approximate age, it is possible to arguethat ogives to ages divisible by five have a systematicdownward bias, because the cumulative age distributionalways stops just short of the highly preferred ages endingin zero or five. To make the point in a different way:cumulative proportions under age 11, 16, 21, 26, 31 etc.would indicate higher fertility than ogives to age 10,15 etc., because the first set would always just include,rather than just exclude, an age containing a greatlyexaggerated number. It appears probable that the ogivesbarely including the ages divisible by five are too largebecause they contain persons really 11 and 12, 16 and 17,21 and 22 etc., reported as 10, 15, 20 etc. On the otherhand, ogives ending just short of ages divisible by fiveare possibly too small, because some persons 8 or 9 arereported as 10, 13 or 14 as 15 etc. To test the extent of thepossible bias caused by this effect of age heaping, aspecial analysis was made of the census of Mexico in1960, when age heaping was extensive. The numbersrecorded in three-year intervals (9-11, 14-16, 19-21, etc.,)around each age divisible by five were reapportioned soas to have the sequence expected in an appropriate stablepopulation, and the ogive recalculated. The effect wasto increase the proportion below each age divisible byfive (except below age fifteen) because some of the personsreported at these preferred ages were reassigned to theyounger quinquennium. Despite the considerable ageheaping, the differences between the birth rates estimatedon the basis of the adjusted C (x) values and those calcu­lated from the unadjusted data were at all ages well withinone per thousand population.

(c) Age-mis-statement and the estimation of fertilityand mortality from special questions on past experience

In chapter II there is a description of methods ofapproximate calculation of fertility and mortality fromthe data obtained in asking women about the number ofchildren they have ever borne, and the number of these

50

surviving. It was noted, because of a tendency towardsomission in the answers given by older women, thatreported parity could be accepted as about correct onlyfor women under thirty. However, it is a plausible hypo­thesis that the age pattern of fertility is correctly indicatedby responses to questions about births in the precedingyear. In the method of fertility estimation devised byWilliam Brass, a comparison at ages 20-24 or 25-29 ofcumulated fertility indicated by births reported for thepreceding year with average numbers of children everborn to these ages provides an appropriate adjustment toreported births for the last year that can give a goodestimate of the fertility schedule. This method, valid inprinciple when age is accurately reported, also workswith tolerable accuracy in the presence of large andfrequent errors in age, provided the errors are notsystematic. But in the Asian-African populations, whereage is often estimated by another person, it appears thatabout half the women in some of the child-bearing agegroups are reported in the next higher group, which meansthat the cumulated experience of those reported as 15-19does not in fact correspond to the history ofthose reportedas twenty. It is not possible to analyse the resultant biaseshere," but merely to warn that the method can be usedonly with the possibility of a wide margin of error inpopulations where massive systematic age-misreporting isapparent. It is much more promising, should the appro­priate questions be asked in a census or survey, in popu­lations subject to milder age distortions, such as in thePhilippines or in Latin America.

Age-misreporting also affects the estimation of childmortality from proportions dead among children everborn to young women. Here again, accurate estimates canbe expected only in populations not subject to grosstransfers of women among ages 15-19, 20-24, 25-29 and30-34. But the sensitivity of the estimated levels of childmortality to such transfers is not great, and one can beconfident, for example, if responses about live and deadchildren are approximately correct, that the adjustedproportion dead reported by women 20-24 is greater than1qo and less than 3qO even if not equal to 2QO'

C. SUGGESTIONS FOR BEST ESTIMATION

A number of rules of thumb have been offered in thisManual, representing, however, not any definitive bestprocedure in each form of estimation, but a preliminarydistillation of the authors' experience with each method.In the first application of an unfamiliar method, it maybe best simply to follow these rules; but in situationswhen many alternative calculations are feasible (surelythe usual situation in making estimates for a singlenational population), all of the possibilities here discussedshould be examined, and a final estimate made onlyafter a critial examination of these alternatives. Theanalyst should be sensitiveto patterns ofage-misreporting,to evidence that the population is not stable, and to thepossibility of systematic omission of events or personswith certain characteristics.

4 See the discussion in Brass et al., op, cit.

A basic feature of demographic analysis underlyingmany of the procedures suggested here, but not alwaysemphasized, is that there are logically necessary or biolo­gically inevitable interrelations among population para­meters, and these relations should be fully exploited inexamining the consistency of estimates. For example,in populations in which circumstances permit a separatevalid estimate of the average number of male and femalebirths in a given decade, it is important to compare theestimated numbers to see if they lie within tolerable limitsof the expected sex ratio at birth. Almost all populationsof non-African origin have 105-107 male births for every100 female births, subject, of course to sampling fluctua­tions. Populations of African origin have a sex ratio atbirth of perhaps 102-104. Thus if a male birth rate (malebirths/male population) and a female birth rate have beencalculated from the age distribution of each sex, it isessential to see whether the estimates are consonant orinconsistent. If the former, there is reason for addedconfidence, if the latter, the data or the methods areinconsistent (a judgement subject to the crucial reser­vation that sampling variance must be allowed for).

One of the principles to be borne in mind in makingsuch checks is the importance of noting what aspectsof the two estimates being confronted are independent.Thus for example a comparison of the level of mortalityindicated by census survival during a decade, and by C(x)in the second census plus the intercensal rate of increaseis a comparison of two inferences from essentially thesame basic data. Deficiencies in the censuses or specialfeatures in the age pattern of mortality would affect thetwo estimates similarly, and perfect agreement would notbe as reassuring as agreement between estimates withwholly independent bases. An example of a more nearlyindependent pair of estimates is the male birth ratecalculated indirectly from the female age distributionplus the sex ratio of the population and an assumednormal sex ratio at birth, and directly from the male agedistribution.

The comparison that is most acceptable as a confir­mation of valid estimation is between figures derivedfrom data of wholly different kinds. For example, in asituation where fertility is constant, data on children everborn by age of woman, if themselves internally consistent,take on added persuasiveness if verified by comparisonwith the cumulation of current fertility rates based onreported births last year. But suppose now that deathrates over five are calculated from census survival, andunder five from proportions surviving among childrenever born, that these rates are combined to form anestimated population death rate, to which the intercensalrate of increase is added to provide a figure for the birthrate. If now the birth rate thus obtained agrees closelywith that derived by the methods outlined earlier in thisparagraph, the confirmation is a strong one. Of course it isalways possible that the agreement is fortuitous. Anadditional principle in judging the importance of agree­ment is its consistency. As a final element in this hypo­thetical example, suppose that the comparison has beenmade in each of a moderate number-ten or more­of subdivisions of the same data system, such as regions orprovinces enumerated in the same national censuses. If

51

the various forms of independent estimation yieldfertilities that agree in the geographical differences theyshow, the reality of the differences is more or less con­clusively established.

The last suggestion for making the best of inaccurateand incomplete data is always to seek the form of esti-

mation least sensitive to unknowable uncertainties-toage patterns of mortality when age specific death ratesare not recorded, to systematic age-misreporting etc.This principle leads to preferring 12 to r as an adjunct toC(x) in estimating the birth rate.

52

Chapter V

DATA USEFUL FOR ESTIMATES OF FERTILITY AND MORTALITY

The importance of measuring current fertility andmortality has led to a number of experimental attempts toobtain current figures by sample registration, by specialsurveys, and by a combination of registration and inter­views. 1 The design of such sample registration andsurveys has been excluded from the discussion in thisManual; the methods outlined here are confined to datathat might be obtained in a census, or large-scale sampledemographic survey. The purpose of this short finalchapter of part one. is to specify explicitly what kindsof questions should be included in censuses or broadpurpose surveys to make reliable estimation of birthand death rates possible.

The same questions would also be a valuable part ofthe design of intensive repeated surveys instituted for thespecific purpose of estimating vital statitics: for example,Brass-style estimates of infant and child mortality wouldbe a useful check on mortality information obtained in asample register, or from frequent interviews.

A. DATA ON AGE

Information about age is an essential part of every formof estimation presented in this Manual. Moreover, theexistence of similar patterns of distortion in age distri­butions in surveys or censuses makes even rough agedistributions the basis of useful estimation. A question onchronological age should be part of every census ordemographic survey, and when the respondent is unableto supply an acceptable figure, the interviewer shouldbe instructed to make an estimate.

1 See notably, C. Chandrasekaran and W.E. Deming, "On aMethod of Estimating Birth and Death Rates and the Extent ofRegistration", Journal ofthe American Statistical Association, vol. 44,March 1949, pages 101·115; Ansley J. Coale, "The Design of anExperimental Procedure for Obtaining Accurate Vital Statistics",in International Union for the Scientific Study of Publications,International Population Conference, New York, 1961 (London, 1963),Vol. II, pages 372-376; Guanabara Demographic Pilot Survey,A joint project of the United Nations and the Government ofBrazil (United Nations publication, Sales No.: 64.XIII.3). Karol J.Krotki, "First Report on the Population Growth Experiment",in International Union for the Scientific Study of Population,International Population Conference, Ottawa, 1963 (Liege, 1964)pages 159·173; Carmen, Arretx G. and Jorge L. Somoza, "SurveyMethods, Based on Periodically Repeated Interviews, Aimed atDetermining Demographic Rates," Demography (Chicago, 1965),vol. II, pages 289-301; cf. also William Brass, "Methods of ObtainingDemographic Measures where Census and Vital Statistics Registra­tion Systems are Lacking or Defective", background paper presentedat the Second World Population Conference (Belgrade, 1965),No. 409.

Tables should be published showing the distributionby the standard five-year age intervals for each sex in eachgeographical unit, and by single years for whole popu­lations covered by the survey. The single-year distributionmakes useful analysis ofage-misreporting possible.

B. DATA ON CHILDREN EVER BORN

In the absence of vital statistics, data on the number ofchildren born in the lifetime of each woman and thenumber of these surviving provide a very useful basis fordetermining fertility and mortality. The following sugges­tions are intended to supply the maximum material forconstructing estimates, and simultaneously to make itpossible both to detect and to minimize biases in theresponses.

1. Questions on fertility histories should ask for thenumber of children born alive who are still living in thehousehold, the number born alive who have left thehousehold, and the number who have died. The questionabout those who have died is the foundation of estimatesof infant and child mortality, and the separate questionsabout those still at home and those who have left minimizea source of omission, especially for older women withgrown children.

2. Women should be asked the sex of each childreported, and males and females should be tabulatedseparately. This procedure makes possible the estimationof male and female child mortality separately. It alsoprovides material for a number of tests ofconsistency-forexample, omissions of children ever born are likely to besex selective, and such a tendency is revealed by a trendin the sex ratio of the reported children ever born as theage of woman increases. Another significant form ofprobable omission is to leave out higher proportions ofdead than of surviving children, especially on the part ofolder women. This form of omission may also be sexselective, resulting in an implausible contrast in estimatedchild mortality by age for the two sexes.

3. Questions about children ever born and childrensurviving should be asked of and tabulated preferablyfor all women, not merely married women. If, for somereason, non-married women must be excluded, thequestions should be asked of and tabulated for allmarried women, not merely "mothers". Interviewersshould be instructed to make an unambiguous entry forevery respondent, especially to enter a zero for women

53

with no children, rather than leaving a blank, whichindicates "no response", rather than "no children".

4. The parity distribution in each five-year age intervalshould be tabulated-i.e., the number of women withno children, with one child, two children etc.-ratherthan just the mean parity or number of children ever born.This tabulation makes possible additional tests of consis­tency, and additional valuable inferences, such as differen­tial mortality in families with different numbers ofchildren.

C. DATA ON THE AGE STRUCTURE OF FERTILITY

In chapter II a method of estimating fertility is describedthat is based on accepting as accurate the number ofchildren ever born reported by younger women as anindication of the level of fertility, and judging the agepattern of fertility from births-by-age reported for thepreceding year. This technique is especially promising inpopulations in which knowledge of approximate age iswidespread, so that the comparison of cumulated fertilitywith reported average parity is not excessively distortedby misreporting of age.

The question that should be asked to reveal the agepattern of fertility is whether each woman bore a childin the year before the survey.Answers should be tabulatedby age, and also by parity-information that is collected

in the same survey if the technique is to be employed.This additional detail creates additional possibilities fordetermining the presumed error in the perception of the"reference period" (one year) that causes mistakes inreporting recent births. For example, the cumulation ofage specific fertility of zero parity women to the ageinterval 30-34 should equal the proportion of womenat age 30-34 having at least one child, and a correctioncan be applied to the births reported for the precedingyear to insure such equality. If this correction factor isabout the same as the one described in chapter II-thecorrection needed to equate cumulative fertility (allparities) with average parity at age 20-24-the credibilityof the adjustment is greatly strengthened.

Another possibility for ascertaining the age structure offertility is to ask every woman whether she is currentlypregnant. Not all pregnancies result in live births; somewomen may not recognize pregnancy in its early stages;and in some populations there may be a tendency forwomen to deny they are pregnant. None of these differ­ences between reported pregnancy and fertility are likelyto be age selective, except in ways that can be estimated;and this question is a promising supplement (or substitute)for a question on births last year. Of course the age of thewoman at the birth of the child would be about fourmonths greater than the mean age during a reportedpregnancy, and an allowance for this difference must bemade in forming a schedule of fertility.

54

Part Two

EXAMPLES OF ESTIMATION

-

Chapter VI

EXAMPLES OF ESTIMATES BASED ON RECORDS OFPOPULATION GROWTH AND DISTRIBUTION BY AGE

A. ESTIMATION OF MORTALITY AND OF THE BIRTH RATE

FROM CENSUS SURVIVAL RATES

An application of the method of estimating mortalityand fertility described in section A.2 of chapter I isillustrated below using information taken from Turkishstatistics, specifically from the national censuses of 1935and 1945. Estimation could have been based on thecensuses of 1935 and 1940, but censuses at five-yearintervals are rare, and the ten-year interval was chosenas more typical.

Required basic data. Distribution of the population byfive-year age groups as recorded in two successive censusestaken several-preferably five or ten-years apart.Columns 2 and 3 of table 15 show the female populationin Turkey in 1935 and 1945, classified by age. Ideally thepopulation should be closed to migration during the

intercensal period and the two censuses should refer tothe same geographical area.

Preliminary adjustment of data. The ideal requirementsas stated in the preceding paragraph are seldom perfectlysatisfied. Deviations from these requirements result inbiases in the final estimates or cause computationalinconveniences. Preliminary adjustments of the basic datacan eliminate or reduce either of these effects. The possi­bility of making such adjustments when they are needed,and their specific nature may differ from case to case:the decision of the analyst concerning the procedures tobe followed should be influenced in each instance by theextent of the deviation from the ideal requirements andby the amount and quality of the information that isavailable for making corrections in the basic data. Thefollowing four types of adjustments may be both com­monly needed and feasible:

TABLE 15. FEMALE POPULATION OF TuRKEY BY AGE IN 1935 AND 1945 (THOUSANDS) AND CENSUSSURVIVAL RATES FOR FIVE-YEAR COHORTS

Population reported Adjusted Adjustedby census population. 1935 population, 1945

For migrationFor "age and boundary For "age

unknowns" changes unknown" Ten-year survivalAge 10,20, 10.21, (col. 2 (col. 4 (col. 3 rates for

Interval 1935 1945 x 1.004.f) x 1.020.f) x 1.00I.f) each cohort"(I) (2) (3) (4) (5) (6) (7)

0-4 .......... J,297 1,185 1,303 1,329 1,1875-9 .......... 1,128 1,242 1,133 1,156 1,244

10-14 .......... 746.0 1,074 749.2 764.2 1,076 .809615-19 .......... 485.9 931.5 488.0 498.0 932.8 .806920-24 .......... 640.2 691.7 643.0 656.1 692.7 .906025·29 .......... 721.3 619.1 724.4 739.3 620.0 1.245030-34 .......... 642.1 699.7 644.9 658.1 700.7 1.068035-39 .......... 509.6 578.4 511.8 522.3 579.2 .783440-44 .......... 473.9 558.0 476.0 485.7 558.8 .849145-49 .......... 314.9 378.5 316.2 322.7 379.0 .725650-54 .......... 384.4 434.1 386.J 394.0 434.7 .895055-59 .......... 195.2 219.4 196.1 200.1 219.7 .680860-64 .......... 297.4 349.2 298.7 304.8 349.7 .887665-69 .......... 105.0 J24.6 105.4 107.6 124.8 .623770-74 .......... 126.1 133.0 126.6 129.2 133.2 .437075 and over .... 118.0 112.3 118.5 120.9 112.5Unknown ...... 35.81 13.11

TOTAL .••••.•.• 8,221 9,344 8,221 8,388 9,344

a Ratio of number of persons in each age interval in 1945 (from Column 6) to numberten years younger in 1935 (from Column 5).

57

(1) Adjustment for persons not classifiedby age. Unlessthe number of persons not classified by age is very small,or their proportion in the total population is very nearlythe same in both censuses, "unknowns" with respect toage should be distributed in a fashion that leaves thedistribution of the total population for the given sex withknown ages unchanged. This is performed by multiplyingthe population classified by age by the ratio:

total populationtotal population - population with ages unknown

Columns 4 and 6 show the female population of Turkeyin 1935 and 1945 after such an adjustment has beenperformed. Column 4, for example, was obtained bymultiplying the population in each group shown inColumn 2 by the factor of 8221/(8221-36) = 1.0044.

(2) Adjustment for boundary changes. Other thingsbeing equal, an increase in the territorial coverage in thesecond census would spuriously inflate the census survivalrates; territorial losses in the intercensal period wouldintroduce an opposite bias. If the two censusesin questiondo not refer to the population of the same territorycomparability must be insured by reckoning the popu­lation in both censuses on an identical territorial basis.If the population involved in the adjustment constitutesa substantial portion of the total population, or if its ageand sex composition is very atypical, it would be highlydesirable to correct the census figures by individual ageand sex groups: e.g., in case of intercensal territorial gainto add to the figures of the first census the population ofthe territory in question (as estimated at the time of thefirst census) as it was actually distributed by sex and age,or to remove the population of the affected territoryfrom the figures of the second census age group by agegroup. When the population involved is small, and itscharacteristics are not strongly deviant from those of therest of the country, a simpler adjustment is adequate.This statement holds true in the case of Turkey for 1935­1945. The territory of that country was increased by theprovince of Hatai in 1939. In 1940 the total populationwithout this province was reported as 17,613,000 and withHatai province as 17,821,000. An adjustment for thisterritorial change may be performed simply by multiplyingthrough the 1935 census figures by the factor of (17821/17613) = 1.0ll8. Alternatively the 1945 census figurescould have been deflated by the factor of .9883.

(3) Adjustment for migration. The considerationsgoverning this adjustment are the same as those under­lying the adjustment for change in territorial coverageoutlined in the preceding paragraph. In Turkey it wasestimated that net immigration during the 1935-1945period amounted to some 150,000 persons. Since nodetailed information is available as to the sex-agecompo­sition of the migrants, and sincethe numbers involved arerelatively small, it may be simply assumed that theirdemographic characteristics (their fertility, mortality,age and sex composition) were the same as those of therest of the population and the adjustment may be per­formed by multiplying through the 1935 figures by thefactor of 1.0085, i.e., by the ratio of the mid-period (1940)population plus the net migratory balance to the mid­period population. Since much of the migration in

question involved movements of whole families it isunlikely that the age and sex composition of the migrantswas highly atypical; therefore the remaining bias due tomigration after this adjustment is undoubtedly small.1

Since statistics on migrants by age and sex are seldomadequate, and since migrants are often concentrated incertain age and sex groups it is obvious that a substantialvolume of migration may strongly bias the mortalityestimates obtained from census survival rates. It shouldbe noted, however, that the availability of certain types ofcensus tabulations may still permit the use of the censussurvival method by identifying population groups thatare more nearly closed to migration than the totalpopulation. An example is the calculation of mortalityestimates for well-specified linguistic, racial or religiousgroups that are not affectedby migration. A possiblymorecommonly feasible application may be the use of tabu­lations of the native population classified by age and sexin two consecutive censuses in population where immi­gration is substantial but out-migration is negligible.

Column 5 of table 15 shows the 1935 population ofTurkey adjusted for comparability with the 1945 figuresgiven in column 6. Column 5 was obtained by multiplyingthrough column 4 by 1.0204-the product of 1.0118(adjustment for territorial coverage) and 1.0085 (adjust­ment for migration).

(4) Adjustment for the length of the intercensal period.Computational convenience and limitations of the data(e.g. lack of classifications by single-year age groups)makes it mandatory to base the calculations on two setsof population figures referring to points of time that areto a very close approximation five, ten or perhaps fifteenyears apart. When this is not the case it is necessary to"move" one of the populations involved over time toestablish the desired time distance between the censuses.No such need arises in the case of the 1935 and 1945Turkish censuses that have reference dates of 20 Octoberand 21 October, respectively. When an adjustment isneeded on this score the simplest procedure to follow is toassume that both the age distribution and the observedintercensal growth rate has been, or will remain, un­changed during the period to be removed from, or addedto, the actual intercensal time distance in order to makethat distance equal the nearest integer multiple of fiveyears. For the sake of an example assume that the twocensuses were taken 8.72 years apart and that the averageyearly increase during this time was r = .022. Underthese circumstances the population registered at the timeof the second census in each age group is to be multipliedby the factor of 1.0286-Le. by e·02 2 x 1.28; 1.28 beingthe additional number of years that would have elapsedbeyond the actual intercensal time distance had the secondcensus been taken exactly ten years after the first one."

1 The use of the 1940 population (or the average intercensalpopulation) in the calculation of the correction factor implies thatthe migration over the decade is assumed to have been approximatelyevenly distributed.

2 Where r (.022) has been calculated as follows:

8.72 r = log e P,+8.72 .P,

58

Computational procedure. The essence of the compu­tations is to find a life table (from among the modeltables) that, employed to project the 1935 population,produces a 1945 population most consistent with therecorded one. The computational steps are:

(1) Apply the ten-year cohort survival rates given intable I.3 of annex I to the 1935 population distributed byage (table 15, column 5) at various "levels" of mortality,"An unnecessarily extravagant procedure would be toproject with all of the tabulated model tables. In practice,it is sufficient to use a range of mortality levelsto produce

3 If the two censuses were fiveyears apart, the survival rates givenin column 7 of table 1.1 would be used. If the interval were fifteenyears, it would be necessary to calculate values of SLx+lS/sLx inthe model life tables.

projections that bracket the recorded numbers above agex in 1945, where x is 10, 15, ... , 50. Levels five to elevenwould have sufficed in this example, but thirteen andfifteen have been employed to illustrate the level ofmortality implicit in the apparent rate of survival of theolder population (over sixty five in 1935). The suggestedprocedure is to begin with a projection using a first guessof the mortality level, and then make projections at otherlevels in a spirit of trial and error. Table 16 shows theprojections at levels five to fifteen;

(2) The projected populations and the adjusted censuspopulations in 1945 are cumulated (from the "top" down)to obtain figures for the number of females above age x,x = 10, 15, 20, ... etc. (see table 17);

(3) By interpolation, find what level of mortality

TABLE 16. THE FEMALE POPULATION OF TuRKEY 1945 (THOUSANDS) AS PROJECTED FROM THE ADJUSTED1935 CENSUS POPULATION WJTH VARIOUS "WEST" MODEL LIPE TABLES

produces a projected population x and over exactlymatching the census population in 1945, x = 10, 15,20 etc. For example, the population over ten in 1945 was6,914,000, the projected populations based on mortalitylevels seven and nine are 6,891,000 and 7,081,000. Thelevel that would duplicate the 1945 census figure is 7.24.The mortality levels and corresponding values of 0 eo, 0esand 12 are given in table 18;

(4) Select the median level among the first nine incolumn 2 of table 18, or level 7.98, as the best singleestimate of the level of mortality among Turkish females,1935-1945;

TABLE 18. INDICES OF MORTALITY IN "WEST" MODEL LIFE TABLESCORRESPONDING TO CENSUS SURVIVAL RATES FOR TURKEY(1935-1945) FROM AGE X AND OVER TO AGE X+ 10 AND OVER

Age x Level 0/ 0"0 °e5 I.mortallty

(1) (2) (3) (4) (5)

0 ................ 7.24 35.62 46.98 .73325 ................ 8.09 37.73 48.32 .7523

10 ................ 10.37 43.41 51.86 .800215 ................ 10.80 44.50 52.53 .809020 ................ 7.98 37.44 48.13 .749725 ................ 5.96 32.39 44.90 .701030 ................ 7.02 35.05 46.62 .728235 ................ 7.21 35.53 46.92 .732440 ................ 9.24 40.61 50.13 .777645 ................ 8.42 38.54 48.83 .759550 ................ 10.44 43.61 51.98 .801855 ................ 7.31 35.79 47.09 .734860 ................ 7.56 36.40 47.47 .740365 ................ 13.81 54.06 58.17 .8772

(5) An estimate of the average intercensal crude deathrate for females can be obtained by calculating the lifetable mortality rates corresponding to level 7.98 (shown incolumn 2 of table 19)4 and multiplying these rates with theaverage intercensal population (or the estimated mid­period-1940-population) given in column 3 of thesame table. The average population is calculated as themean of the reported 1935 and 1945 populations, afteradjustment for ages reported as unknown (columns 4 and6 in table 15, respectively). The result of this operation isthe average yearly number of deaths by age in the inter­censal period, shown in column 4 of table 19. The ratioof the average yearly number of all deaths and theaverage intercensal population gives the estimatedaverage crude death rate, d. In this example d = (231.1//8783) = .0263;

(6) An estimate of the increase of the female populationin Turkey from 1935 to 1945. is provided by the ratio ofthe total populations in these years after adjustments formigration and changing territorial coverage (columns 5and 6 in table 15), i.e., by the ratio of (9344/8388) =

4 The death rate for age 0-4 may be obtained from table I.1 as(/0 - 15)f(lLo +4Ll). The death rate for the population aged 75and over is calculated as 175fT75.

1.1140. The implied annual rate of natural increase is

r = log e 1.1140 = .0108'10 '

TABLE 19. CALCULATION OF THE AVERAGE FEMALE CRUDE DEATH INTuRKEY IN THE PERIOD 1935-1945 CORRESPONDING TO THEMEDIAN LEVEL OF MORTALITY IMPLIED BY CENSUS SURVIVAL RATES

Death rates per Mean population Average annualthousand at age x 1935-1945 (thou- deaths at age x

in median life sands) (from cols. (thousands)Age x table level 7.98 4 and 6 In table 15 (col.2 x col.3)

(I) (2) (3) (4)

0-4 ......... ,. 79.57 1,245 99.065-9 ........... 7.68 1,189 9.132

10-14 ........... 5.97 912.6 5.44815-19 ........... 7.92 710.4 5.62620-24 ........... 9.99 667.8 6.67125-29 ........... 11.25 672.2 7.56230-34 ........... 12.75 672.8 8.57835-39 ........... 14.10 545.5 7.69240-44 ........... 15.33 517.4 7.93245-49 ........... 16.93 347.6 5.88550-54 ........... 22.27 410.4 9.14055-59 ........... 29.00 207.9 6.02960-64 ........... 43.16 324.2 13.9965-69 ........... 59.93 115.1 6.89870-74 ........... 89.77 129.9 11.6675 and over ...... 171.47 115.5 19.80

TOTAL ..•.....•.. 26.31 a 8,783 231.1

a (Sum of col. 4)f(sum of col. 3)•

(7) An estimate of the average annual female birthrate is obtained as the sum of the estimated death rate andrate of natural increase already calculated:

b = .0263 + .0108 = .0371;(8) An estimate of the male birth rate may be obtained

as the product of the female birth rate, the sex ratio atbirth, and the ratio of the average intercensal femalepopulation to the male population. Assuming that thesex ratio at birth was 1.05, and estimating the mid-periodmale population (analogously to the estimation of thefemale population) as 8,692,000, we have

b(males) = .0371 x 1.05 x 8783 = .0393.8692

The average intercensal increase of the male populationwas .0153, as estimated from census figures adjustedfor migration and boundary changes. This gives a maledeath rate of .0393-.0153 = .0240.

Comments. In calculating the crude death rate the agespecific death rates from the estimated life table areweighted by an age distribution that is obviously distortedby misreporting of age. Thus the resulting distribution ofdeaths is also erratic and its detailed features should notbe accepted as a valid description of that distlibution.The effect of age distortions on the calculated total numberof deaths can be expected to be much smaller since theerrors to a large extent are compensating ones. Never-

60

theless the analyst should consider the potential bias dueto this source. In particular if the proportion under agefive is under-reported (either because of omission ofyoung children or overestimation of their ages), theresulting estimate of d (and, given the intercensal r, theestimate of b) will be downward biased. In the givenexample, however, age-misreporting does not appear tohave affected the estimate of d, once a life table has beenobtained. When weightingof the mx values of that tablehas been done by an intercensal population adjustedfor age-mis-reporting (by means of a procedure notdiscussed in this Manual) the resulting d was .0264,instead of .0263 obtained above.

The estimate of childhood mortality is derived in thismethod not from the basic data themselves, but is asimple extrapolation from the estimated adult mortality(or mortality over age five) via the " West" model lifetables. If the pattern of mortality characterizing thisfamily of life tables is not valid for Turkey, the estimatedchildhood mortality, hence the derived d and bareaccordingly biased. This point is discussed in section A.l.aof chapter IV. If, for example, the "South" family ofmodel life tables derived from the experience of otherMediterranean countries more nearly approximates the(unknown) true pattern of Turkish mortality, the deathand birth rates may be as much as .006 higher than theestimates given above. Apart from the argument ofgeographical, and to some extent cultural, closeness tocountries known to be characterized by "South" mortality,there exists some evidence from recent surveys that theage pattern of Turkish mortality is indeed more" southern "than "western".

The above remarks suggest that the crude birth ratejust derived (.0382 for the population as a whole) is lowerthan the actual level. It should be noted however thatduring the period in question the actual level of the birthrate itself must have been appreciably lower than its"normal" level. There are two reasons supporting thisassumption. First, wartime conditions, such as extensivemobilization in the early 1940s probably have depressedfertility. Second, the relative size of the cohorts in theprime child-bearing ages was much below "normal"during the period because of depressed fertility andunusually high mortality due to the Balkan wars and tothe first World War and its troubled aftermath in Turkey.

B. ESTIMATION OF FERTILITY AND MORTALITY BY STABLEPOPULATION ANALYSIS

The method of deriving estimates of fertility andmortality from records of the age distribution and frominformation on the rate of growth under conditions whenthe population may be considered approximately stableis discussed in section B of chapter I. In the presentsection applications of this method are illustrated by threeexamples based on data collected in censuses in Englandand Wales (1871), India (1911) and Brazil (1950). Thesecensuses exemplify three different situations with respectto the quality of the basic data, in particular with respectto the quality of data concerning age.

1. England and Wales, 1871

In section B of chapter I, it was shown that a stablepopulation based on the 1871-1881 English life table andon the rate of natural increase during the same periodmatches very closely the age distribution as actuallyrecorded in the census of 1881. Conversely, an index ofthe recorded age distribution and the rate of growth wereshown to define a model stable population the parametersof which provide an excellent approximation of variousdemographic characteristics of the population, such asthe birth rate or the expectation of life at age zero.However since the values of these parameters were knownfrom direct statistical observations there was little justi­fication of applying stable methods of estimation apartfrom proving the power of the technique under conditionswhen age reporting is highly reliable. The mechanics ofthe application of this method are illustrated in thefollowing paragraphs also by using English data, but undersomewhat less artificial circumstances. Notably stableestimates of population parameters for the periodpreceding 1871 will be derived from the age distribution asreported in the 1871 census and from the rate of growthbetween 1861 and 1871. No official life table has beenprepared for the sixteen-year period preceding 1871, andthe registration of births during that time is known to beslightly more defective than in the 1870s-birth statisticsbecame virtually complete only after legislation in 1874placed the responsibility for registering births upon theparents. 5

Conditions for applying the method. Whether alternativemethods of estimation are available or not, stable esti­mation should be attempted only if a case for the existenceof stability with respect to the relevant demographicconditions can be established. Preferably such a caseshould rest on direct evidence, in particular on theconstancy of the age distribution and of the rate ofpopulation growth. Examination of the distribution byage in the decennial censuses from 1841 through 1871provides a confirmation of approximately stable condi­tions in England and Wales during that period (andattests to the good quality of age reported) althoughmasculinity ratios between ages 30-44 in 1871 are notice­ably smaller than the ones reported in earlier censuses.This appears to reflect the effect of excess male out­migration and suggests that the male population in thiscase is a less satisfactory basis for stable estimates." Asto the rate of growth, the slight fluctuations in the inter­censal rates of increase during the three decades preceding1871 that show no trends are reassuring but, again, cannotbe taken at face value since the population was subject tonet outmigration during the period. Explicit considerationof the effect of migration is clearly necessary. The average

5 cr. D.V. Glass, "A Note on the Under-Registration or Birthsin Britain in the Nineteenth century", Population Studies, vol, V,No. 1 (July 1951), pp. 70-88.

8 No illustration or these points is offered here. For a convenientsource see the historical series in General Register Office, Census1961, England and Wales, Age, Marital Condition and General Tables(London, 1964), pp. 30-32.

61

TABLE 21. FEMALE POPULATION BY AGE, ENGLAND AND WALES,

1871"

TABLE 20. AVERAGE ANNUAL RATE OF INCREASE BY DECADES BETWEEN1841 AND 1871 FOR EACH SEX CALCULATED FROM CENSUS FIGURESAS REPORTED ("INTERCENSAL RATE OF GROWTH") AND AFTERCORRECTION FOR MIGRATION ("NATURAL RATE OF GROWTH"),ENGLAND AND WALES

Required basic data, Apart from numerical evidencenecessary to establish the case for the applicability of thestable method, and, in the present instance, to make acorrection for migration, the required basic data are afive-year distribution of the population by sex in onecensus, and a count of the total population by sex at anearlier point in time to provide a rate of growth. Thelatter information was given in table 20. Table 21 gives

rates of intercensal growth before and after correction fornet outmigration are given in table 20.7

Table 20 shows that there was little change in thenatural rate of growth over the thirty-year period priorto 1871 and the male and female rates were reasonablyclose to each other. (Perfect stability would imply identicalgrowth rates for the two sexes). The effectof the correctionfor migration on the female growth rate is moderate,but is much less so for the male population. On thebasis of the preceding observations estimation of fertilitywill be derived from the female age distribution andgrowth rate only.

the female age distribution in England and Wales up toage forty-five. It is not suggested to go beyond that agefor purposes of stable estimation.

Computational procedure. (I) Obtain values of C(x):proportions up to age x(x, 5, 10, ... ,45) from table 21.These cumulated proportions after rounding are shownin col. 2 of table 22.

(2) From table II in annex II find the parameters of thefemale stable populations characterized by the C(x) valueson the one hand, and by the rate of natural increase for thedecade preceding the census (.0131), on the other hand.This operation may be conveniently executed in thefollowing steps:

(a) Given the value of C(5) as reported, select twostable populations each having the required growth rate(i.e., by interpolating between stable populations tabulatedfor r = .010 and r = .015 to get the female growth rateof .0131) and one of the levels of mortality for whichmodel stable populations are given in table II (i.e.,levels I, 3, ... , 23). Specifically the mortality levels shouldbe so chosen (by means of a rough process of trial anderror) that the C(5) values in the resulting two modelpopulations just bracket the C(5) value in question, i.e.,the ogive at age five in one of them should be just higher,and in the other just lower, than the reported value.The proper levels in this instance are levels nine andeleven. Columns 3.a and 3.b show the values of C(5)in these stable populations and also the parameters (suchas the birth rate) for which estimates are sought;"

(b) Repeat the above procedure for other values ofC(x),i.e., for x = 10, 15, ... ,45, using additional columns ifany of the reported C(x)s are not bracketed by the ogivesof stable populations previously calculated. In the givenexample no such need arises since none of the C(x) valuesimply a mortality level higher than level eleven, or lowerthan level nine, given the growth rate of .0131;

(c) For each value of x find the interpolation factorsthat would be necessary to obtain the reported C(x)­shown in col. 2-from the corresponding values in cols.3.a and 3.b. For example, the reported proportion up toage 10 is .248 which may be expressed as a weightedaverage of the figures for C(10) in cols. 3.a and 3.b;specifically as .27 x .256+ .73 x .245 (cf, annex VI).Applying the same interpolation factors to other popu­lation parameters calculated for the stable populationsin cols. 3.a and 3.b, such as tbe birth rate, one obtainsparameters of the stable population defined by the given rand the observed C(x). For example the birth rate corres­ponding to r = .0131 and C(IO) = .248 is .27 x .0365+.73 x .0329 = .0339. The results of these calculationsare given in cols 4.a through 4.e. Note that one of theparameters calculated is the population death rate. Oncethe birth rate is calculated the death rate may of course beobtained by simply subtracting the specified growthrate from the birth rate.

.0121

.0128

.0131

Naturalrate ofgrowth

13.1711.6410.33

9.409.048.046.986.015.49

19.90100.00

Population(percentage)

Females

.0120

.0118

.0125

Intercensalrate ofgrowth

.0131

.0140

.0138

Naturalrate ofgrowth

Population(thousands)

1,534.81,355.61,203.51,095.71,052.8

937.3813.7700.5639.7

2,319.711,653.3

Males

.0124

.0108

.0124

Intercensalrate ofgrowthPeriod

II Source: See foot-note 1 to the present chapter.

Age

1841-1851 , ,1851-1861 , .1861-1871 ., " , ..

0-4 .5-9 , .

10-14 , ,' ,15-19 .20-24 , ,25-29 .....•.... , .30-34 .. , , , .35-39 " .40-44 ..45 and over ".TOTAL •••••••• , ••••••••

7 The corrections are based on estimates of migration preparedby Glass (op. cit., pp. 85-86, "method c"). The estimated intercensalnet balance in the decade preceding a given census was added to thecensus population and then the intercensal increase was calculatedusing this population and the uncorrected population a decadeearlier.

8 One of the parameters calculated is the gross reproduction rateassociated with iii = 32.1 (the basis for this particular value of iii isdiscussed later). Since table II contains only values of GRR foriii = 27, 29, 31 and 33, it is necessary first to calculate two GRRvalues that bracket the GRR with the correct value of iii, and toobtain this latter quantity by an additional step of interpolation.

62

--------------------------------------

TABLE 22. DERIVATION OF STABLE POPULATION ESTIMATES OF FERTILITY AND MORTALITY BASED ONA REPORTED AGE DISTRIBUTION AND THERATE OF GROWTH. ENGLAND AND WALES, FEMALES, 1871

Values of C (x) andC(x) various parameters in Values of various parameters In

(proportion female stable populo- female stable populations withup to age x) tions with r = .0131 C(x) as shown In col. 2 and with r = .0131

Age x Level 9 Level II Birth Death Level 0/ oeo .ORR (m = 32.1)rate rate mortality

(1) (2) (3.a) (3.b) (4.a) (4.b) (4.c) (4.d) (4.e)

5 .............. .132 .139 .131 .0334 .0202 10.8 44.4 2.3710 .............. .248 .256 .245 .0339 .0208 10.5 43.6 2.4115 .............. .351 .363 .349 .0334 .0203 10.7 44.3 2.3820 .............. .445 .461 .444 .0331 .0200 10.9 44.7 2.3525 .............. .536 .548 .530 .0341 .0210 10.3 43.3 2.4230 .............. .616 .626 .607 .0346 .0215 10.1 42.6 2.4635 .............. .686 .695 .677 .0347 .0216 10.0 42.5 2.4640 .............. .746 .756 .739 .0344 .0213 10.2 42.9 2.4445 .............. .801 .810 .794 .0345 .0214 10.1 42.8 2.45

Birth rate ............ .0365 .0329Death rate ........... .0234 .0198°eo .................. 40.0 45.0GRR(m = 31) ....... 2.52 2.28GRR(m = 33) ....... 2.65 2.39GRR (m = 32.1) ..... 2.59 2.34

(3) Ideally each combination of C(x) and r for a given median among those considered provides the best avail­sex should define the same stable population: the para- able choice. To find this population, rank the estimatedmeters of this model then could be accepted as valid for nine birth rates according to their absolute values, andthe actual population of that sex. In practice however a select the intermediate (the fifth largest) in this series.more or less tightly clustered series of stable populations In the given example the median stable population is theare determined by the various pairs of C(x) and r. The one associated with the reported C(25), giving theprocedure of selecting a single best estimate (or selecting following estimates for the female population: birth rateestimates located within a narrower range than the range = .0341, death rate = .0210, expectation of life at birthof all obtained stable estimates) depends on the nature of = 43.3 years, and gross reproduction rate = 2.42.identifiable errors in the data, especially with respect to (4) Estimates for the male population and for theage misreporting. In the given example the consistency of population as a whole may be obtained from the para­the estimates shown in cols 4.a through 4.e is gratifyingly meters calculated for the females plus the knowledge ofhigh (see figure XX for a graphical representation of the the sex ratio at birth (male births/female births) and theseries of birth rates obtained). This finding tends to sex ratio (males/females) in the total population. Theconfirm the original assumption of stability and the good ratio of registered male births to female births in the fivequality of age reporting. Given these circumstances, and years preceding 1871 was 1.041. The number of maleslacking information that would single out some reported enumerated in 1871 was 11,058.9 thousands. (The femaleC(x) values as particularly reliable, or relatively defective, population was givenin table 21.) The male birth rate thenthe selection of the stable population with an ogive isIcalculated as

f I bi th t sexratio at birth 0341 1.041 0374emae ir raex =. X--= ..sex ratio of the population .949

The total birth rate can be obtained as a population- the two sexesor directly from the female birth rate asweighted average of the rates calculated separately for

. female population ..female birth rate x x (1+sexratloatbuth) = .0341x.513x2.041 = .0357.

total population

(5) By subtracting the appropriate rate of natural table II by finding the level of mortality in the male modelincrease from the estimated male birth rate and total stable population having the estimated male death ratebirth rate death rates for males (.0374- .0138 = .0236) (.0236) and the male rate of natural increase (.0138). Theand for the total population (.0357- .0134 = .0223) are level is 10.0, implying an °eo of 39.7years. Sincemortalityobtained. in England is known to be well described by the model

(6) An estimate of the male expectation of life at birth "West" life tables, the level of mortality in this instance(and/or any desired life table parameter) is obtained from may be estimated also by simply assuming that the

63

--­.".----..._."..".-­BRAZIL,I950

MALES

.050

.040

.045

BIRTHRATE

.035

-----------ENGLAND AND WALES,IB71FEMALES

.030

.025

.020

.015

010

.005

454030252015105oL-_---L.__...l-_--l__-L__.L.-_--!.__....1-__L-._--'-_---J

AGE

Figure Xx. Stable population estimates of the birth rate derived from reported proportions up toage x-C(x)-in censuses of England and Wales, India and Brazil for the year and sex as indicatedand from the average rate of natural increase for the same sex during the ten-year period preceding

each census

relation of male mortality is the same as in the tablesshown in annex I, i.e., that the level of male mortalityis 10.3, as is for the females,hence thatthemale °eois39.9.

(7) Given the sex ratio at birth an estimate of totalfertility can be obtained from the estimated GRR(m = 32.1): TF = 2.42x 2.041 = 4.94 children perwoman.

Estimation of m. The calculation of estimates of thegross reproduction rate as described above presupposesthe existenceof an estimate of the mean age in the scheduleof the age specific fertility rates (m) prevailing in the given

population. Such rates are not available for England andWales for the period in question. Section B.5 of chapter Idescribes two methods for indirect estimation of m;due to lack of data on children ever born only the firstof these methods-based on the reported proportionsmarried-may be applied here. An illustration of thecomputation is given in table 23. The standard age patternof marital fertility (applicable for populations whosebirth control practices are negligible) shown in col. 3is taken from table 1. Note that the absolute magnitudesof the hypothetical age specific fertility rates calculatedin col. 4 have no practical significance. What is relevant

64

for the problem at hand is merely the age pattern of theseimputed rates: the mean age of the fertility schedule iscomputed as the average of the central ages in each ageinterval (col. 5) weighted by the entries in col. 4.

Life table lorEngland and Wales, Stable estimate

1838-1854 from 1871 census

Females 41.85 43.3Males ..... 39.91 39.9

Life table lorEllglandand Wales,

1871-1880

44.6241.35

a ~21-:iJ8~roportion married at age 15-19) = 1.2 -.7 x .032

TABLE. 23 CALCULATION OF m FROM REPORTED PROPORTIONS

MARRIED AND FROM THE STANDARD AGE PATTERN OF MARITALFERTILITY RATES, ENGLAND AND WALES, 1871

Stable estimate .0357Vital registration

1866-1870 03511861-1865 .. . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. .0350

Vital registration corrected for under-registration of births''1866-1870 ' .03571861-1865 ' .0358

15-19 .... .032 1.178" .0377 17.5 .6620-24 .... .343 1.000 .3430 22.5 7.7225-29 .... .624 .935 .5834 27.5 16.0430-34 .... .735 .853 .6270 32.5 20.3835-39 .... .766 .685 .5247 37.5 19.6840-44 .... .758 .349 .2645 42.5 11.2445-49 .... .740 .051 .0377 47.5 1.79

2.4180 77.51

According to vital registration for the sixteen-yearperiod from 1855 to 1870 the average death rate for thetotal population was .0223. This figure is identical to thestable population estimate of the same quantity derivedfrom the age distribution in the census of 1871.

2. India, 1911

Another illustration of the technique of obtaining vitalrates by the stable method based on a census record of theage distribution and an intercensal growth rate is offeredin this section using Indian statistics, notably C(x) fromthe census of 1911 and r for the period 1901-1911. Incontrast to the previous example, information on agedistribution and population growth constitutes virtuallythe only valid basis for establishing vital rates for Indiarelating to the period in question. The usefulness of thatinformation is fortunately greatly enhanced by theapplicability of stable population analysis. The argumentsupporting the assumption of stability rests mainly ontwo considerations. First, the series of decennial censusesin India up to 1911 shows a remarkable degree of stability.This point is sufficiently well-illustrated in figure XI.To be sure, the detailed shape of the reported age distri­butions is far from what would result from sustained pastconstancy of vital rates. But the fact that the markedpeculiarities of the age distribution are reproduced censusafter ce~sus at the same age (as opposed to the same cohort)conclusively proves that the explanation for these peculiar­ities lies in an essentially unchanged pattern of age-mis­reporting rather than in violent past deviations fromstable levels of fertility and mortality.

Second, the series of intercensal growth rates preceding1911 lack any detectable trend away from the horizontal;rather ups are followed by downs in regular succession.I?

Constancy of age distribution and fluctuating growthrates are consistent with indirect or qualitative knowledgeon the main features of the demographic situation in pre­1911 India. Such features are frequent short-term changesin mortality conditions but the absence of lasting im­provement or deterioration in the chances of dying; asustained high level of fertility explained by the lack ofcontraceptive practices and by quasi-universal and earlymarriage; and, finally, the essential closedness of thepopulation with respect to external migration.

Stable conditions notwithstanding, no high precisioncan be expected from estimates derived by stable analysisin the Indian case, primarily because of the defects inage-reporting mentioned earlier. Nevertheless, identi­fication of systematic deviations from the expected stabledistributions, as discussed in detail in section B.c ofchapter I, permits an interpretation of even seeminglyinconsistent series of stable estimates, thus considerablyreducing the apparent range of uncertainty. As was

Col. 4X

col. 5(6)

Medianage(5)

Hypotheticalfertility

rates(col. 2 x col. 3)

(4)

Standardmarital(ertllity

rates(3)

Proportionofmarriedfemales

(2)

Agetnteroal

(I)

77.51Hence, iii = 2,418 = 32.1

Official life tables are available for the periods 1838­1854and 1871-1880,but not for the sixteen years precedingthe census for which the stable estimates derived abovemay be considered as relevant. The stable life tableestimates do, however, suggest a plausible trend of mor­tality change when compared with the official life tablesmentioned. In terms of °eo the following comparison canbe made:

Implicit in the above procedure is the assumption thatall births occur in marriage. However the proportion ofillegitimate births in England at the time amounted tosome 6 per cent of all births. It would be possible toe~tend the calculation just outlined to obtain a hypothe­tical age pattern of fertility that takes into accountillegitimate births as well, e.g., by assuming that illegiti­mate fertility rates by age of mother had the same patternas was recorded in Sweden in the 1860s. Calculations showthat the iii resulting from such assumptions would differlittle from the value obtained in table 23-it would beless by not more than .2 year.

Comments. Some of the estimates obtained above maybe compared with data from vital registration or otherestimates. For example, with respect to the birth rate forthe total population we have:

°Glass, op, cit., p. 85. 10 See Davis, op, cit., pp. 27-28 and 85.

65

TABLE 24. DERIVATION OF STABLE POPULATION ESTIMATES OF FERTILITY AND MORTALITY BASED ONAREPORTED AGE DISTRIBUTION AND THERATE OF GROWTH. INDIA, 1911, FEMALES

Value. ofC(x) and various parameters in femaleC(x) stable population. with r = .0073 and level. of Value. of oarious parameters in female stable population. with C(x) as shoum

(proportion mortality a. indicated in col. 2 and with r = .0073up to age x)India, 1911. Birth Death Level of

Age x female. Levell Level 3 Level 5 Level 7 rate rate mortality oeo ORR (m = 28.2)(1) (2) (3.a) (3.b) (3.e) (3.d) (4.a) (4.b) (4.e) (4.d) (4.e)

5 .0 .••••• •• .141 .151 .139 .0421 .0348 4.7 29.2 2.6810 •• 0 ••••••• .276 .296 .272 .0503 .0430 2.7 24.2 3.1915 . " ....... .375 .383 .360 .0457 .0384 3.7 26.8 2.9120 .......... .455 .457 .435 .0403 .0330 5.2 30.4 2.5725 .......... .548 .573 .546 .0413 .0340 4.9 29.6 2.6330 .......... .640 .653 .625 .0449 .0376 3.9 27.3 2.8535 .......... .725 .754 .724 .0487 .0414 2.8 24.8 3.0940 .......... .782 .784 .759 .0478 .0405 3.2 25.4 3.0345 .......... .847 .861 .837 .0531 .0458 2.2 22.9 3.38

Birth rate .......... .0597 .0484 .0408 .0353Death rate ......... .0524 .0411 .0335 .0280oeo ................ 20.0 25.0 30.0 35.0GRR (m = 27) ...... 3.67 2.98 2.53 2.21GRR (m = 29) '0 ••• 3.88 3.13 2.64 2.30GRR (m = 28.2) ... 3.80 3.07 2.60 2.26

11 Both the age distribution and the growth rate are for thecurrent (post-partition) territory of India. They were calculatedby Mr. S. B. Mukherjee in his" A Study of the Vital Rates in Indiaand West Bengal" (unpublished manuscript, Princeton, 1965) whichhe kindly made available to the authors.

12 See the preceding foot-note for the source of these data.

shown in chapter I-see in particular figures VIII, IXand X-Indian age distributions are characterized bywhat has been described in this Manual as the" African­South Asian" pattern. Rules for analysing such distri­butions were set forth in section BA in chapter 1. Accord­ing to these rules it is preferable to use only the femalepopulation as a basis for estimation as far as the reportedage distribution is concerned. Table 24 shows the deriva­tion of various population parameters from the 1911female age distribution in India (col. 2) and from the1901-1911 average female growth rate (r = .0073).11 Thecomputational procedures underlying this table areexactly analogous to those used and explained in connex­ion with table 22 above. Note that the intermediate stepof calculating parameters for stable populations with theproper growth rate but with approximate ("bracketing")levels of mortality requires in this instance the use of morethan two mortality levels (see cols. 3.a through 3.d) owingto the fact the reported age distribution is less consistentlyclose to one single stable distribution than was the casein the previous example.

The mean age of the fertility schedule (m = 28.2) usedin the computation was estimated from the standardmarital fertility schedule shown in table 1 and from theproportions of married females in India, 1911. Thelatter, for five-year age groups, were as follows: 12

15·19

.818

20-24

.902

25·29

.882

30·34

.814

35·39

.742

40-44

.597

45-49

.522

The procedure of calculation was the same as thatdiscussed in connexion with table 23 above.

Inspection of the sequence of the derived birth rates(column 4.a in table 24), also reproduced in figureXX,shows the same characteristic pattern as was found fromdirect comparisons of the reported population with modelstable populations (cf. figures VIII and IX). This suggeststhat such comparisons are not necessarily required for theidentification of the general character of age-reportingerrors. Once a case for applying the stable method hasbeen established the analysis may proceed directly to thecalculation of birth rates and other parameters implied bythe various pairs of C(x) and r, A judgement on thepattern of age-misreporting, hence on the rules of esti­mation to be applied, then can be based on the results ofthis calculation, in particular on the sequence of thebirth rates obtained for x = 5, 10, ... , 45. In the presentcase the rules call for the acceptance of the parametervalues associated with C(35) as the best single estimates.Given the female rates, parameter values for the malepopulation and for the population as a whole are to beobtained in the same fashion as was shown in the precedingexample, i.e., by using the available information onaverage intercensal growth (in this case .0082 for malesand .0077for the total population), on the sex ratio of thepopulation (1.037), and assuming-in the absence ofinformation to the contrary-that the sex ratio at birthwas 1.05. The male expectation of life and/or other lifetable indices are determined by finding the level ofmortality in the male stable population having thereported male growth rate and the death rate as derivedearlier. Some of the stable estimates resulting from thesecalculations are summarized in table 25.

Naturally all figures in table 25 are to be regarded asrough approximations. Yet, on the basis of knowledge

66

3. Brazil, 1950

TABLE 25. STABLE POPULATION PARAMETERS FOR INDIA, 1911,DERIVED FROM THE FEMALE AGE DISTRIBUTION, THE FEMALE AND

MALE GROWTH RATES, AND THE SEX RATIO OF THE POPULATIONAS REPORTED; AND BY ASSUMING THAT THE SEX RATIO ATBIRTH IS 1.05

of the pattern of age-misreporting it is possible to assertwith a high degree of certainty that the female birth ratewas higher than .046-the estimate associated with C(15).Furthermore, the value derived from C(35) is stronglysupported by the only slightly higher estimate (.050)implied by C(lO) and r, Most likely this latter figure is,or is close to, what may be considered a fair upperestimate of the birth rate. These statements are qualifiedbythe fact that there is no direct evidence confirming thevalidity of the "West" pattern of mortality in the Indiancase. Use of alternative stable population families in theabove calculations would have typically resulted inhigher estimates of the birth and death rates, hence inhigher estimates of total fertility and lower expectationof life. As to the relation of male and female mortalities,the strong masculinity of the population-demonstratedby the other Indian censuses as well-conclusive1yindicates that this particular relation incorporated in themodel life tables is not duplicated in India.

TABLE 26. STABLE POPULATION ESTIMATESOF FERTILITY AND MORTA­LITY BASED ON THE AGE DISTRIBUTION OF THE MALE POPULATIONOF BRAZIL AS REPORTED IN THE CENSUS OF 1950 AND ON r = .0232,THE ANNUAL RATE OF GROWTH OF THAT POPULATION IN THE1940-1950 INTERCENSAL PERIOD

Values of various parameters in male stablepopulations with C (x) as indicated in

column 2 and with r = .0232

C(x)(proportionup to age x)Brazil. 1950,

malesAge x Birth Death Level of

rate rate mortality °eo(1) (2) i;(3.a) (3.b) (3.e) (3.d)

5 .......... .164 .0422 .0190 12.1 45.010 .......... .302 .0430 .0198 11.7 43.915 .......... .424 .0438 .0206 11.3 42.920 .......... .527 .0436 .0204 11.4 43.225 .......... .619 .0441 .0209 ILl 42.430 .......... .698 .0447 .0215 10.9 41.835 .......... .760 .0436 .0204 11.4 43.240 .......... .819 .0447 .0215 10.9 41.845 .......... .867 .0456 .0224 10.5 40.9

of fertility as indicated by 1940 and 1950 census reportson children ever born; on the relative unimportance ofinternational migration; and on the little or no change inmortality prior to 1950as evidencedby reports on propor­tions of children surviving in the 1950and 1940censuses.

The Latin American character of the pattern of age­misreporting is revealed by comparing the actual agedistributions to model stable distributions, or, moredirectly, by calculating estimates of the birth rate for malesand females from C(x) and r, As a result the basic stableanalysis is to be limited to the male population. Table 26shows the parameters implied by the male age distribution(column 2) and the male growth rate (r = .0232). Thecomputations underlying this table were explained abovein connexion with table 22 (cf. also table 24). Naturally,the male model stable populations of annex II wereutilized in this case.

.0490

.0413

Total population

.0493

.04114.0

25.3

Males

.0487

.04142.8

24.83.096.33

Females

Birth rate .Death rate ,Level of mortality .°eo .GRR (m = 28.2) .Total fertility .

Using age distribution data from the Brazilian censusof 1950 jointly with the rate of growth between 1940and1950to derive stable estimates exemplifies the applicationofthe stable method under conditions when age-reporting The series of male birth rates and other parametersis typically "Latin American" in its characteristics. The given in table 26 are located within rather narrow limits.applicability of the method in this instance is supported The median in the series, which is the best single estimate,by somewhat less satisfactory evidence than in the two is associated with C(lS). Assuming a sex ratio at birthpreceding examples because of the lack of an extended of 1.05, and considering that the sex ratio of the popu­series of previous censuses of reasonably good quality. lation (as reported by the census) was .9933, the femaleNevertheless the case for assuming stability is convincing. birth rate is obtained by multiplying the male birth rateIt is based on the close similarity of the 1940 and 1950 by the ratio .9933/1.05. The birth rate for the wholeage distributions; on high and essentially identical levels population is obtained as

male birth rate male population (1 ti t bi th)--_._-- x x +sex ra 10 a If .sex ratio at birth total population

Death rates are calculated by subtracting the rates of 15-49 (which in this case yields an estimated fii of 28.8average intercensal growth (.0238 for females and .0235 years) is open to the objection, serious in the case offor the total population) from the estimated birth rates. Brazil, of ignoring the fertility of women reported as

The value of fii necessary to obtain an estimate of the single but living in de facto unions. Thus the estimate(female)gross reproduction rate can be estimated by both obtained from the relation fii = 2.25 P3/P2 +23.95is to bemethods suggested in chapter I, section B.5. However preferred. The value of P3/P2-the ratio of children everthe method of applying a standard marital fertility born per woman 25-29 and 20-24- was 2.289 accordingschedule to the proportions married among females to the 1950census. Hence we have fii = 29.1. The female

67

TABLE 27. STABLE POPULATION PARAMETERS FOR BRAZIL, 1950,DERIVED FROM THE MALE AGE DISTRIBUTION, THE MALE AND FEMALEGROWTH RATES, AND THE SEX RATIO OF THE POPULATION ASREPORTED; AND BY ASSUMING THAT THE SEX RATIO AT BIRTHIS 1.05

gross reproduction rate, as well as the female expectationof life and other parameters, then can be obtained byreading these values in a female stable population deter­mined by any two of the parameters previously calculated,such as the female death rate and the female rate ofgrowth. Table 27 summarizes the main results.

As was the case in the previous example no vitalstatistics are available with which these estimates could beconfronted. In view of the high consistency of the valuesimplied by the various C(x)s, the major uncertainty withrespect to the goodness of the estimates originates, onceagain, in the choice of the mortality pattern underlyingthe model stable populations utilized: the "West" pattern­and in particular the early childhood mortality impliedby a given adult mortality in that pattern- mayor maynot be a close approximation of Brazil's actual expe­rience.P The application of the census survival method(cf. the discussion in section A in this chapter and inchapter 1) for the Brazilian male population yields an oeovalue of 42.4 that appears to confirm the validity of themortality estimate shown in table 27 (hence, given thestable age distribution, the validity of the birth rateestimate). But this is not pertinent to the problem statedabove, since both methods have essentially the sameweakness in estimating childhood mortality. Unlike inthe case of India, however, census information on childsurvival rates in Brazil supplies a basis for a directestimation of child mortality thus permitting a check on,and improvement of, the estimates shown in table 27.This topic will be taken up in chapters VII and VIIIbelow.

TABLE 28. STABLE POPULATION ESTIMATES OF THE BIRTH RATE (b)AND OF THE GROSS REPRODUCTION RATE (GRR) BASED ON THEAGE DISTRIBUTION OF THE FEMALE POPULATION OF INDIA ASREPORTED IN THE CENSUS OF 1961 AND ON THE ANNUAL RATEOF NATURAL INCREASE OF THAT POPULATION IN THE 1951-1961INTERCENSAL PERIOD (r = .0189)

Birth rate GRR (flj = 28.8)(3.a) (3.b)

Values ofb and of GGR in femalestable populations with C(x) as in

col. 2 and u;lth r = .0189C(x)

(proportionup to age x)India. 1961,

females(2)(1)

Age x

estimates from information on age distribution andgrowth under conditions of approximate constancy offertility and mortality. When mortality has been declining,but other requisites of stability obtain- a situation oftenencountered in the contemporary world- stable analysisis still frequently attempted, the practice being defendedby the argument that the age distribution in such so-calledquasi-stable populations is always close to that of apopulation which is stable in the strict sense, and ischaracterized by the current fertility and mortality of thepopulation in which mortality has been declining.However, such estimates contain a bias which, dependingon the duration and speed of the change in mortality,may be substantial. Section C of chapter I describes amethod by which such a bias can be eliminated or atleast considerably lessened by using information on thenature of the mortality decline. The method is illustratedbelow by the example of two populations; that of Indiain 1961 and of Mexico in 1960.

1. India, 1961

The assumption of essentially stable demographicconditions, on the basis of which estimates for the Indiaof 1911 were derived above, is less defensible after thecensus of 1921. While there are no signs that wouldindicate a change in fertility, the growth of the populationhas been accelerating since the 1920s, undoubtedlyreflecting a more or less steady decline of mortality fromthe high plateau of the 1881-1921 period. Under suchconditions stable estimates should be adjusted to takecare of the effects of that decline.

Computational procedure. (1) Table 28 shows stableestimates of the birth rate and the gross reproduction ratethat are calculated in exactly the same manner explainedin connexion with tables 22 and 24 above. The inputs inthis instance are the 1961 female age distribution.!" and

.0426

.0191

Total populationFemales

.0414

.017612.047.5

2.835.80

Males

.0438

.020611.342.9

Birth rate .Death rate .Level of mortality .°eo .GRR (m = 29.1) ..Total fertility .

C. EsTIMATION OF FERTILITY AND MORTALITY BY STABLEPOPULATION ANALYSIS WHEN THE POPULATION ISQUASI-STABLE

The examples given in the previous section demon­strated the technical details of extracting population

18 An at least qualitatively identifiable source of bias in thesecalculations also arises from the fact that no allowance was madefor external migration in reckoning the growth rate. The rate ofnatural increase may have been perhaps .0002 smaller than theaverage intercensal rate of growth. If so, the birth rates are under­estimated by roughly the same amount, while the error introducedinto the death rates (also underestimated) is about twice as large.

5 ., .................. .154 .0396 2.6410 .................... .303 .0476 3.1815 .................... .412 .0447 2.9820 .................... .493 .0393 2.6225 .................... .583 .0397 2.6430 .................... .668 .0415 2.7635 .................... .738 .0425 2.8340 .................... .794 .0422 2.81

14 Source: Census of India, 1961 Census, Age Tables, p. 54.

68

the rate of female natural increase for 1951-1961, (r= .0189) that was obtained by adjusting the intercensalrate for changed territorial coverage and for net immi­gration. The mean age of the fertility schedule wasestimated from an imputed age specific fertility schedulein the same manner as shown in table 23. The proportionsmarried among females 15-49 that were used in thecalculation are as follows (1961 census data):

The resulting mequals 28.8 years.(2) The preliminary stable estimates of band GRR

(columns 3.a and 3.b in table 28) are to be adjusted for thedistorting effects of changing mortality using the adjust­ments listed in table IIU. Since the preliminary estimatesare based on C(x) and the average rate of growth duringthe ten years preceding the time to which C(x) refers, theappropriate section of that table is its "Part (a)". Toextract the correct adjustment factors from the tabulatedfigures it is first necessary to estimate values of two indi­ces; namely t, the approximate length of time (in years)for which the decline of mortality has been proceeding;and k, a parameter that describesthe speed of the decline.

(3) The value of t may be estimated in this instance as40 years, the time that has elapsed between 1921-thedate up to which growth rates for India showed a regularsequence of ups and downs, and after which accelerationof growth was uninterrupted-and 1961, the date of thelatest census.

15-19

.696

20-24 25-29 30-34

.918 .942 .915

35-39

.871

40-44

.777

45-49

.698

(4) The parameter k can be derived from the averagerate of acceleration of the growth rate itself using theempirical relation k = 17.8x Sr]11t. The absolute increasein the growth rate can be obtained by subtracting fromthe 1951-1961 rate of increase (which may be thought ofas referring to the year 1956), the level of growth thatprevailed, on the average, up to 1921. The latter may beestimated from the ratio of 1921 all-India population tothe same population in 1881. This ratio is 1.1877, therefore,r 19 2 1 = (loge 1.1877)/40 = .00430. The value of !:>.r/I:!.tis then

.0189- .0043 = .0146 = .000417 ;1956-1921 35

hence k = .000417 x 17.8 = .0074.(5) Column 3 in table 29 shows the adjustments as

taken directly from table IIU, which is tabulated fork = .01. For other values of k it is necessary to scale thesefractions up or down in the same proportion that theactual value of k bears to .Ol-i.e., in this instance by.0074/.01 = .74. This is shown in column 4.

(6) Column 4 thus contains proportions to be addedto or subtracted from the preliminary estimates. It isconvenient to transform these adjustments into multipliersby adding 1 to each entry (see column 5). Column 6 givesthe products of these multipliers and the preliminarystable estimates, i.e., the adjusted (quasi-stable) estimatesof the female birth rate and the gross reproduction rate.

(7) Selectionof a singleestimate for band GRR amongthose associated with the various C(x) values should becarried out in the same fashion as was explained andillustrated for "pure" stable estimates in section B of this

TABLE 29. EsTIMATION OF THE BIRTH RATE AND OF THE GROSS REPRODUCTION RATE FOR THE FEMALEPOPULATION OF INDIA, 1961, BY AD1USTMENT OF PRELIMINARY STABLE ESTIMATES OF THESE PARA­METERS (AS CALCULATED IN TABLE 28) FOR THE EFFECTS OF DECLINING MORTALITY

Stable population Adjustments Adjustments Adjustedestimates derived from table III.I, for t = 40, Adjustment estimates

from C(x) and part (a) k = .0074 factors (col. 2 X col. 5)Age x lO-year intercensal r t = 40, k = .01 (col. 3 X .74) o+col. 4)

(1) (2) (3) (4) (5) (6)

Birth rate

5 .0 .••••....• 0. .0396 -.043 -.032 .968 .038310 .............. .0476 -.032 -.024 .976 .046515 .............. .0447 -.004 -.003 .997 .044620 .............. .0393 .026 .019 1.019 .040025 .............. .0397 .051 .034 1.034 .041030 .............. .0415 .073 .054 1.054 .043735 .............. .0425 .092 .068 1.068 .045440 .............. .0422 .114 .084 1.084 .0457

Gross reproduction rate (fii = 28.8)

5 .............. 2.64 -.010 -.007 .993 2.6210 .............. 3.18 .001 .001 1.001 3.1815 .............. 2.98 .031 .023 1.023 3.0520 .............. 2.62 .062 .046 1.046 2.7425 .............. 2.64 .088 .065 1.065 2.8130 .............. 2.76 .111 .082 1.082 2.9935 ••••••••...• 0. 2.83 .130 .096 1.096 3.1040 ........... ,., 2.81 .153 .113 1.113 3.13

69

TABLE 30. ESTIMATES OF VARIOUS POPULATION PARAMETERS FORINDIA, 1961, OBTAINED BY ADJUSTING STABLE ESTIMATES OF THESEPARAMETERS FOR THE EFFECTS OF DECLINING MORTALITY

2. Mexico, 1960

Any attempt to estimate Mexican fertility and mortalityexclusively from census data is bound to be a highlyartificial enterprise since the country has a vital registra-

chapter and of chapter I. In the present instance theestimates derived from C(35) are to be preferred to therest. The male birth rate and the birth rate for the totalpopulation are calculated by assuming a sex ratio atbirth of 1.05 and by accepting the reported masculinityratio of the population (1.062).

(8) Death rates (for the two sexes and for the totalpopulation) are obtained by subtracting the rates ofnatural increase (adjusted intercensal growth rates) fromthe appropriate birth rate estimates. The expectation oflife or any other index of mortality is determined byreading the level of mortality in the stable populations(one for the males, one for the females) determined bythe vital rates calculated earlier. No adjustment of suchestimates for quasi-stability is warranted, or indeeddesirable. The principal parameter values derived byfollowing the above steps of calculation are exhibitedin table 30.

Comments. Application of the stable method to thepopulation of India in 1961 without adjustment fordeclining mortality results in a seriesof birth rate estimatesunlike those typically produced by age distributionssubject to the African-South Asian pattern of age­misreporting. When quasi-stability of the underlyingdemographic conditions is allowed for, however, thepattern familiar from the analysis of the 1911 Indian agedistribution is fully re-established. The apparent range ofuncertainty-apart from any error in the observed r, orin the assumption concerning the pattern of mortality­as to the actual levelof the (female) birth rate is remarkablysmall; what may be tentatively considered maximum andminimum estimates (those associated with C(lO) andC(5), respectively) differ only by about .002 (.0465 versus.0446). Comparisons with the estimates derived for 1911(cf. table 25) show virtually identical gross reproductionrates but appreciably reduced birth rates in 1961. Thechanged relationship of these two indices is of coursea necessary consequence of destabilization. It may benoted that the actual birth rate of the quasi-stable (1961)population is below the intrinsic rate: maintenance of theestimated mortality level and of the GRR would resulteventually in a higher birth rate than the one shown intable 30.

tion system of long tradition and its statistics on birthsand deaths for the past two or three decades at least,are considered virtually complete. Also the assumptionof constant fertility underlying both the stable and thequasi-stable methods discussed here is apparently validonly as a rough approximation. Apart from the violentdemographic disturbance caused by the Mexican revolu­tion in the second decade of the century (the consequencesof which are now less visible than they were in earliercensuses), shifts in the age distribution and trends inpopulation growth in recent decades reflect the influenceof a slight but not negligible increase in Mexican fertilitythat subtly reinforces,and is superimposed on, the domi­nating effect on changes in those variables exerted by thevery rapid decline of mortality since the mid-1930's.There is no practical way to separate such effects in stablepopulation analysis. A straightforward application ofstable methods (including the method of correcting forthe presumed effects of declining mortality) for theanalysis of Mexican data then can be expected to revealsome inconsistencies. The existence of direct informationon vital rates, not used in the stable estimates, offers theadvantage of making explicit the nature of such inconsis­tencies, and should also reveal other biases involved instable analysis that may commonly occur in other appli­cations, yet that are ordinarily not possible to nail downin the absence of independent evidence.

Not surprisingly the tests proposed earlier for detectingerrors in age-reporting when applied to Mexican censusdata reveal the existence of a "Latin American" pattern.Hence the male age distribution is accepted as the mainbasis for stable estimates. Table 31 sets forth the elementsof the calculation leading to a series of estimates of themale birth rate, adjusted for the effects of decliningmortality. The basic data on which these calculations arebuilt are the male age distribution (column 2) and theaverage male growth rate for the period of 1950-1960(.0316). The stable estimates implied by these variables(column 3) are adjusted for quasi-stability in a manneranalogous to the computations shown earlier in table 29.The parameters k and t used in the adjustment processwere obtained by comparing the male growth rate for the1930-1940 intercensa1 period (.0181) with that for 1950­1960. Specifically, !J.r/at = .0316-.0181/20 = .0135/20= .000675; and k = !J.r/atx 17.8 = .0120. Some accele­ration of the growth rate increase probably occurredbefore the 1930s, although very little of such accelerationis suggested by the difference between the r's for 1930­1940 and 1920-1930, or by registered death rates in the1920s and early 1930s. Naturally such comparisons maybe somewhat misleading owing to possible changes incensus coverage and improvement in vital registration.On the other hand an examination of the shifts in the agedistribution of deaths, a statistic not necessarily affectedby moderate omission rates, indicates fluctuating mor­tality before the mid-1930s followed by a clear sustainedupturn. In any event it is evident that far the greatestproportion of the improvement in mortality took placein the twenty-five year period preceding the census of1960. Consequently the value of t was taken as 25 years.

From column 7 of table 31 the median of the birth ratesis selected as the single most acceptable estimate among

.0451

.0259

Total populationMales

.0449

.02549.1

37.5

Females

.0454

.02657.7

36.83.106.36

Birth rate .Death rate .Level of mortality .°eo .GRR (fii = 28.8) .Total fertility .

70

TABLE 31. ESTIMATION OF THE BIRTH RATE FOR THE MALE POPULATION OF MEXICO, 1960, BY ADJUST­MENT OF PRELIMINARY STABLE ESTIMATES OF THAT PARAMETER (DERIVED FROM REPORTED C (X)FOR 1960 AND THE INTERCENSAL GROWTH RATE-r = .0316-FOR 1950-1960) FOR THE EFFECTSOF DECLINING MORTALITY

Birth rate. InC(x) male stable

(proportion populations; Adju.tment. Adiustments Adjustedup to age x), with C(x) a. from table for t ~ 25, Adjustment estimates of

Mexico, 1960, In col. 2 and III.l part (a) k = .012 factors the birth rateAge x male. with r = .0316 t = 25; k = .01 (col. 4 X 1.2) (I +col. 5) (col. 3 x col. 6)

(1) (2) (3) (4) (5) (6) (7)

5 ............ .169 .0383 -.028 -.034 .966 .037010 ............ .325 .0424 -.025 -.030 .970 .041115 ............ .454 .0436 -.004 -.005 .995 .043420 ............ .554 .0422 .021 .025 1.025 .043325 ............ .635 .0405 .053 .064 1.064 .043130 ............ .704 .0395 .081 .097 1.097 .043335 ............ .762 .0385 .099 .119 1.119 .043140 ............ .818 .0394 .106 .127 1.127 .0444

TABLE 32. ESTIMATES OF VARIOUS POPULATION PARAMETERS FORMEXICO, 1960, OBTAINED BY ADJUSTING STABLE ESTIMATES OFTHESE PARAMETERS FOR THE EFFECTS OF DECLINING MORTALITY

estimates shown in table 32 was based on a value of theparameter k obtained from the acceleration of the growthrat~. It is inter~sting to check the consistency of thatestimate of k with the result of an alternative methodalso described in chapter I, section C, that utilizes th~

changing composition of deaths to measure the tempoof mortality change. The index of the age distributione~p~oyed is the proportion of deaths over sixty-fiveWIthin all deaths over age five. This index for Mexicocan be calculated for each year from 1936 on; it showsclear upward trend with relatively minor yearly fluc­tuations. To minimize the effects of the latter it is betterto calculate the index for periods longer than one year.In the following illustration (which is limited to the malepopulation) the average for 1936-1939 and 1956-1959are used; their values are .220 and .322, respectively.This change has occurred in twenty years. If we had anestimate of the °eo at the base period, and if we knew thechange in the expectation of life at birth during thesetwenty years it would be possible to read a tabulated valueof kt in annex table III.3, hence to estimate k. If fertilityis constant, and the age distribution is quasi-stable, it ispossible to obtain just such a base-period value for °eoand a value for !J.°eo using the indices of the age distri­bution of deaths given above. The procedure is as follows:

(1) By means of the tabulation in annex II calculate °eoin the stable populations defined by the 1950-1960

.0422

.0119

Total populationFemale.

.0410

.012015.456.02.905.95

Male.

.0433

.011716.655.5

Birth rate .Death rate .Level of mortality .°eo .GRR (m = 28.8) .Total fertility .

15 The method based on standard marital fertility is not applicablebecause of the prevalence of consensual marriages in Mexico.Usingthe reported Pa/P2 ratio (2.141), m is calculated as 28.8 years.This is the value accepted in the following calculation. Note how­ever, that calculated direct from birth statistics, m is appreciablyhigher - 29.3 years.

those associated with x = 10, 15, ... ,40. C(20) and C(30)are tied for the median position - as a matter of fact allbirth rates implied by C(l5) through C(35) are virtuallyindistinguishable. From the estimated male birth rate themale death rate is calculated by subtracting the growthrate. Mortality indices are then obtainable from thestable population defined by these vital rates. Rates forfemales and for the total population are calculated in asimilar fashion, having first derived the birth rates forthese groups from the male birth rate via the sex ratio atbirth (1.05, assumed) and the sex ratio of the populationas a whole (.995, reported).

A somewhat more roundabout process is to be followedin finding the (female) gross reproduction rate. Thefollowing steps are required here: (a) estimate m;15(b) having selected the median male adjusted birth rate incolumn 7, say the one derived from C(20) and r, find (bythe usual method, i.e., by using the sex ratios at birthand in the population) the stable (unadjusted) femalebirth rate associated with the stable (unadjusted) malebirth rate (Column 3) implied by C(20) and r (results:male birth rate, .0422; female birth rate, .0400); (c) thisfemale birth rate plus the reported female growth rate(.0290) determine a stable population: read the value ofGRR with the appropriate mfrom this stable population(GRR (m = 28.8) = 2.75); (d) adjust the GRR thusobtained for quasi-stability using parameter values asin the earlier calculation (t = 25, k = .012) and selectingthe adjustment factors appropriate for the gross repro­duction rate and for the proper x, in this instance 20.(Adjusted GRR (m = 28.8) = 2.75 x 1.054 = 2.90. Thecorresponding value for total fertility is 5.95. Thiscompares with an estimate of total fertility from numberof children ever born, as P~/P2 which gives 5.99). Theresults of the above calculations are given in table 32.

The correction for quasi-stability that affects all the

71

intercensal r, and (a) C(10) in 1960and (b) C(lS) in 1960.(The results are 57.2 and 54.9.) Calculate the average ofthese two figures (56.0). This gives an estimate of theterminal (end-period) °eo'

(2) Using the same tables calculate the index births/population 15-44(thisindexoffertility isincludedin tableIndefinedby the same parameters as givenin point (1) above(results: .1006 and .1036) and take their average (.1021).

(3) From annex table III.2 obtain ~°eo as the differencebetween two separately calculated estimates of °eo, eachdefined by the average index of births/population 15-49as calculated in point (2), and by the index of the distri­bution of death for the two dates as given above (.322 and.22). The result is: ~oeo = 56.6-40.1 = 16.5.

(4) Calculate an estimate of the base period °eo as thedifference between the end period °eo-see point (I)-andthe ~oeo-see point (3)-i.e., as 56.0-16.5 = 39.5.

Table III.3 can now be used to get kt the value of whichin this example is .1686. Hence k = .1686/20 = .0084.If the adjustment of the preliminary stable estimates iscarried out with this value of k, the procedure is exactlythe same as shown in table 31 with the exception thatcolumn 4 is multiplied by .84, instead of 1.2, to getcolumn 5.

Comments. There is a substantial difference between thetwo independent estimates of k obtained above whichcannot be attributed to the approximate nature of thetechniquesinvolved,or explainedby biases in the reportingof the age of dead persons. A more fundamental cause ofthis difference is that fertility has been increasing, andpart of the acceleration of growth (reflected in the first

estimate of k) is not attributable to mortality decline,which is alone measured by the shift in the age distri­bution of deaths. Fertility increase also biases thelatter measure downwards. Naturally if stable estimatesare corrected only for mortality decline, but the shift inage distribution is reinforced by fertility increase also,the resulting estimates will have a downward bias. Theactual value of k to be used in this instance actuallyshould be larger than .012 (cf. foot-note 20 to chapter I).

Even if the estimates given in table 32 were obtained bya more adequate correction for quasi-stability, theirvalues would still be affected by a more substantial biasowing to the inadequate representation of the true patternof mortality in Mexico by the "West" model life tableswhich underlie the above calculations. Section A.1.b.of chapter I gives a general statement of this problem.In the present instance there is ample evidence from thelife tables prepared for Mexico since 1930that the relationof child mortality (e.g., 15) to "adult" mortality (e.g., 0e10)is much closer to the "South" pattern than it is to the"West". For Mexico this factor alone would cause theestimated birth rates to be some .004 lower than theiractual value, and of course there is a correspondingdistortion in the other parameter values as well. Thisexample thus shows the basic weakness of stable (orquasi-stable) estimates derived from C(x) and r: theirdependence on a well-chosen model life table family.There is often no information available on the truepattern, and no basis for a good choice. This difficultyis however eliminated, and the power of stable techniquesgreatly increased, when censuses provide data on childsurvival. Examples of estimation under such circum­stances are discussed in chapter VIII below.

72

Chapter vnEXAMPLES OF ESTIMATES BASED ON QUESTIONS ABOUT FERTILITY

AND MORTALITY

A. ESTIMATION OF FERTILITY FROM REPORTS ON CHILD­BEARING IN CENSUSES OR SURVEYS

The estimation of fertility from data on (a) births inthe year before a surveyor census, and (b) children everborn, is illustrated by an example based on hypotheticaldata. A synthetic example is used because the onlyinstances of surveys containing the requisite informationare in tropical Africa, and, as is noted in chapter IV, thevalidity of the method is sometimes seriously impaired bythe extensive age-misreporting characteristic of surveysand censuses in Africa. The suitability of the method forestimating fertility in an African population must bedecided after a detailed examination of the quality of thedata in the survey in question-an examination that wouldgo beyond the scope of this manual. The reader is referredto the detailed studies of different populations in thebook on the population of Africa recently completed atthe Office of Population Research. 1

The method exemplified in table 33 is of greatestpotential value for populations (such as in Latin America,the Philippines, and especially in the Republic of Koreaand Thailand) for which age-misreporting is less extreme.The example shows how data from such areas can beemployed when the appropriate questions have beenincluded in a census or survey.

1 Brass, et. al., op. cit.

Total fertilility derived from the age specific fertilityrates reported in the example (column 3) is

7

5 L Ii,1=1

or 5.24. The adjustment of these fertility rates for apossible error in the reference period is achieved bycalculating the average value of cumulative fertility in theage intervals shown in column 2 (F I in column 7), andforming the ratio of reported parity (PI) to cumulatedfertility (FI)' Reported parity at 20-25 (P2) and 25-30 (P3)is assumed to be approximately correct, although thelatter may be affected to some slight degree by omissions.The ratio of P2/F2 is a correction factor that makes thefertility rates consistent with the average number ofchildren ever born reported by women 20-25. In column 9there are age specific fertility rates that have been multi­plied by P2/F2 (1.313). The adjusted estimate of totalfertility is five times the sum of the rates in column 9, or1.313 times the figure derived from the unadjusted fertilityrates. The estimate is 6.88-higher than either the averageparity reported by older women (deficient because ofomissions) or than the cumulation of reported fertility(deficient because of a shortened reference period).

Computationalprocedure. The fertility rates in column 3,based on births reported for the year before the survey,pertain to women one-half year younger than the intervalsin column 2. Therefore the estimation of average cumu-

TABLE 33. THE ESTIMATION OF TOTAL FERTILITY AS P2/F2 TIMES REPORTED TOTAL FERTILITY

Cumulative EstimatedAverage fertility average

number of at beginning Multiplying cumulativebirths In Average of Interval factors for fertility Adju"ted

Exact age preceding number of

(¥Jd)estimating age specific

o/woman year per children average value( I-I )

fertilityInterval at time woman ever born fertility Fi= 1:fj+wi/i rates

(i) of survey (fj}G Pi (Wi) , J-O PilFi (J'i =fi x PI/PI)

(1) (2) (3) (4) (5) (6) (7) (8) (9)

1 15-20 .081 .186 0 1.963 .159 1.170 .106.........2 20-25 .242 1.435 .405 2.842 1.093 1.313 .318.........3 25-30 .261 3.109 1.615 3.011 2.401 1.295 .343.........4 30-35 .238 4.176 2.920 3.121 3.663 1.140 .312.........5 35-40 .166 4.710 4.110 3.247 4.650 1.013 .218.........6 40-45 .043 4.761 4.940 3.548 5.093 .935 .056.........7 45-50 .017 4.503 5.155 4.484 5.231 .861 .022.........

a For age intervals one-half year less than shown in column 2.

73

lative fertility is obtained by the use of multiplying factorsfound by interpolation in annex table IV.I. The key to theinterpolation is fllf2' equal to .335. The multiplyingfactors in column 6 were (5/130) x entries where fl1f2equals .460plus (125/130) x entries wherefl1f2 equals .330.

The entries in column 8 are typical of those found inAfrican surveys least distorted by age-misreporting.Pl/F1 is best ignored because of the intrinsic difficulty ofestimating Fl' P2/F2 and P3/F3 are approximately equal,and either ratio can be taken as a multiplying factor forthe correction of fertility rates for bias in the referenceperiod. The steady decline in PdFI past age thirty is thetypical result of progressively greater omission of childrenever born by older women.

B. ESTIMATION OF MORTALITY FROM REPORTED NUMBERS OFCHILDREN EVER BORN, AND CHILDREN SURVIVING

Preliminary adjustment of data. This method of esti­mation requires responses from a census or survey on thenumber of children ever born alive to each woman, andthe number of children surviving, with average numbersper woman tabulated for the standard five-year ageintervals. Deficiencies in the data for which adjustmentscan sometimesbe made include: (a) a moderate proportionof women for whom no responsesare obtained; (b) askingthe relevant questions (or tabulating responses) only formarried women, or only for women (" mothers") whohave experienced at least one live birth.

(a) Adjustment for non-response. The women for whomchildren ever born are not tabulated are not, ordinarily,representative in their average parity of the age group towhich they belong. There is a widespread tendency toleave a blank space instead of a zero for the response tothe question of the number of children ever born forwomen of zero parity. The evidence of this tendency is astrong positive correlation between the proportion ofwomen reported childless in each age interval with theproportion of non-responses. If all non-responses wereof this sort, they should be counted as zeros in calculatingthe average number of children ever born in each ageinterval. However some of the non-responses representa genuine absence of information, so that the assumptionof zero parity would produce a downward-biasedestimate.EI Badry has proposed 2 a simple but often effective tech­nique for determining approximately which non-responsesrepresent zero parity and which the genuine absence ofinformation (as, for example, when data are supplied by aneighbour). He suggests fitting a straight line of the formy = ax-s-b, where the observed values of yare theproportion of non-respondents in each age interval and xis the proportion reporting zero parity. If the observedrelationship is very closely fitted by the straight line, itmay plausibly be assumed that the fraction of non­responses genuinely associated with the absence of infor­mation is b, and that the proportion of childless women

2 M.A. El Badry, "Failure of enumerators to make entry ofzero: errors in recording childless cases in population censuses".Journal of the American Statistical Association, vol. 56, No. 296,December 1961, pp. 909-924.

recorded as non-respondents is ajI+a. The recommendedmethod of adjusting the non-responses, then, is to makea scatter diagram showingthe proportion of non-responsesin each age interval on one axis, and the proportion ofchildless women on the other. If the resultant points arecloselyfitted by a straight line, extend this line to the zerovalue of the proportion childless. The proportion of non­responses on the straight line at this point (when theproportion childless is hypothetically zero) can be takenas an estimate of the true proportion of non-responses.The recommended arithmetical adjustment of the dataon the average number of children ever born and on theaverage number surviving is the omission from the deno­minator of the estimated number oftrue non-respondents,and the inclusion in the denominator of the estimatednumber of non-respondents who are considered to be infact childless. In most instances almost all of the non­respondents fall in the probably childless category, andlittle or no error is introduced by assuming that all non­responses indicate zero children.

(b) Adjustments when data are tabulated only for marriedwomen or for mothers. In some censuses and surveys,questions about children ever born are asked (or at leasttabulated) only for ever married women, and in others,data are shown only for women who have borne at leastone child. A word about the nature of the resulting biasesis not out of place. First, if illegitimacy is infrequent, theproportions surviving among children ever born will beadequately representative, and the only effectof obtaininginformation solely from ever married women is that it isnecessary to estimate the average parity of all womenin each age group indirectly, usually on the assumptionthat all births occur to married women. If, in fact, illegiti­mate births contribute significantly to the average parityof women 15-19, and not so much to women 20-24, theratio P tiP2 may be underestimated, leading to the selec­tion of adjustment factors in table V.I that tend tooveresti­mate q(2) and q(3). The bias resulting from the highermortality rates experienced by children born to non­married women, on the other hand, tends to cause anunderestimate of mortality, although this latter bias isminimized by the fact that women under twenty bearingchildren while single are often married by the time (atage 20-24 or 25-29) their fertility histories are used toestimate 2qO and 3QO'

The adjustment made when information is given onlyfor ever married women is to determine average parityin the two relevant intervals (15-19 and 20-24) by multi­plying the average parity of ever married women by theratio ever married women/total women. This adjustmentcannot be accepted as valid in populations where a majorproportion (say more than 10 per cent) of the births towomen 15-19 occur to the non-married. When data aresupplied only for "mothers", the adjustment is analogousto that for ever married women, i.e., parity is estimatedby multiplying the average parity of mothers by the ratiomothers/total women. In both instances of limited data,the reported proportion surviving among children everborn is accepted as representative of the experience ofall women. In fact the proportion surviving for evermarried women is higher than for the non-married, andthis bias of course also holds for "mothers" if the

74

TABLE 34. CALCULATION OF lqO. 2qO. aqo. 5qO. lQqO, 15qo, AND 20QO FOR BRAZIL. BASED ON CHILDRENEVER BORN. AND CHILDREN SURVIVING RECORDED IN THE 1950 CENSUS

Average ,",~r Average number Multipliers Proportionof children ofchildren for colUlllll 5 dead by

Interval Ageo! ever born surviving from age x(i) women (P,) (S,) I-S,IP, PIIPa Age x <XQO)

(1) (2)~ t _(4) (5) (6) (7) \8)

1 ........... 15-19 .146 .118 .1918 1.058 1 .2032 ........... 20-24 1.099 .870 .2083 1.050 2 .2193 ........... 25-29 2.516 1.947 .2262 1.016 3 .2304 ........... 30-34 3.883 2.935 .2442 1.019 5 .2495 ........... 35-39 5.065 3.730 .2636 1.029 10 .2716 ........... 40-44 5.778 4.146 .2825 1.007 15 .2847 ........... 45-49 6.212 4.353 .2993 1.006 20 .301

>'.

~~) ') <:JJi ~) Q ;;... ·V"', )< ,

questions have been asked only of married women, andtabulated only for those with at least one child. If, in fact,the questions are asked only of women who have borneat least one child, the probable bias is increased becauseof the likelihood that "mothers" with no surviving child­ren may have been excluded.

Computational procedure. The estimation of mortalityfrom child survival rates is illustrated in table 34 on thebasis of data taken from the 1950 census of Brazil.Columns 3 and 4 show the average number of childrenever born per woman and the average number survivingfor the age intervals shown in column 2. The proportionof non-survivors is shown in column 5; these proportionsare converted into estimates of lqO' 2qO, .... 20qO bymultipliers taken from annex table V.l. P l/P2 is .146/1.099= .133. The multipliers in column 6 were obtained byinterpolating between the columns in table V.I, for whichPl /P2 is .143 and .090, specifically, by adding (.811) x(entries where Pl /P2 = .143) and (.189) x (entries whereP l/P2 = .090). The final estimates are in column 8.

Comments. As is pointed out in the discussion in sectionB of chapter II the level of infant mortality derived fromchild survival reported by women 15-19 should not beregarded seriously, because of the basic weakness of themethod of estimation at this point. A better estimate oflqO is obtained by accepting zqo, and the relationshipbetween 1 qo and 2qo in the model tables. The resultantestimate of infant mortality is .171, if based on the"West" female model tables, and .178 if based on themales, so that an acceptable estimate of lqO is about .175.

The level of mortality indicated by the sequence of esti­mates of "qo implies the following sequence of °eo's forboth sexes (as calculated from annex tables 1.2 and 1.1) :

2qO •••... • • . . . . . . . . • • . . . • • • • • • • • •• 41.2aqo ...•..........................• 42.25QO ••••••••••••••••••••••••••••••• 42.4

lOqO 42.4lIiqO •..••••••••.•..•••••••••••••••• 42.620QO •••••••••••••••••.••••••.•• " .• 42.9

The remarkably consistent sequence of the implied °eovalues (or of the implied mortality levels in general)suggests that the basic data are of very good quality.This is also supported by an examination of the reportedparity distribution itself which shows no obvious signs ofan increasing failure to report children as the age of thereporting women progresses. Naturally these observationsinspire increased confidence in the mortality estimatesderived above. On the other hand, it should be borne inmind that survival rates to increasingly higher agesreflect the mortality experience of an increasingly longerperiod prior to 1950. It is possible, therefore, that thereis a fortuitous element in the high consistency of themortality levels implied by these rates; namely, the effectsof the actually higher mortality of earlier periods to someextent might have been offset by the probable tendencyon the part of older women to omit a higher proportionof their children who are dead than of those who arestill alive.

75

Chapter VIII

EXAMPLES OF ESTIMATION BASED ON CHILD SURVIVALAND AGE DISTRIBUTIONS

A. ESTIMATION OF FERTILITY AND MORTALITY FROM DATAIN A SINGLE CENSUS THAT RECORDS THE AGE COM­POSITION OF THE POPULATION, THE NUMBER OF CHILDRENEVER BORN AND THE NUMBER SURVIVING

The estimation of child mortality from the reportedsurvival of children ever born to women in various ageintervals in the census of Brazil, 1950, (see chapter VII,section B) is combined with the age distribution in theexample given in this section, to provide estimates offertility and over-all mortality by methods described inchapter III, section B. The example begins with the esti­mates of 2qO in table 34., i.e., 2QO for both sexes equal to0.219. The method of estimation (employing the agedistribution of the malepopulation in 1950and the familyof model stable populations) requires determination ofthe level of male mortality. The first step is to ascertainwhether the relations between male and female mortalityin the "West" model life tables can plausibly be assumedto characterize mortality in Brazil. Life tables based onregistered deaths in the Federal District have mortalitydifferences as large as in the " West" tables; sex differencesin mortality in Latin America are almost without excep­tion unfavourable to males up to age two; and the sexratio of the Brazilian population supports the inferencethat sex differences in mortality of about the magnitudefound in the "West" tables have prevailed in Brazil. Thelevel of mortality identified by a 2QO of 0.219 is 10.1(both sexes). The question is whether the sex ratio of theBrazilian population (males/females = .993 for the wholepopulation, and .988 for the native population in 1950)is consistent with the sex differences found at level 10.1in the "West" model life tables. The last entries in eachmodel stable population make it possible to calculatethe sex ratio on the assumption of any stipulated sex

ratio at birth (of, say, 1.05 males/females) and the "West"differences in mortality at the same level. We shallassume growth rates of .015 and .025, almost certainlybracketing the Brazilian rate of increase, here provisionallyassumed unknown. The sex ratio in a tabulated stablepopulation is calculated as follows: (a) assume the sexratio at birth to be 1.000. Note the figure "Pop. size,B(O) = 1" at, for example, mortality level 9, growth rate.015. The figure is 26.095 for females and 24.701 for males,and the sex ratio of the stable population under thesecircumstances is 24.701/26.095 or .9466. If the sex ratioat birth is 1.04 or 1.05, the sex ratio of the stable popu­lation becomes 1.04 x .9466 = .984, or 1.05x .9466 =.994.Futher calculations of this nature provide the figures intable 35.

The sex ratio among more than 200,000 births registeredin the Federal District in 1949-1952 was 1.058, so that asex difference of mortality at least as large as in the"West" model tables is needed to account for the mascu­linity of the Brazilian population, and more especiallythe native population in 1950. Thus it will be assumedthat 2QO for the male population of Brazil just prior to 1950was 0.232, and that the appropriate life table is level10.1. Table 36 shows (columns 4.0, 4.b and 4.c) birth rates,death rates and rates of increase associated with thismortality level and recorded values of C(5), C(lO), ... ,C(45) for Brazilian males, 1950.

Female and total population parameters are estimatedas in chapter VI, section C. The resultant values areshown in table 37.

Note that these estimates indicate slightly higher ferti­lity and mortality than were obtained from the same agedistribution and the intercensal rate of increase. Thefertility estimates presented here are based on evidence

TABLE 35. SEX RATIO IN STABLE POPULATIONS WITH THE "WEST" SEX DIFFERENCES IN MORTALITY,VARIOUS MORTALITY LEVELS, RATES OF INCREASE AND SEX RATIO AT BIRTH

Sex ratio at birth 1.04 Sex ratio at birth 1.05

Rate a!Level 7 Level 9 Levelllincrease Level 7 Level 9 Levelll

.015 ........... .979 .984 .988 .989 .994 .998

.025 ........... .986 .991 .995 .995 1.001 1.005

76

TABLE 36. CALCULATION OF STABLE POPULATION ESTIMATES OF BIRTH AND DEATH RATES BASED ON

A REPORTED AGE DISTRIBUTION AND ON A LEVEL OF MORTALITY DERIVED FROM REPORTED CHILDSURVIVAL RATES, BRAZIL, 1950, MALES.

C(x)VailleS of C(x) IIIIdvarious(proportioll

Values of various parameters ill male stableup to age x) parameters 111 male stableBrazil. /950, populatlolls with a level of populatlolls with C(x) as showlI ill col. 2 IIIId

Age x males mortality of /0./ with mortality level of 10./r = .020 r = .025 Birth rate Death rate Growth rate

(l) (2) (3.a) (3.b) (4.a) (4.b) (4.c)

5 ............ .164 .160 .177 .0444 .0232 .021210 ••••••••••• 0 .302 .293 .321 .0449 .0233 .021615 ............ .424 .410 .444 .0453 .0233 .022220 ............ .527 .514 .550 .0451 .0233 .021825 .0 .••••.•••. .619 .604 .641 .0453 .0233 .022030 ............ .698 .682 .717 .0456 .0233 .022335 ............ .760 .750 .781 .0449 .0233 .021640 ............ .819 .807 .834 .0455 .0233 .022245 ............ .867 .855 .878 .0459 .0233 ,0226

Birth rate ........ .0432 .0484Death rate ....... .0232 .0234

TABLE 37. STABLE POPULATION PARAMETERS FOR BRAZIL, 1950,DERIVED FROM THE MALE AGE DISTRIBUTION, LEVEL OF MORTALITYFROM ESTIMATES 2qO FOR BOTH SEXES, THE SEX RATIO OF THEPOPULATION AS REPORTED, AND ASSUMED SEX RATIO AT BIRTHOF 1.05

subtracting the intercensal rate of natural increase fromthis estimated birth rate. The resulting death rate figuresare little affectedby an age pattern of mortality that doesnot in fact conform to the model life tables (table 38).

Males Females Total populatiollTABLE 38. BIRTH RATES CALCULATED FROM C(x) AND 12 (MALES),

AND DEATH RATES FROM b-r, BRAZIL, 1950

Birth rate .............. .0453 .0428 .0441Intercensal rate of

natural increase .0 ..•• .0232 .0238 .0235Death rate ............. .0221 .0190 .0206

Birth rate .Death rate .Level of mortality .°eo ..ORR (fii = 29.1) .Total fertility .

.0453

.023310.140.0

.0428

.021010.142.8

2.915.97

.0441

.0222 Males Females Total populatlOll

It is not difficult to construct a life table approximatelyconsistent with these rates, embodying mortality aboveagefive derived from C(x) and r, and mortality under agefive from the estimated value of 12, The procedure is asfollows: (a) find the value of sLo and Is consistent withthe estimated /2 (i.e., at mortality level 10.1 for each sex);(b) find the value of °es in the stable population selectedfrom C(x) and r for males, and estimated for femalesfrom the male C(x) and r plus the recorded sex ratio ofthe population, the estimated sex ratio at birth, and thefemaleintercensal rate of natural increase(i.e., at mortalitylevel 11.3 for males and 12.0 for females); (c) calculate°eo as (1/10) (sLo+ls x °es)'

The results are as follows:

Males Females

that is in principle preferable, because fertility estimatesderived from 12 are less affected by variations in the agepattern of mortality. Moreover, the fertility estima~es

obtained in this way are closer to the average pantyreported by women 45-49 years of age; which in t~e

Brazilian census of 1950 appears a remarkably validreport. On the other hand, estimates of the death rate andand of the expectation of life at birth are strongly depen­dent on how closelythe age pattern of mortality in Brazilconforms to the "West" model tables.

B. ESTIMATION OF FERTILITY AND MORTALITY FROM DATAON AGE DISTRIBUTION, THE INTERCENSAL RATE OFINCREASE, AND SURVIVAL RATES OF CHILDREN EVER BORN

If an approximately stable population has been enumer­ated twice in about a decade, and if the second censuscontains information making it possible to calculatechildhood mortality, the birth rate can be estimated (asin the preceding section) in a way that has relatively littlesensitivity to age-mis-statement and unusual age patternsof mortality, and the death rate estimated in turn by

77

°es ..Is .sLo/lo .°eo .

51.3727003.898

41.2

54.4752004.023

44.9

Comments. The analysis of the male age distributionsin Brazil in 1950, combined with the estimated intercensalrate of natural increase, the recorded sex ratio, and dataon child survival leads to estimates of the birth rate ofabout 44 per thousand, a death rate of about 21 perthousand, an expectation of life at birth of about forty­three years, and a total fertility of about six children perwoman passing through the child-bearing years. The onlyindependent information available-the average parityreported by women 45-49-indicates a slightly higher totalfertility of 6.2. It is slightly surprising (and not at all

typical of results found from other censuses) to findreported parity by older women exceeding the estimatederived from the age distribution. Moreover, the averageparity reported by women 45-49 exceeds (p3)2/P2' whichequals 5.76 in this case. But there is enough consistency­in the level of mortality indicated by the succession of"qo estimates (see chapter VII, section B), in the not verywide divergence between estimates based on C(x) and r,and on C(x) and 12, and in the total fertility as estimatedand reported-to lend substantial authority to the approxi­mate values of these parameters.

78

Part Three

ANNEXES

Annex I

MODEL LIFE TABLES *

TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY

LEVEL 1

Age x Ix nmx nqx nLx .Lx+. Tx 0"".Lx

Females

0 .............. 100,000 .4788 .3652 76,264 .5728" 2,000,000 20.001 .............. 63,483 .0790 .2615 210,124 .7886b 1,923,736 30.305 .............. 46,883 .0152 .0731 225,847 .9346 1,713,612 36.55

10 .............. 43,456 .0118 .0572 211,069 .9347 1,487,765 34.2415 ••••• ,0 ••••• 00 40,972 .0154 .0739 197,287 .9175 1,276,696 31.1620 .0 ..••••..•••• 37,943 .0192 .0918 181,008 .9031 1,079,409 28.4525 .............. 34,460 .0216 .1025 163,468 .8914 898,402 26.0730 ••• 0 •••••••••• 30,927 .0245 .1155 145,708 .8797 734,933 23.7635 .............. 27,356 .0268 .1258 128,178 .8708 589,226 21.5440 .............. 23,916 .0286 .1332 111,613 .8633 461,048 19.2845 .............. 20,729 .0303 .1407 96,358 .8418 349,435 16.8650 .............. 17,814 .0393 .1787 81,111 .8019 253,077 14.2155 ....... , ...... 14,631 .0499 .2217 65,045 .7382 171,966 11.7560 •••••••••• 0 ••• 11,387 .0743 .3133 48,017 .6529 106,921 9.3965 •••••••••••• 0. 7,819 .0988 .3962 31,351 .5540 58,904 7.5370 .............. 4,721 .1437 .5285 17,368 .4265 27,553 5.8475 .............. 2,226 .2011 .6691 7,407 .2727c 10,184 4.5880 •••••• 0 ••••••• 737 .2652 2,778 2,777 3.77

Males

0 .............. 100,000 .5827 .4191 71,922 .5287" 1,803,333 18.031 ........... ". 58,093 .0784 .2597 192,420 .7860b 1,731,411 29.805 .............. 43,005 .0140 .0675 207,769 .9417 1,538,991 35.79

10 .............. 40,102 .0099 .0484 195,655 .9435 1,331,223 33.2015 .............. 38,160 .0135 .0651 184,591 .9218 1,135,568 29.7620 ........... ". 35,676 .0193 .0923 170.153 .9024 950,977 26.6625 .............. 32,385 .0218 .1035 153,542 .8889 780,824 24.1130 ••••.••••••. 0. 29,032 .0254 .1196 136,479 .8712 627,282 21.6135 .............. 25,560 .0299 .1392 118,904 .8485 490,803 19.2040 .............. 22,002 .0362 .1658 100,889 .8242 371,898 16.9045 •••••••••••• 0' 18,354 .0415 .1878 83,149 .7929 271,009 14.7750 .............. 14,906 .0522 .2309 65,926 .7519 187,860 12.6055 .............. 11,464 .0626 .2705 49,568 .6953 121,933 10.6460 .............. 8,363 .0853 .3516 34,463 .6136 72,365 8.6565 .............. 5,422 .1128 .4401 21,146 .5178 37,902 6.9970 .............. 3,036 .1546 .5575 10,949 .3963 16,757 5.5275 .............. 1,343 .2193 .7083 4,338 .2530c 5,808 4.3280 .............. 392 .2667 1,470 1,469 3.75

* These tables are extracted from Coale-Demeny, Regional Model Life Tables and StablePopulations (Princeton University Press, Princeton, 1966).

" Pro~ortion surviving from birth to 0-4.b oL50LO.c Tso/T75.

81

TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continuecl)

LEVEL 3

Age x Ix nmx nqx nLx sLx +s Tx °exsLx

Females

0 .............. 100,000 .3807 .3052 80,163 .6371a 2,500,000 25.00I .............. 69,481 .0628 .2155 238,404 .8296b 2,419,837 34.835 .............. 54,506 .0125 .0605 264,280 .9459 2,181,433 40.02

10 ............ " 51,206 .0097 .0473 249,975 .9458 1,917,153 37.44IS .............. 48,783 .0127 .0615 236,422 .9313 1,667,179 34.1820 .............. 45,786 .0159 .0765 220,171 .9192 1,430,756 31.2525 .............. 42,283 .0179 .0855 202,371 .9093 1,210,585 28.6330 .............. 38,666 .0203 .0964 184,008 .8994 1,008,214 26.0835 '0' .••••.•••.• 34,937 .0222 .1053 165,492 .8915 824,206 23.5940 .............. 31,259 .0238 .1l21 147,532 .8844 658,715 21.0745 .............. 27,754 .0254 .1195 130,475 .8649 511,182 18.4250 .............. 24,436 .0331 .1527 112,853 .8298 380,708 15.5855 .............. 20,705 .0422 .1910 93,640 .7731 267,855 12.9460 ••••••••••• II' 16,751 .0628 .2712 72,397 .6964 174,215 10.4065 •••••••• II •••• 12,208 .0843 .3481 50,416 .6034 101,819 8.3470 .............. 7,959 .1232 .4710 30,423 .4817 51,403 6.4675 .............. 4,211 .1746 .6077 14,656 .3014° 20,979 4.9880 .............. 1,652 .2612 6,324 6,323 3.83

Males

0 .............. 100,000 .4595 .3513 76,462 .5982a 2,285,135 22.85I .............. 64,868 .0625 .2144 222,637 .8279b 2,208,674 34.055 .............. 50,957 .0116 .0562 247,627 .9515 1,986,037 38.97

10 .............. 48,093 .0082 .0404 235,61 I .9527 1,738,410 36.15IS .............. 46,151 .0112 .0546 224,456 .9343 1,502,799 32.5620 .............. 43,631 .0161 .0774 209,712 .9182 1,278,343 29.3025 .............. 40,254 .0181 .0867 192,547 .9070 1,068,632 26.5530 ............ ,. 36,765 .0211 .1001 174,630 .8921 876,085 23.8335 •••••••••••• I. 33,087 .0248 .1167 155,785 .8726 701,455 21.2040 .............. 29,227 .0300 .1395 135,941 .8512 545,670 18.6745 ............. , 25,149 .0347 .1596 115,714 .8230 409,729 16.2950 .............. 21,137 .0439 .1979 95,228 .7855 294,015 13.9155 .0 .••••••..••• 16,955 .0533 .2352 74,803 .7329 198,787 11.7360 .............. 12,967 .0730 .3088 54,823 .6573 123,983 9.5665 .............. 8,963 .0974 .3918 36,036 .5659 69,160 7.7270 ........... , .. 5,452 .1347 .5037 20,393 .4481 33,124 6.0875 .............. 2,706 .1922 .6491 9,138 .2822° 12,731 4.7180 .............. 949 .2643 3,593 3,593 3.78

a prohortion surviving from birth to 0-4.b 5L55Lo.c Tso/T75.

82

TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 5

Age x Ix nmx nqx nLx 5Lx+5 Tx °ex5Lx

Females

0 .............. 100,000 .3067 .2557 83,378 .6924a 3,000,000 30.001 .............. 74,427 .0503 .1777 262,817 .8618b 2,916,622 39.195 .............. 61,205 .0103 .0502 298,352 .9552 2,653,805 43.36

10 .............. 58,135 .0080 .0392 284,979 .9549 2,355,453 40.5215 .............. 55,856 .0105 .0512 272,136 .9426 2,070,474 37.0720 .............. 52,998 .0132 .0639 256,525 .9324 1,798,338 33.9325 .............. 49,612 .0148 .0716 239,185 .9240 1,541,813 31.0830 .............. 46,062 .0168 .0807 221,017 .9156 1,302,628 28.2835 .............. 42,345 .0185 .0884 202,364 .9086 1,081,611 25.5440 '" ........... 38,601 .0199 .0948 183,862 .9017 879,247 22.7845 .............. 34,944 .0215 .1021 165,796 .8841 695,385 19.9050 .............. 31,375 .0281 .1313 146,579 .8528 529,589 16.8855 .............. 27,257 .0361 .1657 124,997 .8021 383,009 14.0560 .............. 22,742 .0537 .2365 100,263 .7324 258,012 11.3565 .............. 17,363 .0729 .3083 73,432 .6446 157,750 9.0970 "0 ••••••••••• 12,010 .1075 .4235 47,332 .5277 84,318 7.0275 .............. 6,923 .1544 .5570 24,975 .3248e 36,986 5.3480 .............. 3,067 .2553 12,011 12,011 3.92

Males

0 .............. 100,000 .3684 .2955 80,205 •6580a 2,766,802 27.671 .............. 70,454 .0502 .1771 248,774 .8605b 2,686,597 38.135 .............. 57,976 .0096 .0469 283,083 .9595 2,437,823 42.05

10 .............. 55,258 .0069 .0337 271,627 .9603 2,154,740 38.9915 ........... ". 53,393 .0094 .0460 260,829 .9447 1,883,113 35.2720 .............. 50,938 .0135 .0652 246,394 .9312 1,622,284 31.8525 .............. 47,619 .0151 .0728 229,434 .9219 1,375,890 28.8930 .............. 44,154 .0175 .0839 211,509 .9093 1,146,456 25.9735 .............. 40,449 .0206 .0981 192,329 .8926 934,947 23.1140 .............. 36,482 .0251 .1179 171,663 .8735 742,618 20.3645 .............. 32,183 .0292 .1362 149,954 .8478 570,955 17.7450 ••••••••••• ,0' 27,799 .0373 .1706 127,137 .8133 421,001 15.1555 .. , ........... 23,056 .0460 .2061 103,400 .7641 293,864 12.7560 '0' ••••••••••• 18,304 .0634 .2734 79,008 .6935 190,464 10.4165 .............. 13,299 .0854 .3519 54,795 .6059 111,456 8.3870 .............. 8,619 .1193 .4593 33,198 .4912 56,660 6.5775 .............. 4,660 .1716 .6003 16,307 .3050e 23,462 5.0480 .............. 1,863 .2603 7,155 7,155 3.84

a prohortion surviving from birth to 0-4.b 5L55Lo.e Tso/T75.

83

TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 7

Age x Ix nffix nqx nLx 6Lx+6 Tx °ex6Lx

Females

0 .0 ..•••••••••• 100,000 .2484 .2139 86,099 .7407a 3,500,000 35.001 •••..••••••. 0. 78,614 .0403 .1456 284,252 .8881 b 3,413,901 43.435 ••••••••••••• 0 67,169 :0085 .0414 328,896 .9630 3,129,649 46.59

10 .0 .••••...•••• 64,389 .0066 .0323 316,741 .9627 2,800,754 43.5015 ............... 62,307 .0087 .0425 304,921 .9523 2,484,012 39.8720 .0 .••••••••••• 59,661 .0109 .0532 290,367 .9436 2,179,092 36.5325 .............. 56,486 .0123 .0597 273,999 .9366 1,888,724 33.4430 .............. 53,114 .0140 .0674 256,617 .9294 1,614,726 30.4035 ••••• 0 •••••••• 49,533 .0154 .0741 238,488 .9230 1,358,109 27.4240 ••••• 0 •••••••• 45,862 .0167 .0800 220,133 .9164 1,119,620 24.4145 .............. 42,191 .0183 .0874 201,739 .9003 899,487 21.3250 '0 •••••••••••• 38,504 .0240 .1131 181,633 .8723 697,748 18.1255 .............. 34,149 .0311 .1442 158,433 .8268 516,116 15.1160 .............. 29,224 .0462 .2071 130,989 .7630 357,682 12.2465 .............. 23,171 .0637 .2747 99,942 .6796 226,694 9.7870 .............. 16,806 .0949 .3834 67,921 .5667 126,751 7.5475 '0' ••••••••••• 10,363 .1384 .5142 38,493 .3457" 58,830 5,6880 ............... 5,035 .2476 20,338 20,338 4.04

Males

0 .............. 100,000 .2977 .2482 83,373 .7103a 3,248,436 32.481 .0 .••••••••.••• 75,183 .0403 .1455 271,759 .8868b 3,165,064 42.105 .0 •.••••••••••• 64,242 .0080 .0390 314,946 .9663 2,893,305 45.04

10 .............. 61,737 .0057 .02'81 304,343 .9667 2,578,358 41.7615 .............. 60,000 .0079 .0387 294,202 .9534 2,274,015 37.9020 ............ 0. 57,680 .0113 .0548 280,501 .9422 1,979,813 34.3225 .............. 54,520 .0126 .0610 264,287 .9345 1,699,313 31.1730 .............. 51,195 .0146 .0703 246,981 .9239 1,435,026 28.0335 .............. 47,598 .0172 .0823 228,190 .9094 1,188,045 24.9640 .0 ..•••••••••• 43,678 .0209 .0995 207,526 .8925 959,856 21.9845 .............. 39,332 .0247 .1165 185,206 .8690 752,330 19.1350 .............. 34,751 .0319 .1476 160,935 .8368 567,124 16.3255 .............. 29,623 .0399 .1815 134,677 .7906 406,189 13.7160 .............. 24,248 .0555 .2435 106,477 .7243 271,512 11.2065 ••••••••• 0 •••• 18,343 .0757 .3182 77,123 .6398 165,036 9.0070 .............. 12,506 .1069 .4217 49,345 .5280 87,913 7.0375 .............. 7,232 .1552 .5590 26,052 .3245" 38,567 5.3380 .............. 3,189 .2548 12,515 12,515 3.92

a Proportion surviving from birth to 0-4.b 5L5/5Lo." Tso/T75.

84

TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 9

Age x nmx nqx nLx liLx+li Tx 0""liLx

Females

0 .............. 100,000 .2010 .1777 88,447 .7835a 4,000,000 40.001 .............. 82,226 .0320 .1179 303,316 .91oob 3,911,553 47.575 .............. 72,530 .0069 .0338 356,520 .9698 3,608,237 49.75

10 .............. 70,078 .0054 .0264 345,762 .9694 3,251,718 46.4015 .0 .••••••••••• 68,227 .0071 .0350 335,172 .9606 2,905,956 42.5920 .............. 65,842 .0090 .0440 321,964 .9533 2,570,784 39.0525 ............... 62,944 .0102 .0495 306,933 .9474 2,248,820 35.7330 .............. 59,829 .0115 .0559 290,781 .9412 1,941,886 32.4635 .............. 56,483 .0128 .0618 273,690 .9355 1,651,105 29.2340 .............. 52,993 .0139 .0673 256,043 .9291 1,377,415 25.9945 .............. 49,424 .0155 .0747 237,894 .9144 1,121,372 22.6950 .............. 45,733 .0205 .0975 217,525 .8891 883,478 19.3255 ........... , .. 41,277 .0268 .1257 193,410 .8481 665,953 16.1360 .............. 36,087 .0400 .1818 164,037 .7895 472,543 13.0965 ............... 29,527 .0560 .2457 129,500 .7100 308,507 10.4570 .............. 22,272 .0845 .3488 91,943 .6006 179,007 8.0475 .............. 14,505 .1253 .4772 55,221 .3657c 87,064 6.0080 .............. 7,584 .2382 31,843 31,843 4.20

Males

0 .............. 100,000 .2408 .2074 86,106 .7567a 3,730,053 37.301 .............. 79,263 .0321 .1183 292,227 .9088b 3,643,947 45.975 .............. 69,888 .0065 .0322 343,818 .9722 3,351,720 47.96

10 .............. 67,639 .0047 .0233 334,258 .9722 3,007,902 44.4715 .............. 66,064 .0066 .0324 324,977 .9610 2,673,644 40.4720 0" ••••••••••• 63,926 .0094 .0459 312,305 .9517 2,348,667 36.7425 .............. 60,995 .0104 .0508 297,226 .9454 2,036,363 33.3930 .............. 57,895 .0121 .0585 281,009 .9365 1,739,137 30.0435 .............. 54,509 .0142 .0688 263,172 .9240 1,458,128 26.7540 .............. 50,760 .0175 .0837 243,182 .9088 1,194,955 23.5445 .............. 46,512 .0209 .0994 220,999 .8872 951,774 20.4650 ., ............ 41,887 .0273 .1277 196,068 .8572 730,775 17.4555 .............. 36,540 .0348 .1602 168,066 .8136 534,706 14.6360 .............. 30,686 .0489 .2177 136,730 .7510 366,640 11.9565 .............. 24,006 .0676 .2891 102,678 .6693 229,910 9.5870 .............. 17,066 .0967 .3893 68,718 .5598 127,231 7.4675 .............. 10,421 .1418 .5234 38,471 .3425c 58,514 5.6280 .............. 4,967 .2478 20,043 20,043 4.04

a pror:ortion surviving from birth to 0-4.b IIL55Lo.c Tso/T75.

85

TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 11

A.g~ x Ix nmx nCb: nLx IiLx+1i Tx 0""IiLx

Females

0 .............. 100,000 .1615 .1461 90,502 .8219G 4,500,000 45.001 .0 .••••••••••. 85,388 .0250 .0937 320,442 .9288b 4,409,498 51.645 .............. 77,389 .0055 .0272 381,683 .9758 4,089,056 52.84

10 .............. 75,285 .0043 .0212 372,430 .9752 3,707,373 49.2515 · ............. 73,687 .0058 .0284 363,207 .9679 3,334,942 45.2620 .............. 71,596 .0073 .0360 351,543 .9618 2,971,735 41.5125 .............. 69,022 .0083 .0405 338,115 .9569 2,620,192 37.9630 .............. 66,224 .0094 .0459 323,525 .9516 2,282,017 34.4635 .0' ••••••••••• 63,186 .0105 .0510 307,872 .9465 1,958,552 31.0040 .............. 59,963 .0116 .0562 291,388 .9402 1,650,680 27.5345 .............. 56,592 .0131 .0636 273,969 .9267 1,359,291 24.0250 .............. 52,996 .0175 .0837 253,884 .9039 1,085,322 20.4855 .............. 48,558 .0232 .1095 229,493 .8669 831,439 17.1260 .............. 43,239 .0347 .1596 198,946 .8127 601,946 13.9265 .............. 36,339 .0495 .2203 161,682 .7367 403,000 11.0970 ·............. 28,333 .0758 .3184 119,112 .6304 241,319 8.5275 .............. 19,311 .1144 .4448 75,083 .3856 122,207 6.3380 ·............. 10,722 .2275 47,125 47,124 4.40

Males

0 .............. 100,000 .1940 .1717 88,499 .7983G 4,211,576 42.121 .............. 82,835 .0252 .0944 310,632 .9274b 4,123,076 49.785 .............. 75,015 .0053 .0262 370,157 .9773 3,812,445 50.82

10 .0 .•••••.••••• 73,048 .0038 .0190 361,763 .9771 3,442,288 47.1215 ........... '" 71,657 .0054 .0268 353,478 .9677 3,080,525 42.9920 .............. 69,7~4 .0078 .0380 342,042 .9601 2,727,047 39.1125 .0 ..•••...•••• 67,083 .0086 .0419 328,380 .9550 2,385,005 35.5530 .0 ..•••..••••. 64,269 .0099 .0482 313,607 .9476 2,056,625 32.0035 .............. 61,173 .0117 .0569 297,166 .9368 1,743,018 28.4940 ••••••••••••• 0 57,693 .0145 .0698 278,395 .9231 1,445,852 25.0645 .............. 53,665 .0177 .0845 256,985 .9032 1,167,457 21.7650 .0 •••••••••••• 49.129 .0233 .1102 232,105 .8750 910,472 18.5355 .............. 43.713 .0305 .1416 203,093 .8337 678,366 15.5260 '0' ••••••••••• 37,524 .0432 .1951 169,318 .7743 475,273 12.6765 .............. 30,203 .0607 .2636 131,109 .6951 305,955 10.1370 .............. 22,241 .0881 .3610 91,133 .5879 174,847 7.8675 .............. 14,213 .1306 .4922 53,575 .3600" 83,714 5.8980 .............. 7,217 .2395 30,139 30,139 4.18

G pr0ft0rtion surviving from birth to 0-4.b liLs sLo." Tso/T76.

86

TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 13

Age x Ix nmx nqx nLx .Lx+6 Tx OCx

6Lx

Females

0 .............. 100,000 .1282 .1183 92,310 .8566a 5,000,000 50.001 •••••••• 0 ••••• 88,169 .0188 .0717 335,996 .9453b 4,907,690 55.665 .............. 81,848 .0043 .0214 404,871 .9810 4,571,694 55.86

10 .............. 80,100 .0034 .0166 397,178 .9804 4,166,823 52.0215 .............. 78,771 .0046 .0226 389,403 .9743 3,769,645 47.8620 ••••••••••••• 0 76,990 .0059 .0289 379,397 .9693 3,380,242 43.9125 .............. 74,769 .0066 .0327 367,738 .9652 3,000,845 40.1430 .............. 72,326 .0076 .0370 354,934 .9608 2,633,107 36.4135 ............. . 69,647 .0085 .0415 341,009 .9561 2,278,173 32.7140 .............. 66,756 .0095 .0464 326,031 .9500 1,937,165 29.0245 .............. 63,656 .0111 .0538 309,724 .9375 1,611,134 25.3150 .............. 60,234 .0149 .0717 290,374 .9170 1,301,410 21.6155 .............. 55,916 .0200 .0953 266,259 .8834 1,011,036 18.0860 .............. 50,587 .0301 .1401 235,225 .8332 744,777 14.7265 .............. 43,503 .0439 .1980 195,982 .7603 509,552 11.7170 .............. 34,890 .0683 .2918 149,002 .6566 313,570 8.9975 .............. 24,711 .1051 .4163 97,836 .4055" 164,568 6.6680 .............. 14,424 .2162 66,732 66,732 4.63

Males

0 .............. 100,000 .1538 .1394 90,659 .8375a 4,711,432 47.111 .............. 86,058 .0186 .0708 328,086 •9449b 4,620,773 53.695 .............. 79,961 .0042 .0206 395,689 .9822 4,292,687 53.69

10 .............. 78,315 .0030 .0149 388,655 .9816 3,896,998 49.7615 ••••••••••••• 0 77,147 .0044 .0219 381,507 .9735 3,508,343 45.4820 .............. 75,456 .0063 .0311 371,407 .9674 3,126,836 41.4425 .•••...••... 0. 73,107 .0069 ,0341 359,299 .9634 2,755,429 37.6930 .............. 70,613 .0080 .0391 346,160 .9573 2,396,130 33.9335 •••••••••••• 0' 67,851 .0095 .0464 331,379 .9482 2,049,970 30.2140 .0 •••.•••••••• 64,700 .0119 .0575 314,197 .9358 1,718,590 26.5645 ••••••••••••• 0 60,979 .0148 .0712 294,038 .9176 1,404,393 23.0350 '0 •••••••••••• 56,636 .0198 .0944 269,814 .8912 1,110,356 19.6155 .0 •••••••••••• 51,289 .0266 .1247 240,455 .8519 840,541 16.3960 .0 •••••••••••• 44,893 .0383 .1748 204,850 .7954 600,086 13.3765 .............. 37,047 .0547 .2408 162,938 .7183 395,236 10.6770 ............. . 28,128 .0807 .3357 117,033 .6129 232,298 8.2675 ••••• 0 •••••••• 18,685 .1210 .4644 71,730 .3777" 115,265 6.1780 .............. 10,007 .2299 43,536 43,535 4.35

a Pr0r.0rtion surviving from birth to 0-4.b 5L5 sLo." Tso/T7S.

87

TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 15

Age x Ix nmx nqx nLx aLx+a Tx 0""aLx

Females

0 .............. 100,000 .0996 .0934 93,745 .8890a 5,500,000 55.001 ••••••••••••• 0 90,661 .0129 .0500 350,729 .9613b 5,406,255 59.635 .............. 86,127 .0032 .0157 427,251 .9860 5,055,527 58.70

10 .............. 84,773 .0025 .0122 421,284 .9852 4,628,276 54.6015 '0 •••••••••••• 83,740 .0035 .0174 415,061 .9800 4,206,992 50.2420 .............. 82,284 .0046 .0227 406,751 .9757 3,791,931 46.0825 .............. 80,416 .0053 .0259 396,874 .9724 3,385,180 42.1030 .0 ••••••• ,. "0 78,333 .0060 .0294 385,907 .9686 2,988,306 38.1535 .............. 76,029 .0068 .0334 373,805 .9643 2,602,399 34.2340 .............. 73,493 .0078 .0382 360,445 .9581 2,228,595 30.3245 ,0 •••••••••••• 70,686 .0094 .0458 345,343 .9464 1,868,150 26.4350 .............. 67,452 .0128 .0619 326,821 .9274 1,522,807 22.5855 .............. 63,276 .0175 .0840 303,101 .8966 1,195,986 18.9060 .0.0 •••••••••• 57,964 .0266 .1246 271,765 .8492 892,885 15.4065 .............. 50,742 .0398 .1808 230,771 .7783 621,120 12.2470 .............. 41,567 .0629 .2716 179,610 .6765 390,349 9.3975 "0 •••••••• "0 30,277 .0984 .3948 121,501 .4235c 210,740 6.9680 .............. 18,323 .2053 89,239 89,238 4.87

Males

0 .............. 100,000 .1203 .1114 92,539 .8720a 5,183,094 51.831 .............. 88,864 .0132 .0511 343,443 .9594b 5,090,555 57.295 "0 •••••••• "0 84,327 .0032 .0159 418,287 .9862 4,747,112 56.29

10 .............. 82,988 .0024 .0117 412,515 .9854 4,328,825 52.1615 ••••• 0 •••••••• 82,018 .0036 .0176 406,474 .9787 3,916,310 47.7520 .............. 80,571 .0051 .0250 397,814 .9739 3,509,837 43.5625 ............... 78,554 .0055 .0272 387,439 .9709 3,112,023 39.6230 .............. 76,421 .0063 .0310 376,180 .9659 2,724,584 35.6535 .............. 74,051 .0076 .0373 363,357 .9579 2,348,404 31.7140 .............. 71,292 .0096 .0471 348,068 .9464 1,985,047 27.8445 ••••••••••••• 0 67,936 .0125 .0604 329,422 .9290 1,636,979 24.1050 .............. 63,833 .0172 .0822 306,046 .9035 1,307,556 20.4855 ••••.•••••• '0. 58,585 .0237 .1120 276,526 .8657 1,001,510 17.1060 ., ............ 52,025 .0347 .1595 239,386 .8112 724,984 13.9465 .............. 43,729 .0504 .2238 194,182 .7355 485,599 11.1170 .............. 33,944 .0753 .3170 142,825 .6315 291,416 8.5975 .............. 23,186 .1142 .4440 90,190 .3930c 148,592 6.4180 .............. 12,890 .2207 58,402 58,401 4.53

a Proportion surviving from birth to 0-4.b sLs/sLo.c Tso/T7s.

88

TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 17

Age x I" nm" nq" nLx 5Lx+5 T" °e"5Lx

Females

0 .............. 100,000 .0745 .0707 94,785 .9171 a 6,000,000 60.001 ••••••••••••• 0 92,934 .0085 .0332 363,755 .9744b 5,905,215 63.545 .............. 89,854 .0022 .0110 446,805 .9902 5,541,460 61.67

10 .............. 88,868 .0017 .0085 442,445 .9805 5,094,655 57.3315 .............. 88,110 .0025 .0125 437,800 .9855 4,652,210 52.8020 .............. 87,010 .0033 .0165 431,463 .9822 4,214,410 48.4425 •••••••• 0 ••••• 85,575 .0039 .0191 423,798 .9795 3,782,947 44.2130 •••• '0 •••••••• 83,944 .0044 .0219 415,125 .9763 3,359,149 40.0235 .............. 82,106 .0052 .0255 405,299 .9722 2,944,024 35.8640 .............. 80,014 .0061 .0302 394,022 .9660 2,538,725 31.7345 .0 ...•.•..•... 77,595 .0077 .0379 380,623 .9550 2,144,704 27.6450 .............. 74,655 .0108 .0523 363,504 .9378 1,764,080 23.6355 .... , ......... 70,747 .0151 .0727 340,876 .9097 1,400,576 19.8060 •••••••••••• ,0 65,603 .0231 .1093 310,088 .8652 1,059,700 16.1565 .............. 58,432 .0356 .1633 268,300 .7968 749,612 12.8370 .............. 48,888 .0574 .2508 213,785 .6969 481,312 9.8575 ............ ,. 36,626 .0917 .3729 148,989 .4431 c 267,528 7.3080 .0 ...•...•.••• 22,970 .1938 118,539 118,538 5.16

Males

0 .............. 100,000 .0918 .0862 93,882 .9021 a 5,647,390 56.471 • ••••••• 0 ••••• 91,379 .0089 .0350 357,189 .9717b 5,553,508 60.785 .0 .•..••...••• 88,184 .0024 .0118 438,323 .9897 5,196,319 58.93

10 .............. 87,145 .0018 .0088 433,805 .9887 4,757,996 54.6015 , •••••• 0 ••••• 0 86,377 .0028 .0138 428,910 .9834 4,324,192 50.0620 " ............ 85,187 .0039 .0195 421,781 .9798 3,895,282 45.7325 .............. 83,526 .0042 .0209 413,271 .9777 3,473,501 41.5930 • •••••• 0 •••••• 81,782 .0048 .0238 404,055 .9737 3,060,231 37.4235 .............. 79,839 .0059 .0289 393,429 .9669 2,656,176 33.2740 .............. 77,532 .0076 .0375 380,399 .9563 2,262,747 29.1945 • .•....••... 0. 74,627 .0103 .0501 363,783 .9400 1,882,348 25.2250 .............. 70,886 .0146 .0704 341,950 .9156 1,518,565 21.4255 .0 ..•••...••.. 65,894 .0209 .0994 313,100 .8794 1,176,615 17.8660 ·............. 59,346 .0311 .1442 275,333 .8270 863,514 14.5565 ., ............ 50,788 .0461 .2066 227,706 .7531 588,181 11.5870 ••• 0 •••••••••• 40,295 .0700 .2978 171,474 .6505 360,475 8.9575 .............. 28,295 .1073 .4231 111,545 .4098c 189,000 6.6880 ............... 16,323 .2107 77,456 77,456 4.75

a Pro,;ortion surviving from birth to 0-4.b sLs sLo.c Tso/T7s.

89

TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVIlL OF MORTALITY (continued)

LEVEL 19

Age x Ix nmx nqx nLx &Lx+& Tx °ex&Lx

Females

0 .0 •••••••••••• 100,000 .0520 .0499 96,004 .9428a 6,500,000 65.00I ............ " 95,006 .0048 .0190 375,407 .9851 b 6,403,996 67.415 ........... ". 93,201 .0014 .0069 464,405 .9939 6,028,589 64.68

10 .0 ...••••.•••• 92,561 .0011 .0054 461,567 .9932 5,564,184 60.1115 .............. 92,065 .0017 .0082 458,440 .9904 5,102,618 55.4220 .............. 91,311 .0022 .0111 454,021 .9879 4,644,178 50.8625 .0 •••••••••••• 90,298 .0026 .0130 448,547 .9859 4,190,157 46.4030 .............. 89,121 .0031 .0152 442,221 .9833 3,741,610 41.9835 .0 .•••••••••.• 87,767 .0037 .0183 434,829 .9795 3,299,389 37.5940 .............. 86,164 .0046 .0228 425,918 .9735 2,864,560 33.2545 .............. 84,203 .0062 .0303 414,639 .9635 2,438,642 28,9650 .............. 81,653 .0088 .0430 399,491 .9480 2,024,003 24.7955 .0 •••••••••••• 78,144 .0127 .0615 378,702 .9228 1,624,511 20.7960 .............. 73,337 .0197 .0940 349,452 .8815 1,245,809 16.9965 .............. 66,444 .0314 .1455 308,047 .8158 896,357 13.4970 .............. 56,775 .0518 .2294 251,317 .7182 588,310 10.3675 .............. 43,751 .0848 .3499 180,485 .4664c 336,993 7.7080 .............. 28,442 .1817 156,509 156,509 5.50

Males

0 .0 •••••••••••• 100,000 .0661 .0629 95,117 .93ooa 6,122,821 61.231 .............. 93,713 .0053 .0210 369,858 .9826b 6,027,704 64.325 .............. 91,744 .0016 .0081 456,871 .9929 5,657,846 61.67

10 .............. 91,004 .0012 .0062 453,620 .9918 5,200,975 57.1515 .............. 90,444 .0020 .0102 449,920 .9878 4,747,355 52.4920 .............. 89,524 .0029 .0143 444,411 .9853 4,297,435 48.0025 .............. 88,240 .0030 .0151 437,879 .9840 3,853,024 43.6730 .............. 86,912 .0034 .0171 430,851 .9809 3,415,145 39.2935 .............. 85,429 .0043 .0211 422,637 .9753 2,984,293 34.9340 .............. 83,626 .0058 .0284 412,201 .9658 2,561,657 30.6345 .............. 81,255 .0082 .0402 398,115 .9508 2,149,456 26.4550 .............. 77,991 .0121 .0587 378,509 .9277 1,751,341 22.4655 ••••••••••••• 0 73,412 .0181 .0867 351,159 .8933 1,372,832 18.7060 .............. 67,051 .0275 .1286 313,696 .8433 1,021,673 15.2465 .............. 58,427 .0417 .1889 264,546 .7713 707,977 12.1270 '" ........... 47,392 .0645 .2779 204,036 .6705 443,431 9.3675 .............. 34,223 .1004 .4011 136,796 .4286c 239,394 7.0080 .............. 20,495 .1998 102,599 102,599 5.01

a ~proh0rtion surviving from birth to 0-4.b liLli liLo.c Tso/T7li.

90

91

TABLE 1.1. "WEST" MODEL LIFE TABLES ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 23

Age x Ix nmx n'b: nLx 5Lx+5 Tx Oex

5Lx

Females

0 ........ , ..... 100,000 .0154 .0152 98,629 .9840" 7,500,000 75.001 .............. 98,484 .0006 .0024 393,346 .9979b 7,401,371 75.155 .............. 98,248 .0003 .0013 490,926 .9988 7,008,025 71.33

10 .............. 98,123 .0002 .0011 490,355 .9986 6,517,099 66.4215 .............. 98,019 .0004 .0018 489,662 .9979 6,026,745 61.4920 .............. 97,846 .0005 .0025 488,610 .9971 5,537,082 56.5925 '0' ••••••••••• 97,598 .0006 .0032 487,207 .9963 5,048,472 51.7330 .............. 97,285 .0008 .0041 485,417 .9950 4,561,265 46.8935 .0 ............ 96,882 .0012 .0058 483,004 .9927 4,075,848 42.0740 '0' ••••••••••• 96,319 .0018 .0089 479,464 .9883 3,592,844 37.3045 .............. 95,466 .0029 .0146 473,851 .9813 3,113,380 32.6150 ........... '" 94,074 .0046 .0228 465,006 .9707 2,639,529 28.0655 .............. 91,928 .0073 .0360 451,369 .9531 2,174,523 23.6660 .............. 88,619 .0120 .0582 430,193 .9211 1,723,154 19.4465 .............. 83,458 .0212 .1008 396,265 .8652 1,292,961 15.4970 .............. 75,048 .0378 .1727 342,839 .7759 896,696 11.9575 .............. 62,087 .0668 .2862 266,015 .5197" 553,857 8.9280 .............. 44,318 .1540 287,843 287,842 6.50

Males

0 .............. 100,000 .0219 .0214 98,080 .9774" 7,118,759 71.191 .............. 97,856 .0009 .0034 390,618 .9967b 7,020,680 71.755 .............. 97,521 .0005 .0022 487,062 .9979 6,630,062 67.99

10 .............. 97,303 .0004 .0019 486,055 .9972 6,143,000 63.1315 .............. 97,119 .0007 .0037 484,692 .9956 5,656,944 58.2520 .............. 96,758 .0010 .0051 482,547 .9949 5,172,252 53.4625 .............. 96,261 .0010 .0051 480,084 .9946 4,689,705 48.7230 .............. 95,773 .0011 .0057 477,500 .9934 4,209,621 43.9535 .............. 95,227 .0015 .0075 474,348 .9905 3,732,121 39.1940 .............. 94,512 .0023 .0116 469,820 .9843 3,257,773 34.4745 .............. 93,416 .0040 .0199 462,430 .9734 2,787,952 29.8550 .............. 91,556 .0068 .0334 450,149 .9553 2,325,522 25.4055 .............. 88,503 .0116 .0564 430,029 .9273 1,875,373 21.1960 .............. 83,508 .0188 .0900 398,759 .8849 1,445,344 17.3165 .............. 75,995 .0307 .1427 352,874 .8200 1,046,585 13.7770 .............. 65,154 .0503 .2235 289,364 .7259 693,711 10.6575 .............. 50,591 .0817 .3394 210,034 .4806" 404,347 7.9980 .............. 33,422 .1720 194,314 194,313 5.81

" Proh0rtion surviving from birth to 0-4.b liLu liLo." Tso/T7li.

92

TABLE 1.2. VALUES OF THE FUNCTION Ix (SURVIVORS TO AGE X) FOR X = 1,2, 3 AND 5 IN "WEST" MODEL LIFE TABLES AT VARIOUS LEVELS OFMORTALITY, FOR FEMALES, FOR MALES, AND FOR BOTH SEXES ASSUMING THAT THE SEX RATIO AT BIRTH IS 1.05

(/0 = 100,000)

Level h 12 18 15 Level 11 12 18 15

Females Males (continued)

1 '" .. , ....... , .. 63,483 55,000 51,199 46,883 13 ................ 86,058 82,912 81,534 79,9613 .. ...... .... .... 69,481 61,829 58,399 54,506 15 ................ 88,864 86,523 85,498 84,3275 .... ............ 74,427 67,671 64,643 61,205 17 ................ 91,379 89,790 89,056 88,1847 ................ 78,614 72,765 70,145 67,169 19 ................ 93,713 92,796 92,338 91,7749 ................ 82,226 7,1,271 75,051 72,530 21 . " ............. 95,909 95,508 95,285 94,989

11 ............. , .. 85,388 81,300 79,468 77,389 23 ................ 97,856 97,719 97,636 97,52113 ................ 88,169 84,939 83,492 81,84815 ................ 90,661 88,364 87,324 86,12717 ................ 92,934 91,419 90,709 89,854 Both sexes19 ................ 95,006 94,143 93,724 93,20121 ................ 96,907 96,559 96,385 96,160 1 . ............... 60,722 52,597 48,996 44,89723 ................ 98,484 98,377 98,321 98,248 3 . ............... 67,118 59,709 56,425 52,688

5 ................ 72,392 65,798 62,877 59,5517 ................ 76,857 71,112 68,567 65,670

Males 9 ................ 80,709 75,813 73,646 71,17711 ................ 84,080 80,019 78,220 76,173

1 ................ 58,093 50,308 46,898 43,005 13 . ............... 87,088 83,901 82,489 80,8813 ................ 64,868 57,690 54,546 50,957 15 . ............... 89,740 87,421 86,389 85,2055 ................ 70,454 64,015 61,195 57,976 17 . ............... 92,137 90,584 89,862 88,9997 ................ 75,183 69,537 67,064 64,242 19 . ............... 94,144 93,453 93,011 92,4559 ................ 79,263 74,425 72,307 69,888 21 . ............... 96,396 96,020 95,822 95,560

11 ................ 82,835 78,800 77,032 75,015 23 . ............... 98,162 98,040 97,970 97,876

93

TABLE 1.3. TEN-YEAR SURVIVAL RATES IN "WEST" MODEL LIPB TABLES AT VARIOUS LEVELS OF MORTALITY

Level Level Level Level Level Level Level Level Level Level Level LevelAge I 3 5 7 9 II 13 15 17 19 21 23

Females

0-4 to 10-14 ............ .7370 .7847 .8232 .8552 .8826 .9063 .9273 .9478 .9649 .9791 .9906 .99675-9 to 15-19 ............ .8736 .8946 .9121 .9271 .9401 .9516 .9618 .9715 .9798 .9872 .9934 .9974

10-14 to 20-24 ............ .8576 .8808 .9001 .9167 .9312 .9439 .9552 .9655 .9752 .9837 .9912 .996415-19 to 25-29 ............ .8286 .8560 .8789 .8986 .9158 .9309 .9444 .9562 .9680 .9784 .9878 .995020-24 to 30-34 ............ .8050 .8358 .8616 .8838 .9032 .9203 .9355 .9488 .9621 .9740 .9848 .993525-29 to 35-39 ............ .7841 .8178 .8461 .8704 .8917 .9106 .9273 .9419 .9563 .9694 .9814 .991430-34 to 40-44 ............ .7660 .8018 .8319 .8578 .8805 .9007 .9186 .9340 .9492 .9631 .9762 .987735-39 to 45-49 ............ .7518 .7884 .8193 .8459 .8692 .8899 .9083 .9239 .9391 .9536 .9673 .981040-44 to 50-54 ............ .7267 .7649 .7972 .8251 .8496 .8713 .8906 .9067 .9225 .9380 .9528 .969845-49 to 55-59 ............ .6750 .7177 .7539 .7853 .8130 .8377 .8597 .8777 .8956 .9133 .9308 .952550-54 to 60-64 ............ .5920 .6415 .6840 .7216 .7541 .7836 .8101 .8315 .8530 .8747 .8964 .925155-59 to 65-69 ............ .4820 .5384 .5875 .6308 .6696 .7045 .7361 .7614 .7871 .8134 .8402 .877960-64 to 70-74 ............ .3617 .4202 .4721 .5185 .5605 .5987 .6335 .6609 .6894 .7192 .7501 .796965-69 to 75-79 ............ .2362 .2907 .3401 .3852 .4264 .4644 .4992 .5265 .5553 .5859 .6183 .671370+ to 80+ .. , ......... .1008 .1230 .1425 .1605 .1789 .1951 .2128 .2286 .2463 .2660 .2879 .3210

Males

0-4 to 10-14............. .7402 .7877 .8257 .8570 .8835 .9064 .9281 .9462 .9617 .9756 .9871 .99465-9 to 15-19 ............ .8844 .9064 .9214 .9341 .9452 .9549 .9642 .9718 .9185 .9848 .9904 .9951

10-14 to 20-24 ............ .8697 .8901 .9071 .9217 .9343 .9455 .9556 .9644 .9723 .9197 .9865 .992815-19 to 25-29 ............ .8318 .8578 .8796 .8983 .9146 .9290 .9418 .9532 .9635 .9733 .9822 .990520-24 to 30-34 ............ .8021 .8327 .8584 .8805 .8998 .9169 .9320 .9456 .9580 .9695 .9800 .989525-29 to 35-39 ............ .7744 .8091 .8383 .8634 .8854 .9049 .9223 .9378 .9520 .9652 .9771 .988130-34 to 40-44 ............ .7392 .7785 .8116 .8402 .8654 .8877 .9077 .9253 .9415 .9567 .9706 .983935-39 to 45-49 ............ .6993 .7428 .7797 .8116 .8397 .8648 .8873 .9066 .9246 .9420 .9579 .974940-44 to 50-54 ............ .6535 .7005 .7406 .1755 .8063 .8337 .8587 .8793 .8989 .9183 .9365 .958145-49 to 55-59 ............ .5961 .6464 .6895 .7272 .7605 .7903 .8178 .8394 .8607 .8820 .9028 .929950-54 to 60-64 ............ .5228 .5757 .6214 .6616 .6974 .7295 .7592 .7822 .8052 .8288 .8523 .885855-59 to 65-69 ............ .4266 .4817 .5299 .5727 .6109 .6456 .6776 .7022 .7213 .7533 .7801 .820660-64 to 70-74 ..... ....... .3177 .3720 .4202 .4634 .5026 .5382 .5713 .5966 .6228 .6504 .6794 .725765-69 to 15-79 ............ .2052 .2536 .2976 .3378 .3747 .4086 .4402 .4645 .4899 .5171 .5463 .595270+ to 80+ ............ .0877 .1085 .1263 .1424 .1575 .1724 .1874 .2004 .2149 .2314 .2500 .2801

94

Annex II

MODEL STABLE POPULATIONSTABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY

LEVEL 1Females (Oeo = 20.00 years)

Annual rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0292 .0335 .0381 .0430 .0482 .0535 .0590 .0647 .0704 .0763 .0822 .0882 .09421-4 .......................... .0824 .0935 .1051 .1171 .1294 .1420 .1547 .1674 .1800 .1926 .2049 .2171 .22905-9 .......................... .0926 .1027 .1129 .1231 .1330 .1426 .1519 .1607 .1690 .1768 .1840 .1906 .1965

10-14 .......................... .0910 .0984 .1055 .1122 .1182 .1237 .1285 .1326 .1360 .1387 .1408 .1422 .143015-19 .......................... .0894 .0943 .0986 .1022 .1051 .1073 .1087 .1094 .1094 .1088 .1077 .1061 .104120-24 .......................... .0862 .0887 .0905 .0915 .0917 .0913 .0902 .0885 .0864 .0838 .0809 .0778 .074425-29 .......................... .0819 .0822 .0817 .0806 .0788 .0765 .0737 .0706 .0672 .0635 .0598 .0561 .052330-34 ... , ...................... .0767 .0751 .0729 .0701 .0668 .0633 .0594 .0555 .0515 .0475 .0437 .0399 .036335-39 ..... " ................... .0709 .0677 .0641 .0601 .0559 .0516 .0473 .0431 .0390 .0351 .0314 .0280 .024940-44 .......................... .0649 .0605 .0558 .0510 .0463 .0417 .0373 .0331 .0292 .0257 .0224 .0195 .016945-49 .......................... .0589 .0535 .0482 .0430 .0380 .0334 .0291 .0252 .0217 .0186 .0158 .0134 .011350-54 .......................... .0522 .0462 .0406 .0353 .0305 .0261 .0222 .0187 .0157 .0131 .0109 .0090 .007455-59 . " ....................... .0440 .0380 .0325 .0276 .0232 .0194 .0161 .0133 .0109 .0088 .0072 .0058 .004560-64 .......................... .0341 .0288 .0240 .0199 .0163 .0133 .0108 .0086 .0069 .0055 .0043 .0034 .002765-69 .......................... .0234 .0193 .0157 .0127 .0101 .0081 .0064 .0050 .0039 .0030 .0023 .0018 .001470-74 .......................... .0136 .0109 .0087 .0068 .0053 .0041 .0032 .0024 .0018 .0014 .0011 .0008 .0006

I,Q 75-79 ... , .......... , ........... .0061 .0048 .0037 .0028 .0022 .0016 .0012 .0009 .0007 .0005 .0004 .0003 .0002VI 80+ .......................... .0024 .0018 .0014 .0010 .0008 .0006 .0004 .0003 .0002 .0002 .0001 .0001 .0001

Age Proportion under given age

1 ............................ .0292 .0335 .0381 .0430 .0482 .0535 .0590 .0647 .0704 .0763 .0822 .0882 .09425 ............................ .1115 .1269 .1432 .1601 .1776 .1955 .2137 .2320 .2504 .2688 .2871 .3053 .3232

10 ............................ .2041 .2297 .2561 .2832 .3106 .3381 .3656 .3928 .4195 .4456 .4711 .4958 .519715 ............................ .2951 .3281 .3617 .3953 .4288 .4618 .4941 .5253 .5555 .5843 .6119 .6380 .6627

20 ............................ .3845 .4225 .4603 .4976 .5339 .5691 .6027 .6347 .6649 .6932 .7196 .7441 .766925 ............................ .4707 .5112 .5508 .5891 .6257 .6604 .6929 .7232 .7512 .7770 .8005 .8219 .841330 ............................ .5526 .5934 .6325 .6697 .7045 .7369 .7666 .7938 .8184 .8405 .8603 .8780 .893635 ............................ .6293 .6685 .7054 .7397 .7713 .8001 .8261 .8493 .8699 .8881 .9040 .9179 .929940 ............................ .7003 .7362 .7695 .7998 .8272 .8517 .8734 .8924 .9089 .9232 .9355 .9459 .9548

45 ............................ .7652 .7967 .8253 .8509 .8736 .8934 .9107 .9255 .9382 .9489 .9579 .9654 .971750 ............................ .8241 .8502 .8735 .8939 .9116 .9268 .9398 .9507 .9599 .9675 .9737 .9789 .983055 ............................ .8763 .8964 .9140 .9292 .9420 .9529 .9620 .9695 .9756 .9806 .9847 .9879 .9905

60 ............................ .9203 .9344 .9465 .9568 .9653 .9723 .9781 .9827 .9865 .9895 .9918 .9937 .995165 ............................ .9544 .9632 .9705 .9766 .9816 .9856 .9888 .9914 .9934 .9949 .9962 .9971 .9978

Parameter ofstable populations

Birth rate ........................ .0380 .0438 .0500 .0566 .0635 .0707 .0782 .0859 .0937 .1018 .1100 .1182 .1266Death rate ....................... .0480 .0488 .0500 .0516 .0535 .0557 .0582 .0609 .0637 .0668 .0700 .0732 .0766GRR (27) ....................... 2.33 2.66 3.03 3.44 3.93 4.46 5.07 5.75 6.52 7.39 8.36 9.45 10.68GRR (29) ....................... 2.39 2.75 3.16 3.64 4.18 4.79 5.49 6.28 7.18 8.20 9.36 10.67 12.16GRR (31) ....................... 2.45 2.85 3.31 3.84 4.45 5.16 5.96 6.89 7.96 9.18 10.57 12.17 13.99GRR (33) ....................... 2.51 2.95 3.47 4.07 4.77 5.58 6.52 7.62 8.90 10.38 12.09 14.07 16.36Average age ..................... , 29.2 27.3 25.5 23.8 22.3 20.8 19.4 18.2 17.0 16.0 15.0 14.1 13.3Births/population 15-44 ............ .081 .093 .108 .124 .143 .164 .188 .215 .245 .279 .318 .361 .410

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 3Females (Oeo = 25.00 years)

A1/IIUiIlrat« of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0241 .0279 .0321 .0365 .0412 .0461 .0511 .0564 .0617 .0672 .0727 .0783 .08401-4 .......................... .0733 .0840 .0954 .1072 .1194 .1320 .1447 .1575 .1703 .1831 .1957 .2082 .22045-9 .......................... .0850 .0953 .1057 .1162 .1266 .1367 .1466 .1560 .1650 .1734 .1812 .1885 .1951

10-14 .......................... .0846 .0924 .1000 .1072 .1139 .1200 .1254 .1302 .1343 .1377 .1403 .1423 .143715-19 .......................... .0841 .0896 .0946 .0989 .1024 .1053 .1074 .1087 .1093 .1093 .1087 .1075 .105920-24 .......................... .0823 .0856 .0881 .0898 .0908 .0910 .0905 .0893 .0876 .0854 .0829 .0799 .076825-29 .......................... .0795 .0806 .0809 .0805 .0793 .0776 .0752 .0725 .0693 .0659 .0624 .0587 .055030-34 .......................... .0760 .0752 .0736 .0714 .0686 .0654 .0619 .0581 .0543 .0503 .0464 .0426 .038935-39 .......................... .0719 .0693 .0662 .0626 .0587 .0546 .0504 .0461 .0420 .0380 .0342 .0306 .027340-44 .......................... .0674 .0634 .0590 .0544 .0498 .0452 .0406 .0363 .0322 .0284 .0249 .0218 .018945-49 .......................... .0626 .0575 .0522 .0470 .0419 .0370 .0325 .0283 .0245 .0211 .0181 .0154 .013050-54 ....................... '" .0570 .0510 .0451 .0396 .0345 .0297 .0254 .0216 .0183 .0153 .0128 .0106 .008855-59 .......................... .0497 .0434 .0375 .0321 .0272 .0229 .0191 .0158 .0130 .0107 .0087 .0070 .005760-64 .......................... .0404 .0344 .0290 .0242 .0200 .0164 .0134 .0108 .0087 .0069 .0055 .0043 .003465-69 .......................... .0296 .0245 .0202 .0164 .0133 .0106 .0084 .0066 .0052 .0041 .0031 .0024 .001970-74 .......................... .0188 .0152 .0122 .0097 .0076 .0059 .0046 .0035 .0027 .0021 .0015 .0012 .000975-79 .......................... .0095 .0075 .0059 .0045 .0035 .0027 .0020 .0015 .0011 .0008 .0006 .0004 .0003

\0 80+ ........................... .0043 .0033 .0025 .0019 .0014 .0011 .0008 .0006 .0004 .0003 .0002 .0001 .00010\

Age Proportion under given age

I ............................ .0241 .0279 .0321 .0365 .0412 .0461 .0511 .0564 .0617 .0672 .0727 .0783 .08405 ............................ .0974 .1120 .1274 .1437 .1606 .1780 .1958 .2139 .2320 .2503 .2685 .2865 .3044

10 ............................ .1824 .2072 .2331 .2599 .2872 .3147 .3424 .3699 .3970 .4237 .4497 .4750 .499515 ............................ .2670 .2996 .3331 .3670 .4010 .4347 .4678 .5001 .5313 .5613 .5900 .6173 .643220 ............................ .3511 .3893 .4277 .4659 .5035 .5400 .5752 .6088 .6406 .6706 .6987 .7248 .749125 ............................ .4334 .4748 .5158 .5557 .5942 .6310 .6656 .6981 .7283 .7561 .7815 .8048 .825930 ............................ .5129 .5555 .5967 .6362 .6736 .7085 .7409 .7706 .7976 .8220 .8439 .8635 .880835 ............................ .5890 .6306 .6703 .7076 .7422 .7739 .8028 .8287 .8518 .8723 .8903 .9061 .919740 ............................ .6608 .7000 .7365 .7702 .8009 .8285 .8531 .8748 .8938 .9103 .9245 .9367 .947045 ............................ .7282 .7633 .7955 .8247 .8507 .8737 .8938 .9112 .9261 .9387 .9494 .9584 .965950 ............................ .7908 .8208 .8477 .8716 .8926 .9107 .9263 .9395 .9506 .9598 .9675 .9738 .979055 ............................ .8478 .8717 .8929 .9112 .9270 .9405 .9517 .9611 .9688 .9752 .9803 .9844 .987860 ............................ .8975 .9151 .9303 .9433 .9542 .9633 .9708 .9770 .9819 .9858 .9890 .9915 .993465 ............................ .9379 .9495 .9593 .9675 .9742 .9798 .9842 .9878 .9906 .9928 .9945 .9958 .9968

Parameter of stable populations

Birth rate ........................ .0299 .0347 .0400 .0456 .0516 .0579 .0644 .0712 .0782 .0853 .0926 .0999 .1074Death rate ....................... .0399 .0397 .0400 .0406 .0416 .0429 .0444 .0462 .0482 .0503 .0526 .0549 .0574GRR (27) ....................... 1.88 2.15 2.46 2.80 3.19 3.62 4.12 4.68 5.31 6.01 6.81 7.70 8.71GRR (29) ....................... 1.92 2.21 2.54 2.93 3.36 3.86 4.43 5.07 5.81 6.64 7.58 8.65 9.86GRR (31) ....................... 1.95 2.27 2.64 3.07 3.56 4.13 4.78 5.53 6.39 7.34 8.50 9.79 11.26GRR (33) ....................... 1.99 2.34 2.75 3.22 3.78 4.43 5.18 6.06 7.08 8.26 9.63 11.22 13.06Average age ...................... 31.3 29.2 27.3 25.5 23.8 22.2 20.7 19.3 18.0 16.9 15.8 14.9 14.0Births/population 15-44 ............ .065 .075 .087 .100 .115 .132 .151 .173 .198 .226 .258 .293 .333

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)~...,..",~.

<,LEVEL 5

Females (Oco = 30.00 years)

AIUfUDl rate ofincrease

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0205 .0240 .0278 .0319 .0362 .0408 .0455 .0505 .0555 .0607 .0659 .0713 .07671-4 .......................... .0662 .0766 .0876 .0992 .1113 .1238 .1365 .1494 .1623 .1753 .1881 .2008 .21325-9 .......................... .0787 .0889 .0995 .1101 .1208 .1314 .1416 .1515 .1610 .1700 .1783 .1861 .1933

10-14 .......................... .0790 .0871 .0950 .1026 .1098 .1164 .1224 .1277 .1324 .1363 .1395 .1420 .143815-19 .......................... .0793 .0853 .0907 .0956 .0997 .1031 .1058 .1077 .1088 .1092 .1090 .1083 .106920-24 .......................... .0786 .0824 .0855 .0879 .0894 .0902 .0902 .0896 .0883 .0864 .0842 .0815 .078525-29 .......................... .0770 .0788 .0797 .0799 .0793 .0780 .0761 .0737 .0708 .0677 .0642 .0607 .057030-34 .......................... .0748 .0746 .0737 .0720 .0697 .0669 .0636 .0601 .0563 .0525 .0486 .0448 .041035-39 .......................... .0720 .0701 .0675 .0643 .0607 .0568 .0527 .0486 .0444 .0403 .0364 .0327 .029340-44 .......................... .0688 .0653 .0613 .0570 .0525 .0479 .0433 .0389 .0347 .0308 .0271 .0237 .020745-49 .......................... .0652 .0603 .0553 .0501 .0450 .0401 .0354 .0310 .0269 .0233 .0200 .0171 .014550-54 .......................... .0606 .0547 .0489 .0432 .0378 .0329 .0283 .0242 .0205 .0173 .0145 .0121 .010055-59 .......................... .0543 .0478 .0417 .0359 .0307 .0260 .0218 .0182 .0151 .0124 .0101 .0082 .006660-64 .......................... .0458 .0393 .0334 .0281 .0234 .0193 .0158 .0129 .0104 .0083 .0066 .0053 .004265-69 .......................... .0353 .0295 .0245 .0201 .0163 .0131 .0105 .0083 .0066 .0051 .0040 .0031 .002470-74 .......................... .0239 .0195 .0158 .0126 .0100 .0079 .0061 .0047 .0036 .0028 .0021 .0016 .001275-79 .......................... .0133 .0106 .0083 .0065 .0050 .0038 .0029 .0022 .0017 .0012 .0009 .0007 .0005

\0 80+ .......................... .0068 .0052 .0040 .0030 .0023 .0017 .0013 .0009 .0007 .0005 .0003 .0002 .0002-...I

Age Proportion under given age

1 ............................ .0205 .0240 .0278 .0319 .0362 .0408 .0455 .0505 .0555 .0607 .0659 .0713 .07675 ............................ .0867 .1006 .1154 .1311 .1475 .1646 .1820 .1998 .2178 .2359 .2540 .2720 .2899

10 ............................ .1654 .1895 .2148 .2412 .2683 .2959 .3237 .3514 .3789 .4059 .4324 .4582 .483215 ............................ .2444 .2765 .3098 .3438 .3781 .4123 .4461 .4791 .5112 .5422 .5718 .6001 .627020 ............................ .3237 .3618 .4006 .4394 .4778 .5154 .5518 .5868 .6200 .6514 .6809 .7084 .733925 ............................ .4022 .4442 .4861 .5272 .5672 .6056 .6421 .6763 .7083 .7379 .7650 .7899 .812430 ............................ .4792 .5230 .5658 .6071 .6465 .6836 .7182 .7500 .7791 .8055 .8293 .8505 .869435 ............................ .5541 .5976 .6395 .6791 .7162 .7505 .7818 .8101 .8355 .8580 .8779 .8953 .910540 ............................ .6261 .6677 .7069 .7434 .7769 .8073 .8345 .8587 .8799 .8983 .9143 .9280 .939745 ............................ .6949 .7329 .7682 .8004 .8294 .8552 .8779 .8976 .9146 .9291 .9414 .9518 .960450 ............................ .7601 .7933 .8235 .8505 .8744 .8953 .9132 .9286 .9415 .9524 .9614 .9689 .975055 ............................ .8207 .8480 .8723 .8937 .9122 .9281 .9415 .9528 .9621 .9697 .9759 .9809 .985060 ............................ .8750 .8958 .9140 .9297 .9429 .9541 .9634 .9709 .9771 .9821 .9860 .9892 .991665 ............................ .9208 .9352 .9474 .9578 .9664 .9735 .9792 .9838 .9875 .9904 .9927 .9944 .9958

Parameter of stable populations

Birth rate ... " .. " ............... .0245 .0281 .0333 .0383 .0437 .0493 .0552 .0613 .0676 .0741 .0807 .0874 .0943Death rate ....................... .0345 .0337 .0333 .0333 .0337 .0343 .0352 .0363 .0376 .0391 .0407 .0424 .0443GRR (27) ....................... 1.60 1.82 2.08 2.37 2.70 3.08 3.50 3.98 4.51 5.12 5.80 6.56 7.42GRR (29) ....................... 1.61 1.86 2.15 2.47 2.84 3.26 3.74 4.29 4.91 5.62 6.42 7.33 8.36GRR (31) ....................... 1.63 1.90 2.21 2.57 2.99 3.47 4.01 4.65 5.37 6.21 7.16 8.25 9.50GRR(33) ....................... 1.65 1.94 2.29 2.68 3.15 3.69 4.33 5.06 5.92 6.91 8.06 9.39 10.94Average age ...................... 33.1 31.0 28.9 26.9 25.1 23.4 21.8 20.3 19.0 17.7 16.6 15.5 14.6Births/population 15-44 ............ .054 .063 .073 .084 .097 .111 .128 .146 .168 .191 .218 .249 .283

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 7Females (Oeo = 35.00 years)

A1IIlUQl rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0179 .0211 .0246 .0284 .0325 .0368 .0413 .0460 .0508 .0557 .0608 .0659 .07111-4 .......................... .0605 .0705 .0812 .0926 .1046 .1170 .1297 .1426 .1556 .1686 .1816 .1944 .20715-9 .......................... .0732 .0834 .0940 .1048 .1157 .1265 .1371 .1474 .1573 .1667 .1755 .1837 .1914

10-14 .......................... .0741 .0823 .0905 .0984 .1060 .1130 .1195 .1253 .1304 .1347 .1384 .1413 .143515-19 .......................... .0750 .0813 .0871 .0924 .0970 .1009 .1041 .1064 .1080 .1089 .1091 .1086 .107620-24 .......................... .0751 .0794 .0830 .0858 .0879 .0892 .0897 .0894 .0885 .0870 .0850 .0826 .079825-29 .......................... .0745 .0768 .0783 .0790 .0789 .0781 .0766 .0745 .0719 .0690 .0657 .0622 .058630-34 .......................... .0733 .0737 .0733 .0721 .0703 .0678 .0649 .0616 .0580 .0542 .0504 .0465 .042835-39 .............. , ........... .0716 .0703 .0681 .0654 .0621 .0585 .0546 .0505 .0464 .0423 .0383 .0345 .031040-44 .......................... .0695 .0665 .0629 .0589 .0546 .0501 .0456 .0411 .0368 .0328 .0290 .0255 .022345-49 .......................... .0670 .0625 .0576 .0526 .0476 .0426 .0378 .0333 .0291 .0252 .0217 .0186 .015950-54 .......................... .0634 .0577 .0519 .0462 .0407 .0356 .0308 .0264 .0225 .0191 .0160 .0134 .011155-59 .......................... .0581 .0516 .0453 .0393 .0338 .0288 .0243 .0203 .0169 .0140 .0114 .0093 .007660-64 .......................... .0505 .0437 .0374 .0317 .0266 .0221 .0182 .0148 .0120 .0097 .0077 .0062 .004965-69 .......................... .0405 .0342 .0286 .0236 .0193 .0156 .0125 .0100 .0079 .0062 .0048 .0038 .002970-74 .......................... .0290 .0238 .0194 .0156 .0125 .0099 .0077 .0060 .0046 .0035 .0027 .0020 .001575-79 .......................... .0172 .0138 .0110 .0086 .0067 .0052 .0040 .0030 .0023 .0017 .0012 .0009 .0007

\Q 80+ .......................... .0097 .0075 .0058 .0044 .0034 .0025 .0019 .0014 .0010 .0007 .0005 .0004 .000300

Age Proportion under given age

1 ............................ .0179 .0211 .0246 .0284 .0325 .0368 .0413 .0460 .0508 .0557 .0608 .0659 .07115 ............................ .0783 .0915 .1058 .1210 .1370 .1537 .1709 .1885 .2064 .2244 .2424 .2604 .2782

10 ............................ .1515 .1749 .1998 .2258 .2527 .2802 .3081 .3359 .3637 .3910 .4179 .4441 .469615 ............................ .2256 .2573 .2903 .3242 .3587 .3932 .4275 .4612 .4940 .5258 .5563 .5854 .613120 ............................ .3006 .3385 .3774 .4166 .4557 .4942 .5316 .5677 .6021 .6347 .6653 .6940 .720725 ............................ .3757 .4179 .4604 .5025 .5436 .5834 .6213 .6571 .6906 .7217 .7504 .7766 .800530 ............................ .4501 .4947 .5386 .5814 .6225 .6614 .6978 .7316 .7625 .7907 .8161 .8389 .859235 ............................ .5235 .5684 .6120 .6536 .6928 .7293 .7627 .7931 .8205 .8449 .8665 .8854 .901940 ............................ .5951 .6386 .6801 .7190 .7549 .7877 .8173 .8436 .8669 .8872 .9048 .9199 .932945 ............................ .6646 .7051 .7430 .7779 .8095 .8378 .8629 .8848 .9037 .9200 .9338 .9454 .955250 ............................ .7316 .7676 .8006 .8305 .8571 .8804 .9007 .9180 .9328 .9452 .9555 .9640 .971155 ............................ .7950 .8253 .8525 .8767 .8978 .9160 .9314 .9445 .9553 .9642 .9715 .9774 .982260 ............................ .8531 .8769 .8978 .9160 .9316 .9448 .9557 .9648 .9722 .9782 .9830 .9868 .989865 ............................ .9036 .9206 .9352 .9477 .9582 .9668 .9739 .9796 .9842 .9879 .9907 .9920 .9946

Parameter ofstable populations

Birth rate ........................ .0206 .0244 .0286 .0331 .0375 .0430 .0484 .0541 .0599 .0659 .0720 .0783 .0847Death rate ....................... .0306 .0294 .0286 .0281 .0279 .0280 .0284 .0291 .0299 .0309 .0320 .0333 .0347GRR (27) ....................... 1.34 1.59 1.82 2.08 2.37 2.70 3.07 3.49 3.96 4.49 5.09 5.76 6.52GRR (29) ....................... 1.40 1.62 1.87 2.15 2.47 2.84 3.26 3.74 4.29 4.91 5.61 6.41 7.31GRR (31) ....................... 1.41 1.64 1.91 2.23 2.59 3.00 3.48 4.03 4.67 5.39 6.23 7.18 8.27GRR (33) ....................... 1.42 1.67 1.97 2.31 2.71 3.18 3.73 4.37 5.11 5.97 6.97 8.13 9.47Average age ...................... 34.7 32.5 30.3 28.3 26.3 24.5 22.8 21.2 19.8 18.4 17.2 16.1 15.1Births/population 15-44 ............ .047 .055 .063 .073 .084 .097 .111 .128 .146 .167 .191 .217 .247

.._-'--_. -_."-,._.

"--

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTAUTY (continued)

LEVEL 9Females (Oeo = 40.00 years)

--Annual rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0158 .0188 .0221 .0257 .0295 .0336 .0379 .0424 .0471 .0518 .0567 .0617 .06671-4 .......................... .0557 .0653 .0758 .0870 .0988 .1111 .1238 .1367 .1498 .1629 .1760 .1890 .20185-9 .......................... .0684 .0786 .0891 .1000 .1111 .1221 .1330 .1436 .1538 .1636 .1728 .1814 .1894

10-14 .......................... .0698 .0781 .0864 .0946 .1024 .1098 .1167 .1229 .1284 .1332 .1372 .1405 .143115-19 .......................... .0711 .0776 .0838 .0894 .0945 .0988 .1024 .1051 .1071 .1084 .1089 .1087 .108020-24 .......................... .0718 .0765 .0805 .0838 .0863 .0880 .0890 .0891 .0886 .0874 .0856 .0834 .080825-29 .......................... .0720 .0747 .0767 .0779 .0783 .0779 .0767 .0750 .0727 .0699 .0668 .0635 .060030-34 ............ , ............. .0717 .0726 .0727 .0720 .0705 .0684 .0658 .0627 .0593 .0556 .0518 .0480 .044335-39 ................ , ......... .0709 .0701 .0684 .0661 .0632 .0598 .0560 .0521 .0480 .0439 .0400 .0361 .032440-44 .......................... .0698 .0672 .0640 .0603 .0562 .0519 .0474 .0430 .0387 .0345 .0306 .0270 .023645-49 .................... '" .,. .0681 .0640 .0595 .0546 .0497 .0447 .0399 .0352 .0309 .0269 .0233 .0200 .017150-54 .............. '" . " ...... .0655 .0600 .0544 .0487 .0432 .0379 .0330 .0284 .0243 .0207 .0174 .0146 .012255-59 .......................... .0612 .0547 .0484 .0423 .0365 .0313 .0265 .0223 .0186 .0154 .0127 .0104 .008460-64 ..... '" .... , ............. .0546 .0476 .0410 .0350 .0295 .0246 .0204 .0167 .0136 .0110 .0088 .0070 .005665-69 .......................... .0453 .0385 .0324 .0269 .0221 .0180 .0145 .0116 .0092 .0073 .0057 .0044 .003470-74 .......................... .0338 .0280 .0230 .0186 .0150 .0119 .0093 .0073 .0056 .0043 .0033 .0025 .001975-79 .............. " .......... .0213 .0173 .0138 .0109 .0085 .0066 .0051 .0039 .0029 .0022 .0016 .0012 .0009

\0 80+ .......................... .0131 .0103 .0080 .0061 .0046 .0035 .0026 .0019 .0014 .0010 .0007 .0005 .0004\0

Age Proportion under given age

1 ............................ .0158 .0188 .0221 .0257 .0295 .0336 .0379 .0424 .0471 .0518 .0567 .0617 .06675 ............................ .0715 .0842 .0979 .1127 .1284 .1484 .1617 .1791 .1968 .2147 .2327 .2506 .2685

10 ............................ .1400 .1627 .1871 .2127 .2394 .2668 .2947 .3227 .3507 .3783 .4055 .4321 .457915 ............................ .2097 .2408 .2735 .3073 .3419 .3767 .4114 .4456 .4791 .5115 .5427 .5725 .601020 ............................ .2809 .3185 .3573 .3968 .4363 .4755 .5137 .5507 .5862 .6198 .6516 .6813 .709025 ............................ .3527 .3949 .4378 .4806 .5227 .5635 .6027 .6399 .6747 .7072 .7372 .7647 .789830 ............................ .4247 .4697 .5145 .5585 .6009 .6414 .6794 .7148 .7474 .7771 .8040 .8282 .849835 ............................ .4963 .5423 .5872 .6305 .6715 .7098 .7452 .7775 .8067 .8328 .8559 .8762 .894040 ............................ .5673 .6124 .6556 .6966 .7346 .7696 .8012 .8296 .8547 .8767 .8958 .9123 .926545 ............................ .6370 .6796 .7197 .7568 .7908 .8214 .8487 .8726 .8933 .9112 .9264 .9393 .950150 ............................ .7052 .7436 .7791 .8115 .8405 .8662 .8885 .9078 .9243 .9381 .9497 .9593 .967255 ............................ .7707 .8036 .8335 .8602 .8837 .9041 .9215 .9363 .9486 .9588 .9671 .9739 .979460 ............................ .8319 .8583 .8819 .9025 .9202 .9354 .9481 .9586 .9672 .9742 .9798 .9843 .987865 ............................ .8865 .9059 .9229 .9374 .9497 .9600 .9684 .9753 .9808 .9852 .9886 .9913 .9934

Parameter ofstable populations

Birth rate .. '" ................... .0175 .0212 .0250 .0291 .0336 .0383 .0433 .0486 .0540 .0597 .0654 .0713 .0773Death rate ....................... .0278 .0262 .0250 .0241 .0236 .0233 .0233 .0236 .0240 .0247 .0254 .0263 .0273ORR (27) ....................... 1.24 1.42 1.63 1.86 2.12 2.41 2.75 3.12 3.55 4.02 4.56 5.17 5.85ORR (29) ....................... 1.25 1.44 1.66 1.91 2.20 2.53 2.91 3.34 3.83 4.38 5.01 5.73 6.54ORR,(3l) ....................... 1.25 1.46 1.70 1.97 2.30 2.67 3.09 3.58 4.15 4.79 5.54 6.39 7.36ORR (33) ....................... 1.25 1.47 1.73 2.04 2.40 2.81 3.30 3.86 4.52 5.28 6.17 7.20 8.39Average age .................... " 36.2 33.9 31.6 29.5 27.4 25.5 23.7 22.0 20.5 19.1 17.8 16.7 15.6Births/population 15-44 ...... ~ ..... .042 .048 .056 .065 .075 .086 .099 .114 .130 .149 .170 .194 .221

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LBVEL OF MORTALITY (continued)

LEVEL 11Females (Oeo = 45.00 years)

AIUUIOl rate of increase

-.010 -.005 .000 .005 .010 .0/5 .020 .025 .030 .035 .(}4() .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0142 .0170 .0201 .0235 .0272 .oJ 11 .0352 .0396 .0440 .0487 .0534 .0582 .06311-4 .......................... .0516 .0610 .0712 .0822 .0939 .1061 .1187 .1316 .1447 .1579 .1711 .1842 .19715-9 .......................... .0643 .0743 .0848 .0957 .1069 .1181 .1292 .1401 .1506 .1606 .1702 .1792 .1875

10-14 .......................... .0660 .0743 .0828 .0911 .0992 .1069 .1141 .1206 .1265 .1316 .1360 .1396 .142515-19 .......................... .0676 .0743 .0807 .0867 .0920 .0967 .1006 .1038 .1061 .1077 .1086 .1087 .108220-24 .......................... .0688 .0738 .0781 .0818 .0847 .0868 .0881 .0887 .0884 .0875 .0860 .0840 .081625-29 .......................... .0696 .0727 .0751 .0767 .0175 .0175 .0767 .0753 .0732 .0707 .0677 .0645 .061130-34 .......................... .0700 .0714 .0719 .0716 .0706 .0688 .0664 .0635 .0603 .0568 .0531 .0493 .045535-39 .. , ....................... .0700 .0696 .0684 .0665 .0639 .0607 .0572 .0534 .0494 .0453 .0413 .0375 .033740-44 .......................... .0697 .0676 .0648 .0614 .0575 .0533 .0490 .0446 .0402 .0360 .0320 .0283 .024945-49 .......................... .0689 .0651 .0609 .0563 .0514 .0465 .0417 .0370 .0326 .0284 .0247 .0213 .018250-54 .......................... .0671 .0619 .0564 .0508 .0453 .0400 .0349 .0302 .0260 .0221 .0187 .0157 .013155-59 .......................... .0637 .0574 .0510 .0448 .0390 .0335 .0286 .0241 .0202 .0168 .0138 .0114 .009360-64 .......................... .0581 .0510 .0442 .0379 .0321 .0270 .0224 .0185 .0151 .0122 .0098 .0079 .006265-69 .......................... .0496 .0425 .0359 .0300 .0248 .0203 .0165 .0132 .0105 .0083 .0065 .0051 .004070-74 .......................... .0384 .0321 .0265 .0216 .0174 .0139 .0110 .0086 .0067 .0052 .0039 .0030 .002375-79 .......................... .0255 .0207 .0167 .0133 .0104 .0081 .0063 .0048 .0036 .0027 .0020 .0015 .0011

..- 80+ .......................... .0170 .0134 .0105 .0081 .0062 .0047 .0035 .0026 .0019 .0014 .0010 .0007 .000500

Age Proportion under given age

1 ............................ .0142 .0170 .0201 .0235 .0272 .0311 .0352 .0396 .0440 .0487 .0534 .0582 .06315 ............................ .0658 .0780 .0913 .1057 .1210 .1371 .1539 .1711 .1887 .2065 .2245 .2424 .2602

10 ............................ .1301 .1523 .1761 .2014 .2279 .2552 .2831 .3112 .3393 .3672 .3947 .4215 .447715 ............................ .1961 .2266 .2589 .2926 .3271 .3621 .3971 .4318 .4658 .4988 .5306 .5611 .590220 ............................ .2637 .3009 .3396 .3792 .4191 .4588 .4978 .5356 .5719 .6065 .6392 .6698 .698425 ............................ .3325 .3747 .4177 .4610 .5038 .5457 .5859 .6242 .6604 .6940 .7252 .7539 .780030 ............................ .4021 .4474 .4929 .5377 .5814 .6231 .6626 .6995 .7336 .7647 .7930 .8184 .841135 ............................ .4721 .5188 .5648 .6094 .6519 .6919 .7290 .7630 .7938 .8215 .8460 .8677 .886740 ............................ .5421 .5884 .6332 .6758 .7158 .7527 .7862 .8164 .8432 .8668 .8874 .9052 .920445 ............................ .6117 .6559 .6979 .7372 .7733 .8060 .8352 .8610 .8835 .9028 .9194 .9335 .945350 ............................ .6806 .7211 .7588 .7934 .8247 .8525 .8769 .8980 .9160 .9313 .9441 .9547 .963555 ............................ .7476 .7829 .8152 .8443 .8700 .8925 .9118 .9282 .9420 .9534 .9628 .9705 .976660 ............................ .8114 .8403 .8662 .8891 .9090 .9260 .9404 .9523 .9622 .9702 .9766 .9818 .985965 ............................ .8695 .8913 .9104 .9270 .9411 .9530 .9628 .9708 .9772 .9824 .9865 .9897 .9921

Parameter of stable populations

Birth rate ........................ .0156 .0188 .0222 .0260 .0302 .0346 .0393 .0443 .0494 .0547 .0602 .0658 .0715Death rate ....................... .0256 .0238 .0222 .0210 .0202 .0196 .0193 .0193 .0194 .0197 .0202 .0208 .0215ORR (27) ....................... 1.13 1.29 1.48 1.69 1.92 2.19 2.50 2.84 3.23 3.66 4.16 4.71 5.33ORR (29) ....................... 1.13 1.30 1.50 1.73 2.00 2.30 2.64 3.03 3.47 3.98 4.55 5.20 5.94ORR (31) ....................... 1.12 1.31 1.53 1.78 2.07 2.41 2.79 3.24 3.75 4.34 5.01 5.78 6.68ORR (33) ....................... 1.12 1.32 1.56 1.83 2.15 2.53 2.97 3.47 4.07 4.76 5.56 6.48 7.56Average age ...................... 37.6 35.2 32.9 30.6 28.5 26.4 24.6 22.8 21.2 19.7 18.4 17.2 16.1Births/population 15-44 ............ .038 .044 .051 .059 .068 .078 .090 .103 .118 .135 .155 .177 .201

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 13Females (Oeo = 50.00 years)

A1UlIUII rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .()4() .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0129 .0155 .0185 .0217 .0252 .0290 .0330 .0372 .0415 .0460 .0506 .0553 .06011-4 .......................... .0481 .0572 .0672 .0780 .0895 .1016 .1l42 .1271 .1402 .1534 .1667 .1799 .19305-9 .......................... .0607 .0705 .0810 .0919 .1031 .1144 .1257 .1368 .1476 .1579 .1678 .1770 .1857

10-14 .......................... .0625 .0709 .0794 .0879 .0962 .1041 .11l6 .1184 .1246 .1301 .1348 .1387 .141915-19 .......................... .0645 .0713 .0779 .0841 .0897 .0947 .0990 .1025 .1052 .1070 .1082 .1086 .108320-24 .......................... .0660 .0712 .0759 .0799 .0832 .0856 .0873 .0881 .0882 .0875 .0863 .0845 .082225-29 .......................... .0673 .0708 .0735 .0755 .0767 .0770 .0765 .0754 .0736 .0712 .0685 .0654 .062030-34 ......... '" .............. .0683 .ovoo .0710 .0711 .0704 .0689 .0668 .0642 .061l .0577 .0541 .0504 .046635-39 .................... , ..... .0690 .0690 .0682 .0666 .0643 .0615 .0581 .0544 .0505 .0466 .0426 .0387 .034940-44 .......................... .0693 .0676 .0652 .0621 .0585 .0545 .0503 .0459 .0416 .0374 .0333 .0295 .026045-49 .......................... .0692 .0659 .0619 .0576 .0529 .0480 .0432 .0385 .0340 .0298 .0259 .0224 .019250-54 .......................... .0682 .0633 .0581 .0526 .0471 .0418 .0367 .0319 .0274 .0234 .0199 .0168 .014055-59 ...... , ... " .... , ......... .0658 .0595 .0533 .0471 .041l .0355 .0304 .0258 .0217 .0180 .0149 .0123 .010060-64 .......................... .061l .0539 .0470 .0406 .0346 .0291 .0243 .0201 .0165 .0134 .0108 .0087 .006965-69 .......................... .0535 .0461 .0392 .0330 .0274 .0225 .0183 .0148 .01l8 .0094 .0074 .0058 .004570-74 . .. . .. . . . . . . . .... . . . . . .. . .0428 .0359 .0298 .0244 .0198 .0159 .0126 .0099 .0077 .0060 .0046 .0035 .002675-79 .......................... .0295 .0242 .0196 .0156 .0124 .0097 .0075 .0057 .0044 .0033 .0025 .0018 .0014- 80+ .......................... .0214 .0170 .0133 .0103 .0079 .0060 .0045 .0034 .0025 .0018 .0013 .0009 .0007

0- Age Proportion under given age

1 ............................ .0129 .0155 .0185 .0217 .0252 .0290 .0330 .0372 .0415 .0460 .0506 .0553 .06015 ............................ .0610 .0727 .0857 .0997 .1l47 .1306 .1471 .1642 .1817 .1994 .2173 .2352 .2531

10 ............................ .1217 .1432 .1666 .1916 .2178 .2450 .2728 .3010 .3293 .3574 .3851 .4122 .438715 ............................ .1842 .2142 .2461 .2795 .3140 .3491 .3844 .4195 .4539 .4874 .5198 .5509 .580620 ............................ .2487 .2855 .3240 .3636 .4037 .4439 .4834 .5219 .5591 .5945 .6280 .6595 .688925 ............................ .3147 .3567 .3998 .4435 .4869 .5295 .5707 .6101· .6472 .6820 .7143 .7440 .771130 ............................ .3820 .4275 .4734 .5190 .5636 .6065 .6472 .6854 .7208 .7533 .7827 .8093 .833135 ............................ .4503 .4975 .5444 .5901 .6340 .6754 .7141 .7496 .7819 .81l0 .8369 .8597 .879840 ............................ .5192 .5665 .6126 .6567 .6983 .7369 .7722 .8041 .8325 .8575 .8794 .8984 .914745 ............................ .5885 .6341 .6778 .7188 .7568 .7914 .8225 .8500 .8741 .8949 .9127 .9279 .940750 ............................ .6578 .7000 .7397 ..7764 .8097 .8394 .8657 .8885 .9081 .9247 .9386 .9503 .959955 ............................ .7260 .7634 .7978 .8290 .8568 .8812 .9023 .9203 .9355 .9481 .9585 .9670 .973960 ............................ .7917 .8229 .8510 .8761 .8979 .9168 .9327 .9461 .9572 .9662 .9735 .9793 .983965 ............................ .8528 .8768 .8981 .9166 .9325 .9459 .9571 .9662 .9736 .9796 .9843 .9880 .9908

Parameter of stable populations

Birth rate ........................ .0139 .0168 .0200 .0236 .0275 .0316 .0361 .0408 .0457 .0507 .0559 .0613 .0667Death rate ....................... .0239 .0218 .0200 .0186 .0175 .0166 .0161 .0158 .0157 .0157 .0159 .0163 .0167GRR (27) ....................... 1.04 1.19 1.36 1.55 1.77 2.02 2.30 2.62 2.98 3.38 3.83 4.35 4.92GRR (29) ....................... 1.03 1.19 1.38 1.59 1.83 2.1l 2.42 2.78 3.19 3.66 4.19 4.79 5.47GRR (31) ....................... 1.03 1.20 1.40 1.63 1.89 2.20 2.56 2.96 3.43 3.97 4.59 5.30 6.12GRR (33) ....................... 1.02 1.20 1.42 1.67 1.96 2.30 2.70 3.17 3.71 4.34 5.07 5.92 6.91Average age ...................... 38.9 36.4 34.0 31.6 29.4 27.3 25.3 23.5 21.8 20.3 18.9 17.6 16.5Births/population 15-44 ............ .034 .040 .046 .054 .062 .072 .082 .095 .109 .124 .142 .163 .185

,/

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 15Females (Oeo = 55.00 years)

Annual rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age intervalProportion in age interval

Under 1 ....................... .0118 .0143 .0170 .0201 .0235 .0271 .0310 .0350 .0392 .0436 .0481 .0527 .0573

1-4 .......................... .0452 .0540 .0638 .0744 .0857 .0977 .1102 .1230 .1362 .1494 .1627 .1760 .18925-9 .......................... .0576 .0673 .0777 .0886 .0998 .1112 .1227 .1339 .1449 .1555 .1656 .1751 .1840

10-14 .......................... .0597 .0680 .0766 .0852 .0936 .1018 .1094 .1165 .1230 .1287 .1337 .1379 .1413

15-19 .......................... .0618 .0687 .0755 .0819 .0877 .0930 .0976 .1013 .1043 .1065 .1078 .1085 .1084

20-24 .......................... .0637 .0691 .0740 .0782 .0818 .0846 .0865 .0876 .0880 .0876 .0865 .0849 .0828

25-29 .......................... .0653 .0691 .0722 .0744 .0759 .0765 .0764 .0755 .0739 .0717 .0691 .0661 .062930-34 .......................... .0668 .0689 .0702 .0706 .0702 .0691 .0672 .0648 .0618 .0586 .0550 .0514 .0476

35-39 .......................... .0680 .0684 .0680 .0667 .0647 .0621 .0589 .0554 .0516 .0476 .0436 .0397 .035940-44 .......................... .0689 .0676 .0655 .0627 .0593 .0555 .0514 .0471 .0428 .0385 .0345 .0306 .0270

45-49 .......................... .0694 .0664 .0628 .0586 .0541 .0493 .0445 .0398 .0353 .0310 .0270 .0234 .020150-54 .......................... .0691 .0645 .0594 .0541 .0487 .0433 .0381 .0333 .0287 .0246 .0209 .0177 .014855-59 .......................... .0674 .0613 .0551 .0489 .0430 .0373 .0320 .0272 .0229 .0192 .0159 .0131 .010760-64 .......................... .0635 .0564 .0494 .0428 .0366 .0310 .0260 .0215 .0177 .0144 .0117 .0094 .0075

65-69 .............. , ........... .0567 .0491 .0420 .0354 .0296 .0244 .0200 .0161 .0129 .0103 .0081 .0064 .004970-74 .......................... .0464 .0392 .0327 .0269 .0219 .0176 .0141 .0111 .0087 .0067 .0052 .0040 .003075-79 ...... , ................... .0330 .0272 .0221 .0178 .0141 .0111 .0086 .0066 .0050 .0038 .0029 .0021 .0016

.- 80+ .......................... .0258 .0206 .0162 .0126 .0097 .0074 .0056 .0041 .0031 .0022 .0016 .0012 .00080N

AgeProportion under given age

I ............................ .0118 .0143 .0170 .0201 .0235 .0271 .0310 .0350 .0392 .0436 .0481 .0527 .0573

5 ............................ .0570 .0683 .0808 .0945 .1092 .1248 .1411 .1580 .1754 .1930 .2108 .2287 .246510 ............................ .1146 .1356 .1585 .1831 .2090 .2360 .2638 .2920 .3203 .3485 .3764 .4038 .430515 ............................ .1742 .2036 .2351 .2683 .3026 .3378 .3732 .4085 .4433 .4772 .5101 .5417 .5718

20 ............................ .2361 .2724 .3106 .3501 .3904 .4308 .4708 .5099 .5476 .5837 .6179 .6502 .680325 ............................ .2998 .3414 .3845 .4283 .4722 .5154 .5573 .5975 .6356 .6713 .7045 .7350 .7630

30 ............................ .3651 .4105 .4567 .5028 .5481 .5919 .6337 .6729 .7095 .7430 .7736 .8012 .825935 ............................ .4319 .4794 .5268 .5734 .6183 .6610 .7009 .7377 .7713 .8016 .8286 .8525 .873640 ............................ .4999 .5478 .5948 .6401 .6830 .7230 .7598 .7931 .8228 .8492 .8722 .8922 .9095

45 ............................ .5688 .6154 .6603 .7028 .7423 .7785 .8111 .8402 .8656 .8877 .9067 .9228 .936550 ............................ .6383 .6819 .7231 .7614 .7964 .8279 .8557 .8800 .9009 .9187 .9337 .9462 .956655 ............................ .7073 .7464 .7825 .8155 .8451 .8712 .8938 .9132 .9296 .9433 .9546 .9639 .9714

60 ............................ .7747 .8077 .8377 .8645 .8881 .9085 .9259 .9405 .9526 .9625 .9705 .9770 .982165 ............................ .8382 .8640 .8871 .9073 .9247 .9395 .9518 .9620 .9703 .9769 .9822 .9864 .9896

Parameter ofstable populations

Birth rate ........................ .0125 .0152 .0182 .0215 .0252 .0291 .0334 .0378 .0425 .0473 .0523 .0574 .0627

Death rate ................ ·· ..... .0225 .0202 .0182 .0165 .0152 .0141 .0134 .0128 .0125 .0123 .0123 .0124 .0127GRR (27) ....................... 0.96 1.10 1.26 1.44 1.64 1.87 2.14 2.43 2.76 3.14 3.56 4.04 4.57GRR (29) ....................... 0.95 1.10 1.27 1.47 1.69 1.95 2.24 2.58 2.96 3.39 3.88 4.44 5.07GRR (31) ....................... 0.95 1.10 1.29 1.50 1.75 2.05 2.36 2.74 3.17 3.67 4.24 4.90 5.66GRR (33) ....................... 0.94 1.11 1.30 1.53 1.80 2.12 2.49 2.92 3.42 4.00 4.67 5.46 6.37Average age ...................... 40.0 37.4 35.0 32.6 30.5 28.1 26.1 24.2 22.4 20.8 19.4 18.0 16.8Births/population 15-44 ............ .032 .037 .043 .050 .057 .066 .076 .088 .101 .115 .132 .151 .172

"TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 17Females (Oeo = 60.00 years)

Annual rate ofincrease

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0108 .0131 .0158 .0187 .0220 .0255 .0292 .0331 .0372 .0415 .0459 .0503 .05491-4 .......................... .0425 .0511 .0606 .0710 .0823 .0941 .1066 .1194 .1325 .1459 .1592 .1726 .18595-9 .......................... .0546 .0642 .0745 .0853 .0966 .1081 .1196 .1311 .1422 .1530 .1634 .1731 .1823

10-14 ...................... " .. .0569 .0652 .0737 .0824 .0910 .0993 .1072 .1145 .1212 .1272 .1325 .1369 .140615-19 .......................... .0591 .0661 .0730 .0795 .0856 .0911 .0960 .1000 .1033 .1057 .1073 .1082 .108320-24 .......................... .0613 .0668 .0719 .0764 .0803 .0833 .0856 .0870 .0876 .0874 .0866 .0851 .083225-29 .......................... .0633 .0673 .0706 .0732 .0750 .0759 .0761 .0754 .0740 .0721 .0696 .0668 .063630-34 .......................... .0652 .0676 .0692 .0700 .0699 .0690 .0674 .0652 .0624 .0593 .0558 .0522 .048535-39 .......................... .0669 .0676 .0675 .0666 .0649 .0625 .0596 .0562 .0525 .0486 .0446 .0407 .036940-44 .......................... .0683 .0674 .0657 .0632 .0600 .0564 .0524 .0482 .0439 .0396 .0355 .0316 .027945-49 .......................... .0694 .0668 .0634 .0595 .0552 .0505 .0458 .0411 .0365 .0322 .0281 .0244 .021050-54 .......................... .0697 .0654 .0606 .0554 .0501 .0448 .0396 .0346 .0300 .0258 .0220 .0186 .015655-59 .............. " .......... .0687 .0629 .0568 .0507 .0447 .0389 .0336 .0286 .0242 .0203 .0169 .0139 .011460-64 .......................... .0657 .0586 .0517 .0450 .0387 .0329 .0276 .0230 .0190 .0155 .0126 .0101 .008165-69 .......................... .0598 .0520 .0447 .0380 .0318 .0264 .0216 .0176 .0141 .0113 .0089 .0070 .005570-74 .............. " .......... .0501 .0425 .0356 .0295 .0241 .0195 .0156 .0123 .0097 .0075 .0058 .0044 .003475-79 .......................... .0367 .0304 .0248 .0200 .0160 .0126 .0098 .0076 .0058 .0044 .0033 .0025 .0018.... 80+ .......................... .0311 .0250 .0198 .0154 .0119 .0091 .0069 .0051 .0038 .0028 .0020 .0015 .0011

0w

Age Proportion under given age

1 ............................ .0108 .0131 .0158 .0187 .0220 .0255 .0292 .0331 .0372 .0415 .0459 .0503 .05495 ............................ .0533 .0642 .0764 .0898 .1042 .1196 .1357 .1525 .1698 .1873 .2051 .2229 .2407

10 ............................ .1079 .1284 .1509 .1751 .2008 .2277 .2554 .2836 .3120 .3404 .3685 .3961 .423115 ............................ .1648 .1936 .2246 .2575 .2918 .3270 .3626 .3981 .4333 .4676 .5009 .5330 .563620 ............................ .2239 .2597 .2976 .3370 .3774 .4181 .4585 .4981 .5365 .5733 .6082 .6412 .672025 ............................ .2852 .3265 .3695 .4135 .4577 .5014 .5441 .5851 .6241 .6607 .6948 .7263 .755130 ............................ .3485 .3938 .4401 .4867 .5327 .5774 .6201 .6605 .6981 .7328 .7644 .7931 .818835 ............................ .4136 .4614 .5093 .5567 .6026 .6464 .6876 .7257 .7606 .7921 .8203 .8453 .867340 ............................ .4805 .5290 .5769 .6233 .6675 .7089 .7471 .7818 .8130 .8407 .8649 .8860 .904245 ............................ .5489 .5964 .6425 .6864 .7275 .7653 .7995 .8300 .8569 .8803 .9004 .9176 .932150 ............................ .6183 .6632 .7060 .7460 .7827 .8158 .8453 .8711 .8934 .9125 .9285 .9420 .953155 ............................ .6880 .7286 .7666 .8014 .8238 .8606 .8849 .9057 .9234 .9382 .9505 .9606 .968860 ............................ .7567 .7915 .8234 .8521 .8775 .8995 .9184 .9344 .9476 .9585 .9674 .9745 .980265 ............................ .8224 .8501 .8751 .8971 .9161 .9324 .9461 .9574 .9666 .9740 .9800 .9846 .9883

Parameter of stable populations

Birth rate ...................... ,. .0113 .0138 .0167 .0198 .0233 .0271 .0311 .0354 .0399 .0445 .0494 .0543 .0594Death rate ....................... .0213 .0188 .0167 .0148 .0133 .0121 .0111 .0104 .0099 .0095 .0094 .0093 .0094ORR (27) ....................... 0.90 1.03 1.18 1.35 1.54 1.76 2.00 2.28 2.59 2.95 3.34 3.79 4.30ORR (29) ....................... 0.89 1.03 1.19 1.37 1.58 1.82 2.10 2.41 2.77 3.17 3.63 4.16 4.75ORR (31) ....................... 0.88 1.03 1.20 1.40 1.63 1.89 2.20 2.55 2.96 3.43 3.96 4.58 5.29ORR (33) ....................... 0.87 1.03 1.21 1.43 1.68 1.97 2.31 2.71 3.18 3.72 4.35 5.08 5.93Average age . . . . . . . . . . . . . . . . . . . . . . 41.1 38.5 36.0 33.5 31.1 28.9 26.8 24.8 23.0 21.3 19.8 18.4 17.2Births/population 15-44 ...... '" ... .030 .034 .040 .046 .053 .062 .071 .082 .094 .108 .124 .141 .151

TABLE I!. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 19Females (Oeo = 65.00 years)

Annual rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age intervalProportion in age interval

Under 1 ....................... .0100 .0122 .0148 .0176 .0207 .0241 .0278 .0316 .0356 .0398 .0441 .0485 .0530

1-4 .......................... .0401 .0484 .0578 .0680 .0791 .0909 .1033 .1161 .1293 .1426 .1561 .1695 .1829

5-9 .......................... .0518 .0613 .0714 .0823 .0935 .1051 .1167 .1283 .1397 .1507 .1612 .1712 .180610-14 .......................... .0542 .0624 .0710 .0797 .0884 .0969 .1050 .1126 .1195 .1257 .1312 .1359 .1398

15-19 ............... , .......... .0566 .0636 .0705 .0772 .0835 .0893 .0944 .0987 .1022 .1048 .1067 .1078 .1081

20-24 .......................... .0589 .0646 .0698 .0746 .0787 .0820 .0846 .0862 .0871 .0872 .0865 .0852 .083425-29 .......................... .0612 .0654 .0690 .0719 .0740 .0752 .0756 .0752 .0741 .0723 .0700 .0672 .0642

30-34 .......................... .0634 .0661 .0680 .0691 .0694 .0688 .0674 .0654 .0628 .0598 .0565 .0529 .049335-39 .......................... .0655 .0666 .0669 .0663 .0649 .0627 .0600 .0568 .0532 .0494 .0455 .0416 .0377

40-44 .......................... .0675 .0669 .0655 .0633 .0604 .0570 .0532 .0491 .0448 .0406 .0365 .0325 .0288

45-49 .......................... .0691 .0668 .0638 .0601 .0560 .0515 .0468 .0422 .0376 .0332 .0291 .0253 .021850-54 .......................... .0699 .0660 .0615 .0565 .0513 .0460 .0408 .0358 .0312 .0268 .0229 .0194 .0164

55-59 .......................... .0697 .0641 .0583 .0522 .0463 .0405 .0350 .0300 .0254 .0214 .0178 .0147 .012160-64 .......................... .0676 .0607 .0538 .0470 .0406 .0327 .0292 .0244 .0202 .0165 .0134 .0108 .0087

65-69 ........................ " .0627 .0549 .0474 .0404 .0340 .0283 .0233 .0190 .0153 .0122 .0097 .0076 .006070-74 .......................... .0537 .0459 .0387 .0322 .0264 .0215 .0172 .0137 .0108 .0084 .0065 .0050 .0038

75-79 .......................... .0406 .0338 .0278 .0225 .0181 .0143 .0112 .0087 .0066 .0051 .0038 .0029 .0021.....0376 .0303 .0241 .0189 .0146 .0112 .0085 .0064 .0047 .0035 .0025 .0018

~80+ .......................... .0013

AgeProportion under given age

1 ............................ .0100 .0122 .0148 .0176 .0207 .0241 .0278 .0316 .0356 .0398 .0441 .0485 .0530

5 ............................ .0501 .0606 .0725 .0856 .0998 .1150 .1310 .1477 .1649 .1824 .2002 .2180 .235810 ............................ .1019 .1219 .1440 .1679 .1934 .2201 .2478 .2760 .3046 .3331 .3614 .3892 .4165

15 ............................ .1561 .1843 .2150 .2476 .2818 .3170 .3528 .3886 .4241 .4589 .4926 .5251 .5563

20 ............................ .2126 .2479 .2855 .3249 .3653 .4063 .4471 .4873 .5262 .5637 .5993 .6329 .664425 ............................ .2715 .3125 .3554 .3995 .4440 .4883 .5317 .5735 .6133 .6509 .6858 .7182 .7478

30 ............................ .3327 .3779 .4244 .4714 .5180 .5635 .6073 .6487 .6874 .7231 .7558 .7854 .812035 ............................ .3961 .4440 .4924 .5405 .5874 .6323 .6747 .7141 .7502 .7830 .8123 .8384 .861340 ............................ .4616 .5106 .5593 .6068 .6522 .6950 .7347 .7709 .8034 .8323 .8578 .8799 .899045 ............................ .5291 .5775 .6248 .6701 .7127 .7521 .7879 .8199 .8482 .8729 .8942 .9124 .927850 ............................ .5981 .6443 .6886 .7302 .7687 .8035 .8347 .8621 .8858 .9061 .9233 .9377 .9497

55 ............................ .6681 .7103 .7501 .7867 .8200 .8496 .8755 .8979 .9170 .9330 .9462 .9572 .966060 ............................ .7378 .7745 .8083 .8390 .8662 .8901 .9105 .9279 .9424 .9543 .9640 .9719 .978165 ............................ .8054 .8352 .8621 .8860 .9068 .9247 .9398 .9523 .9626 .9709 .9775 .9827 .9868

Parameter of stable populations

Birth rate ............... ···· ..... .0104 .0127 .0154 .0184 .0217 .0253 .0292 .0333 .0377 .0422 .0469 .0517 .0566Death rate ....................... .0204 .0177 .0154 .0134 .0117 .0103 .0092 .0083 .0077 .0072 .0069 .0067 .0066ORR (27) ....................... 0.85 0.97 1.12 1.28 1.46 1.66 1.89 2.16 2.45 2.79 3.17 3.59 4.07ORR (29) ....................... 0.84 0.97 1.12 1.30 1.49 1.72 1.98 2.28 2.61 3.00 3.43 3.93 4.49ORR (31) ....................... 0.83 0.97 1.13 1.32 1.53 1.78 2.07 2.41 2.79 3.23 3.74 4.32 4.99ORR (33) ....................... 0.82 0.96 1.14 1.34 1.57 1.85 2.17 2.55 2.99 3.50 4.09 4.78 5.58Average age ...................... 42.2 39.6 38.0 34.4 32.0 29.7 27.5 25.4 23.5 21.8 20.3 18.8 17.5Births/population 15-44 ............ .028 .032 .038 .044 .050 .058 .067 .077 .089 .102 .117 .113 .152

"

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTAUTY (continued)

LEVEL 21Females (Oeo = 70.00 years)

Annual rate of lncrease

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

---Age interval Proportion in age interval

..under 1 ....................... .0093 .0115 .0139 .0167 .0197 .0230 .0266 .0304 .0343 .0385 .0427 .0471 .05151-4 ... ~~ ..................... .0378 .0460 .0551 .0652 .0762 .0879 .1002 .1131 .1262 .1397 .1532 .1667 .18015-9 .......................... .0492 .0585 .0686 .0793 .0906 .1022 .1140 .1257 .1373 .1485 .1592 .1694 .1790

10-14 ............. , ... " ....... .0516 .0598 .0684 .0771 .0859 .0946 .1028 .1106 .1178 .1242 .1299 .1349 .139015-19 ............... , .......... .0540 .0611 .0681 .0750 .0815 .0874 .0927 .0973 .1010 .1039 .1060 .1073 .107820-24 .......................... .0565 .0623 .0678 .0727 .0771 .0807 .0834 .0854 .0865 .0868 .0863 .0852 .083525-29 .......................... .0590 .0634 .0673 .0704 .0728 .0743 .0750 .0748 .0739 .0723 .0702 .0676 .064630-34 .......................... .0615 .0645 .0667 .0681 .0687 .0684 .0673 .0655 .0631 .0602 .0570 .0535 .049935-39 .......................... .0640 .0654 .0660 .0658 .0647 .0628 .0603 .0572 .0537 .0500 .0462 .0423 .038540-44 .......................... .0663 .0662 .0651 .0633 .0607 .0574 .0538 .0498 .0456 .0414 .0373 .0333 .029545-49 .......................... .0684 .0665 .0639 .0605 .0566 .0523 .0477 .0431 .0385 .0341 .0299 .0261 .022650-54 .......................... .0699 .0663 .0621 .0573 .0523 .0471 .0420 .0369 .0322 .0278 .0238 .0202 .017155-59 .......... , ............... .0704 .0651 .0595 .0536 .0477 .0419 .0364 .0312 .0266 .0224 .0187 .0155 .012760-64 .......................... .0692 .0625 .0556 .0489 .0424 .0364 .0308 .0258 .0214 .0176 .0143 .0116 .009365-69 .......................... .0653 .0575 .0500 .0428 .0362 .0303 .0250 .0204 .0165 .0132 .0105 .0083 .006570-74 .......................... .0574 .0493 .0417 .0349 .0288 .0235 .0189 .0151 .0119 .0093 .0072 .0055 .004275-79 ...... , ........ , ..... " ... .0446 .0374 .0309 .0252 .0203 .0161 .0127 .0098 .0076 .0058 .0044 .0033 .0024-0 80+ .......................... .0455 .0368 .0294 .0231 .0180 .0138 .0105 .0079 .0058 .0043 .0031 .0023 .0016

VI

Age Proportion under given age

1 ............................ .0093 .0115 .0139 .0167 .0197 .0230 .0266 .0304 .0343 .0385 .0427 .0471 .05155 ............................ .0471 .0574 .0690 .0819 .0959 .1109 .1268 .1434 .1606 .1781 .1959 .2138 .2317

10 ............................ .0963 .1159 .1376 .1612 .1865 .2132 .2408 .2692 .2978 .3266 .3551 .3832 .410615 ............................ .1479 .1757 .2059 .2383 .2724 .3077 .3437 .3798 .4156 .4508 .4850 .5180 .549620 ............................ .2020 .2368 .2741 .3133 .3539 .3951 .4364 .4770 .5166 .5547 .5910 .6253 .657525 ............................ .2585 .2990 .3418 .3860 .4310 .4758 .5198 .5624 .6031 .6415 .6174 .7105 .741030 ............................ .3175 .3625 .4091 .4565 .5038 .5501 .5948 .6373 .6770 .7139 .7476 .7781 .805635 ............................ .3790 .4270 .4758 .5246 .5724 .6185 .6621 .7028 .7401 .1741 .8046 .8317 .855540 ............................ .4429 .4924 .5419 .5904 .6371 .6813 .7223 .7599 .7939 .8241 .8507 .8739 .894045 ............................ .5093 .5586 .6070 .6537 .6978 .7378 .7161 .8091 .8395 .8655 .8880 .9013 .923550 ............................ .5777 .6251 .6709 .7142 .7544 .7910 .8238 .8528 .8780 .8996 .9180 .9333 .946155 ............................ .6416 .6914 .1330 .1715 .8061 .8381 .8658 .8898 .9102 .9275 .9418 .9536 .963260 ............................ .1179 .1566 .7924 .8251 .8543 .8800 .9022 .9210 .9368 .9498 .9605 .9691 .915965 ............................ .7811 .8190 .8481 .8740 .8967 .9163 .9329 .9468 .9582 .9674 .9148 .9806 .9852

Parameter of stable populations

Birth rate ........................ .0095 .0117 .0143 .0172 .0204 .0238 .0276 .0316 .0358 .0402 .0448 .0495 .0543Death rate ....................... .0195 .0167 .0143 .0122 .0104 .0088 .0076 .0066 .0058 .0052 .0048 .0045 .0043GRR (27) ....................... 0.81 0.93 1.06 1.22 1.39 1.58 1.81 2.06 2.34 2.66 3.02 3.43 3.88GRR (29) ....................... 0.80 0.92 1.01 1.23 1.42 1.64 1.88 2.17 2.49 2.85 3.27 3.14 4.28GRR (31) ....................... 0.78 0.92 1.07 1.25 1.45 1.69 1.97 2.28 2.65 3.01 3.55 4.10 4.74GRR (33) ....................... 0.77 0.91 1.07 1.21 1.49 1.75 2.06 2.41 2.83 3.31 3.88 4.53 5.29Average age ...................... 43.4 40.7 38.0 35.4 32.8 30.4 28.2 26.0 24.1 22.3 20.7 19.2 11.9Births/population 15-44 ............ .026 .031 .036 .041 .048 .055 .064 .073 .084 .097 .111 .127 .145

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 23Females (Oeo = 75.00 years)

Annual rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0087 .0108 .0132 .0159 .0189 .0221 .0257 .0295 .0334 .0375 .0418 .0461 .05061-4 .......................... .0355 .0434 .0524 .0624 .0734 .0851 .0975 .1103 .1236 .1371 .1508 .1644 .17805-9 .......................... .0463 .0555 .0655 .0762 .0875 .0993 .1112 .1231 .1348 .1462 .1572 .1676 .1774

10-14 ................ , ......... .0486 .0568 .0654 .0742 .0832 .0920 .1005 .1085 .1159 .1226 .1285 .1337 .138015-19 .......................... .0511 .0581 .0653 .0723 .0790 .0852 .0908 .0956 .0996 .1028 .1051 .1066 .107320-24 .......................... .0536 .0595 .0651 .0704 .0750 .0789 .0820 .0842 .0855 .0861 .0859 .0849 .083425-29 .......................... .0561 .0608 .0650 .0684 .0711 .0730 .0739 .0741 .0734 .0721 .0701 .0676 .064830-34 .......................... .0588 .0621 .0647 .0665 .0674 .0674 .0667 .0651 .0630 .0603 .0572 .0538 .050335-39 ............. " ........... .0615 .0634 .0644 .0645 .0638 .0623 .0600 .0572 .0539 .0503 .0466 .0427 .038940-44 .......................... .0642 .0645 .0639 .0625 .0602 .0573 .0539 .0501 .0461 .0419 .0379 .0339 .030145-49 .......................... .0667 .0654 .0632 .0602 .0566 .0526 .0482 .0437 .0392 .0348 .0306 .0267 .023250-54 .......................... .0688 .0658 .0620 .0576 .0529 .0479 .0428 .0378 .0331 .0287 .0246 .0210 .017755-59 .......................... .0702 .0655 .0602 .0546 .0488 .0431 .0376 .0324 .0277 .0234 .0196 .0162 .013460-64 .......................... .0704 .0640 .0574 .0507 .0443 .0381 .0324 .0273 .0227 .0187 .0153 .0124 .009965-69 .......................... .0681 .0604 .0528 .0456 .0388 .0326 .0270 .0222 .0180 .0145 .0115 .0091 .007170-74 ........................ , . .0620 .0536 .0457 .0385 .0319 .0261 .0212 .0169 .0134 .0105 .0082 .0063 .004875-79 .......................... .0505 .0426 .0355 .0291 .0236 .0188 .0149 .0116 .0089 .0068 .0052 .0039 .0029.... 80+ .......................... .0588 .0479 .0384 .0304 .0237 .0182 .0139 .0104 .0078 .0057 .0042 .0030 .00220

0\

Age Proportion under given age

1 ............................ .0087 .0108 .0132 .0159 .0189 .0221 .0257 .0295 .0334 .0375 .0418 .0461 .05065 ............................ .0442 .0542 .0656 .0783 .0922 .1072 .1231 .1398 .1570 .1747 .1926 .2106 .2285

10 ............................ .0905 .1096 .1311 .1545 .1798 .2065 .2343 .2629 .2918 .3209 .3497 .3782 .406015 ............................ .1391 .1664 .1964 .2288 .2629 .2984 .3348 .3714 .4077 .4435 .4783 .5118 .544020 ............................ .1902 .2246 .2617 .3011 .3419 .3837 .4255 .4670 .5073 .5463 .5834 .6184 .651325 ............................ .2437 .2841 .3269 .3714 .4169 .4625 .5075 .5511 .5929 .6323 .6692 .7034 .734730 ............................ .2999 .3449 .3918 .4399 .4880 .5355 .5814 .6252 .6663 .7044 .7393 .7710 .799535 ............................ .3587 .4070 .4566 .5064 .5554 .6030 .6481 .6903 .7293 .7647 .7965 .8248 .849740 ............................ .4202 .4704 .5210 .5709 .6192 .6652 .7081 .7475 .7832 .8150 .8431 .8675 .888745 ............................ .4844 .5349 .5849 .6334 .6795 .722b .7620 .7976 .8293 .8570 .8809 .9014 .918850 ............................ .5511 .6003 .6481 .6936 .7361 .7751 .8103 .8413 .8685 .8918 .9115 .9282 .941955 ............................ .6199 .6661 .7101 .7512 .7890 .8230 .8531 .8792 .9016 .9204 .9362 .9491 .959760 ............................ .6902 .7315 .7702 .8058 .8378 .8661 .8907 .9116 .9292 .9438 .9557 .9653 .973065 ............................ .7605 .7955 .8276 .8565 .8821 .9042 .9231 .9389 .9519 .9625 .9710 .9777 .9830

Parameter of stable populations

Birth rate ........................ .0086 .0109 .0133 .0161 .0192 .0226 .0263 .0302 .0344 .0387 .0432 .0478 .0526Death rate ....................... .0186 .0159 .0133 .0111 .0092 .0076 .0063 .0052 .0044 .0037 .0032 .0028 .0026ORR (27) ....................... 0.78 0.90 1.03 1.17 1.34 1.53 1.75 1.99 2.26 2.57 2.92 3.32 3.76ORR (29) ....................... 0.71 0.89 1.03 1.19 1.37 1.58 1.82 2.09 2.40 2.76 3.16 3.62 4.14ORR (31) ....................... 0.75 0.88 1.03 1.20 1.40 1.63 1.90 2.20 2.55 2.96 3.43 3.96 4.57ORR (33) ....................... 0.74 0.88 1.03 1.22 1.43 1.68 1.98 2.32 2.72 3.19 3.73 4.36 5.10Average age ..................... 44.9 42.11 39.3 36.46 33.9 31.4 29.0 26.8 24.7 22.8 21.1 19.6 18.2Births/population 15-44 ........... .025 .030 .034 .040 .046 .053 .062 .071 .082 .094 .107 .123 .140

or

TABl.E II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 1Males (oeo = 18.03 years)

AII1IIUl1 rate 01inuease

-.010 -.005 ooo .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under I ....................... .0309 .0352 .0399 .0448 .0499 .0553 .0608 .0665 .0722 .0781 .0841 .0901 .09621-4 .......................... .0847 .0954 .1067 .1I83 .1303 .1424 .1547 .1670 .1793 .1915 .2035 .2154 .22715-9 .......................... .0956 .1054 .1I52 .1249 .1345 .1438 .1527 .1611 .1692 .1766 .1836 .1900 .1958

10-14 .......................... .0947 .1018 .1085 .1I48 .1205 .1256 .1301 .1339 .1371 .1396 .1415 .1428 .143615-19 .......................... .0939 .0984 .1024 .1056 .1081 .IC99 .1110 .1I15 .1113 .II06 .1093 .1076 .105520-24 .......................... .0910 .0930 .0944 .0949 .0948 .0940 .0926 .0907 .0883 .0856 .0825 .0792 .075725-29 ............. " ........... .0863 .0861 .0851 .0835 .0814 .0787 .0756 .0722 .0686 .0648 .0610 .0571 .053230-34 .......................... .0806 .0785 .0757 .0724 .0688 .0649 .0608 .0567 .0525 .0484 .0444 .0405 .036835-39 .......................... .0739 .0701 .0659 .0615 .0570 .0525 .0479 .0436 .0394 .0354 .0316 .0282 .025040-44 .......................... .0659 .0610 .0559 .0509 .0460 .0413 .0368 .0326 .0287 .0252 .0220 .0191 .016545-49 .......................... .0571 .0515 .0461 .0409 .0361 .0316 .0275 .0237 .0204 .0174 .0148 .0126 .010650-54 ...... " .................. .0476 .0419 .0366 .0317 .0272 .0232 .0197 .0166 .0139 .0116 .0096 .0080 .006555-59 ................ , ......... .0376 .0323 .0275 .0232 .0195 .0162 .0134 .0110 .0090 .0073 .0059 .0048 .003860-64 .......................... .0275 .0230 .0191 .0157 .0129 .0104 .0084 .0068 .0054 .0043 .0034 .0027 .002165-69 ...... " .................. .0177 .0145 .0117 .0094 .0075 .0059 .0047 .0037 .0028 .0022 .0017 .0013 .001070-74 ........ , ...... , .......... .0097 .0077 .0061 .0048 .0037 .0029 .0022 .0017 .0013 .0010 .0007 .0005 .000475-79 ........... , ............ , . .0040 .0031 .0024 .0018 .0014 .0011 .0008 .0006 .0004 .0003 .0002 .0002 .0001....80+ ....... , .................. .0014 ooסס. oo0סס. .0011 .0008 .0006 .0004 .0003 .0002 .0002 .0001 .0001 .0001

-l

Age Proportion under given age

I ............................ .0309 .0352 .0399 .0448 .0499 .0553 .0608 .0665 .0722 .0781 .0841 .0901 .09625 ............................ .1155 .1307 .1466 .1631 .1802 .1977 .2155 .2335 .2515 .2696 .2876 .3055 .3232

10 ............................ .2111 .2361 .2618 .2881 .3147 .3415 .3682 .3946 .4207 .4462 .4712 .4955 .519015 ............................ .3058 .3378 .3703 .4028 .4352 .4671 .4982 .5285 .5578 .5859 .6127 .6383 .662620 ............................ .3997 .4363 .4727 .5084 .5433 .5770 .6093 .6400 .6691 .6965 .7221 .7459 .768125 ............................ .4907 .5293 .5670 .6034 .6381 .6710 .7019 .7307 .7574 .7820 .8046 .8251 .843830 ............................ .5770 .6154 .6522 .6869 .7195 .7497 .7775 .8030 .8260 .8469 .8655 .8822 .897035 ....... ' .................... .6576 .6939 .7278 .7593 .7883 .8146 .8384 .8596 .8785 .8952 .9099 .9227 .933940 ............................ .7315 .7639 .7938 .8209 .8453 .8671 .8863 .9032 .9179 .9306 .9415 .9509 .958945 ................. , .......... .7974 .8249 .8497 .8718 .8913 .9084 .9231 .9358 .9466 .9558 .9635 .9700 .975450 ............................ .8545 .8764 .8958 .9128 .9274 .9399 .9506 .9595 .9670 .9732 .9784 .9826 .986055 .......... , ................. .9021 .9183 .9324 .9444 .9546 .9632 .9703 .9761 .9809 .9848 .9880 .9905 .992560 ............................ .9397 .9506 .9599 .9676 .9741 .9794 .9837 .9872 .9899 .9922 .9939 .9953 .996465 ............................ .9672 .9736 .9790 .9834 .9869 .9898 .9921 .9939 .9953 .9964 .9973 .9979 .9985

Parameter of stable populations

Birth rate .. '" ................... .0427 .0489 .0555 .0624 .0698 .0774 .0854 .0936 .1020 . II05 .1I93 .1281 .1371Death rate ....... , ............... .0527 .0539 .0555 .0514 .0598 .0624 .0654 .0686 .0720 .0755 .0793 .0831 .0871ORR (27) ............ , .......... 2.49 2.84 3.23 3.68 4.19 4.76 5.41 6.14 6.96 7.88 8.92 10.08 11.39ORR (29) ....................... 2.55 2.94 3.38 3.89 4.46 5.12 5.86 6.71 7.67 8.76 9.99 11.39 12.97ORR (31) ....................... 2.62 3.05 3.54 4.11 4.77 5.52 6.38 7.37 8.51 9.81 11.30 13.00 14.95ORR (33) ........ , .............. 2.70 3.17 3.72 4.36 5.11 5.98 6.99 8.17 9.53 1I.11 12.94 15.06 17.51Average age ...................... 27.8 26.1 24.5 23.0 21.5 20.2 18.9 17.7 16.7 15.7 14.8 13.9 13.2Births/population 15-44 ............ .087 .100 .116 .133 .153 .175 .201 .230 .262 .299 .340 .386 .438

~

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 3Males (oeo = 22.85 years)

A1UIUal rate of Increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age intervalProportion in age interval

Under 1 ....................... .0254 .0293 .0335 .0379 .0426 .0474 .0525 .0577 .0631 .0685 .0741 .0797 .08541-4 .......................... .0759 .0864 .0974 .1090 .1209 .1330 .1454 .1579 .1704 .1828 .1952 .2073 .21935-9 .......................... .0883 .0983 .1084 .1185 .1285 .1383 .1478 .1569 .1656 .1737 .1813 .1883 .1948

10-14 .......................... .0883 .0959 .1031 .1100 .1163 .1221 .1272 .1317 .1356 .1387 .1412 .1431 .144415-19 .......................... .0884 .0936 .0982 .1022 .1054 .1079 .1097 .1108 .1112 .1110 .1102 .1089 .107120-24 .......................... .0869 .0897 .0918 .0931 .0937 .0935 .0927 .0913 .0894 .0870 .0843 .0812 .077925-29 .......................... .0838 .0844 .0843 .0834 .0818 .0797 .0770 .0740 .0706 .0671 .0633 .0595 .055730-34 .............. '" ......... .0799 .0785 .0764 .0737 .0706 .0670 .0632 .0592 .0551 .0511 .0470 .0431 .039435-39 .......................... .0750 .0718 .0682 .0642 .0599 .0555 .0510 .0466 .0423 .0382 .0344 .0307 .027340-44 ............... · ... ······ . .0688 .0643 .0595 .0546 .0497 .0449 .0403 .0359 .0318 .0280 .0245 .0214 .018645-49 .......................... .0615 .0561 .0506 .0453 .0403 .0355 .0310 .0270 .0233 .0200 .0171 .0145 .012350-54 .......................... .0532 .0473 .0417 .0364 .0315 .0271 .0231 .0196 .0165 .0138 .0115 .0096 .007955-59 .......................... .0440 .0381 .0327 .0279 .0235 .0197 .0164 .0136 .0112 .0091 .0074 .0060 .004860-64 .................. · .. · .. · . .0339 .0286 .0240 .0199 .0164 .0134 .0109 .0088 .0070 .0056 .0044 .0035 .002865-69 .......................... .0234 .0193 .0158 .0128 .0103 .0082 .0065 .0051 .0040 .0031 .0024 .0018 .001470-74 .......................... .0139 .0112 .0089 .0071 .0055 .0043 .0033 .0025 .0019 .0015 .0011 .0008 .000675-79 .......................... .0066 .0051 .0040 .0031 .0024 .0018 .0013 .0010 .0007 .0006 .0004 .0003 .0002.... 80+ .......................... .0027 .0021 .0016 .0012 .0009 .0006 .0005 .0003 .0002 .0002 .0001 .0001 .0001

000

AgeProportion under given age

1 ............................ .0254 .0293 .0335 .0379 .0426 .0474 .0525 .0577 .0631 .0685 .0741 .0797 .08545 ............................ .1013 .1157 .1309 .1468 .1634 .1805 .1979 .2156 .2334 .2513 .2692 .2870 .3047

10 ............................ .1896 .2139 .2393 .2653 .2919 .3188 .3457 .3725 .3990 .4250 .4505 .4754 .499515 ............................ .2779 .3098 .3424 .3753 .4082 .4409 .4729 .5042 .5346 .5638 .5918 .6185 .643820 ............................ .3063 .4034 .4406 .4774 .5136 .5488 .5826 .6150 .6457 .6747 .7019 .7273 .750925 ............................ .4532 .4931 .5324 .5705 .6073 .6423 .6754 .7063 .7351 .7617 .7862 .8085 .828930 ............................ .5370 .5775 .6166 .6539 .6891 .7220 .7524 .7803 .8058 .8288 .8495 .8681 .884635 ............................ .6170 .6561 .6930 .7276 .7597 .7890 .8156 .8396 .8609 .8799 .8966 .9112 .924040 ............................ .6920 .7279 .7612 .7918 .8196 .8445 .8666 .8862 .9033 .9181 .9309 .9419 .951345 ............................ .7607 .7921 .8207 .8464 .8693 .8894 .9069 .9221 .9351 .9461 .9555 .9633 .969950 ............................ .8232 .8482 .8713 .8917 .9095 .9249 .9380 .9491 .9584 .9661 .9726 .9779 .982255 ............................ .8755 .8955 .9130 .9281 .9410 .9519 .9611 .9686 .9749 .9800 .9841 .9874 .990160 ............................ .9195 .9336 .9457 .9560 .9646 .9717 .9775 .9822 .9860 .9891 .9915 .9934 .994965 ............................ .9534 .9623 .9697 .9759 .9810 .9851 .9884 .9910 .9931 .9947 .9960 .9969 .9977

Parameter ofstable populations

Birth rate ........................ .0331 .0382 .0438 .0497 .0559 .0625 .0694 .0764 .0837 .0912 .0988 .1066 .1145Death rate .......... ··.· .. ··.···· .0431 .0432 .0438 .0447 .0459 .0475 .0494 .0514 .0537 .0562 .0588 .0616 .0645ORR (27) ....................... 1.99 2.27 2.59 2.95 3.36 3.82 4.34 4.92 5.58 6.33 7.17 8.11 9.16ORR (29) ....................... 2.02 2.33 2.68 3.09 3.55 4.07 4.69 5.35 6.12 6.99 7.98 9.11 10.38ORR (31) ....................... 2.06 2.40 2.79 3.24 3.76 4.36 5.04 5.83 6.74 7.77 8.96 10.32 11.87ORR (33) ....................... 2.10 2.47 2.91 3.41 4.00 4.68 5.48 6.41 7.48 8.73 10.18 11.85 13.79Average age ...................... 29.8 28.0 26.2 24.5 23.0 21.5 20.1 18.8 17.7 16.6 15.6 14.7 13.8Births/population 15-44 ............ .069 .079 .091 .105 .121 .139 .160 .183 .209 .239 .272 .309 .351

·"

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 5Males (OeD = 27.67 years)

Annual rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0217 .0252 .0290 .0331 .0374 .0419 .0467 .0516 .0566 .0618 .0670 .0723 .07771-4 .......................... .0689 .0791 .0899 .1013 .1131 .1253 .1377 .1503 .1629 .1755 .1881 .2005 .21285-9 .......................... .0820 .0920 .1023 .1127 .1231 .1333 .1432 .1528 .1620 .1706 .1788 .1863 .1933

10-14 ., ........................ .0827 .0905 .0982 .1055 .1123 .1186 .1243 .1294 .1338 .1374 .1404 .1428 .144515-19 .......................... .0835 .0891 .0943 .0988 .1026 .1057 .1080 .1096 .1106 .1108 .1104 .1095 .108020-24 .......................... .0829 .0863 .0891 .0910 .0922 .0926 .0923 .0914 .0899 .0879 .0854 .0826 .079525-29 . '" ...................... .0812 .0824 .0829 .0826 .0817 .0800 .0778 .0751 .0720 .0687 .0651 .0614 .057630-34 .......................... .0787 .0779 .0764 .0743 .0716 .0684 .0649 .0611 .0572 .0531 .0491 .0452 .041435-39 . " ......... '" ........... .0752 .0726 .0695 .0659 .0619 .0577 .0534 .0490 .0447 .0406 .0366 .0328 .029340-44 .......................... .0706 .0665 .0620 .0574 .0526 .0478 .0431 .0386 .0344 .0304 .0267 .0234 .020445-49 .......................... .0648 .0595 .0542 .0489 .0437 .0387 .0341 .0298 .0258 .0223 .0191 .0163 .013950-54 ............. " ........... .0577 .0518 .0460 .0404 .0352 .0305 .0261 .0223 .0189 .0159 .0133 .0110 .009255-59 ............ '" ........... .0494 .0432 .0374 .0321 .0273 .0230 .0192 .0160 .0132 .0108 .0088 .0072 .005860-64 .......................... .0397 .0338 .0286 .0239 .0198 .0163 .0133 .0108 .0087 .0069 .0055 .0044 .003465-69 ....... " ................. .0289 .0240 .0198 .0162 .0131 .0105 .0083 .0066 .0052 .0040 .0031 .0024 .001970-74 ...... , ...... , ............ .0184 .0149 .0120 .0095 .0075 .0059 .0046 .0035 .0027 .0021 .0016 .0012 .000975-79 .......................... .0095 .0075 .0059 .0046 .0035 .0027 .0020 .0015 .0011 .0008 .0006 .0005 .0003..... 80+ .......................... .0044 .0034 .0026 .0020 .0015 .0011 .0008 .0006 .0004 .0003 .0002 .0002 .00010

\0

Age Proportion under given age

1 ............................ .0217 .0252 .0290 .0331 .0374 .0419 .0467 .0516 .0566 .0618 .0670 .0723 .07775 ............................ .0905 .1043 .1189 .1344 .1505 .1672 .1844 .2018 .2195 .2373 .2551 .2729 .2905

10 ............................ .1725 .1963 .2212 .2471 .2736 .3005 .3276 .3546 .3815 .4079 .4339 .4592 .483815 ............................ .2552 .2868 .3194 .3525 .3859 .4191 .4519 .4840 .5152 .5454 .5743 .6020 .628320 ............................ .3387 .3760 .4137 .4513 .4885 .5248 .5599 .5937 .6258 .6562 .6847 .7115 .736425 ............................ .4216 .4623 .5027 .5423 .5806 .6174 .6523 .6851 .7157 .7440 .7701 .7941 .815830 ............................ .5028 .5448 .5856 .6249 .6623 .6974 .7301 .7602 .7877 .8127 .8352 .8555 .873535 ............................ .5814 .6227 .6621 .6993 .7339 .7658 .7949 .8213 .8449 .8658 .8844 .9007 .914940 ............................ .6566 .6953 .7316 .7652 .7958 .8235 .8483 .8703 .8896 .9064 .9210 .9335 .944245 ............................ .7272 .7618 .7936 .8225 .8484 .8714 .8915 .9089 .9240 .9368 .9477 .9569 .964550 ............................ .7920 .8214 .8478 .8714 .8921 .9101 .9256 .9387 .9498 .9591 .9668 .9732 .978455 ............................ .8497 .9731 .8938 .9118 .9273 .9406 .9517 .9610 .9687 .9750 .9801 .9842 .987660 ............................ .8991 .9163 .9312 .9439 .9546 .9636 .9709 .9770 .9819 .9858 .9889 .9914 .993465 ............................ .9387 .9501 .9597 .9678 .9744 .9799 .9842 .9878 .9906 .9927 .9945 .9958 .9968

Parameter of stable populations

Birth rate ........................ .0269 .0313 .0361 .0413 .0469 .0527 .0588 .0651 .0717 .0784 .0853 .0923 .0994Death rate ....................... .0369 .0363 .0361 .0363 .0369 .0377 .0388 .0401 .0417 .0434 .0453 .0413 .0494GRR (27) ....................... 1.67 1.90 2.17 2.48 2.82 3.21 3.65 4.15 4.71 5.34 6.05 6.85 7.75GRR (29) ....................... 1.69 1.95 2.24 2.58 2.97 3.41 3.91 4.48 5.13 5.87 6.71 7.65 8.73GRR(3l) ....................... 1.71 1.99 2.32 2.69 3.13 3.63 4.20 4.86 5.62 6.49 7.48 8.62 9.93GRR (33) ....................... 1.73 2.04 2.40 2.82 3.30 3.87 4.53 5.30 6.20 7.24 8.44 9.83 11.45Average age ...................... 31.6 29.6 27.7 25.9 24.2 22.7 21.2 19.8 18.5 17.4 16.3 15.3 14.4Births/population 15-44 ............ .057 .066 .076 .088 .101 .116 .134 .153 .175 .200 .228 .260 .296

J

TABLB II. "WEST" MODBL STABLE POPULATIONS ARRANGBD BY LEVEL OF MORTALITY (continued)

LEVEL 7Males (oeo = 32.48 years)

Annual raze of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0189 .0221 .0257 .0295 .0335 .0378 .0423 .0470 .0518 .0567 .0617 .0668 .07201-4 .......................... .0631 .0730 .0837 .0949 .1066 .1187 .1312 .1438 .1565 .1693 .1820 .1947 .20715-9 .......................... .0765 .0866 .0970 .1075 .1181 .1286 .1389 .1489 .1585 .1676 .1762 .1842 .1917

10-14 .......................... .0777 .0858 .0937 .1013 .1086 .1153 .1215 .1270 .1318 .1360 .1394 .1422 .144215-19 .......................... .0790 .0850 .0906 .0955 .0998 .1034 .1063 .1083 .1097 .1103 .1103 .1097 .108620-24 .......................... .0792 .0831 .0863 .0888 .0905 .0915 .0917 .0912 .0900 .0883 .0861 .0835 .080625-29 .......................... .0784 .0803 .0814 .0816 .0811 .0800 .0781 .0758 .0730 .0698 .0664 .0629 .059230-34 .......................... .0771 .0769 .0760 .0744 .0721 .0693 .0661 .0625 .0587 .0548 .0508 .0469 .043135-39 ............. , ............ .0748 .0729 .0702 .0670 .0634 .0594 .0552 .0510 .0467 .0425 .0385 .0346 .031040-44 .......................... .0716 .0680 .0639 .0595 .0548 .0501 .0455 .0409 .0365 .0324 .0286 .0251 .021945-49 .......................... .0671 .0622 .0570 .0518 .0466 .0415 .0367 .0322 .0281 .0243 .0209 .0179 .015350-54 .......................... .0613 .0554 .0495 .0439 .0385 .0335 .0289 .0247 .0210 .0177 .0149 .0124 .010355-59 ........ , ................. .0540 .0475 .0415 .0358 .0306 .0260 .0219 .0182 .0151 .0125 .0102 .0083 .006760-64 .......................... .0448 .0385 .0328 .0276 .0230 .0191 .0156 .0127 .0103 .0083 .0066 .0052 .004165-69 .... , ..................... .0341 .0286 .0237 .0195 .0159 .0128 .0102 .0081 .0064 .0050 .0039 .0030 .002370-74 .......................... .0230 .0188 .0152 .0122 .0097 .0076 .0059 .0046 .0035 .0027 .0021 .0015 .001275-79 ......... " .. , ............ .0127 .0102 .0080 .0063 .0049 .0037 .0028 .0021 .0016 .0012 .0009 .0007 .0005- 80+ .......................... .0065 .0050 .0039 .0029 .0022 .0016 .0012 .0009 .0006 .0005 .0003 .0002 .0002-0

Age Proportion under given age

1 ............................ .0189 .0221 .0257 .0295 .0335 .0378 .0423 .0470 .0518 .0567 .0617 .0668 .07205 ............................ .0820 .0952 .1093 .1244 .1401 .1566 .1735 .1908 .2083 .2260 .2438 .2615 .2791

10 ............................ .1585 .1818 .2063 .2319 .2582 .2852 .3124 .3397 .3668 .3936 .4200 .4457 .470815 ............................ .2363 .2675 .3000 .3332 .3668 .4005 .4339 .4667 .4986 .5296 .5594 .5879 .615020 ............................ .3153 .3526 .3905 .4287 .4666 .5039 .5401 .5750 .6083 .6399 .6697 .6976 .723625 ............................ .3945 .4357 .4769 .5175 .5572 .5954 .6318 .6662 .6984 .7283 .7558 .7812 .804230 ............................ .4729 .5160 .5582 .5992 .6383 .6753 .7099 .7420 .7714 .7981 .8223 .8440 .863435 ............................ .5500 .5929 .6343 .6736 .7105 .7447 .7760 .8045 .8301 .8529 .8731 .8909 .906540 ............................ .6248 .6658 .7045 .7406 .7739 .8041 .8313 .8554 .8768 .8954 .9116 .9255 .937545 ............................ .6964 .7338 .7684 .8001 .8287 .8542 .8767 .8963 .9133 .9278 .9402 .9506 .959450 ............................ .7635 .7959 .8254 .8519 .8753 .8957 .9134 .9286 .9414 .9522 .9611 .9686 .974755 ............................ .8249 .8513 .8750 .8957 .9137 .9292 .9423 .9533 .9624 .9699 .9760 .9810 .985060 ............................ .8788 .8989 .9164 .9315 .9444 .9552 .9641 .9715 .9775 .9823 .9862 .9893 .991765 ............................ .9236 .9374 .9492 .9591 .9674 .9742 .9798 .9842 .9878 .9906 .9928 .9945 .9958

Parameter ofstable populations

Birth rate ........................ .0225 .0265 .0308 .0354 .0404 .0457 .0513 .0570 .0630 .0692 .0755 .0820 .0885Death rate ....................... .0325 .0315 .0308 .0304 .0304 .0307 '.0313 .0320 .0330 .0342 .0355 .0370 .0385GRR (27) ....................... 1.45 1.65 1.89 2.16 2.46 2.80 3.18 3.62 4.11 4.66 5.28 5.98 6.76GRR (29) ....................... 1.46 1.68 1.94 2.23 2.57 2.95 3.39 3.89 4.45 5.10 5.83 6.65 7.59GRR (31) •...................... 1.47 1.71 1.99 2.32 2.69 3.12 3.62 4.19 4.85 5.61 6.47 7.46 8.59GRR (33) ....................... 1.48 1.74 2.05 2.41 2.83 3.32 3.89 4.55 5.32 6.21 7.25 8.45 9.85Average age ...................... 33.2 3I.I 29.1 27.2 25.4 23.7 22.1 20.7 19.3 18.1 16.9 15.9 14.9Births/population 15-44 ............ .049 .057 .066 .076 .088 .101 .116 .133 .152 .174 .198 .226 .257

.}'

TABLE II "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 9Males (Oeo= 37.30 years)

Annual rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0168 .0198 .0231 .0267 .0305 .0346 .0389 .0433 .0480 .0527 .0576 .0625 .06751-4 .......................... .0583 .0680 .0783 .0894 .1010 .1131 .1255 .1382 .1510 .1639 .1768 .1895 .20225-9 .......................... .0718 .0818 .0922 .1028 .1l36 .1244 .1350 .1453 .1552 .1647 .1737 .1821 .1899

10-14 .......................... .0733 .0815 .0896 .0975 .1051 .1122 .1l87 .1247 .1299 .1344 .1383 .1414 .143815-19 .......................... .0750 .0812 .0871 .0925 .0972 .1012 .1045 .1070 .1087 .1097 .1l01 .1098 .108920-24 .......................... .0757 .0801 .0837 .0867 .0888 .0902 .0908 .0907 .0899 .0885 .0866 .0842 .081525-29 .......................... .0758 .0781 .0797 .0804 .0804 .0797 .0782 .0762 .0737 .0707 .0675 .0640 .060430-34 ............ , ....... " .... .0753 .0757 .0753 .0742 .0723 .0699 .0669 .0636 .0599 .0561 .0522 .0483 .044535-39 .................... '" ... .0741 .0727 .0706 .0678 .0644 .0607 .0567 .0525 .0483 .0441 .0400 .0361 .032440-44 .......................... .0720 .0689 .0652 .0611 .0566 .0520 .0474 .0428 .0384 .0342 .0303 .0267 .023345-49 .......................... .0688 .0642 .0592 .0541 .0490 .0439 .0390 .0344 .0301 .0261 .0225 .0193 .116550-54 .......................... .0642 .0584 .0526 .0468 .0413 .0361 .0313 .0269 .0230 .0194 .0164 .0137 .011455-59 .................... " .... .0578 .0513 .0451 .0392 .0337 .0287 .0243 .0203 .0169 .0140 .01l5 .0094 .007660-64 .......................... .0495 .0428 .0367 .0311 .0261 .0217 .0179 .0146 .0119 .0096 .0077 .0061 .004865-69 .......................... .0390 .0330 .0275 .0228 .0186 .0151 .0121 .0097 .0077 .0060 .0047 .0037 .002870-74 .......................... .0275 .0226 .0184 .0149 .0119 .0094 .0074 .0057 .0044 .0034 .0026 .0020 .001575-79 .................... " .... .0162 .0130 .0103 .0081 .0063 .0049 .0037 .0028 .0021 .0016 .0012 .0009 .0006..- 80+ .......................... .0089 .0070 .0054 .0041 .0031 .0023 .0017 .0013 .0009 .0007 .0005 .0003 .0002..-..-

Age Proportion under given age

1 ............................ .0168 .0198 .0231 .0267 .0305 .0346 .0389 .0433 .0480 .0527 .0576 .0625 .06755 ............................ .0751 .0877 .1014 .1l61 .1315 .1477 .1644 .1815 .1990 .2166 .2343 .2520 .2597

10 ............................ .1468 .1695 .1936 .2189 .2452 .2721 .2994 .3268 .3542 .3813 .4080 .4341 .459615 ............................ .2202 .2510 .2832 .3164 .3502 .3843 .4181 .4515 .4841 .5158 .5463 .5755 .603420 ............................ .2951 .3322 .3703 .4089 .4474 .4855 .5226 .5585 .5928 .6255 .6563 .6853 .712325 ............................ .3709 .4123 .4541 .4956 .5363 .5767 .6134 .6492 .6827 .7140 .7429 .7695 .793830 ............................ .4466 .4904 .5338 .5760 .6167 .6553 .6916 .7253 .7564 .7847 .8104 .8335 .854235 ............................ .5219 .5661 .6091 .6502 .6890 .7252 .7585 .7889 .8163 .8409 .8626 .8818 .898640 ............................ .5961 .6389 .6796 .7179 .7534 .7859 .8152 .8414 .8646 .8850 .9027 .9180 .931145 ............................ .6681 .7078 .7448 .7790 .8101 .8379 .8626 .8843 .9031 .9192 .9330 .9446 .954450 ............................ .7369 .7719 .8041 .8331 .8590 .8818 .9016 .9186 .9331 .9453 .9555 .9640 .970955 ............................ .8011 .8303 .8566 .8800 .9003 .9179 .9329 .9455 .9561 .9648 .9719 .9777 .982460 ............................ .8589 .8817 .9017 .9191 .9340 .9466 .9572 .9659 .9730 .9788 .9834 .9871 .990065 ............................ .9084 .9245 .9384 .9502 .9601 .9683 .9751 .9805 .9849 .9883 .9910 .9932 .9948

Parameter of stable populations

Birth rate ........................ .0194 .0229 .0268 .0311 .0356 .0405 .0456 .0510 .0566 .0623 .0682 .0742 .0804Death rate ....................... .0294 .0279 .0268 .0261 .0256 .0255 .0256 .0260 .0266 .0273 .0282 .0292 .0304GRR (27) •...................... 1.29 1.47 1.68 1.92 2.19 2.49 2.84 3.22 3.66 4.16 4.71 5.34 6.04GRR (29) ....................... 1.29 1.49 1.72 1.98 2.28 2.62 3.01 3.45 3.96 4.53 5.18 5.92 6.76GRR (31) ....................... 1.29 1.51 1.76 2.05 2.38 2.76 3.20 3.71 4.29 4.96 5.73 6.61 7.62GRR (33) ....................... 1.30 1.53 1.80 2.12 2.49 2.92 3.42 4.00 4.68 5.47 6.39 7.45 8.69Average age . . • . . . . . . . . . . . . . . . . . .. 34.7 32.5 30.4 28.4 26.5 24.7 23.0 21.5 20.0 18.7 17.5 16.4 15.4Births/population 15-44 ..... '" .... .043 .050 .058 .061 .Oll .089 .103 .1l8 .135 .154 .176 .201 .229

..•_-----~.~~----------------------~.

~

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 11Males (Oeo = 42.12 years)

Annual rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0151 .0179 .0210 .0244 .0281 .0320 .0361 .0404 .0449 .0495 .0542 .0590 .06391-4 .......................... .0542 .0636 .0738 .0846 .0962 .1082 .1206 .1333 .1462 .1591 .1721 .1850 .19785-9 .......................... .0676 .0775 .0879 .0986 .1095 .1205 .1313 .1419 .1522 .1620 .1713 .1800 .1882

10-14 .......... , ......... " .... .0694 .0776 .0859 .0940 .1018 .1092 .1161 .1224 .1280 .1329 .1371 .1405 .143215-19 .......................... .0713 .0778 .0839 .0896 .0947 .0990 .1027 .1055 .1077 .1090 .1096 .1096 .109020-24 ......... '" ., ............ .0725 .0772 .0812 .0845 .0871 .0889 .0899 .0901 .0897 .0885 .0869 .0847 .082125-29 .... , ..................... .0732 .0760 .0780 .0792 .0796 .0792 .0781 .0764 .0741 .0714 .0683 .0649 .061430-34 .......................... .0735 .0744 .0745 .0737 .0723 .0702 .0675 .0644 .0609 .0572 .0534 .0495 .045735-39 .......................... .0732 .0723 .0706 .0681 .0651 .0617 .0579 .0538 .0497 .0455 .0414 .0375 .033740-44 .......................... .0721 .0694 .0661 .0623 .0581 .0536 .0490 .0445 .0400 .0358 .0318 .0280 .024645-49 .......................... .0700 .0657 .0610 .0561 .0510 .0459 .0410 .0362 .0318 .0277 .0240 .0207 .017750-54 .......................... .0664 .0608 .0551 .0494 .0438 .0385 .0335 .0289 .0247 .0210 .0178 .0149 .012455-59 " ........................ .0611 .0546 .0482 .0421 .0365 .0312 .0265 .0223 .0186 .0154 .0127 .0104 .008560-64 .......................... .0536 .0467 .0402 .0343 .0289 .0242 .0200 .0164 .0134 .0108 .0087 .0069 .005565-69 .............. , ..... '" ... .0436 .0370 .0311 .0259 .0213 .0174 .0140 .0112 .0089 .0070 .0055 .0043 .003370-74 .......................... .0319 .0264 .0216 .0175 .0141 .0112 .0088 .0069 .0053 .0041 .0031 .0024 .001875-79 " ........ " .. , ........... .0197 .0159 .0127 .0101 .0079 .0061 .0047 .0036 .0027 .0020 .0015 .0011 .0008

..... 80+ .......................... .0118 .0092 .0072 .0055 .0042 .0031 .0023 .0017 .0013 .0009 .0007 .0005 .0003.....N

Age Proportion under given age

1 ............................ .0151 .0179 .0210 .0244 .0281 .0320 .0361 .0404 .0449 .0495 .0542 .0590 .06395 ............................ .0693 .0815 .0948 .1091 .1242 .1402 .1567 .1737 .1910 .2086 .2263 .2440 .2616

10 ............................ .1368 .1590 .1827 .2077 .2338 .2607 .2880 .3156 .3432 .3706 .3976 .4240 .449815 ............................ .2062 .2366 .2686 .3017 .3356 .3699 .4042 .4380 .4712 .5035 .5346 .5645 .593120 ............................ .2775 .3144 .3525 .3913 .4303 .4689 .5068 .5436 .5789 .6125 .6443 .6742 .702125 ............................ .3500 .3916 .4337 .4758 .5174 .5578 .5967 .6337 .6685 .7010 .7312 .7589 .784230 ............................ .4232 .4675 .5117 .5550 .5970 .6370 .6748 .7101 .7426 .7724 .7994 .8238 .845635 ............................ .4967 .5419 .5861 .6288 .6692 .7072 .7423 .7744 .8035 .8296 .8528 .8733 .891340 ............................ .5699 .6142 .6567 .6969 .7344 .7689 .8002 .8282 .8532 .8751 .8943 .9109 .925045 ............................ .6420 .6836 .7228 .7592 .7924 .8225 .8492 .8727 .8932 .9109 .9260 .9388 .949650 ............................ .7120 .7493 .7838 .8152 .8434 .8684 .8902 .9090 .9251 .9386 .9500 .9595 .967355 ............................ .7784 .8102 .8389 .8646 .8872 .9068 .9236 .9379 .9498 .9597 .9678 .9744 .979760 ............................ .8395 .8648 .8872 .9068 .9237 .9381 .9502 .9602 .9684 .9751 .9805 .9848 .988265 ............................ .8931 .9114 .9274 .9410 .9526 .9622 .9701 .9766 .9818 .9859 .9892 .9917 .9937

Parameter of stable populations

Birth rate ........................ .0169 .0202 .0237 .0277 .0319 .0364 .0412 .0463 .0515 .0569 .0625 .0682 .0740Death rate ....................... .0269 .0252 .0237 .0227 .0219 .0214 .0212 .0213 .0215 .0219 .0225 .0232 .0240GRR (27) ....................... 1.16 1.33 1.52 1.74 1.98 2.26 2.57 2.93 3.32 3.77 4.23 4.85 5.49GRR (29) ....................... 1.16 1.34 1.55 1.79 2.06 2.37 2.72 3.12 3.58 4.10 4.69 5.36 6.12GRR (31) ....................... 1.16 1.36 1.58 1.84 2.14 2.48 2.88 3.34 3.87 4.47 5.17 5.96 6.87GRR (33) ....................... 1.16 1.37 1.61 1.89 2.23 2.61 3.06 3.59 4.20 4.91 5.74 6.70 7.81Average age ...................... 36.1 33.8 31.6 29.5 27.5 25.6 23.9 22.2 20.7 19.3 18.1 16.8 15.8Births/population 15-44 ............ .039 .045 .052 .060 .070 .080 .093 .106 .122 .140 .160 .182 .207

... "

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 13Males (Oeo = 47.11 years)

Annual rate 0/ increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age intervalProportion in age interval

Under 1 ....................... .0136 .0163 .0192 .0225 .0260 .0297 .0337 .0379 .0422 .0467 .0512 .0559 .0606

1-4 .......................... .0506 .0597 .0696 .0804 .0917 .1037 .1160 .1287 .1417 .1547 .1678 .1808 .1937

5-9 .......................... .0638 .0736 .0840 .0948 .1058 .1169 .1279 .1388 .1493 .1594 .1690 .1781 .1865

10-14 ., ........................ .0659 .0741 .0825 .0908 .0988 .1065 .1137 .1203 .1262 .1314 .1359 .1397 .1427

15-19 .......................... .0680 .0746 .0810 .0869 .0923 .0970 .1010 .1042 .1066 .1083 .1092 .1095 .1091

20-24 .......................... .0696 .0745 .0788 .0825 .0855 .0876 .0889 .0895 .0893 .0885 .0871 .0851 .0827

25-29 .......................... .0707 .0739 .0763 .0779 .0786 .0786 .0779 .0764 .0744 .0719 .0690 .0657 .0623

30-34 .......................... .0716 .0730 .0735 .0732 .0721 .0703 .0679 .0650 .0617 .0581 .0544 .0506 .0467

35-39 .......................... .0721 .0716 .0703 .0683 .0656 .0624 .0588 .0549 .0508 .0467 .0426 .0387 .0349

40-44 .......................... .0719 .0696 .0667 .0632 .0592 .0549 .0504 .0459 .0415 .0372 .0331 .0293 .0257

45-49 .......................... .0707 .0668 .0624 .0576 .0527 .0477 .0427 .0379 .0334 .0292 .0254 .0219 .0188

50-54 .......................... .0682 .0629 .0573 .0516 .0460 .0406 .0355 .0307 .0264 .0225 .0190 .0160 .0134

55-59 .......................... .0639 .0574 .0510 .0448 .0390 .0335 .0286 .0242 .0202 .0168 .0139 .0114 .0093

60-64 .......................... .0572 .0502 .0435 .0373 .0316 .0265 .0220 .0182 .0148 .0120 .0097 .0078 .0062

65-69 .......................... .0479 .0409 .0346 .0289 .0239 .0196 .0159 .0127 .0102 .0080 .0063 .0049 .0038

70-74 .......................... .0361 .0301 .0248 .0202 .0163 .0130 .0103 .0081 .0063 .0048 .0037 .0028 .0021

75-79 .... , ..................... .0233 .0189 .0152 .0121 .0095 .0074 .0057 .0044 .0033 .0025 .0019 .0014 .0010.... 80+ .......................... .0150 .0118 .0092 .0071 .0054 .0041 .0031 .0023 .0017 .0012 .0009 .0006 .0005....\,U

AgeProportion under given age

I ............................ .0136 .0163 .0192 .0225 .0260 .0297 .0337 .0379 .0422 .0467 .0512 .0559 .0606

5 ............................ .0642 .0760 .0889 .1028 .1177 .1334 .1498 .1666 .1839 .2014 .2190 .2367 .2543

10 ............................ .1280 .1496 .1729 .1976 .2235 .2503 .2777 .3054 .3331 .3607 .3880 .4147 .4408

15 ............................ .1938 .2237 .2554 .2884 .3223 .3568 .3914 .4256 .4593 .4922 .5239 .5544 .5835

20 ............................ .2618 .2983 .3363 .3753 .4146 .4538 .4923 .5298 .5660 .6004 .6331 .6638 .6926

25 ............................ .3313 .3728 .4152 .4578 .5000 .5414 .5813 .6193 .6553 .6889 .7202 .7489 .7753

30 ............................ .4021 .4467 .4914 .5356 .5787 .6200 .6591 .6958 .7297 .7608 .7891 .8147 .8376

35 ............................ .4737 .5196 .5649 .6088 .6507 .6903 .7270 .7607 .7914 .8189 .8435 .8652 .8843

40 ............................ .5458 .5913 .6352 .6771 .7164 .7527 .7858 .8156 .8422 .8656 .8861 .9039 .9192

45 ............................ .6177 .6609 .7019 .7403 .7756 .8076 .8362 .8616 .8837 .9028 .9192 .9332 .9449

50 ............................ .6884 .7277 .7643 .7979 .8282 .8552 .8789 .8995 .9171 .9320 .9446 .9550 .9637

55 ............................ .7566 .7906 .8216 .8495 .8742 .8958 .9144 .9302 .9435 .9545 .9636 .9711 .9771

60 ............................ .8205 .8480 .8726 .8943 .9132 .9294 .9430 .9544 .9637 .9714 .9775 .9825 .9864

65 ............................ .8777 .8982 .9161 .9316 .9448 .9559 .9650 .9725 .9786 .9834 .9872 .9902 .9926

Parameter of stable populations

Birth rate ........................ .0150 .0179 .0212 .0249 .0288 .0331 .0376 .0423 .0472 .0524 .0577 .0631 .0686

Death rate ....................... .0250 .0229 .0212 .0199 .0188 .0181 .0176 .0173 .0172 .0174 .0177 .0181 .0186

ORR (27) ....................... 1.06 1.22 1.39 1.59 1.81 2.07 2.35 2.68 3.04 3.46 3.92 4.45 5.03

ORR (29) ....................... 1.06 1.22 1.41 1.63 1.88 2.16 2.48 2.85 3.27 3.74 4.29 4.90 5.60

ORR (31) ....................... 1.05 1.23 1.43 1.67 1.94 2.26 2.62 3.04 3.52 4.07 4.71 5.43 6.27

ORR (33) ....................... 1.05 1.24 1.46 1.71 2.02 2.37 2.78 3.25 3.81 4.46 5.21 6.08 7.09

Average age ...................... 37.3 35.0 32.7 30.6 28.5 26.5 24.6 22.9 21.3 19.9 18.6 17.4 16.2

Births/population 15-44 ............ .035 .041 .048 .055 .064 .073 .084 .097 .111 .128 .146 .166 .190

,

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 15Males (Oeo = 51.83 years)

Annual rate 0/ increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0125 .0150 .0179 .0210 .0243 .0280 .0318 .0358 .0400 .0444 .0489 .0534 .05811-4 .......................... .0476 .0565 .0663 .0768 .0881 .0999 .1123 .1249 .1379 .1510 .1641 .1772 .19025-9 .......................... .0607 .0704 .0807 .0915 .1025 .1138 .1249 .1360 .1467 .1571 .1669 .1762 .1850

10-14 .......................... .0629 .0712 .0796 .0880 .0962 .1041 .1115 .1183 .1245 .1300 .1348 .1388 .142115-19 ., ........... " ........... .0652 .0719 .0784 .0846 .0902 .0951 .0994 .1029 .1056 .1076 .1087 .1092 .109020-24 .......................... .0670 .0722 .0768 .0807 .0839 .0864 .0880 .0889 .0890 .0884 .0871 .0853 .083125-29 .......................... .0686 .0721 .0748 .0767 .0778 .0781 .0776 .0764 .0746 .0722 .0695 .0664 .063030-34 .......................... .0701 .0717 .0726 .0726 .0718 .0703 .0682 .0655 .0623 .0589 .0552 .0515 .047735-39 .......................... .0711 .0710 .0701 .0684 .0660 .0630 .0596 .0558 .0518 .0477 .0437 .0397 .035940-44 .......................... .0716 .0698 .0672 .0639 .0601 .0560 .0516 .0472 .0427 .0384 .0343 .0304 .026745-49 ........... , .............. .0713 .0677 .0636 .0590 .0541 .0492 .0442 .0394 .0348 .0305 .0265 .0229 .019750-54 .......................... .0696 .0645 .0590 .0534 .0478 .0424 .0372 .0323 .0278 .0238 .0202 .0170 .1)14355-59 ........................ , . .0661 .0597 .0534 .0471 .0411 .0355 .0304 .0258 .0216 .0180 .0149 .0123 .010060-64 .......................... .0602 .0530 .0462 .0398 .0339 .0285 .0238 .0197 .0161 .0131 .0106 .0085 .006865-69 ............. " ..... '" ... .0513 .0441 .0375 .0315 .0261 .0215 .0175 .0141 .0113 .0089 .0070 .0055 .004370-74 ................... '" .... .0397 .0333 .0276 .0226 .0183 .0147 .0116 .0091 .0071 .0055 .0042 .0032 .0024

.... 75-79 ............. '" .......... .0263 .0215 .0174 .0139 .0110 .0086 .0066 .0051 .0039 .0029 .0022 .0016 .0012.... 80+ .......................... .0181 .0144 .0113 .0087 .0067 .0051 .0038 .0028 .0021 .0015 .0011 .0008 .0006.J>,

Age Proportion under given age

1 ............................ .0125 .0150 .0179 .0210 .0243 .0280 .0318 .0358 .0400 .0444 .0489 .0534 .05815 ............................ .0601 .0715 .0841 .0978 .1124 .1279 .1441 .1608 .1779 .1954 .2130 .2306 .2483

10 ............................ .1208 .1419 .1648 .1892 .2150 .2416 .2690 .2968 .3246 .3524 .3799 .4069 .433215 ............................ .1837 .2131 .2444 .2772 .3112 .3457 .3805 .4151 .4492 .4825 .5147 .5457 .575320 ............................ .2488 .2850 .3228 .3618 .4013 .4409 .4799 .5180 .5548 .5900 .6234 .6549 .684325 ............................ .3159 .3572 .3996 .4425 .4853 .5273 .5679 .6069 .6438 .6784 .7105 .7402 .767430 ............................ .3845 .4292 .4743 .5192 .5630 .6053 .6455 .6833 .7184 .7506 .7800 .8066 .830535 ............................ .4546 .5009 .5469 .5918 .6348 .6756 .7137 .7488 .7807 .8095 .8353 .8581 .878140 ............................ .5257 .5720 .6170 .6602 .7008 .7386 .7732 .8045 .8325 .8573 .8789 .8977 .914045 ............................ .5973 .6417 .6842 .7241 .7610 .7946 .8249 .8517 .8753 .8957 .9132 .9281 .940750 ............................ .6686 .7094 .7477 .7830 .8151 .8438 .8691 .8911 .9101 .9262 .9397 .9511 .960455 ............................ .7382 .7739 .8068 .8365 .8630 .8862 .9063 .9234 .9379 .9500 .9599 .9681 .974760 ............................ .8044 .8337 .8601 .8836 .9041 .9217 .9367 .9492 .9595 .9680 .9749 .9804 .984765 ............................ .8645 .8867 .9063 .9233 .9379 .9502 .9605 .9688 .9757 .9811 .9854 .9888 .9915

Parameter ofstable populations

Birth rate ....................... · .0135 .0162 .0193 .0227 .0264 .0304 .0347 .0392 .0439 .0488 .0539 .0591 .0643Death rate ....................... .0235 .0212 .0193 .0177 .0164 .0154 .0147 .0142 .0139 .0138 .0139 .0141 .0143ORR (27) ....................... 0.99 1.13 1.29 1.47 1.68 1.92 2.19 2.49 2.83 3.21 3.65 4.13 4.68ORR (29) ....................... 0.98 1.13 1.31 1.51 1.74 2.00 2.30 2.64 3.03 3.47 3.97 4.55 5.19ORR (31) ....................... 0.97 1.13 1.32 1.54 1.79 2.08 2.42 2.81 3.25 3.76 4.35 5.03 5.80ORR (33) ....................... 0.96 1.14 1.34 1.58 1.85 2.18 2.55 3.00 3.51 4.10 4.80 5.60 6.54Average age ...................... 38.4 36.0 33.7 31.4 29.3 27.2 25.3 23.5 21.9 20.4 19.0 17.7 16.6Births/population 15-44 ............ .033 .038 .044 .051 .059 .068 .078 .090 .103 .118 .135 .154 .176

.\,

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 17Males (Oeo = 56.47 years)

Annual rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age intervalProportion in age interval

Under 1 ....................... .0115 .0139 .0166 .0196 .0229 .0264 .0301 .0340 .0381 .0424 .0467 .0512 .05581-4 .......................... .0450 .0537 .0632 .0737 .0848 .0966 .1089 .1216 .1346 .1477 .1609 .1741 .1872

5-9 .......................... .0578 .0674 .0776 .0884 .0995 .1108 .1221 .1333 .1443 .1548 .1649 .1745 .1834

10-14 .......................... .0601 .0684 .0768 .0853 .0937 .1017 .1094 .1164 .1229 .1286 .1336 .1379 .1414

15-19 .......................... .0625 .0693 .0759 .0823 .0881 .0933 .0978 .1016 .1046 .1068 .1082 .1089 .1089

20-24 ...................... ··· . .0646 .0699 .0747 .0789 .0824 .0851 .0871 .0882 .0885 .0881 .0871 .0855 .0834

25-29 .......................... .0665 .0702 .0732 .0754 .0768 .0774 .0772 .0762 .0747 .0725 .0699 .0669 .0636

30-34 .......................... .0684 .0704 .0715 .0719 .0714 .0702 .0683 .0658 .0628 .0595 .0559 .0522 .0484

35-39 .......................... .0700 .0702 .0697 .0683 .0662 .0634 .0602 .0565 .0526 .0486 .0446 .0406 .0367

40-44 ................... · ...... .0711 .0696 .0674 .0644 .0608 .0569 .0526 .0482 .0438 .0395 .0353 .0313 .0277

45-49 ........................ · . .0715 .0683 .0644 .0601 .0554 .0505 .0455 .0407 .0361 .0317 .0276 .0239 .0206

50-54 .......................... .0707 .0658 .0606 .0551 .0495 .0440 .0387 .0338 .0292 .0250 .0213 .0180 .0151

55-59 .............. , ........... .0680 .0618 .0554 .0492 .0431 .0374 .0321 .0273 .0230 .0192 .0159 .0131 .0108

60-64 .......................... .0629 .0557 .0488 .0422 .0361 .0305 .0255 .0212 .0174 .0142 .0115 .0092 .0074

65-69 .......................... .0547 .0472 .0403 .0340 .0284 .0234 .0191 .0155 .0124 .0098 .0078 .0061 .0047

70-74 .......................... .0433 .0365 .0304 .0250 .0203 .0163 .0130 .0103 .0080 .0062 .0048 .0037 .0028

- 75-79 ............... , .......... .0296 .0243 .0198 .0158 .0126 .0099 .0077 .0059 .0045 .0034 .0026 .0019 .0014- 80+ .......................... .0219 .0174 .0137 .0107 .0082 .0062 .0047 .0035 .0026 .0019 .0014 .0010 .0007VI

AgeProportion under given age

1 ............................ .0115 .0139 .0166 .0196 .0229 .0264 .0301 .0340 .0381 .0424 .0467 .0512 .0558

5 ............................ .0565 .0676 .0799 .0933 .1077 .1229 .1390 .1556 .1727 .1901 .2076 .2253 .2430

10 ..............,.............. .1143 .1350 .1575 .1816 .2072 .2337 .2611 .2889 .3169 .3449 .3726 .3998 .4264

15 ............................ .1744 .2033 .2343 .2669 .3008 .3355 .3705 .4054 .4398 .4735 .5062 .5377 .5678

20 ............................ .2369 .2726 .3103 .3492 .3889 .4288 .4683 .5070 .5444 .5803 .6144 .6465 .6767

25 ............................ .3014 .3425 .3849 .4281 .4713 .5139 .5554 .5952 .6329 .6684 .7015 .7320 .7601

30 ............................ .3680 .4127 .4581 .5035 .5481 .5913 ..6325 .6714 .7076 .7409 .7714 .7989 .8237

35 ............................ .4363 .4830 .5297 .5754 .6195 .6615 .7008 .7372 .7704 .8004 .8273 .8511 .8721

40 ............................ .5063 .5533 .5993 .6437 .6857 .7249 .7610 .7937 .8230 .8491 .8719 .8917 .9089

45 ............................ .5774 .6229 .6667 .7081 .7465 .7818 .8136 .8419 .8669 .8885 .9072 .9231 .9365

50 ............................ .6490 .6912 .7311 .7681 .8019 .8323 .8592 .8827 .9029 .9202 .9348 .9470 .9571

55 ............................ .7196 .7570 .7917 .8232 .8514 .8763 .8979 .9164 .9321 .9452 .9561 .9650 .9722

60 ............................ .7877 .8188 .8471 .8723 .8945 .9136 .9300 .9437 .9551 .9644 .9720 .9781 .9830

65 ............................ .8506 .8745 .8958 .9145 .9305 .9441 .9555 .9649 .9725 .9786 .9835 .9873 .9903

Parameter ofstable populations

Birth rate ........................ .0122 .0148 .0177 .0209 .0245 .0283 .0324 .0367 .0412 .0459 .0508 .0558 .0609

Death rate ....................... .0222 .0198 .0177 .0159 .0145 .0133 .0124 .0117 .0112 .0109 .0108 .0108 .0109

ORR (27) ....................... 0.92 1.06 1.20 1.38 1.58 1.80 2.05 2.34 2.66 3.02 3.43 3.88 4.40

ORR (29) ....................... 0.91 1.06 1.22 1.41 1.62 1.87 2.15 2.47 2.84 3.25 3.73 4.26 4.87

ORR (31) ....................... 0.90 1.06 1.23 1.44 1.67 1.95 2.26 2.62 3.04 3.52 4.07 4.70 5.42

ORR (33) ....................... 0.90 .106 1.25 1.47 1.72 2.03 2.38 2.79 3.27 3.82 4.47 5.22 6.10

Average age ...................... 39.5 37.0 34.6 32.3 30.1 28.0 26.0 24.1 22.4 20.9 19.4 18.1 16.9

Births/population 15-44 ............ .030 .035 .041 .047 .055 .063 .073 .084 .097 .111 .127 .145 .165

f

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 19Males (Oeo = 61.23 years)

Annual rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0107 .0130 .0155 .0184 .0215 .0249 .0286 .0324 .0364 .0406 .0449 .0493 .05371-4 .......................... .0425 .0510 .0604 .0707 .0817 .0934 .1057 .1184 .1314 .1446 .1579 .1712 .18445-9 ........................... .0505 .0644 .0746 .0854 .0965 .1079 .1193 .1307 .1418 .1526 .1629 .1727 .1819

10-14 .......................... .0574 .0656 .0741 .0827 .0911 .0994 .1072 .1145 .1212 .1272 .1324 .1369 .140715-19 .......................... .0598 .0667 .0735 .0800 .0860 .0915 .0962 .1002 .1035 .1059 .1076 .1084 .108620-24 .............. " .......... .0621 .0676 .0726 .0770 .0808 .0838 .0860 .0874 .0880 .0878 .0870 .0855 .083625-29 .......................... .0643 .0683 .0715 .0740 .0757 .0766 .0767 .0760 .0746 .0726 .0702 .0673 .064130-34 .......................... .0666 .0689 .0704 .0710 .0709 .0699 .0683 .0660 .0632 .0600 .0565 .0529 .049135-39 .. " .......... , ........... .0686 .0693 .0690 .0680 .0661 .0636 .0606 .0571 .0533 .0494 .0454 .0414 .037540-44 .......................... .0704 .0693 .0673 .0646 .0614 .0576 .0535 .0492 .0448 .0404 .0362 .0323 .028545-49 .......................... .0715 .0686 .0650 .0609 .0564 .0516 .0467 .0419 .0372 .0328 .0287 .0249 .021550-54 ....... , ................. , .0714 .0669 .0618 .0565 .0510 .0455 .0402 .0352 .0305 .0262 .0223 .0189 .015955-59 .. '" ......... " .... " .... .0697 .0636 .0574 .0511 .0450 .0392 .0337 .0288 .0243 .0204 .0169 .0140 .011560-64 .......................... .0654 .0583 .0512 .0445 .0382 .0325 .0273 .0227 .0187 .0153 .0124 .0100 .008065c69 .......................... .0580 .0504 .0432 .0366 .0307 .0254 .0208 .0169 .0136 .0108 .0086 .0067 .005270-74 .......................... .0470 .0398 .0333 .0275 .0225 .0182 .0145 .0115 .0090 .0070 .0054 .0041 .003175-79 . '" .......... " .......... .0331 .0274 .0223 .0180 .0143 .0113 .0088 .0068 .0052 .0039 .0030 .0022 .0016.... 80+ .......................... .0265 .0212 .0168 .0131 .0101 .0077 .0058 .0043 .0032 .0024 .0017 .0012 .0009....

0'1

Age Proportion under given age

1 ............................ .0107 .0130 .0155 .0184 .0215 .0249 .0286 .0324 .0364 .0406 .0449 .0493 .05375 ............................ .0532 .0640 .0759 .0891 .1033 .1184 .1343 .1508 .1679 .1852 .2028 .2205 .2381

10 ............................ .1082 .1284 .1506 .1744 .1998 .2263 .2536 .2815 .3097 .3378 .3657 .3932 .420015 ............................ .1655 .1940 .2246 .2571 .2909 .3257 .3609 .3961 .4309 .4650 .4982 .5301 .560720 ............................ .2254 .2607 .2981 .3371 .3769 .4171 .4571 .4963 .5344 .5709 .6057 .6386 .669325 ............................ .2875 .3283 .3707 .4141 .4577 .5009 .5431 .5837 .6224 .6588 .6927 .7241 .752930 ............................ .3518 .3965 .4422 .4881 .5334 .5775 .6198 .6597 .6970 .7314 .7629 .7914 .817135 ............................ .4184 .4654 .5126 .5592 .6043 .6475 .6880 .7257 .7601 .7914 .8194 .8443 .866240 ............................ .4870 .5347 .5816 .6271 .6705 .7111 .7486 .7828 .8135 .8408 .8648 .8857 .903845 ............................ .5574 .6039 .6489 .6918 .7318 .7687 .8021 .8319 .8583 .8812 .9010 .9179 .932350 ............................ .6289 .6725 .7140 .7527 .7882 .8203 .8488 .8738 .8955 .9140 .9297 .9428 .953755 ............................ .7003 .7394 .7758 .8091 .8392 .8658 .8890 .9090 .9260 .9420 .9520 .9617 .969660 ............................ .7699 .8030 .8331 .8602 .8842 .9050 .9228 .9378 .9503 .9606 .9690 .9757 .981165 ............................ .8353 .8612 .8844 .9047 .9224 .9374 .9500 .9605 .9690 .9759 .9813 .9857 .9891

Parameter ofstable populations

Birth rate ........................ .0112 .0136 .0163 .0184 .0228 .0264 .0304 .0345 .0389 .0434 .0481 .0530 .0579Death rate ....................... .0212 .0186 .0163 .0144 .0128 .0114 .0104 .0095 .0089 .0084 .0081 .0080 .0079GRR (27) ....................... 0.87 1.00 1.14 1.31 1.49 1.70 1.94 2.21 2.51 2.85 3.24 3.67 4.16GRR (29) ....................... 0.86 1.00 1.15 1.33 1.53 1.76 2.03 2.33 2.68 3.07 3.52 4.02 4.60GRR (31) ....................... 0.85 0.99 1.16 1.35 1.57 1.83 2.13 2.47 2.86 3.31 3.83 4.43 5.11GRR (33) ....................... 0.84 0.99 1.17 1.37 1.62 1.90 2.23 2.62 3.07 3.59 4.20 4.90 5.73Average age ...................... 40.6 38.1 35.6 33.2 30.9 28.7 26.7 24.7 23.0 21.3 19.9 18.5 17.3Births/population 15-44 ...... , ..... .028 .033 .038 .045 .052 .060 .069 .079 .091 .104 .119 .137 .156

, ,,~----- -----~-------

-'-') .. ",_,'_r

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 21Males (Oeo = 66.02 years)

Annual rate 0/ increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age interval Proportion in age interval

Under 1 ....................... .0099 .0121 .0146 .0174 .0205 .0238 .0274 .0311 .0351 .0392 .0434 .0477 .05221-4 .......................... .0403 .0485 .0578 .0679 .0788 .0905 .1028 .1155 .1285 .1418 .1551 .1685 .18185-9 .......................... .0523 .0617 .0718 .0825 .0936 .1051 .1167 .1282 .1395 .1505 .1610 .1710 .1804

10-14 .......................... .0548 .0630 .0715 .0801 .0887 .0971 .1051 .1126 .1195 .1258 .1312 .1359 .139915-19 .......................... .0573 .0642 .0711 .0777 .0839 .0896 .0946 .0989 .1023 .1050 .1069 .1080 .108320-24 .......................... .0597 .0653 .0705 .0752 .0792 .0824 .0849 .0865 .0874 .0874 .0868 .0855 .083725-29 " ............ '" ... " .... .0622 .0663 .0698 .0726 .0746 .0757 .0761 .0756 .0745 .0727 .0704 .0676 .064530-34 .......................... .0647 .0673 .0691 .0701 .0702 .0695 .0681 .0660 .0634 .0604 .0570 .0534 .049735-39 .......................... .0671 .0681 .0682 .0675 .0659 .0637 .0609 .0576 .0539 .0500 .0461 .0421 .038340-44 .......................... .0694 .0686 .0671 .0647 .0617 .0581 .0541 .0499 .0456 .0413 .0371 .0331 .029345-49 .......................... .0711 .0686 .0653 .0615 .0572 .0525 .0477 .0429 .0383 .0338 .0296 .0257 .022250-54 .......................... .0718 .0676 .0628 .0576 .0522 .0468 .0415 .0364 .0316 .0273 .0233 .0197 .016655-59 ...................... , ... .0709 .0651 .0590 .0528 .0467 .0408 .0353 .0302 .0256 .0215 .0179 .0148 .012260-64 .......................... .0676 .0605 .0535 .0467 .0403 .0343 .0290 .0242 .0200 .0164 .0133 .0107 .008665-69 ............... '" ........ .0611 .0534 .0460 .0392 .0330 .0274 .0225 .0183 .0148 .0118 .0094 .0074 .005870-74 .......................... .0508 .0432 .0364 .0302 .0248 .0201 .0161 .0128 .0101 .0078 .0061 .0046 .003575-79 " ........................ .0369 .0307 .0251 .0204 .0163 .0129 .0101 .0078 .0060 .0045 .0034 .0026 .0019.... 80+ .......................... .0321 .0258 .0205 .0161 .0124 .0095 .0072 .0054 .0040 .0029 .0021 .0016 .0011....

~

Age Proportion under given age

1 ............................ .0099 .0121 .0146 .0174 .0205 .0238 .0274 .0311 .0351 .0392 .0434 .0477 .05225 ............................ .0502 .0607 .0724 .0853 .0993 .1143 .1301 .1466 .1636 .1809 .1985 .2162 .2340

10 ............................ .1025 .1223 .1442 .1678 .1930 .2194 .2468 .2748 .3031 .3314 .3595 .3872 .414315 ............................ .1573 .1853 .2156 .2479 .2817 .3165 .3519 .3874 .4226 .4571 .4907 .5231 .554220 ............................ .2146 .2495 .2867 .3256 .3656 .4061 .4465 .4863 .5249 .5621 .5976 .6311 .662525 ............................ .2743 .3148 .3572 .4007 .4448 .4885 .5314 .5728 .6123 .6496 .6844 .7166 .746230 ............................ .3364 .3811 .4270 .4733 .5193 .5643 .6075 .6484 .6868 .7223 .7547 .7842 .810835 ............................ .4011 .4484 .4961 .5434 .5896 .6338 .6756 .7145 .7502 .7826 .8117 .8376 .860540 ............................ .4682 .5165 .5643 .6109 .6555 .6975 .7365 .7720 .8041 .8327 .8578 .8798 .898845 ............................ .5376 .5851 .6313 .6756 .7172 .7556 .7906 .8220 .8497 .8740 .8949 .9128 .928050 ............................ .6087 .6537 .6967 .7371 .7743 .8081 .8383 .8649 .8880 .9077 .9245 .9386 .950355 ............................ .6805 .7213 .7595 .7947 .8266 .8550 .8798 .9013 .9196 .9350 .9478 .9583 .966960 ............................ .7514 .7863 .8185 .8475 .8733 .8958 .9151 .9315 .9452 .9565 .9657 .9731 .979165 ............................ .8190 .8469 .8720 .8942 .9135 .9301 .9441 .9557 .9652 .9729 .9790 .9839 .9877

Parameter of stable populations

Birth rate ........................ .0102 .0125 .0152 .0181 .0213 .0248 .0286 .0326 .0369 .0413 .0459 .0506 .0554Death rate ....................... .0202 .0175 .0152 .0131 .0113 .0098 .0086 .0076 .0069 .0063 .0059 .0056 .0054ORR (27) ....................... 0.83 0.95 1.09 1.24 1.42 1.62 1.84 2.10 2.39 2.71 3.08 3.50 3.96ORR (29) ....................... 0.81 0.94 1.09 1.26 1.45 1.67 1.92 2.21 2.54 2.91 3.34 3.82 4.37ORR (31) ....................... 0.80 0.94 1.10 1.28 1.49 1.73 2.01 2.34 2.71 3.14 3.63 4.20 4.84ORR (33) ....................... 0.79 0.93 1.10 1.30 1.53 1.79 2.11 2.47 2.90 3.39 3.97 4.64 5.42Average age ...................... 41.7 39.1 36.6 34.1 31.7 29.5 27.3 25.3 23.5 21.8 20.3 18.9 17.6Births/population 15-44 .... '" ... " .027 .031 .036 .042 .049 .057 .065 .075 .086 .099 .113 .130 .148

{

TABLE II. "WEST" MODEL STABLE POPULATIONS ARRANGED BY LEVEL OF MORTALITY (continued)

LEVEL 23Males (Oeo= 71.19 years)

AnllUll1 rate of increase

-.010 -.005 .000 .005 .010 .015 .020 .025 .030 .035 .040 .045 .050

Age intervalProportion in age interval

Under 1 ....................... .0092 .0114 .0138 .0165 .0195 .0228 .0263 .0301 .0340 .0381 .0423 .0466 .05101-4 .......................... .0377 .0458 .0549 .0649 .0758 .0874 .0997 .1125 .1256 .1390 .1524 .1660 .17945-9 .......................... .0492 .0584 .0684 .0791 .0903 .1019 .1136 .1253 .1368 .1480 .1588 .1690 .1786

10-14 .......................... .0516 .0598 .0683 .0770 .0858 .0943 .1026 .1104 .1175 .1240 .1297 .1347 .138815-19 .......................... .0541 .0611 .0681 .0749 .0813 .0873 .0926 .0971 .1009 .1038 .1059 .1072 .107820-24 .......................... .0566 .0624 .0678 .0727 .0770 .0806 .0834 .0853 .0864 .0868 .0863 .0852 .083625-29 .......................... .0592 .0636 .0674 .0706 .0729 .0744 .0751 .0749 .0740 .0725 .0703 .0677 .064830-34 .......'................... .0619 .0649 .0671 .0685 .0690 .0687 .0676 .0658 .0634 .0605 .0573 .0538 .050235-39 .......................... .0647 .0661 .0666 .0663 .0652 .0633 .0607 .0576 .0542 .0504 .0466 .0427 .038840-44 .......................... .0673 .0671 .0660 .0641 .0614 .0581 .0544 .0504 .0462 .0419 .0378 .0337 .029945-49 .......................... .0697 .0677 .0650 .0615 .0575 .0531 .0485 .0438 .0391 .0347 .0304 .0265 .023050-54 .......................... .0713 .0676 .0632 .0584 .0532 .0480 .0427 .0376 .0328 .0283 .0243 .0206 .017455-59 .......................... .0716 .0662 .0604 .0544 .0484 .0425 .0369 .0317 .0270 .0227 .0190 .0157 .012960-64 .......................... .0698 .0629 .0560 .0492 .0427 .0366 .0310 .0259 .0215 .0177 .0144 .0116 .009365-69 .......................... .0649 .0571 .0496 .0425 .0359 .0300 .0248 .0203 .0164 .0131 .0104 .0082 .006470-74 .......................... .0560 .0480 .0406 .0340 .0280 .0228 .0184 .0147 .0116 .0090 .0070 .0054 .0041

- 75-79 .......................... .0427 .0357 .0295 .0240 .0193 .0154 .0121 .0094 .0072 .0055 .0042 .0031 .0023- 80+ .......................... .0423 .0342 .0273 .0215 .0167 .0128 .0097 .0073 .0054 .0040 .0029 .0021 .001500

AgeProportion under given age

1 ............................ .0092 .0114 .0138 .0165 .0195 .0228 .0263 .0301 .0340 .0381 .0423 .0466 .05105 ............................ .0470 .0571 .0686 .0814 .0953 .1102 .1260 .1425 .1596 .1770 .1947 .2126 .2304

10 ............................ .0961 .1155 .1371 .1605 .1856 .2121 .2396 .2678 .2964 .3251 .3535 .3816 .409015 ............................ .1478 .1753 .2053 .2375 .2714 .3065 .3422 .3782 .4139 .4491 .4832 .5162 .547920 ............................ .2019 .2364 .2734 .3124 .3527 .3937 .4348 .4753 .5148 .5529 .5892 .6235 .655725 ............................ .2585 .2988 .3412 .3851 .4298 .4744 .5182 .5607 .6013 .6396 .6755 .7087 .739330 ............................ .3177 .3624 .4087 .4557 .5027 .5488 .5933 .6356 .6753 .7121 .7458 .7764 .804035 ............................ .3796 .4273 .4757 .5242 .5716 .6174 .6608 .7013 .7386 .7726 .8031 .8302 .854240 ............................ .4443 .4933 .5424 .5905 .6368 .6807 .7215 .7590 .7928 .8230 .8496 .8729 .893045 ............................ .5116 .5604 .6084 .6545 .6982 .7388 .7760 .8094 .8390 .8650 .8874 .9066 .923050 ............................ .5813 .6282 .6733 .7160 .7557 .7919 .8244 .8531 .8781 .8996 .9178 .9332 .945955 ............................ .6526 .6958 .7366 .7744 .8090 .8399 .8671 .8907 .9109 .9279 .9421 .9538 .963360 ............................ .7242 .7620 .7970 .8288 .8573 .8824 .9040 .9224 .9379 .9506 .9611 .9695 .976265 ............................ .7940 .8249 .8530 .8780 .9000 .9189 .9350 .9484 .9594 .9683 .9755 .9811 .9856

Parameter ofstable populations

Birth rate ....................... .0094 .0116 .0141 .0169 .0200 .0234 .0271 .0310 .0352 .0395 .0440 .0486 .0534Death rate ....................... .0194 .0166 .0141 .0119 .0100 .0084 .0071 .0060 .0052 .0045 .0040 .0036 .0034GRR (27) ....................... 0.79 0.91 1.04 1.19 1.36 1.55 1.77 2.02 2.30 2.61 2.96 3.36 3.81GRR (29) ....................... 0.78 0.90 1.04 1.21 1.39 1.60 1.85 2.12 2.44 2.80 3.20 3.67 4.20GRR (31) ....................... 0.71 0.90 1.05 1.22 1.42 1.66 1.93 2.24 2.59 3.00 3.48 4.02 4.64GRR (33) ....................... 0.75 0.89 1.05 1.24 1.46 1.71 2.01 2.36 2.77 3.24 3.79 4.43 5.18Average age ...................... 43.1 40.5 37.9 35.3 32.8 30.4 28.2 26.1 24.1 22.4 20.7 19.3 17.9Births/population 15-44 ............ .026 .030 .035 .040 .047 .054 .062 .072 .083 .095 .109 .125 .142

!fl..)II "

-- , -----, --

AnnexID

TABLES FOR ADJUSTING STABLE ESTIMATES FOR THEEFFECTS OF DECLINING MORTALITY

TABLE III.l. PROPORTIONS TO BE ADDED TO STABLE ESTIMATES OF THE BIRTH RATE AND OF THE GROSSREPRODUCTION RATE TO CORRECT FOR THE EFFECTS OF DECLINING MORTALITY, AT VARIOUS DURATIONS(t YEARS) OF THAT DECLINE AND ASSUMING THAT k = .01

Part (a). For use to correct stable estimates derived from C(x), andfrom ten-year intercensal growth rate

xt 5 10 15 20 25 30 35 40

Birth rate

5 .0 ••••••.••..••••••• -.016 -.030 -.027 -.025 -.028 -.033 -.039 -.04310 .................... -.008 -.015 -.019 -.023 -.025 -.027 -.029 -.03215 '0 •••••••••••••••••• .000 .012 .011 .001 -.004 -.006 -.004 -.00420 •••••••••••••••• 0 ••• .005 .032 .043 .034 .021 .018 .022 .02625 '0' ••••••••••••••••• .006 .040 .064 .066 .053 .043 .047 .05130 .................... .006 .043 .073 .085 .081 .072 .069 .07335 .................... .007 .044 .016 .094 .099 .096 .094 .09240 .................... .006 .043 .076 .096 .106 .112 .116 .114

Gross reproduction rate

5 ••••••••••• 0 •••••••• -.017 -.032 -.035 -.014 -.005 -.006 -.005 -.01010 .................... -.009 -.017 -.027 -.011 -.002 .005 .006 .00115 .0 .•.••••••..••••••• .000 .010 .013 .012 .019 .026 .032 .03120 ••••••••••••••••••• 0 .004 .029 .045 .045 .045 .051 .059 .06225 .0 •••••••••••••••••• .005 .037 .065 .078 .078 .077 .085 .08830 .................... .005 .040 .074 .096 .106 .107 .108 .11135 .................... .006 .041 .017 .105 .124 .131 .133 .13040 ••••••• 0.0 •••••••• 0. .005 .040 .016 .108 .131 .147 .156 .153

Part (b). For use to correct stable estimates derived from C(x), andfromfive-year intercensal growth rate

xt 5 10 15 20 25 30 35 40

Birth rate

5 .................... -.003 -.007 -.010 -.012 -.013 -.014 -.016 -.018I 10 .................... .010 .017 .005 -.002 -.004 -.000 -.001 .000.. 15 .021 .045 .039 .024 .019 .021 .028 .031••••••••••••••• 0 •• 0.

20 ••••••••••••• 0 •••••• .025 .062 .070 .055 .045 .045 .054 .05925 .................... .025 .068 .089 .085 .075 .068 .075 .08230 .................... .025 .069 .096 .102 .102 .095 .096 .10135 .................... .023 .066 .098 .109 .118 .117 .118 .11940 .................... .022 .063 .094 .108 .123 .129 .137 .138

Gross reproduction rate

5 .................... -.005 -.009 -.004 .005 .017 .021 .019 .01510 ................... , .007 .015 .012 .016 .027 .034 .038 .03415 ... ,. ................ .018 .042 .045 .042 .050 .057 .065 .06620 .................... .022 .058 .076 .073 .076 .081 .092 .09625 .................... .023 .065 .094 .103 .108 .106 .114 .12030 .................... .022 .065 .102 .120 .135 .133 .136 .14035 .................... .021 .063 .103 .127 .150 .155 .158 .15840 .................... .019 .059 .100 .126 .156 .167 .178 .117

119

Part (d). For use to correct stable estimates derived from C(x), and from average mortality in thefive years preceding the census

xt 5 10 15 20 25 30 35 40

Birth rate

5 .................... -.001 -.000 -.000 -.001 -.001 -.000 -.001 -.00210 .................... .006 .013 .008 .005 .005 .008 .009 .00915 ................. , .. .0Il .026 .024 .018 .016 .019 .022 .024 x20 .................... .013 .035 .040 .034 .028 .031 .035 .03925 .................... .014 .038 .049 .049 .044 .043 .046 .05130 •••••••••••••• 0 ••••• .014 .040 .054 .059 .059 .058 .059 .06335 .0 ••••••••••••••• "0 .014 .040 .057 .065 .070 .072 .073 .07440 .................... .014 .039 .057 .067 .075 .082 .087 .088

I.'Y-

Gross reproduction rate

5 .................... -.003 -.003 .004 .016 .026 .032 .032 .03010 .................... .004 .012 .014 .022 .033 .042 .044 .04215 .................... .009 .026 .031 .037 .046 .054 .059 .05920 .................... .012 .035 .048 .054 .060 .057 .073 .07625 .................... .012 .039 .058 .071 .077 .081 .087 .09130 .................... .013 .041 .063 .082 .094 .099 .101 .10435 .................... .013 .041 .066 .088 .105 .Il6 .117 .Il840 .................... .013 .040 .066 .090 .Il2 .127 .134 .134

120

I•

TABLB 111.2. VALUES OF TIm INDEX "DBATHS AT AGE 65 AND OVER/DEATHS AT AGB 5 AND OVER" IN "WEST" MODBL STABLB POPULATIONSGIVEN THE RATIO OF ANNUAL BIRTHS TO THE POPULATION AGED 15·44, AND TIm EXPECTATION OF LIFB AT BIRTH

Expectation Ratio ofbirths to population aged 15-44of life

.050 .060 .070 .080 .090 .100 .110 .120 .130 .140at birth .040

Females

20 .378 .326 .287 .255 .225 .241 .181 .164 .148 .136 .12425 .408 .396 .309 .259 .241 .216 .193 .174 .157 .143 .10330 .439 .378 .333 .294 .259 .232 .207 .186 .167 .152 .13935 .469 .406 .359 .318 .281 .251 .224 .201 .180 .163 .12040 .502 .437 .388 .344 .304 .272 .243 .219 .196 .178 .13145 .536 .470 0418 .373 .332 .298 .266 .240 .216 .196 .14450 .572 .506 .454 .370 .364 .328 .295 .266 .240 .205 .16255 .607 .542 .490 .406 .399 .362 .327 .297 .269 .246 .18460 .650 .589 .538 .492 .447 .410 .373 .342 .312 .287 .26465 .699 .644 .598 .554 .510 .473 .436 .404 .373 .346 .32170 .757 .711 .671 .633 .594 .560 .526 .495 .464 .437 .41175 .830 .797 .768 .739 .709 .682 .654 .627 .601 .577 .554

Males

18.0 ....... .310 .268 .233 .206 .183 .164 .149 .134 .122 .110 .10222.9 ....... .341 .294 .255 .226 .201 .179 .162 .145 .132 .120 .10927.7 ....... .375 .323 .282 .247 .219 .179 .177 .159 .145 .130 .11532.5 ....... .408 .354 .309 .273 .242 .214 .193 .173 .157 .142 .13037.3 ....... .434 .381 .335 .296 .263 .235 .212 .191 .173 .157 .14242.1 ....... .471 .414 .366 .325 .291 .261 .234 .211 .191 .173 .15747.1 ....... .509 .451 .402 .359 .326 .288 .259 .234 .213 .193 .17651.8 ....... .542 0485 .435 .392 .350 .318 .289 .263 .238 .214 .19956.5 ....... .581 .526 .477 .433 .391 .357 .326 .298 .272 .246 .22861.2 ....... .628 .576 .529 .485 .443 .408 .375 .346 .360 .291 .27266.0 ....... .682 .634 .590 .547 .510 .476 .443 .413 .384 .360 .33671.2 ....... .759 .718 .682 .622 .615 .582 .599 .524 .498 .472 .447

121

TABLE I1I.3. VALUES OF THE PRODUCT kt IN "WEST" MODEL LIFE TABLES FOR GIVEN INCREASES IN EXPECTATION OF LIFE AT BIRTH DURING A

PERIOD OF t YEARS FROM INDICATED LEVELS OF oeO AT THE BEGINNING OF THE PERIOD

Increase Expectation of life at birth at the beginning ofperiod oft yearsInOeo 20 24 28 32 36 40 44 48 52 56 60

Females

2 .0575 .0456 .0374 .0314 .0268 .0233 .0204 .0185 .0165 .0144 .01274 .1085 .0868 .0715 .0603 .0518 .0451 .0397 .0383 .0318 .0278 .02476 .1541 .1242 .1029 .0872 .0751 .0655 .0582 .0548 .0461 .0405 .03608 .1953 .1583 .1319 .1121 .0969 .0848 .0780 .0701 .0596 .0525 .0468

10 .2327 .1897 .1587 .1354 .1173 .1033 .0909 .0845 .0723 .0639 .056212 .2668 .2187 .1837 .1572 .1366 .1231 .1098 .0979 .0843 .0746 .063614 .2982 .2455 .2070 .1777 .1551 .1396 .1242 .1106 .0956 .0840 .070316 .3272 .2704 .2287 .1969 .1749 .1548 .1376 .1226 .1064 .091418 .3540 .2937 .2492 .2154 .1913 .1692 .1503 .1339 .1158 .098120 .3789 .3155 .2684 .2352 .2066 .1827 .1623 .1447 .123222 .4022 .3360 .2870 .2517 .2210 .1954 .1736 .1541 .129924 .4240 .3552 .3067 .2670 .2345 .2074 .1844 .161526 .4445 .3738 .3232 .2814 .2472 .2187 .1938 .168228 .4637 .3935 .3385 .2948 .2592 .2295 .201230 .4823 .4100 .3529 .3075 .2705 .2389 .207932 .5020 .4253 .3663 .3195 .2813 .246234 .5185 .4397 .3790 .3308 .2907 .253036 .5338 .4531 .3910 .3416 .298038 .5482 .4658 .4024 .3510 .304840 .5616 .4778 .4131 .358442 .5743 .4892 .4225 .365144 .5863 .4999 .429946 .5977 .5093 .436648 .6084 .516750 .6178 .523452 .625254 .6319

Males

2 .0602 .0478 .0392 .0329 .0281 .0244 .0215 .0197 .0171 .0150 .01324 .1136 .0909 .0749 .0632 .0542 .0472 .0437 .0381 .0331 .0291 .02576 .1613 .1300 .1078 .0913 .0786 .0687 .0634 .0552 .0481 .0423 .03758 .2044 .1658 .1381 .1174 .1014 .0909 .0818 .0712 .0622 .0548 .0469

10 .2436 .1987 .1662 .1418 .1230 .1106 .0989 .0862 .0754 .0666 .055012 .2794 .2290 .1924 .1646 .1451 .1290 .1149 .1003 .0879 .0759 .061914 .3122 .2571 .2167 .1862 .1648 .1461 .1299 .1136 .0997 .0840 .0676 ~

16 .3426 .2832 .2396 .2083 .1832 .1621 .1440 .1260 .1091 .090918 .3707 .3076 .2611 .2280 .2003 .1771 .1572 .1378 .1172 .096720 .3968 .3304 .2832 .2464 .2163 .1912 .1697 .1472 .124122 .4212 .3520 .3029 .2635 .2313 .2044 .1815 .1553 .129824 .4440 .3741 .3213 .2795 .2454 .2169 .1908 .162226 .4655 .3938 .3384 .2945 .2586 .2287 .1989 .167928 .4876 .4122 .3544 .3086 .2711 .2380 .205830 .5074 .4293 .3694 .3218 .2829 .2461 .211632 .5257 .4453 .3835 .3343 .2923 .253034 .5429 .4603 .3968 .3461 .3003 .258836 .5589 .4744 .4093 .3554 .307338 .5739 .4876 .4210 .3635 .313040 .5880 .5001 .4304 .370442 .6012 .5119 .4385 .376244 .6137 .5213 .445446 .6255 .5294 .451148 .6348 .536350 .6429 .542052 .649854 .6555

122

123

Annex IV

TABLES FOR ESTIMATING CUMULATED FERTILITYFROM AGE SPECIFIC FERTILITY RATES

TABLE IV.1. MULTIPLYING FACTORS Wi FOR ESTIMATING THE AVERAGE VALUE OVER FIVE-YEAR AGE GROUPS OF CUMULATED FERTILITY (Fi)ACCORDING TO THE FORMULA

I-I

F I = 5 L jj+wJ;}=o

(WHEN/o = o./l = AGE SPECIFIC FERTILITY RATE FOR AGES 14.5 TO 19.5,fa = FOR AGES 19.5 TO 24.5 ETC.)

Age Interval Exact limits of(i) age interval Multiplying factors we for values off1/fs and iii asIndicated in lower part of table

1 ................... 15-20 1.120 1.310 1.615 1.950 2.305 2.640 2.925 3.1702 ................... 20-25 2.555 2.690 2.780 2.840 2.890 2.925 2.960 2.9853 ................... 25-30 2.925 2.960 2.985 3.010 3.035 3.055 3.075 3.0954 ................... 30-35 3.055 3.075 3.095 3.120 3.140 3.165 3.190 3.2155 ................... 35-40 3.165 3.190 3.215 3.245 3.285 3.325 3.375 3.4356 ................... 40-45 3.325 3.375 3.435 3.510 3.610 3.740 3.915 4.1507 ................... 45-50 3.640 3.895 4.150 4.395 4.630 4.840 4.985 5.000

/l/f2 .036 .113 .213 .330 .460 .605 .764 .939iii 31.7 30.7 29.7 28.7 27.7 26.7 25.7 24.7

TABLE IV.2. MULTIPLYING FACTORS W FOR ESTIMATING THE AVERAGE VALUE OVER FIVE-YEAR AGE GROUPS OF CUMULATED FERTILITY (F,)ACCORDING TO THE FORMULA

i-I

FI=5 L Jj+wJi}=o

(WHEN/o = O.11 = AGE SPECIFIC FERTILITY RATE FOR AGES 15 TO 20,12 = FOR AGES 20 TO 25 ETC.)

Age Interval Exact limits ofMultiplying factors we for values offl/f2 and iii as Indicated In lower part of table(i) age Interval

1 ................... 15-20 .335 .680 1.030 1.390 1.760 2.130 2.460 2.7542 ................... 20-25 2.025 2.170 2.265 2.330 2.380 2.420 2.455 2.4853 ................... 25-30 2.420 2.455 2.485 2.510 2.535 2.560 2.580 2.6054 ................... 30-35 2.560 2.580 2.605 2.625 2.650 2.675 2.700 2.7305 ................... 35-40 2.675 2.700 2.730 2.760 2.800 2.845 2.895 2.9606 ................... 40-45 2.845 2.895 2.960 3.040 3.145 3.285 3.470 3.7207 ................... 45-50 3.195 3.455 3.720 3.980 4.240 4.495 4.750 5.000

/l/h .036 .113 .213 .330 .460 .605 .764 .939iii 32.2 31.2 30.2 29.2 28.2 27.2 26.2 25.2

124

'-)

Annex V

TABLES FOR ESTIMATING MORTALITY FROM CHILD SURVIVORSIllP RATES

TABLE V.2. MULTIPLYING FACTORS FOR ESTIMATING THE PROPORTION OF CHILDREN BORN ALIVE WHO DIE BY AGE a - q(a) - FROMTHE PROPORTION DEAD AMONG ClDLDREN EVER BORN REPORTED BY WOMEN CLASSIFIED IN TEN-YEAR AGB INTERVALS

Mortality measure Exacts limits ofage Multiplying factors to obtain q (a) sholD1l in col. 1 from proportion ofchildrenestimated Interval ofwomen reported as deadby women ofages specified In col. 2; for values ofPli PI' ffi,

(1) (2) and iii' as specified in lower part of table

{

q(2) ................. 15-25 0.982 1.000 1.021 1.045 1.072 1.l05 1.144 1.193q(5) ................. 25-35 0.990 1.004 1.018 1.033 1.048 1.064 1.081 1.099q(l5) ................ 35-45 0.977 0.993 1.009 1.024 1.040 1.056 1.071 1.086q(25) ................ 45-55 0.990 1.008 1.025 1.043 1.062 1.080 1.099 1.118q(35) ................ 55-65 0.990 1.007 1.025 1.043 1.061 1.080 1.099 1.119

\...Pl/P2 0.387 0.330 0.268 0.205 0.143 0.090 0.045 0.014

m 24.7 25.7 26.7 27.7 28.7 29.7 30.7 31.7m' 24.2 25.2 26.2 27.2 28.2 29.2 30.2 31.2

~

125

/'

ADnex VI

A NOTE ON INTERPOLATION

The tables presented in annexes I to V contain entries at smallenough intervals of the argument (e.g., small enough differences inmortality between model life tables, and small enough differencesin rate of increase in model stable populations incorporating thesame life table) for intervening values to be approximated satis­factorily by linear interpolation. This procedure assumes that eachrelevant variable follows a straight line in the interval betweenentries.

In the use of most of the tables in the annexes, the interpolationrequired is a simple determination of an intervening value, and thereis scarcely justification for comment on such a standard procedure.However, for completeness the requisite steps will be described.

Consider as an example the use of interpolation in finding a"West" female model life table consistent with the enumeratedfemalepopulationover ten at the end ofadecade, and the enumeratedpopulation at all ages at the beginning. Assume that the projectionof the initial population by the survival factors in annex I (table 1.3)at mortality level 7 (Oeo = 35 years for females) produced a popula­tion ten and over 0.8201 times the initial population; that projectionby the mortality factors at level 9 (Oeo = 40 years for females)produced a population over ten 0.8416 times the initial population,and the recorded female population over ten in the later census was0.8312 times the initial population. How is the level of mortality(and the expectation of life at birth) found?

The difference between the proportion surviving according tomodel level 9 and model level 7 is 0.8416 - 0.8201, or 0.0215.The difference between the proportion surviving according to modellevel 9 and the recorded proportion surviving is 0.8416 - 0.8312,or 0.0104. In descending linearly from level 9 towards level 7, onemust traverse a fraction .0104/.0215 = .484 to arrive at a level thatwould yield the reported proportion surviving. Thus the estimatedmortality level is 9 -.484(2), or 8.032. The expectation of life inthis model life table is 40 -.484(5), or 37.58 years. An equivalentprocedure that is simpler on a desk calculator is to multiply thevalue at one end of the interval for which interpolation is neededby the fraction f (here equal to .484), and the value at the other endof the interval by 1 -.f (here .516), and add the products. This issimpler because the addition ofthe products is achieved automaticallythrough cumulative multiplication. A rule for making what can bea confusing choice is: multiply by the larger number (ofland 1 - f)the nearer value. Thus the estimated level of mortality in the presentexample is 9(.516) + 7(.484) = 8.032. Nine is multiplied by thelarger fraction (by .516) because the recorded proportion surviving(0.8312) is closer to the proportion surviving according to level 9(0.8416) than level 7 (0.8201). Once these multipliers have beendetermined (i.e., f and 1 - 0, they may be applied to any of thelife table functions given in annex I for model tables 7 and 9 - toprovide estimates of l» or qx at every age, for example.

If the observed quantity is beyond the last tabulated value, sothat extrapolation instead ofinterpolation is required, the mechanicsof calculation are not modified. Suppose the fraction survivingaccording to level 1 were 0.6831, according to level 3 0.7125, andobserved 0.6715. The difference between the given value and thatfor level 3 would be 0.0410, between levelland level 3 would be0.0294. Hence f is .0410/.0294 = 1.395, and 1 - f is -0.395. The

estimated expectation oflife is thus (1.395) (20) -(.395) (25) = 18.03.Extrapolation should not ordinarily be attempted more than one­half interval beyond the tabulated values.

A more complicated form of interpolation is needed in usingthe model stable populations given in annex II. Model stablepopulations with mortality not exactly coincident with one of themodel life tables reproduced in annex I can be found by interpolatingbetween two pages, if the rate of increase happens to. be -.005,0.00, 0.005, or one of the values at the head of the columns on eachpage. Similarly, model stable populations that happen to havemortality at exactly one of the levels given in annex I, but a rate ofincrease not divisible by .005 can be located by interpolation betweencolumns on one ofthe pages in annex II. But in general, it is necessaryto locate the model stable population best fitting an observedpopulation by first interpolating between columns on each of twolevels of mortality - for example, to locate stable populationswith a given rate of increase, but a C(35) that is higher (at onemortality level) and lower (at the adjacent mortality level) than inthe given population. Then an additional interpolation determinesthe fractional mortality level in the model stable population agreeingwith the given population in C(35) as well as r.

Thus suppose that one wants to find the female model stablepopulation with r = 0.0136, and C(35) = .7150. Inspection of thetables in annex II indicates that a value in the neighbourhood of.7150 is found a little more than halfway between the columnsheaded r = .010 and r = .015 at mortality level 7. Interpolationthen shows that when r = .0136, C(35) is .72 (.7293) + .28 (.6928)= .7191 at level 7. This value is slightly higher than .7150; C(35) islower in a stable population with the same r and higher oeo' so thatthe next step is to find that at level 9, when r = .0136, C(35) is.72 (.7098) + .28 (.6715) = .6991. This procedure has found twomodel stable populations, each with the requisite r, one with aC(35) above .7150, and the other with C(35) below .7150. Thedifference between the two values of C(35) is .0200, and the lowersubtracted from .7150 is .0159. The interpolation factor (f) is thus.0159/.0200, or .795. Hence the estimated mortality level is (.795) (7)+ (.205) (9) = 7.41, and 0eo is (.795) (35.0) + (.205) (40.0) = 36.0.Once these interpolation factors have been determined, they may beused to find any other desired parameters of the model stablepopulation with C(35) = .7150 and r = .0136. Consider the estima­tion of the birth rate. First the birth rate is found for the stablepopulation at levels 7 and 9 with r = .0136 (the first is .72 (.04304)+ .28 (.03791) = .04160, and the second is.72 (.03832) + .28 (.03357)= .03699); then the birth rate at level 7.41 is found «.795) (.04160)+ (.205) (.03699) = .04065). The estimation ofthe gross reproductionrate for an estimated mean age of the fertility schedule of, say,28.1 years requires additional interpolations. The procedure to befollowed is to interpolate (with factors of .55 and .45) between theGRR's for m= 27 and 29 in the model stable populations at levels7and 9 with r = .010 and .015 to establish GRR for m = 28.1 inthese four tabulated stable populations, and then to employ stepsanalogous to those used in estimating the birth rate.

These examples have all been selected from model life table andstable populations, but the same principles and procedures applyto other tables equally well.

f,

126

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