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A modification for use in destabilizedpopulations of brass's technique forestimating completeness of deathregistrationLinda G. MartinPublished online: 08 Nov 2011.

To cite this article: Linda G. Martin (1980) A modification for use in destabilized populationsof brass's technique for estimating completeness of death registration, Population Studies: AJournal of Demography, 34:2, 381-395

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A Modification for Use in Destabilized Populations of Brass's Technique for Estimating Completeness of

Death Registration

L I N D A G. M A R T I N * t

BRASS'S TECHNIQUE

A method of estimating completeness of death registration using only data on deaths and popu- lation by age and sex for a particular period has been developed by Brass) The basis of his tech- nique can be viewed as an application of the identity r = b - d which holds in all closed populations, where b, d and r stand for birth rate, death rate and rate of increase respectively.

This relation also holds for the sub-set of a population composed of those above a given

age a: r(a +) = b(a +) - d ( a +) (1) or

d(a +) = b(a +) --r(a +), (2)

where b(a +) is the birth rate above age a, if achievement of age a is regarded as being 'born' into the population age a and over; r(a +) is the growth rate and d(a +) the death rate of the popu- lation aged a and over. Equation (2) can be re-written

Da na - - r(a +), (3)

~va - N a

where na is the number of persons reaching age a per year, and Na and Da are respectively the number at risk and the number reported dying at ages a and above per year. The various rates in the equations may be interpreted either as annual rates, i.e. representing the population's age distribution, growth, and deaths in a single year, or as average rates for a population over an inter- censal period, of ten years.

Given data on the population by age at one time, the number of persons reaching age a in a year may be approximated as follows:

sea-s + sPa n a - (4)

10

where sPa-s and sP a are respectively the populations aged a - 5 to a and a to a + 5 at the time under consideration. If, on the other hand, one is interested in the experience of the population over an intercensal period of t years, then the average n a representative of the period may be

calculated as: sPa- s (0) + sPa(t) (5)

na = 10

where 0 indicates data from the first census and t indicates data from the later census. For this case, Na may be calculated as the average of the Na's at the two dates.

* Linda G. Martin is a Research Associate at the East-West Population Institute and an Assistant Professor of Economics at the University of Hawaii.

t I would like to thank Ansley Coale, Bryan Boulier, and James Trussell for their advice and encouragement throughout my research on this topic. Elizabeth Gould, Andrew Mason, Robert Retherford, and the referee made useful suggestions in the preparation of the paper.

W. Brass, 'Estimating Mortality from Deficienl Registration Data,' in Methods for Estimating Fertility and Mortality from Limited and Defective Data (University of North Carolina: POPLAB Occasional Publication, 1975).

381

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382 LINDA G. MARTIN

If deaths in a population are not accurately measured, Equation (2) may be modified to:

fd(a +) = b(a +) - r ( a +), (6)

where d(a +) is the death rate at age a and over calculated using the reported number as opposed to the true number of deaths, and ]'is the ratio of true to reported deaths. Rearranging terms,

d(a +) = v{b(a +) --r(a +)}, (7)

where v = 1/for the ratio of reported to true deaths. In a stable population r(a +) in the above equations may be replaced by r, the rate of

natural increase per year for the whole population. Equation (7) then becomes:

d(a +) = v{b(a +)- - r} . (8)

In a stable population the age distribution does not change and, therefore, all age groups must be growing at the same speed as the whole population.

To assess the completeness of death registration, Brass suggests on the basis of Equation (8) that d(a +) be plotted against the b(a +) for each age. If the points fall on a straight line, its slope will provide an estimate of completeness, v. The growth rate, r, can be ignored, since it is constant and thus does not affect the slope of the line.

There are, however, several conditions necessary before Equation (8) can apply. First, measurement of completeness of death registration by such a procedure is based on the assump- tion that completeness is the same at each age. Secondly, as in the case of the simpler identity, Equation (8) holds only in a closed population. Therefore, it does not hold for populations in which there is considerable migration. Thirdly, it is assumed that there is no misreporting of age either of the population or of deaths, nor any differential coverage by age in the population count. Finally, the assumption of stability is crucial. In a population that is not stable, i.e., where fertility and/or mortality have been changing, the rate of natural increase may be different for each age group, so that r(a +), the growth rate of the population age a and above, must be used instead of r in the equations.

Failure o f the Stability Assumption o f Brass's Technique

As regards the last caveat, Brass asserts that deviations from stability probably do not introduce much bias into the estimate of the completeness of death registration, especially if only mortality is changing. 2 However, investigation of simulated populations destabilized in just such a manner casts doubt on the robustness of the procedure? For example, beginning with a stable female population from the Coale-Demeny tables 4 with West mortality level 5 (expectation of life at birth, eo ~ = 30), a gross reproduction rate of 2.5, and a mean age of childbearing of 29, and allowing mortality to change at a rate of 1.17 levels every five years, we fred that Brass's procedure produces estimates of completeness considerably lower than the assumed level of 100%. In par- ticular, using a variant of Bartlett's technique s to calculate nine different estimates of the slope of

2 Ibid., p. 119. The populations over time were simulated using a modified version of the FIVSIN population projection

package which is documented in F. C. Shorter with progIamming assistance from D. Pasta, Computational Methods for Population Pro/ections: With Particular Reference to Development Planning (New York' The Population Council, 1974).

4 A. J. Coale and P. Demeny, Regional Model Life Tables and Stable Populations (Princeton: Pzinceton University Press, 1966).

s The slope of the line formed by the b(a +) and d(a +) points was estimated using a procedure employed by Barclay, et al., in 'A Reassessrnent of the Demography of Traditional Rural China,' Population Index, October 1976, pp. 606--635. It is based on a technique presented in 1949 by M. S. Bartlett in 'Fitting a Straight Line When Both Variables Are Subject to Error,' Biometrics 5, pp. 207-212. The observations are divided into three ~roups after being ordered according to the most reliably measured variable, which in most populations is d(a +). The death rate above age a is more reliable than the birth rate since the former is the quotient of two

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A MODIFICATION OF BRASS'S TECHNIQUE 383

the line for each duration of mortality change yields median estimates of completeness of 95.47 per cent after ten years of mortality change, 89.74 per cent after 20 years, and 88.87 per cent after 30 years. This example is in no way exceptional.

In Table 1 are listed 20 simulations of female populations with original West mortality levels of 5, 8, 11, and 14, gross reproduction rates of 2.5 and 3.5; mean age of childbearing 29; and various speeds of decrease in mortality. The estimates of the completeness of death registration, when it is in fact perfect, resulting from use of Brass's technique are presented under the 'unmodified' headings of the table. Clearly, use of the method in situations in which mortality has been declining leads to underestimation of the completeness of death registration. The degree of underestimation varies with original mortality level, as well as with the duration and speed of the mortality change. 6

Also in Table 1 in parentheses are the ranges of the nine estimates of the slope calculated for each case. The range indicates the degree of linearity of the curve whose slope is being estimated. The smaller the range, the closer are the points to a straight line, and thus the more certain is the estimate of the completeness of death registration.

Pattern of r(a +)

Thus, when mortality has been changing, the variation of the r(a +)'s by age must be taken into account in estimating completeness of death registration, that is, r(a +) should be subtracted from b(a +) for each age group before plotting the b(a +)'s and d (a +)'s. Unfortunately, it is not usually possible to measure r(a +) very accurately, especially in the types of populations for which Brass's technique was originally developed. Accurate measurement of r(a +) requires enumerations on two different occasions with the same completeness of enumeration. For example, a common way of calculating r(a +) over a period of t years is:

r(a +) = ~loge {toPa(t)/toPa(O)} (9)

where wPa is the population aged a and over. In countries with poor death registration systems complete enumeration of the population is also often lacking. Therefore, to estimate complete- ness of death registration in a population in which mortality has been changing, it would be helpful to have an indirect way of estimating the r(a +)'s from the limited data available.

Fortunately, the r(a +)'s of a simulated destabilized population when plotted by age follow a very smooth J-shaped pattern with the minimum almost always occurring at an age equal to the duration of the change in mortality. Figure 1 shows the r(a +)'s by age after 20 years of mortality change for the destabilized population used as an example in the last section. Given this J-pattern,

cumulated sums, D a and N a, whereas the numerator of the latter is not formed by cumulation and thus is more sensitive to measurement error for a particular age group. The median values of d(a +) and b (a +) in the highest group and the lowest group are calculated, and the completeness is estimated by the slope of the line drawn between these median points. Of course, the estimate of completeness will vary depending on what age )ange of observations is used. The results presented here represent the median values of the slope estimates made with starting ages 15, 20, and 25, and final ages 55, 60, and 65, i.e., the median of nine estimates of the slope for each case at each duration of mortality change.

Although the figures from the simulated populations are not subject to error, for convenience of com- puter programming the above technique was used for analyzing data from both simulated and real populations. The results for the simulated populations would have been only slightly different if ordinary least squares had been used to estimate the slope of the line.

6 Another, contemporaneous investigation of this issue has been carried out by H. M. Rachad in The Esti- mation of Adult Mortality from Defective Data', Ph.D. dissertation, Faculty of Medicine, University of London, London School of Hygiene and Tropical Medicine, 1978. Rachad works with the logit life table model system and also develops a modification of Brass's technique for use in populations where mortality has been changing, but does not recommend its actual application as compared with that of the original technique.

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384 LINDA G. MARTIN

Table 1. Median and Range of Estimates of Completeness of Death Registration for Selected Female Populations with Constant Change in Mortality Level Using the Modified and Unmodified

Brass Techniques

Duration of Mortality Change

10 Years 20 Years 30 Years

Case a Modified Unmodified Modified Unmodified Modified Unmodified

0525033 100.16% 98.79 100.06 96.83 100.47 96.40 (0.27) (0.39) (0.22) (0.17) (0.20) (1.23)

0535033 99.99 98.60 100.00 96.77 100.57 96.50 (0.26) (0.10) (0.14) (0.25) (0.27) (1.50)

0525117 100.14 95.47 100.91 89.74 101.30 88.87 (0.21) (0.53) (0.41) (0.70) (0.78) (4.38)

0535117 99.85 95.20 100.69 89.51 100.81 88.94 (0.31) (0.25) (0.43) (0.80) (1.74) (4.97)

0525248 100.19 90.79 100.50 80.90 103.65 81.03 (0.18) (0.47) (0.34) (1.44) (4.18) (9.00)

0535248 99.76 90.35 100.01 80.26 104.31 86.71 (0.47) (0.49) (0.49) (1.68) (8.27) (11.61)

0825045 99.99 98.63 100.15 96.31 100.12 95.64 (0.10) (0.25) (0.12) (0.33) (0.11) (1.45)

0835045 100.53 99.11 100.46 96.53 100.27 95.70 (0.39) (0.26) (0.10) (0.41) (0.08) (1.58)

0825163 99.97 94.60 99.72 87.93 100.90 87.60 (0.12) (0.52) (0.22) (1.15) (1.43) (5.16)

0835163 100.47 94.92 99.86 87.81 101.27 87.59 (0.45) (0.32) (0.28) (1.33) (3.11) (6.46)

0825311 99.11 90.05 100.25 79.86 b b (0.21) (0.83) (0.36) (1.77)

0835311 99.54 90.24 100.31 79.45 b b (0.44) (0.67) (0.42) (2.06)

1125062 99.56 98.01 99.66 95.41 99.93 95.20 (0.15) (0.17) (0.17) (0.37) (0.11) (1.68)

1135062 100.43 98.84 99.82 95.45 99.95 95.09 (0.33) (0.54) (0.07) (0.34) (0.49) (2.07)

1125214 99.05 93.76 99.73 86.66 99.38 80.83 (0.18) (0.54) (0.20) (1.15) (0.42) (5.14)

1135214 99.88 94.40 99.85 86.43 99.73 80.33 (0.48) (0.97) (0.22) (1.20) (2.29) (6.75)

1425079 100.17 98.67 99.91 95.38 99.54 94.12 (0.22) (0.33) (0.36) (0.61) (0.16) (1.85)

1435079 100.42 98.87 99.90 95.27 99.73 94.13 (0.48) (0.68) (0.24) (0.33) (0.30) (2.05)

1425200 100.16 95.64 99.76 86.79 b b (0.16) (0.50) (0.76) (1.64)

1435200 100.41 95.73 99.70 86.34 b b (0.50) (0.96) (0.27) (1.50)

a Seven-digit case numbers are coded as follows: Digits 1 and 2 Coale-Demeny mortality level of original stable population Digits 3 and 4 gross reproduction rate, e.g., 25 implies GRR = 2.5 Digits 5, 6, 7 speed at which the mortality level is changing measured as the number of levels traversed in five

yeats, e.g., 033 implies that it is increasing 0.33 levels every f'rce years; 117 implies 1.17 levels every five years

b No estimates for these cases at the indicated durations were possible since the Coale-Demeny model life tables do not provide for increases in mortality level beyond 24.

it is easy to see why use of Brass's technique in populations destabilized by mortal i ty change leads

to underestimates o f completeness o f death registration. Because bo th b(a +) and d(a +) increase with age after childhood in most populations and because r(a +) is much bigger at some of the older ages than at younger ages and has not been subtracted from b(a +) before the latter is

p lot ted with d(a +) for that age, the curve is pulled down at the upper ages. Consequently, the

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A MODIFICATION OF BRASS'S TECHNIQUE 385

.025

.020

A §

z .

,015

.010

. 0 0 8 I t I I I I I I I [ I I I I I

0 10 20 30 40 50 60 70 75 Age

Figure 1. r(a +) After 20 Years of Mortality Change for the Female Population with Original West Mortality Level 5, Original Gross Reproduction Rate = 2.5, and Mortality Changing 1.17 Levels Every Five Years

completeness of death registration, as estimated by the slope of the plot of d(a +) against b (a +), is underestimated.

The reason for the J-shaped pattern of the r(a +)'s is that as mortality changes in most populations and in the simulated destabilized populations analyzed here, survival probabilities in childhood and at the older ages improve more than at other ages. Coale has observed that when the age-specific death rates of one model life table are subtracted from those of another of the same family and sex, the differences follow a U-shaped pattern by age. 7 This pattern can be characterized by three parameters: one representing the relatively slight average change in rates between ages 5 and 45, another representing the extent to which mortality change between ages 0 to 5 exceeds that between 5 and 45, and a third representing the rate at which mortality change at the older ages which exceeds the average change between 5 and 45 increases with age above 45. This common U-shaped pattern leads directly to the observed J-pattern of the r(a +)'s.

The basis for the J-pattern can be seen more clearly in mathematical terms, following Coale's analysis. Suppose mortality has been decreasing only above age 47.5 years. Assume that the change is linear, so that the annual rate of decrease in mortality at ages above 47.5 is B .y , where

7 A. J. Coale, The Growth and Structure of Human Populations (Princeton: Princeton University Press, 1972), pp. 34-36.

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386 LINDA G. MARTIN

y = (a --47.5). Then, writing m(a, t) for the death rate at age a and time t and assuming that mortality starts to decline at time zero,

m(a,t) = m ( a , O ) - B y t (10)

for a t> 47.5. Given this type of mortality change, the number of persons aged a at time t can be expressed as follows: s

n(a,t) = n (a ,O)exp{r t+(B /2 ) (y2 t - y3 /3 ) } for t ~>y (11)

= n(a,O)exp{r t+(B/2)(y t 2 - t 3 / 3 ) } for t < y (12)

where r is the growth rate in the original stable population and also the growth rate at time t of the population aged less than 47.5, and where y = a - 47.5 for a/> 47.5 and y = 0 for a < 47.5. In instantaneous terms, r(a, t), the growth rate of the population age a at time t, can be expressed as:

d log e n (a, t) r(a, t) - (13)

dt

r(a,t) = r+By2/2 for t /> y (14)

= r + B y t - - B f l / 2 for t < y (15) or

= r + B y 2 / 2 - - B ( y - - t ) 2 / 2 for t < y . (16)

Thus, if the onset of mortality decline occurred sufficiently long ago (t>~y), the difference between r(a, t) andthe growth rate in the stable population is simply an increasing function of y, given B. If the onset of mortality decline was more recent, the older ages for which t < y will have benefited from a decline for a period shorter than t years. Accordingly, the growth rates for the populations at those older ages will be smaller than they would have been had the mortality decline started earlier. So if t <y , r(a, t) - r is positively related to t, as well as to y.

To obtain r(a +, t) an average of the r(a, t)'s for ages a and above, weighted by the appro- priate number of persons at age a at time t, must be taken. Thus, r(a +, t) > r for all ages, even if mortality were decreasing only above age 47.5, as is the case in this example. The growth rate above age a at time t would, however, increase monotonically with age.

A similar analysis can be carried out by assuming improvement in survival only in childhood. Putting together the two experiments with restricted mortality change, the J-pattern of the r(a +)'s found in simulated destabilized populations can be better understood. Excessive declines in mortality at the younger and older ages, relative to that at ages between 5 and 45, lead to the left and right arms of the J, with the minimum r(a +) at an age close to the duration of the mortality change.

Relation o f r(a +) to the Speed and Duration o f Mortality Change

Given the above discussion, it is not surprising that it is possible to use information on the speed and duration of mortality change to estimate the pattern of the r(a +)'s for populations for which direct calculation of the r(a +)'s is not recommended. The question that arises is how to measure the speed of the change in populations in which mortality has declined in a less linear and more realistic way than in the above mathematical experiment. Compare the change in mortality in moving from the female West Coale-Demeny model life table at level 3 (eo ~ = 25) to the one at level 7 (eo ~ = 35) with the change in mortality implied by moving from level 7 to level 11 (e ~ = 45). Figure 2 presents a graph of the changes in age-specific death rates for each case. Both

8 For the derivation of the formulae see Linda G. Martin, 'Measuring Completeness of Death Registration in Destabilized Populations', Ph.D. Dissertation, Department of Economics, Princeton University, 1977, pp. 16-20.

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A MODIFICATION OF BRASS'S TECHNIQUE 387

�9 F

.10

A

el

.0!

_ _ L 1 L I I I 1 I 1 I I I I l I

10 20 30 40 50 60 Age

70 80

Figure 2. Change in Age-Specific Death Rates When Moving from One Female West Model Life Table to Another

mortality changes lead to the expected U-pattern of the change in death rates by age. The speed of mortality change at the older ages relative to the middle ages can be approximated by the slope of the right-hand branch of the U, measured as follows:

ASM6S -- AsM4s /3 = 20T (17)

where AsM~s is the change over a period of T years in the death rate for those aged 65 to 70 and AsMa s is the change for those aged 45 to 50. 9 The difference between the two changes is divided b y 20T, since division by T expresses/3 in units of annual changes in mortality over a period of T years, and division by 20 accounts for the difference in age groups (45-50) to (65-70). It is

9 The slope of the right-hand branch may be estimated using observations for ages other than 45 to 50 and 65 to 70, which will indeed, be necessary in some applications where data are restricted or unreliable at those ages.

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388 LINDA G. MARTIN

important to note that fl summarizes relative mortality change only at the upper ages and may or may not be related to the change in childhood mortality taking place concurrently. In many cases, however, it is possible to approximate the speed of mortality change for the population as a whole using only r, the parameter of change in older adult mortality. 1~

Continuing to use the examples of mortality change presented in Figure 2, fl = 67.2 x 10 -6 for the change from level 3 to level 7 over a ten-year period and 45.1 x 10 -6 for the change from level 7 to level 11. Thus, equal changes in expectation of life at birth may be associated with quite different values of ft.

With this measure of the speed of mortality change and with information on the duration of the change, it is possible to estimate the pattern of r(a +) relative to r(d +), the growth rate above the age equal to the duration of the mortality change in the destabilized population and usually the minimum value of r(a +). Accordingly, the variable

r*(a +) = r(a +)--r(d +). (18)

will be of interest. An estimate of r*(a +) for each age will be sufficient to adjust Brass's procedure for use in populations that have experienced mortality decline. Using r*(a +) rather than r(a +) in Equation (7) does not affect the slope of the line, since r(d +) is independent of a. By first sub- tracting r*(a +) from b(a +) rather than subtracting r(a +), the completeness of death registration can still be assessed by plotting those observations against the d(a +) observations and estimating the slope.

In order to quantify the relation between r*(a +), r, and the duration of mortality change, populations in which mortality has been changing at a constant rate/3 have been investigated. Altogether, 36 populations have been generated for each sex. The original stable populations are drawn from the Coale-Demeny West tables with mortality levels 5, 8, 11 and 14; and gross repro- duction rates of 2.5, 3.0, and 3.5. Three values o f t are used for each level and gross reproduction rate (106/3 = 10, 30 and 50). The subsequent mortality levels implied by the value of ~3 for each case are interpolated from the model life tables.

Using the above populations, the growth rates above age a and the differences between them and the growth rate above the age equal to the duration of the mortality change have been calculated for each population at each duration. The relation between r*(a+) and /3 for each age and each duration of mortality change can now be estimated using ordinary least squares. Remarkably, r*(a+) can be very closely approximated as a simple linear function o f t . That is,

r*(a +) = c + g(3 (19)

where c is a constant and g is a coefficient, derived for each age and each duration of mortality change. In Table 2 the constants, coefficients, and associated R2's for the female populations are presented for each duration and each age. Table 3 contains the same information for the male populations.

It is interesting to note the pattern of the R2's by age for a given duration. As age increases they first decline and then increase, with the minimum at ages close to the duration of the mor- tality change. In general, the amount of variation in r*(a +) explained by/3 is small at the early ages, because, as mentioned earlier, ~ summarizes change in mortality at the older ages only and is not necessarily related to the changes in childhood survival. Fortunately, the value of r*(a +) is usually very small at the ages where least is explained and, hence, error in its estimation will not affect the estimate of completeness.

10 For a discussion of the mathematical relation between changes in child and adult mortality in West female model life tables, see Linda G. Martin, op. cit. in footnote 8, pp. 33-36.

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Tab

le 2

. Reg

ress

ion

Res

ults

for

Fem

ale

Pop

ulat

ions

wit

h C

onst

ant {

3. (r

* (a

+)

= c

+ g

(3)

Dur

atio

n =

10

Yea

rs

Dur

atio

n =

15

Yea

rs

Dur

atio

n =

20

Yea

rs

Age

c

g R2

C

g

R2

C

g R

2

10

-0.0

00

00

1

11.2

9166

2 0.

8810

-0

.00

00

13

34

.763

168

0.91

23

15

-0.0

00

02

1

0.89

5833

0.

3428

-0

.00

00

03

6

.88

15

78

0.

7848

20

-0

.00

00

22

6.

3124

97

0.95

64

-0.0

00

02

2

6.12

4995

0.

9559

25

-

0.00

0031

11

.854

160

0.97

92

-0.0

00

01

8

16.6

0415

6 0.

9811

-

0.00

0019

10

.763

158

0.98

15

;>

30

-0.

0000

38

17.4

5832

8 0.

9844

-

0.00

0029

27

.541

656

0.98

36

-0.

0000

10

25.9

9781

8 0.

9876

a:

35

-0.

0000

44

23.5

0000

0 0.

9847

-

0.00

0037

38

.958

328

0.98

50

-0.

0000

20

42.2

1492

0 0.

9876

0

40

-

0.00

0071

30

.312

500

0.98

45

-0

.00

00

40

51

.625

000

0.98

38

-0.

0000

29

59.7

6754

8 0.

9869

~

45

-0.

0000

80

38.8

5417

2 0.

9869

-

0.00

0067

67

.000

000

0.98

34

-0.

0000

29

80.2

4780

3 0.

9859

"I

j

50

-

0.00

0101

51

.041

656

0.98

95

-0

.00

00

70

87

.999

924

0.98

55

-0.

0000

51

106.

9626

92

0.98

74

(=i

55

-0.

0001

50

68.1

0415

6 0.

9884

-

0.0

00

08

0

117.

3957

37

0.98

90

-0.

0000

39

143.

8245

70

0.99

04 ~

60

-

0.00

0207

90

.604

095

0.99

02

-0.0

00

10

8

156.

9582

98

0.99

21

-0.

0000

34

194.

7544

10

0.9

93

0

0 65

-

0.00

0319

11

8.29

1611

0.

9928

-

0.00

0141

20

7.24

9908

0.

9942

-

0.00

0052

26

1.86

1816

0.

9946

Z

70

-

0.00

0536

15

0.58

3282

0.

9829

-

0.0

00

23

2

268.

2915

04

0.99

56

-0.

0000

86

346.

1557

62

0.99

51

0 75

-

0.00

1209

18

9.62

4924

0.

9506

-

0.00

0461

34

4.31

2256

0.

9914

-

0.00

0219

4

53

.61

84

08

0.

9941

"I

j

1:1:1

Dur

atio

n =

25

Yea

rs

Dur

atio

n =

30

Yea

rs

Dur

atio

n =

35

Yea

rs

:::c

;>

Age

C

g

R2

C

g R2

R

2

CIl

C

g

CIl

til

10

0.

0000

34

54.9

1069

0 0.

9150

0.

0000

73

67.6

1833

2 0

.92

90

0.

0001

04

70.1

1293

0 0.

9149

....

, 15

0.

0000

29

25.0

0595

1 0.

8884

0.

0000

76

43.3

6026

0 0.

8971

0.

0001

09

50.4

2477

4 0.

9027

~

20

0.00

0016

2.

0357

13

0.34

18

0.0

00

03

0

18.2

0431

5 0.

8439

0.

0001

09

30.5

9678

7 0.

8457

::I:

: 25

0.

0000

09

-1.1

61

29

1

0.17

26

0.00

0053

9.

4408

65

0.64

56

Z

30

-0.

0000

17

15.0

4761

2 0.

9827

0.

0000

21

-6.0

43

01

3

0.8

39

0

Z

35

-0.

0000

02

34.9

9403

4 0.

9856

-0

.00

00

11

18

.946

243

0.98

60

c::

40

-

0.00

0005

56

.613

068

0.98

42

0.00

0008

43

.451

630

0.9

90

2

-0.0

00

02

9

25.6

8817

1 0.

9831

tT

l

45

-0.

0000

07

81.2

0233

2 0.

9836

0

.00

00

12

71

.026

932

0.99

01

-0.

0000

25

58.9

6241

8 0.

9864

5

0

0.00

0001

11

1.99

3958

0.

9849

0.

0000

23

104.

6505

89

0.9

90

2

-0.

0000

28

98.3

1187

4 0.

9883

55

-

0.00

0013

15

3.41

0614

0.

9876

0.

0000

45

148.

6290

59

0.99

11

-0.

0000

21

147.

7742

46

0.99

07

60

0.

0000

11

210.

2677

15

0.99

08

0.0

00

04

2

208.

3280

33

0.99

28

0.00

0007

21

2.39

7980

0.

9926

65

0.

0000

25

286.

8218

29

0.99

31

0.00

0076

28

9.02

7100

0

.99

38

-

0.00

0008

29

8.72

6074

0.

9944

70

-

0.00

0001

38

6.81

5186

0.

9939

0.

0000

93

396.

5058

59

0.99

41

-0.

0000

04

414.

8872

07

0.99

53

75

-0.

0000

76

518.

1425

78

0.99

32

0.0

00

03

4

542.

8767

09

0.99

41

-0.

0000

51

574.

6938

48

0.99

57

IoN

0

0

\0

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w ~

Tab

le 3

. Reg

ress

ion

Res

ults

for

Mal

e P

opul

atio

ns w

ith

Con

stan

t fI

(r*

(a +

) =

c +

g(3)

Dur

atio

n =

10 Y

ears

D

urat

ion

= 15

Yea

rs

Dur

atio

n =

20 Y

ears

--

----

----

----

----

----

----

---

Age

c

g R

2

C

g R

2

C

g R

2

10

-0.

0000

09

13.8

7499

3 0.

8794

-

0.00

0029

42

.975

876

0.90

87

15

-0.

0000

23

1.85

4166

0.

6481

-

0.00

0017

7.

8771

94

0.6

78

9

20

-0.

0000

20

9.74

9993

0.

9632

-

0.00

0012

9.

0624

97

0.96

74

25

-0.

0000

22

18.2

2915

6 0.

9758

0.

0000

06

24.2

9915

6 0.

9822

-

0.00

0005

15

.956

139

0.98

03

30

-0.

0000

29

26.9

1665

6 0.

9812

0.

0000

16

40.2

4998

5 0.

9858

0.

0000

20

38.3

9694

2 0.

9896

35

-

0.00

0028

36

.312

485

0.98

61

0.00

0021

57

.520

813

0.98

89

0.00

0039

62

.776

321

0.99

21

40

-

0.00

0037

47

.000

000

0.98

80

0.00

0031

76

.729

141

0.99

16

0.00

0049

89

.535

065

0.9

94

0

45

-0.

0000

51

58.6

8748

5 0.

9887

0.

0000

31

98.3

7495

4 0.

9937

0.

0000

65

119.

4439

17

0.9

95

2

50

-0.

0000

59

72.5

0000

0 0.

9891

0.

0000

25

123.

3748

93

0.99

45

0.00

0070

15

3.88

1577

0.

9960

55

-

0.00

0082

89

.416

641

0.98

84

0.00

0033

15

3.45

8282

0.

9949

0.

0000

77

194.

4538

88

0.9

96

4!:

:::

60

-

0.00

0123

11

1.14

5752

0.

9876

0.

0000

30

191.

7499

24

0.99

55

0.00

0096

24

5.35

9665

0.

9964

Z

65

-

0.00

0197

13

9.37

4954

0.

9850

0.

0000

26

241.

1457

52

0.99

58

0.00

0108

31

1.10

5225

0

.99

68

~

70

-0.

0004

27

173.

6249

69

0.97

81

-0.

0000

12

303.

5832

52

0.99

57

0.00

0114

39

4.91

8945

0.

9971

' 0

75

-

0.00

0978

21

5.47

9111

0.

9600

-

0.00

0261

38

3.37

4756

0.

9933

0.

0000

48

505.

5898

44

0.9

96

5· a:::

D

urat

ion

= 25

Yea

rs

Dur

atio

n =

30 Y

ears

D

urat

ion

= 35

Yea

rs

>

:;:c

Age

c

g R

2

C

K

R2

C

g

R2

::l Z

10

0.

0000

11

65.9

9403

4 0.

9099

0.

0000

36

76.3

6027

5 0.

9283

0.

0000

81

73.1

2503

1 0.

8937

15

0.

0000

10

29.2

7378

8 0.

8411

0.

0000

65

46.9

3014

5 0.

8458

0.

0000

78

49.1

6668

7 0

.86

00

20

0.

0000

09

0.28

5714

0.

0042

0.

0000

31

16.0

0538

6 0.

6422

0.

0000

78

26.0

0001

5 0.

7723

25

0.

0000

05

-6.

1344

10

0.81

28

0.00

0041

1.

7916

67

0.04

95

30

-0.

0000

09

23.2

1427

9 0.

9923

0.

0000

14

-13

.74

99

98

0

.95

83

35

0.

0000

18

53.8

9283

8 0.

9945

-

0.00

0006

30

.973

129

0.99

55

40

0.00

0038

87

.607

056

0.99

50

0.00

0036

70

.188

202

0.99

58

-0.

0000

20

40.5

0001

5 0

.99

38

45

0.

0000

49

125.

0118

56

0.99

57

0.00

0061

11

3.82

7988

0.

9954

0.

0000

03

90.7

9168

7 0.

9941

50

0.

0000

66

167.

6963

20

0.99

60

-0.

0000

12

168.

7527

77

0.99

44

0.00

0020

14

6.83

3267

0.

9941

55

0.

0000

82

217.

6665

04

0.99

64

0.00

0106

22

0.18

8248

0.

9966

0.

0000

22

211.

8749

08

0.99

53

60

0.

0000

89

278.

9582

52

0.99

67

0.00

0133

28

9.57

0068

0.

9973

0.

0000

43

289.

6667

48

0.99

65

65

0.00

0127

35

6.38

6475

0.

9970

0.

0001

60

376.

2690

43

0.99

78

0.00

0063

38

6.25

0000

0.

9976

70

0.

0001

39

456.

6901

86

0.99

72

0.00

0203

48

7.55

3955

0.

9979

0.

0000

63

509.

1250

00

0.99

76

75

0.00

0077

59

1.94

6289

0.

9965

0.

0001

68

639.

6079

10

0.99

75

0.00

0050

67

3.62

5000

0.

9965

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A MODIFICATION OF BRASS'S TECHNIQUE 391

Modification of Brass's Technique

Given the establishment of the relation between the r*(a +)'s and the speed and duration of mortality change, it is possible to modify Brass's technique for use in populations destabilized by mortality change. First, using information on the duration and speed of mortality change and the coefficients and constants in Tables 2 and 3, estimate the r*(a +)'s. Then subtract the r*(a +)'s from the b(a +)'s and plot the results against the d(a +)'s. Finally, calculate the slope of the line formed by the observations to obtain an estimate of the completeness of death registration.

Such an indirect procedure for estimating the r*(a +)'s and estimating the completeness of death registration has been tested in the 36 female populations with mortality changing at a constant rate/~ that were used originally to estimate the relation between r*(a +) and/3. In all cases estimates of completeness of death registration using this method are much closer to 100 per cent than those obtained by Brass's unmodified technique. Table 4 presents a summary of the results. For each duration of change, the range of the final median estimates of completeness for the 36 populations using first the modified and then the unmodified techniques is shown. Obviously, the allowance for the change in mortality leads to more accurate estimates of the completeness of death registration than those obtained using Brass's unmodified technique.

Table 4. Summary of Test Results. Range of Median Estimates of Completeness of Death Regis- tration for 36 Female Populations with Constant ~ Using the Modified (M) and Unmodified (U)

Techniques

Duration (in years) Technique Range

10 M 99.35 to 100.55 U 90.14 to 99.15

15 M 98.37 to 101.08 U 80.17 to 97.66

20 M 98.45 to 101.38 U 75.17 to 96.88

25 M 97.33 to 101.14 U 74.36 to 96.46

30 M 98.03 to 100.95 U 76.14 to 96.38

35 M 97.53 to 101.23 U 75.76 to 96.96

Testing the Modified Technique in Populations Where/3 Is Not Constant

What happens where mortality has been changing in a way that cannot be summarized by a constant rate /~? The modified technique has been applied to the cases of the 20 female popu- lations cited in Table 1 with constant increases of e ~ , but where/3 is not constant. Table 5 shows for each population and each duration of mortality change the value of/3 implied by the entire duration of the mortality change, as well as that implied by only the previous ten years' change in mortality. Shown in Table 1 under the heading 'Modified' are the estimates of completeness of death registration resulting from use of the modification and ~ for the entire duration. In all cases, use of the modified technique produces an estimate of completeness of death regis- tration closer to 100 per cent than does the unmodified method. Furthermore, in 51 of the 56 instances the range of estimates using the modification is smaller, indicating less variability in the estimation of the slope of the line.

The cases where the modification works least well are those in which there are large deviations from the assumption that/3 is constant, on which the coefficients and constants used in the modification are based. For example, in case 0525248 at duration 30 there has been consider- able change in the rate at which mortality has been changing as indicated by the values of/3 over

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392 LINDA G. MARTIN

Table 5. Relation of Change in Mortality Level and/3 in Female West Destabilized Populations

Average 13 x 10 ~ for the Entire Duration of Mortality Change (13 X l0 s for the Ten Years Prior to the Given Duration of Mortality Change)

Start ,x Level/ Level Five Years Duration: 10 Duration: 20 Duration: 30

5 0.33 11.5 9.5 10.0 (11.5) (7.4) (11.0)

5 1.17 33.7 31.0 28.1 (33.7) (28.2) (22.2)

5 2.48 65.2 52.2 45.5 (65.2) (39.2) (32.3)

8 0.45 10.6 10.2 9.8 (10.6) (9.7) (10.7)

8 1.63 35.0 29.6 26.8 (35.0) (24.2) (21.1)

8 3.11 57.4 48.7 - (57.4) (40.0)

11 0.62 10.8 10.3 9.4 (10.8) (9.9) (7.5)

11 2.14 31.9 29.9 33.8 (31.9) (27.9) (41.6)

14 0.79 9.9 10.1 10.0 (9.9) (10.3) (9.8)

14 2.00 25.4 27.6 - (25.4) (29.8)

for ten-year periods in Table 5. Use of the average for the entire duration leads to an overestimate of the r* (a +) ' s and of the completeness of death registration (103.65 per cent). On the other hand, use of the most recent ten-year period for estimating/3 would give an underestimate (95.88 per cent, not shown in the tables). Even in this extreme case, however, Brass's modified technique gives a considerably more accurate estimate than does the unmodified.

An Example of the Use of the Modification of Brass's Technique in a Population Destabilized by Mortality Change: Costa Rican Females in 1963

The female population of Costa Rica in 1963 provides an excellent case for applying the modified technique. Four sets of information are needed for the application: (1) the age-sex distribution of deaths for at least one period; (2) the age-sex distribution of the population at approximately the middle of that period; (3) an estimate of how long mortality has been changing; and (4) an estimate of the rate at which adult mortality has been changing. Estimation of the last two param- eters will require knowledge of the age distribution of deaths and population for two periods. In this analysis, the census statistics for the years 1950 and 1963 are used. 11 The deaths for 1962 to 1964 are averaged to provide a more reliable estimate of the deaths in 1963.12 It is not possible to use the average number of deaths during 1949 to 1951 for the 1950 figure, since death statistics by age and sex are available only for 1950.13

The change in adult female death rates between 1950 and 1963 follows the typical U-shaped pattern by age. Because of irregularities in the curve three estimates of/3 are made. One is based

on Equation (17) above and results in a/3 of 41.9 x 10-6; another uses the average of AsM4o,

11 Census figures for 1950 were taken from Demographic Yearbook 1952 (New York: United Nations, 1952), p. 135, and for 1963 are from Demographic Yearbook 1966 (New York: United Nations, 1967), pp. 146-147.

1~ Death data for 1962, 1963, and 1964 were taken from Demographic Yearbook 1966, pp. 370-371. 13 Death statistics for 1950 were taken from Demographic Yearbook 1957 (New York: United Nations,

1957), pp. 228-229.

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A MODIFICATION OF BRASS'S TECHNIQUE 393

AsM4s and AsMso as the lower point and the average of AsM6o, AsM6s and AsMTo as the upper point, and results in/3 -- 42.8 x 10 -6 ; a third uses the same lower point as the second, but the

average of AsMss, AsM6o and AsM6s as the upper point to obtain/3 = 32.0 x 10 -6.14 The first estimate, the median of the three, is used in the following analysis, is

The duration o f the mortality change is ascertained by calculating the values of r(a +) for the intercensal period, plotting them by age, and observing at what age the minimum occurs.

Although the r(a +)'s obtained directly from the data are very unreliable and should not be used in the estimation of completeness, they can give a fairly good idea of the duration o f mortality change. 16 Figure 3 shows the r(a +)'s for the periods 1940 to 1950 and 1950 to 1963 for Costa

Rican females. The minimum occurs at age 10 for the period ending in 1950 and at ages 15 or 20 for the period ending in 1963. Since investigation of r(a +)'s for Costa Rican males indicates a

duration of mortality change of 20 years for 1963, that figure is used for the females as well.

Given the 1963 population and death statistics, together with the estimates of the speed and duration o f mortality change, it is possible to calculate the d(a +)'s and b (a +) 's and to estimate the r*(a +) 's by using the results of the regression analysis of simulated populations. 17 Using the modified technique, i.e. allowing for mortality change, leads to an estimate o f 96 per cent completeness o f death registration. When the unmodified technique is applied to the data for Costa Rican females in 1963, an estimate of completeness of 79 per cent is obtained. The former estimate when used to adjust mortality statistics for construction of a life table implies a South mortality level o f 18.46 (e ~ --63.65 years); the latter implies a South mortality level of 16.51 (e ~ -- 58.78 years). Given the potential for error in various steps along the way, it is not possible to state conclusively that one estimate is better than another, but the preceding analysis lends greater credence to the modified estimate. Moreover, support for the higher expectation o f life is provided by an independent analysis of Costa Rican mortality by ArriagaJ a

14 The difference between the upper and lower points in the third case should be divided by 15 multiplied by the time elapsed, rather than by the product of 20 and the time elapsed as in the other two cases.

, s It should be noted that the mortality statistics used in the estimation of B have not been adjusted for underregistration and there is no assurance that completeness of death registration has not changed over the period examined. A more elaborate procedure would be to make an initial adjustment of the mortality rates for incompleteness at each time and then to estimate/~ from the adjusted death rates, but it is not clear that such an effort would be rewarded by increased accuracy. It is shown in Linda G. Martin, op. cit. in footnote 8, pp. 102-106, that an error as large as 33 per cent in the estimation of B is not necessarily intolerable in a typical destabilized population. In the case of such an error, use of the modified technique still leads to more accurate estimation of the completeness of death registration than use of the unmodified technique. Similarly, a five-year error in the estimate of duration of mortality change is not critical.

~ If the r(a +)'s calculated from the census are subtracted from the b(a +)'s and the results plotted against the d(a +)'s, a clearly absurd median completeness estimate of 137 per cent is obtained.

17 In the calculations using both the modified and the unmodified techniques, the b(a +)'s for the 1963 Costa Rican female population are smoothed by using the b(a +)'s from a Coale-Demeny model stationary population. Such smoothing does not lead to large changes in the median estimate of completeness, but rather straightens the plot of the b(a +)'s and d(a +)'s and makes it easier to assess the slope of the line.

Schedules of b (a +)'s by age for stable populations, including stationary ones, and for populations destabilized by mortality change show much the same monotonically increasing pattern, similar to an exponential. When the b (a +)'s from various stationary populations with different mortality levels are plotted against each other by age, a straight line is formed, with the slope of the line approaching unity as the mor- tality levels of the populations compared are closer together. Similarly, when b(a +)'s from a stationary popu- lation are plotted against b(a +)'s from a destabilized population, a straight line results, although a slope close to unity does not necessarily result when the mortality levels are the same.

In practice, the choice of the stationary population by which to smooth the b (a +)'s of a real destabilized population can be made by regressing the b(a +)'s from the real population on those from various stationary populations using only those observations which form a fakly straight line. Select the estimated b(a +)'s to be used in the completeness aoalysis from the regression with a high coefficient of determination and a slope close to unity. For a more complete discussion of the procedure, see Linda G. Martin, op. cit. in footnote 8, pp. 106-113.

,s E. E. Arriaga, New Life Tables for Latin American Populations in the Nineteenth and Twentieth Cen- turies (Berkeley: Institute of International Studies, University of California, 1968).

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394 LINDA G. MARTIN

.05

.04

§

.03

1950 to 1963

1940 to 195

, 0 2 I ! I I 1 t I 1 I I I I 0 10 20 30 40 50 60

Age

Figure 3. r(a +) for Costa Riean females for the periods 1940 to 1950 and 1950 to 1963. Sources: Figures for 1950 and 1963 are from sources indicated in the text. Population data for 1940 are from United States Depart- ment of Commerce, Bureau of the Census, Costa Rica: Summary of Biostatisties (Washington: U.S. Government, 1944), p. 44.

CONCLUSION

It has been demonstrated that the applicability o f Brass's method of estimating completeness o f death registration is sensitive to the fulfillment o f one o f the assumptions upon which the method is based. In particular, the circumstances of recent mortality change may be a reason for caution in use o f the method.

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A MODIFICATION OF BRASS'S TECHNIQUE 395

The modification of the technique for mortality change leads to estimates of completeness in simulated populations that are more accurate than those using the original technique. The modification, however, relies on fairly good knowledge of the duration and rate of change of mortality, which may not be available in real populations. Furthermore, its applicability is limited to populations in which the rate of change of mortality has not varied too greatly.

In addition, the estimation of completeness by means of either the original technique or its modification can be hampered in real populations by an unreliable age distribution. Age mis- reporting or differential coverage by age in the population count can lead to large variations in the estimates of completeness. Moreover, the other assumptions upon which Brass's technique is based, i.e. no fertility change, no migration, no differential coverage by age in death registration, and no age misreporting of deaths, may not hold in practice.

Despite its limitations, the modification of Brass's technique for estimating completeness of death registration for use in destabilized populations can be useful. The straightness of the plot of the d(a +)'s and {b(a +)--r*(a +)}'s gives an idea of how well the modification works. Where possible, comparison should be made with mortality estimates based on other methods of analysis. 19 In the worst of all possible situations, use of both the modified and the unmodified techniques provides a range of estimates within which the true completeness of death registration most probably lies.

~9 A possible alternative that does not use population age-distribution statistics has been suggested by S. Preston in 'Estimating the Completeness of Death Registration,' mimeograph, 1978. The sensitivity of the technique to the failure of the stability assumption upon which it is based is not known, although a few exploratory calculations indicate that it may be more robust than Brass's technique.

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