verth's application of event history on clinical tests

STATISTICAL ANALYSIS OF SURVIVAL DATA

Department of Biostatistics

May 11 2004University of Aarhus

Michael Vth

STATISTICAL ANALYSIS OF SURVIVAL DATA

IN CLINICAL RESEARCH 4The main topic in the third period is analysis of aggregated survival time data, i.e. data in which a record reflects the survival experience of several individuals in a given time and/or age period. Such data are often encountered in epidemiological studies and the methods presented below are essentially identical to methods used to analyze incidence rates and mortality rates in epidemiology.A comprehensive coverage of analysis of aggregated survival time data is beyond the scope of this course, but the main approaches will be presented and exemplified.

One additional topic related to survival time data with individual records are also presented today: Calculation of the expected survival curve based on life tables for an external reference population

ANALYSIS OF SURVIVAL TIME DATA - RELATION TO METHODS USED IN EPIDEMIOLOGYThe statistical methods described in period 1 and 2 have focused on mortality rates and modeled how the rates depend on prognostic factors.

A clinical study of a group of patients followed until death or some common closing date may be viewed as an epidemiological study of a fixed cohort. The methods for analysis of survival time data are closely related to methods used for analysis of incidence and mortality rates in epidemiological cohort studies.

In epidemiology event rates are computed as

The time scale and/or age scale is often split in a number of intervals (e.g. 5-years intervals) and separate rates are computed for each time/age interval.The effect of an exposure on the occurrence of the event can be expressed as a rate ratio, which can be estimated at a crude rate ratio or stratified on age/time categories as well as other risk factors. Poisson regression is used for more comprehensive analysis.

In the Cox regression analysis the hazard rate is unspecified, i.e. no restriction is imposed on the way the hazard rate depends on time and the shape of the estimated baseline functions - hazard rate and survival function is determined completely by the data.

Alternatively, a parametric description of the hazard rate may be postulated and the unknown parameters are then estimated from the data, typically as maximum-likelihood estimates

A simple parametric model the exponential distribution

The simplest possible parametric model for the hazard rate is assuming an unknown, constant rate. The distribution of life times with a constant hazard rate is called the exponential distribution. In this case we have that

The maximum-likelihood estimate of the constant hazard rate is

The standard error of is estimated by

A 95% confidence intervals for the unknown rate is usually obtained by computing a symmetric confidence interval for ln(rate) and transforming this interval back to the original scale.One may show that

,

so a 95% confidence interval for the constant hazard rate has

lower bound

upper bound

where

Note: The individual survival times are not needed to compute the estimate, the standard errors and the confidence limits. They can all be obtained from directly from the aggregated data d and s.Example: Survival with malignant melanoma

Consider the data used in Exercise 12. First a patient identification number is generated (this is needed for some of the commands) then the data are defined as survival time data gen id=_n

stset survtime , failure(status==1) id(id) noshow scale(365.25)stptime The stptime generates the following output. The calculations are based on the formulas aboveCohort | person-time failures rate [95% Conf. Interval]

---------+----------------------------------------------------

total | 1208.2793 57 .04717452 .0363884 .0611578

Separate rates for each category of a covariate are also availablestptime , by(sex) sex | person-time failures rate [95% Conf. Interval]

---------+----------------------------------------------------

female | 787.46886 28 .03555696 .0245506 .0514976

male | 420.8104 29 .06891465 .0478903 .0991689

---------+----------------------------------------------------

total | 1208.2793 57 .04717452 .0363884 .0611578

Only one variable is allowed in the by option. To get separate rates for intervals of follow-up time use the option at(), which may be combined with the by option, e.g.

stptime , at(2(2)8)

stptime , at(2(2)8) by(sex)

Output from the first commandCohort | person-time failures rate [95% Conf. Interval]

---------+----------------------------------------------------

(0 - 2]| 387.25394 15 .03873427 .0233516 .0642502

(2 - 4]| 338.72005 21 .0619981 .0404232 .095088

(4 - 6]| 241.72895 14 .05791611 .034301 .0977896

(6 - 8]| 131.79329 5 .0379382 .0157909 .0911477

> 8 | 108.78303 2 .01838522 .0045981 .0735122

---------+----------------------------------------------------

total | 1208.2793 57 .04717452 .0363884 .0611578

Apparently, the mortality rate initially increases to reach a plateau and then decreases.

The underlying life time distribution is no longer exponential when the rate is computed for different follow-up time intervals. The rate is now piecewise constant. Distributions with piecewise constant hazard rate constitute a flexible class of distributions, which are just as easy to work with as the exponential distribution, which usually provides a too crude picture of the distribution of lifetimes.The distributions are characterized by the value of the hazard rate in each of a number of disjoint intervals:

For interval j from let

the number of events

the total time at risk

Knowledge of the statistics permits calculation of all relevant estimates and test statistics.For each interval the value of the hazard rate is estimated by

,the corresponding standard error becomes

,and 95% confidence intervals are also obtained as before. A distribution with piecewise constant hazard rate can be viewed as a parametric version of the life table and we may in fact estimate the survival function in a way very similar to the one used when computing the life table estimate of the survival function. The probability of surviving the jth interval given alive at the beginning of the interval is estimated by

and the probability of surviving from time 0 until the end of the jth interval is then estimated by

Distributions with piecewise constant hazards provide the link between the methods used for analysis of survival data and the method used for analysis epidemiological cohort studies.

Survival analysis methodology uses individual records whereas the epidemiological analysis usually is based on a multi-way table of aggregated data.

Example. Survival with malignant melanoma..The STATA command

stptime , at(2(2)8) by(sex)

produces the following output

sex | person-time failures rate [95% Conf. Interval]

-------+------------------------------------------------------

female |

(0 - 2]| 243.64408 5 .02052174 .0085417 .0493041

(2 - 4]| 221.13621 11 .0497431 .0275477 .0898214

(4 - 6]| 159.99452 8 .05000171 .0250057 .0999839

(6 - 8]| 86.639288 2 .02308422 .0057733 .0923008

> 8 | 76.054757 2 .02629684 .0065768 .1051463

-------+------------------------------------------------------

male |

(0 - 2]| 143.60986 10 .0696331 .0374664 .1294164

(2 - 4]| 117.58385 10 .0850457 .0457592 .1580614

(4 - 6]| 81.734428 6 .07340848 .0329795 .1633984

(6 - 8]| 45.154004 3 .06643929 .0214281 .2059996

> 8 | 32.728268 0 0 . .

-------+------------------------------------------------------

total | 1208.2793 57 .04717452 .0363884 .0611578

To compare the survival for males and females controlling for follow-up time (categorized in five time intervals) using standard epidemiological methods only the 2x5 person-time and 2x5 failures are needed. STATA has the command stsplit and collapse which can be used to form an aggregated data set. This new data set can then be analyzed by a series of commands for analysis of aggregated data.

First the individual records are split after 2, 4, 6, and 8 years of follow-up.stsplit timecat , at(2(2)8)

We then define the new variables died and risktime from the system variables _d, _t0 and _t, which for each interval gives the event count (0 or 1), the start time and the end time of the interval:

gen died=_d

gen risktime=_t-_t0

All the individual contributions are then aggregated (i.e. summed) in a two-way table of sex versus time interval and the result is saved in a new file

collapse (sum) died risktime , by(timecat sex)

save e:\kurser\survival\melatimesex.dta

To see the context of the new file write

use e:\kurser\survival\\melatimesex.dta

list

. list

+------------------------------------+

| sex timecat died risktime |

|------------------------------------|

1. | female 0 5 243.6441 |

2. | male 0 10 143.6099 |

3. | female 2 11 221.1362 |

4. | male 2 10 117.5838 |

5. | female 4 8 159.9945 |

|------------------------------------|

6. | male 4 6 81.73443 |

7. | female 6 2 86.63929 |

8. | male 6 3 45.154 |

9. | female 8 2 76.05476 |

10. | male 8 0 32.72827 |

+------------------------------------+

Note that timecat takes the lower limit of the interval as category value.

To compute the mortality rate ratio for males versus females stratified on categories of follow-up time writeir died sex risktime , by(timecat)

The command syntax is

ir event-variable exposure-variable time-at-risk-variable ,by(stratum-variable)The exposure variable must have two categories and only one stratum variable is allowed. Output. ir died sex risktime , by(timecat)

timecat | IRR [95% Conf. Interval] M-H Weight

-------------+------------------------------------------------

0 | 3.388889 1.055401 12.63597 1.85567

2 | 1.702619 .6482636 4.416032 3.828909

4 | 1.463415 .4185228 4.809543 2.710744

6 | 2.9 .3322018 34.72104 .6818182

8 | 0 0 12.26261 .6055046

-------------+------------------------------------------------

Crude | 1.936121 1.111589 3.377279 (exact)

M-H combined | 1.936666 1.147481 3.268616

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

Test of homogeneity (M-H) chi2(4) = 1.60 Pr>chi2 = 0.8096

Essentially the same results is obtained by a Cox regression analysis of the original data set with individual records--------------------------------------------------------------

_t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Inter]

--------+-----------------------------------------------------

sex | 1.939011 .5140979 2.50 0.013 1.153182 3.260339--------------------------------------------------------------

EXAMPLE: THE LIFE SPAN STUDYThe mortality of approximately 100,000 survivors after the atomic bombing of Hiroshima and Nagasaki has been followed since October 1950 in a still on-going study called the Life Span Study (LSS). The table below gives aggregated data on cancer mortality from 1971 to 1990 in Hiroshima, only two dose categories are considered here.

Age at

exposureSexDose (Gy)No. in 1950Cancer deathsRisk time

(in 1000 y)

0-19M1392236.77

0.005369610567.33

F1406227.12

0.00542217179.03

20-39M1219372.92

0.005177421823.93

F1506697.82

0.005427226671.48

40-M1431401.59

0.005335523313.50

F1376372.47

0.005405225527.19

LSS: Mortality, all cancers combined, Hiroshima, 1971-90The STATA file hiro7190.dta contains these data. The variable names are agex, sex, dose, cases, and pyr.

The following STATA commands may be used to estimate the effect of exposure stratified on age-at-exposure and sex

use e:\kurser\survival\hiro7190.dta

//combine agex and sex to a single variate

egen agexsex=group(agex sex) //avoid rounding errorreplace pyr=pyr*1000

ir cases dose pyr , by(agexsex)

Output group(agex sex) IRR [95% Conf. Interval] M-H Weight

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

1 | 2.178505 1.323522 3.445787 9.593117

2 | 3.43935 2.029444 5.615747 5.867905 3 | 1.390929 .9539309 1.977719 23.70801

4 | 2.371075 1.792351 3.100524 26.23102

5 | 1.457608 1.01505 2.045349 24.5507

6 | 1.597253 1.099477 2.26114 21.23567

-------------+----------------------------------------------- Crude | 1.955327 1.688804 2.255824 M-H combined | 1.852352 1.607143 2.134972-------------------------------------------------------------

Test of homogeneity (M-H) chi2(5) = 15.53 Pr>chi2 = 0.0083Note: The last stratum variable is moving fastest, i.e. 1 ~ 0-19 M, 2 ~ 0-19 F, 3 ~ 20-39 M, etc.

The rate ratio for exposure effect is almost 2 and highly significant. The test of homogeneity (identical stratum-specific rate ratios) is, however, also statistically significant, indicating that the effect of exposure depends on sex and/or age at exposure of the survivor.

A further investigation of this effect modification requires a more refined method of analysis, a so-called Poisson regression analysis. POISSON REGRESSION

In the Poisson regression analysis the number of events in a given cell of the multi-way table is treated as a Poisson variable with mean equal to . Comment: A Poisson distribution with mean is the limiting distribution of a binomial distribution (n,p) as the n goes to infinity and p tends to zero such that the mean is fixed . Poisson distributions are often used to model occurrence of random events.In a Poisson regression model the rate in a given cell is modeled as a product of factors reflecting the effect of the category levels of the variables defining the multi-way table.

Example: Cancer mortality in the LSSThe LSS data above form a 3x2x2 table with agex, sex, and dose as classifying variables. A Poisson regression model specifies multiplicative structure for mortality rate in the cell given by agex=i, sex=j, dose=k (i = 0,1,2, j = 0,1, k = 0,1)If a reference category is chosen for each of the classifying variables (e.g. i = j = k = 0), the Poisson regression model with no interaction assumes that the rates satisfy

The parameters are rate ratios. The parameter represents the rate ratio of the mortality in the second age-at-exposure category relative to the first category when controlling for sex and dose as independent risk factors.

Poisson models are usually specified as additive models for the ln(rate). Using dummy variables we have

The constant, the parameter , is the ln(rate) in the reference cell and the other -parameters are logarithms of rate ratios. Models with interaction terms may also be used.Poisson regression with STATAThe following commands fit the above Poisson regression model to the LSS data in hiro7190.dta,

Note that the output from the default version (the first command) gives the log-linear parameter estimates. To get rate ratios the option irr must be added (second version)xi:poisson cases i.sex i.agex i.dose ,

exposure(pyr) nolog

xi:poisson cases i.sex i.agex i.dose ,

exposure(pyr) nolog irr

Output

. xi:poisson cases i.sex i.agex i.dose , exposure(pyr) nolog

i.sex _Isex_1-2 (naturally coded; _Isex_1 omitted)

i.agex _Iagex_1-3 (naturally coded; _Iagex_1 omitted)

i.dose _Idose_0-1 (naturally coded; _Idose_0 omitted)

Poisson regression Number of obs = 12

LR chi2(4) = 1150.09

Prob > chi2 = 0.0000

Log likelihood = -47.475645 Pseudo R2 = 0.9237

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

cases | Coef. Std. Err. z P>|z| [95% Conf. Int.]

---------+----------------------------------------------------

_Isex_2 | -.6606352 .054719 -12.07 0.000 -.767882 -.55339_Iagex_2 | 1.529529 .0798769 19.15 0.000 1.37297 1.68609_Iagex_3 | 2.29477 .0796431 28.81 0.000 2.13867 2.45087_Idose_1 | .6165749 .0725606 8.50 0.000 .474358 .75879 _cons | -6.357755 .0712617 -89.22 0.000 -6.49743 -6.2181 pyr | (exposure)

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

. xi:poisson cases i.sex i.agex i.dose , exposure(pyr) irr

******************* first part as above **********************

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

cases | IRR Std. Err. z P>|z| [95% Conf. Int.]

-------------+------------------------------------------------

_Isex_2 | .5165231 .0282636 -12.07 0.000 .463995 .574998_Iagex_2 | 4.616002 .368712 19.15 0.000 3.94707 5.39830

_Iagex_3 | 9.922157 .7902317 28.81 0.000 8.48816 11.5984_Idose_1 | 1.852572 .1344237 8.50 0.000 1.60698 2.13569 pyr | (exposure)

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

The reference group is unexposed males, age 0-19 in August 1945. Note that the constant term is not printed when rate ratios are requested.Parameter estimates are maximum-likelihood estimates. The dose effect is extremely significant and almost identical to the one found previously, on page 18 we had 1.852352. As expected the mortality depends also on sex and age-at-exposure.Does the model fit the data?The table below compares observed count with expected count predicted by the Poisson model fitted above. We have

, where .

unexp.exposed

Age at exposure0-1920-3940-0-1920-3940-

Malesobserved105218233233740

expected116.7191.5232.221.743.350.7

Femalesobserved71266255226937

expected70.8295.4241.511.859.940.6

Illustration: For exposed males aged 20-39 at exposure we have e.g.

The usual goodness-of-fit test becomes 22.27 with 12 5 = 7 degrees of freedom giving a p = 0.0023. STATAs command poisgof computes this statistic and the corresponding likelihood ratio test poisgof

poisgof , pearson

Output

. poisgof

Goodness-of-fit chi2 = 20.61145

Prob > chi2(7) = 0.0044

. poisgof , pearson


Prob > chi2(7) = 0.0023

The fit of the model can be improved by adding interaction terms. The following output shows the result of a series of such model fits. Only the output from the final model is shown. Note the first model, which includes the agex*sex interaction, corresponds to the stratified analysis above.

. quietly

xi:poisson cases i.sex*i.agex i.dose , exposure(pyr) irr

. poisgof


Prob > chi2(5) = 0.0105

. quietly xi:poisson cases i.sex*i.agex i.dose i.dose*i.sex , exposure(pyr) irr

. poisgof


Prob > chi2(4) = 0.0496

. quietly xi: poisson cases i.sex*i.agex i.dose i.dose*i.sex i.dose*i.agex , exposure(pyr) irr

. poisgof


Prob > chi2(2) = 0.3886

. poisson //with no argument the previous fit is displayedPoisson regression Number of obs = 12 LR chi2(9) = 1168.81

Prob > chi2 = 0.0000


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

case | IRR Std. Err. z P>|z| [95% Conf. Inter]

------------+-------------------------------------------------

_Isex_2| .588025 .082229 -3.80 0.000 .447058 .773442 _Iagex_2| 5.79417 .660672 15.41 0.000 4.63376 7.24516 _Iagex_3| 11.3722 1.28203 21.57 0.000 9.11773 14.184_IsexXage_~2| .715692 .11442 -2.09 0.036 .52317 .97907_IsexXage_~3| .891059 .143171 -0.72 0.473 .65034 1.22088 _Idose_1| 2.27988 .411002 4.57 0.000 1.60127 3.24609_IdosXsex_~2| 1.43131 .210778 2.44 0.015 1.07246 1.91022_IdosXage_~2| .679969 .135972 -1.93 0.054 .45949 1.00624_IdosXage_~3| .557839 .115872 -2.81 0.005 .371279 .838141 pyr| (exposure)

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

. testparm *sXa* //testing no dose by age interaction ( 1) [cases]_IdosXage_1_2 = 0

( 2) [cases]_IdosXage_1_3 = 0

chi2( 2) = 7.90

Prob > chi2 = 0.0193

. testparm *xXa* //testing no sex by age interaction ( 1) [cases]_IsexXage_2_2 = 0

( 2) [cases]_IsexXage_2_3 = 0

chi2( 2) = 5.74

Prob > chi2 = 0.0566

Comments

The final model is consistent with the data, but gives a rather complex description.

The dose effect is modified by both sex (larger rate ratio for females) and age-at-exposure (the dose effect decreases with age-at-exposure).

Having 10 estimated parameters the final model is only slightly simpler than the saturated model (i.e. the model with 12 freely varying rates).Note alsoThe goodness-of-fit test is not very reliable in large tables with many small counts. In such circumstances one may instead compare a given model with a much richer model that e.g. includes a lot of interaction parameters.

FROM SURVIVAL TIME DATA

TO POISSON REGRESSION ANALYSIS

In the LSS example the cancer mortality rate in each of the 12 groups was constant during the follow-up from 1971 to 1990. This is a highly unrealistic model, since it is well-known that cancer mortality rates increase dramatically with age.

In analyses of data from large, epidemiological cohort studies the dependence of rates on age and calendar time is usually described by piecewise constant hazard rates models. This gives much more realistic models with a better correction for confounding effects of age and/or calendar time. The analysis of such models is based on event counts and risk times in a multi-way table and in this context the method of analysis is usually denoted Poisson regression, since the analysis is formally identical (i.e. gives the same maximum likelihood estimates) to a regression model for counts described by Poisson distributions.

Individual data records are initially aggregated to form the multi-way table of event counts and person-years-at-risk, see the figure below.

In the analysis of the LSS data multi-way tables with 3000-8000 cells are routinely used. These tables are e.g. formed by a cross-classification on age (5-years intervals), calendar time (5-years intervals), sex, city and dose (8-12 categories) and separate analyses are carried out for the most common cancer types.

For each entry in the multi-way table a crude rate can be estimated as D/S = events/risktime. In the analysis the dependence of these rates on the classifying factors are studied using Poisson regression models very similar to Cox regression models.

Main difference: the unspecified baseline hazard of the Cox regression model is replaced by a piecewise constant hazard. In Poisson regression models effects of categorical covariates used as classifying factors are described by rate ratios. Both models with internal reference rates and models with external reference rates are available.

The analysis requires software that can

1. Form the multi-way table of counts and person-years,

2. Perform a Poisson regression analysis of the aggregated data.

Software: Forming the table: EPICURE, SAS, STATA (but not SPSS)

Poisson regression: EPICURE, EGRET, SAS, STATA, S-Plus, Genstat, GLIM, Statistix etc. (SPSS: not really).

FORMING EVENT-RISKTIME TABLES WITH STATA

The STATA commands stsplit and collapse are used to transform survival time data with individual records into a multi-way event-risktime table to be analyzed with Poisson regression.

A few examples illustrate some of the possibilities. The manual presents many more the stsplit is a highly versatile command Example 1. Splitting on time in study

In a clinical trial the data are usually described bystset time , failure(status==1) noshowto split the data at 1,2,3, and 5 years of follow-up write

stsplit timecat , at(1,2,3,5)

and data are split in 5 time categories.

Example 2: Splitting on ageIf the survival time data are defined by

stset outdate , failure(status==1) enter(time indate) origin(time bdate) scale(365.25) noshow

the time scale is age in years and we may consider using stsplit agecat , at(10(10)70)Example 3 Splitting on age and time in study

The data considered in example 2 can be split on both age and time in study with the commandsstset outdate , failure(status==1) enter(time indate) origin(time bdate) scale(365.25) noshowstsplit agecat , at(10(10)70)stsplit timecat , at(5(5)25) from(time indate)After the data have been split the multi-way table is formed by the commands

gen event=_d

gen risktime=_t-_t0

collapse (sum) event risktime , by(varlist)

save newfilenameuse newfilenamexi: poisson event varlist1 , exposure(risktime) other options

etc.

where varlist is a subset of the variables defining the multi-way table and interaction terms.Note:

The data do not have to be collapsed to do Poisson regression, but data may become very large if split on several time scales in many intervals and collapsing the data may speed up computation. Also consider deleting unnecessary variables first.

POISSON REGRESSIONMALIGNANT MELANOM DATA

To compare the results from a Cox regression analysis with those from a Poisson regression model of the same covariates consider use "E:\kurser\survival\melanoma.dta"

* generate a person id number

gen id=_n

* define data as survival time data

stset survtime , ///

failure(status==1) noshow scale(365.25) id(id)

* for later comparison we fit the following

* Cox model

xi:stcox i.sex i.invasion i.ecells ///

i.ulcerat , nolog

* now be split on follow-up time

stsplit timecat , at(2(2)8)

gen died=_d

gen risktime=_t-_t0

collapse (sum) risktime died , ///

by(timecat sex invasion ecells ulcerat)

* and save the multi-way table

save e:\kurser\survival\data\mmtable.dta

use e:\kurser\survival\mmtable.dta

* fit the corresponding

* poisson regression model

xi:poisson died ///

i.sex i.invasion i.ecells i.ulcerat ///

i.timecat , exposure(risktime) irrApart from the baseline hazard rate the two models are identical and both give results as rate ratios.

Selected Output

Cox regression -- no ties

No. of subjects = 205 Number of obs = 205

No. of failures = 57

Time at risk = 1208.279261

LR chi2(5) = 44.51

Log likelihood = -260.94353 Prob > chi2 = 0.0000

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

_t |Haz. Ratio Std. Err. z P>|z| [95% Conf. Int]

------------+-------------------------------------------------

_Isex_1| 1.87870 .509345 2.33 0.020 1.10429 3.19618_Iinvasion_1| 2.14216 .711768 2.29 0.022 1.11693 4.10845

_Iinvasion_2| 2.78566 1.09658 2.60 0.009 1.28781 6.02569 _Iecells_1| 1.79241 .547121 1.91 0.056 .985399 3.2603 _Iulcerat_1| 2.75137 .88215 3.16 0.002 1.4677 5.15780--------------------------------------------------------------Poisson regression Number of obs = 109

LR chi2(9) = 50.08

Prob > chi2 = 0.0000


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

died | IRR Std. Err. z P>|z| [95% Conf. Int]

------------+-------------------------------------------------

_Isex_1| 1.85962 .504960 2.28 0.022 1.09217 3.16635_Iinvasion_1| 2.1712 .719682 2.34 0.019 1.13382 4.15761_Iinvasion_2| 2.71461 1.06812 2.54 0.011 1.25541 5.86988 _Iecells_1| 1.81955 .555404 1.96 0.050 1.00033 3.3097 _Iulcerat_1| 2.76710 .885904 3.18 0.001 1.47743 5.18254 _Itimecat_2| 1.88015 .638056 1.86 0.063 .966772 3.65645 _Itimecat_4| 1.79353 .668697 1.57 0.117 .863671 3.72451 _Itimecat_6| 1.27918 .662529 0.48 0.635 .463521 3.53018 _Itimecat_8| .613017 .462909 -0.65 0.517 .139541 2.6930 risktime| (exposure)

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

The ratio between corresponding estimates varies between 0.974 and 1.015, so estimated rate ratios are indeed very similar in the two models. This is not surprising since a piecewise constant hazard rate based on 5 time intervals is rather flexible and it is therefore possible to approximate the shape of a wide range of baseline hazard rate functions.

USING POPULATION MORTALITY RATES IN THE ANALYSIS

Main types of problems

SYMBOL 183 \f "Symbol" \s 18 \hComparison of mortality (or survival) in a study group with that of an external reference population for which the mortality is known from e.g. published life tables.

SYMBOL 183 \f "Symbol" \s 18 \hComparison of the excess mortality (relative to an external reference group) found in two or several subgroups of a study.

First carefully consider:

Why introduce an external reference population? Is it really necessary or just a "convenient" way to correct for age or sex?

Also consider:

Which external reference population should be used? The whole country? The county? The individuals in the working force? etc.

Which endpoint? All causes of death or specific causes that are expected to be particularly relevant?

Here mainly a discussion of "how to do it" without taking a random sample from the background population.

The statistical methods which include the mortality of the background population can roughly be divided in two groups:

RELATIVE SURVIVAL

The statistical methods in this group include:

SYMBOL 183 \f "Symbol" \s 18 \hThe expected survival curve

SYMBOL 183 \f "Symbol" \s 18 \h"Crude", "corrected" and "relative" survival

SYMBOL 183 \f "Symbol" \s 18 \hExcess mortality parameters are usually describing additive effects on the mortality rate.

RELATIVE MORTALITY

The statistical methods in this group include:

SYMBOL 183 \f "Symbol" \s 18 \hThe expected number of deaths

SYMBOL 183 \f "Symbol" \s 18 \hThe person-year method

SYMBOL 183 \f "Symbol" \s 18 \hStandardized mortality ratios (SMR)

SYMBOL 183 \f "Symbol" \s 18 \hPoisson regression with external rates

SYMBOL 183 \f "Symbol" \s 18 \hExcess mortality parameters are usually describing multiplicative effects on the mortality rate.

FIRST:

What kind of information is available about the mortality of the "normal" population? - and how can it be utilized?

NATIONAL LIFE TABLES AND MORTALITY STATISTICS

Sources:

Most countries regularly - typically once a year - publish a cross-sectional population life table. A standard lay-out and terminology are used.

In Denmark:

Publications from The National Bureau of Statistics (Danmarks Statistik) including "Statistisk rbog", "Befolkningens bevgelser" contain life tables for the Danish population based on one year period or five year periods for each sex and single year age intervals from 0 to 99 year.

Life tables since 1981 can be found on the website

http://www.statistikbanken.dk/

which also gives access to other tables with mortality statistics - select the link to Population and elections (Befolkning og valg)A typical life table is included on the last page.

Sundhedsstyrelsen publishes information on cause of death (based on the death certificates) each year in "Ddsrsagerne i Danmark". Cancer incidence rates are available from Krftens Bekmpelse.THE COLUMNS OF THE LIFE TABLE

For each sex and single year age intervals from 0 to 99 years:

Age-specific mortality proportion (Aldersklassens ddshyppighed):

The probability of dying at the age of

x years given alive on the x year birthday.

The table gives

Survival function (Overlevende):

The probability for a new-born of surviving

until the x year birthday.

The table gives

Expected remaining lifetime (Middellevetid)

The expected remaining lifetime for a x

year old from the x year birthday.

Interrelationships:

COMPUTING MORTALITY RATES

FROM THE LIFE TABLEIf the national mortality rate is piecewise constant on 1 year intervals, i.e. for x in 1 year intervals, the following relation is true

The (total) mortality rate can therefore be obtained from the first or the second column of the life table as

NotesThe age-specific mortality proportion is a probability and has no dimension, whereas the mortality rate has dimension per time unit. In epidemiology both are often denoted the mortality rate. The mortality rate is always numerically larger than the corresponding age-specific mortality proportion, but apart from extremely old ages the discrepancy is very small.The plots below show the ratio plotted against the proportion and against age for each sex for the 2000-01 Danish life table.

The plots indicate that is essentially correct for ages below 60 and that an improved approximation can be obtained as

.CAUSE-SPECIFIC MORTALITY

Simple -and reasonably accurate estimates of cause-specific mortality rates can be derived from the relation

where the cause-specific mortality rate, the proportion of deaths from the specified cause at age x, and the total mortality rate.Estimation of the total mortality rate has already been described.For each sex and age in 5 year intervals the proportion can be estimated from tables of number of deaths by cause published each year in "Causes of death in Denmark" (Ddsrsagerne i Danmark) - or on the website mention above - as

,where the "age interval" refers to the five age interval containing age x.

In each 5 year interval total mortality rates (one for each of the 5 years) are then multiplied by this estimate to give the corresponding cause-specific mortality rates .

THE EXPECTED SURVIVAL CURVE, RELATIVE SURVIVAL AND CORECTED SURVIVAL

The expected survival curve:

Typical area of application: A clinical follow-up study.

Here: classical version based on grouped survival times

Follow-up yearAlive at start of yearDuring the year of follow-upmodified number at risk

deadcensored

1. year

2. year

3. year

The modified number at risk is obtained as

For each follow-up year the mortality proportion is estimated by

and the corresponding (conditional) survival proportion is

The usual life table estimate of the survival function is

The probability of surviving until the end of period i

.

Computation of the corresponding "expected" survival curve involves the following steps:

First follow-up year :

Consider the individuals alive at the start of the year. For let

The probability according to the published life

table of surviving one year for an individual of

the same sex and age as individual j.

The average expected survival probability for the first year:

Second follow-up year :

Consider the individuals alive at the start of the year. For let

The probability according to the published life

table of surviving one year for an individual of

the same sex and age as individual j.

The average expected survival probability for the second year:

For each of the following years of follow-up an average expected survival probability etc. are similarly computed.

After i year of follow-up the expected survival curve takes the value:

The corrected survival curve is defined as the ratio of the estimated (crude) survival to the expected survival:

.

Note that the corrected survival curve is not necessarily decreasing!

For each follow-up interval the relative survival is defined as the ratio of the estimated conditional survival probability to the corresponding conditional expected survival probability:

.

Software for calculation of expected survival curvesTo my knowledge none of the commercial statistical software packages are able to compute expected survival, corrected survival and relative survival, but several public-domain products are available. See e.g.http://www.cancerregistry.fi/surv2/

A locally developed PASCAL program is available from Department at Biostatistics.THE STATISTICAL MODEL BEHIND EXPECTED SURVIVAL AND CORRECTED SURVIVAL

The mortality rate for patient j at time t is the sum of two terms: the background mortality for a person of the same age and sex and the excess mortality "caused" by the disease in question:

,where is the population mortality rate and the excess mortality rate.

Let

Then

is an estimate of

is an estimate of,

The corrected survival can therefore be viewed as an estimate of

,the survival function corresponding to the excess mortality rate .

RELATIVE MORTALITY, THE PERSON-YEAR METHOD

Notation:

mortality rate in the study group

mortality rate in the reference population

Survival function and integrated mortality rate in the reference population are denoted and .

A simple statistical model:

Assume that the mortality rate in the study group is proportional to that of the reference group:

Two situations:

1.Age a is chosen as the underlying time t. With the model becomes

2.Follow-up time is chosen as the underlying time scale. If e denote the age at entry the model becomes

The dependence on sex is suppress below.

The parameter is the mortality rate ratio or the relative mortality. In epidemiology the estimate of is usually called the standardized mortality ratio (SMR).If the mortality in the study group is higher (lower) than the mortality in the reference population.

Generalizations: The relative mortality may depend on e.g. sex, age-at-entry, follow-up time or risk factors, which are known for each individual.

Estimation of the relative mortality

Data: A record for each individual with:

Age at entry in the study

Age at exit from the study

Status at exit (dead or alive).

The maximum likelihood estimate of SYMBOL 113 \f "Symbol" becomes

The numerator D:

D = the observed number of deaths during follow-up.

The denominator E

E = the expected number of deaths during follow up. This is a convenient terminology, but not quite correct. E is rather number of deaths to be expected with the observed follow-up times.

Confidence intervals for the relative mortality

An approximate 95% confidence interval for SYMBOL 113 \f "Symbol" is obtained by using

A symmetric confidence interval for ln() is transformed back to a asymmetric confidence interval for :

,

where

Null hypothesis:

The mortality in the study group is identical to the mortality in the reference population, i.e. .

The expected value of is 0 on the null hypothesis

and can be estimated by E.

These results lead to the following test statistic

which on the null hypothesis is approximately a variate on 1 degree of freedom. Large values provide evidence against the null hypothesis.

THE EXPECTED NUMBER OF DEATHS

Above the expected number of deaths was computed as the sum of individual contributions of the form

Since the mortality rate is assumed constant on a number of age intervals (typically 1 year or 5 year intervals) we have

,hvor is the time the individual spends in the age category x, i.e. the individuals contribution to the time-at-risk in age category x. Often it is simpler to calculate the expected number of deaths by first computing total time-at-risk in each age category, multiply by the age-specific mortality rate, and sum contributions from each age category, i.e.

where the person-years at risk in age category x. The following figure illustrates the two different ways to calculation of

The "expected" number of deaths E depends on the survival times and is therefore a random variable (i.e. subject to random variation) and not really an expected number (i.e. a constant).

If the mortality in the study group is identical to the mortality in the reference population, i.e. if one may show that

,

i.e. on the average the "expected" number of death is equal to the expected number of death!!!

Example: Mortality for patients diagnosed with manic-depressive psychosis (Weeke, Juel & Vth, J. Affective Disorders 1987; 13: 287-292).

Data:

Patients admitted to a psychiatric hospital for the first time in the period April 1, 1970 - March 31, 1972 and followed until March 31, 1977.

Number of patients N = 2168.

17 patients were lost to follow-up due to emigration and were censored on date of emigration.

Results:All patientsMalesFemales

Observed309159150

"Expected"176.5573.34103.21

1.752.171.45

99.4100.021.2

95% confidence intervals for the relative mortality:

All patientsMalesFemales

Method above1.57 - 1.971.86 - 2.531.24 - 1.71

"exact" Poisson1.56 - 1.961.84 - 2.531.24 - 1.71

The results above are often derived from a Poisson model for the observed number of deaths assuming that the "expected" number E is fixed. Confidence interval based on this Poisson model is referred to as "exact" confidence intervals.

Here we can compare the excess mortality among men with the excess mortality among women.

Null hypothesis: The relative mortality does not depend on the sex of the patient:.

Test statistic:

On the null hypothesis the following test statistic is approximately distributed as a variate on 1 degree of freedom

We find

giving p = 0.00044.

USING STATA TO COMPUTE STANDARDIZED MORTALITY RATIOS

The STATA command stptime can compute the SMR relative to a set of reference rates read from a separate file. Example: Diabetes mortality data

The STATA file diabetes.dta contains data on mortality for patients with diabetes from Green & Hougaard, Diabetologia 1984; 26: 190-194, see Exercise 8 for a variable description.Here we compare the mortality in this group with the mortality in the general population represented by a life table based on the calendar years 1976-1980. For simplicity 10 years age intervals are used in the rate file. The file kvrater7680-10.dta contains the following female mortality rates (per 1000 years)

agecatrate

0.461

10.125

20.205

30.417

401.204

502.757

606.128

7016.824

8050.955

Note that agecat gives the lower bound of the age interval.

The rates are computed from the life table as

A similar file, marater7680-10.dta contains the mortality rates for males. Both files must be sorted on agecat before saving them.Note: stptime only allows one set of rates, so a combined analysis is not possible unless the same set of rates are applied to both men and women.

The following commands produce expected number of deaths and SMR for each sex separately.

* defining the survival time data with

* age as time scale

gen exitage=entryage+futime/365.25

stset exitage ,

///

failure(status==1) entry(time entryage)

///

id(id) noshow* calculations for females

stptime if(sex==0) ,

///

smr(agecat rate)

///

using(E:\kurser\survival\kvrater7680-10.dta)///

at(30(10)80) trim per(1000)

* calculations for males

stptime if(sex==1) ,

///

smr(agecat rate)

///

using(E:\kurser\survival\marater7680-10.dta)///

at(30(10)80) trim per(1000)

The option trim specifies that follow-up time less than 30 or greater than 90 are to be excluded from the computationsOutput. stptime if(sex==0) , smr(agecat rate) using(E:\kurser\survival\aarhus2003\data\kv

> rater7680-10.dta) at(30(10)80) trim per(1000)

| observed expected

Cohort |person-time failures failures SMR [95% Conf.Inter]

---------+----------------------------------------------------

(30 - 40]| 646.53937 11 .26951 40.815 22.6032 73.6995

(40 - 50]| 676.54623 8 .814799 9.8184 4.91015 19.6329(50 - 60]| 787.27589 22 2.17029 10.137 6.67464 15.3951(60 - 70]| 959.54 55 5.87991 9.3539 7.18152 12.1834(70 - 80]| 723.50156 87 12.1722 7.1475 5.79286 8.81881---------+----------------------------------------------------

total | 3793.403 183 21.3067 8.5889 7.43041 9.92792. stptime if(sex==1) , smr(agecat rate) using(E:\kurser\survival\aarhus2003\data\ma

> rater7680-10.dta) at(30(10)80) trim per(1000)

| observed expected

Cohort |person-time failures failures SMR [95% Conf.Inter]

---------+----------------------------------------------------

(30 - 40]| 954.56259 17 .652274 26.063 16.2021 41.9243(40 - 50]| 957.03772 22 1.6278 13.515 8.89909 20.5258(50 - 60]| 970.16295 37 4.54827 8.135 5.89412 11.2277

(60 - 70]| 800.0821 63 9.53987 6.6039 5.15889 8.45355(70 - 80]| 462.71036 82 13.747 5.9649 4.80402 7.40634---------+----------------------------------------------------

total | 4144.5557 221 30.1153 7.3385 6.43202 8.37266For both women and men the mortality is considerably higher than the mortality in the general population.

The SMR is slightly larger for women, but a clear trend with age is seen in both sexes, so the overall SMR is less relevant. The command strate has also options for simple comparisons with external rates.

POISSON REGRESSION WITH EXTERNAL REFERENCE RATES USING STATA

A computation of a standardized mortality ratio for a group of individuals or patients is a rather crude comparison with the mortality in a reference population.Often further insight can be gained by studying how the relative mortality depends on a number of covariates. Such models, SMR regression models, are conveniently expressed as a Poisson regression model for the aggregated data. The relevant parameters are estimated by choosing the expected number of deaths as time-at-risk. The following example illustrates the approach using STATA.Example: Diabetes mortality dataThe data in diabetes.dta from Green&Hougaard (1987) is first split on 5-year age categories and collapsed in a multi-way event time table with sex, agecat, and dxcat (age-at-diagnosis) as classifying factors (output omitted)egen dxacat=cut(dxage) , at(0,20,40,60,120)

gen exitage=entryage+futime/365.25

stset exitage ,

///

failure(status==1) entry(time entryage) id(id) noshow

stsplit agecat , at(5(5)95)gen died=_d

gen risktime=_t-_t0collapse (sum) died risktime , by(sex agecat dxacat)

save e:\kurser\survival\diabetes-coll-agecat2.dtaThe national mortality rates (per 1000 years) on 20 five-years intervals for each sex are placed in the file mort7680.dta. The file contains data in variables sexage and mrate, where sexage takes the values from 1 to 40:sexagecatsexage

female0-41

female5-92

female10-143

etc.

male90-9439

male95-9940

Apart from now using 5-years intervals mrate is computed from the life table as before. The file mort7680.dta must be sorted on sexage before it is saved.

The reference rates are now appended to the multi-way event time table using the commandsuse e:\kurser\survival\diabetes-coll-agecat2.dta

egen sexage=group(sex agecat)

sort sexage

merge sexage

///

using e:\kurser\survival\mort7680.dtasave e:\kurser\survival\ diabetes-coll-agecat3.dtaWe have now added a column, mrate, to the file. The new column contains the reference rate (per 1000 years) in the appropriate sex and age category.

In a Poisson regression model number of events in a cell of the multi-way table is treated as a Poisson variate with mean raterisktime (see page 19). The present table has sex, agecat and dxacat as classifying factors and a total of 93 non-empty entries with event, risktime and mrate in each cellA Poisson regression model with external reference rates specifies multiplicative structure for mortality rate in the cell given by sex=i, agecat=j, dxacat=k (i = 0,1, j = 0,5,..,95, k = 0,20,40,60)

,

where is the relative mortality in the i,j,k-cell and is the sex and age specific reference rate.

The model therefore specifies that the number of events in the i,j,k-cell has mean

,

where is the expected number of deaths in the cell according the reference rates. If we use instead of risktime in the Poisson regression we have a regression model for the relative mortality and may fit models like e.g.

A couple of examples illustrate the possibilities.

The irr option is not used since the constant term is not displayed if this option is used. Rather inconvenient.gene expected = risktime*mrate/1000

xi: poisson died , exposure(expected) nolog

Output

Poisson regression Number of obs = 93

LR chi2(0) = 0.00

Prob > chi2 = .


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

died | Coef. Std. Err. z P>|z| [95% Conf. Inte]

---------+----------------------------------------------------

_cons | 1.911351 .0451294 42.35 0.000 1.82290 1.99980expected | (exposure)

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

The coefficient is equal to ln(SMR) so

A similar calculation gives the 95% confidence interval for the SMR:Lower limit = exp(1.82290) = 6.190

Upper limit = exp(1.99980) = 7.388

Next see if the relative mortality depends on sexxi: poisson died i.sex , exposure(expected) nolog

OutputPoisson regression Number of obs = 93 LR chi2(1) = 0.64

Prob > chi2 = 0.4236


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

died | Coef. Std. Err. z P>|z| [95% Conf. Inter]

---------+----------------------------------------------------

_Isex_1 | .072242 .0903129 0.80 0.424 -.104768 .249252

_cons |1.874631 .064957 28.86 0.000 1.747318 2.001945

expected | (exposure)

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

The constant is ln(SMR) for females and the coefficient for sex is . The SMR for males and females are not significantly different (the ln(ratio) is close to 0). We find

and therefore

Confidence limits can be computed similarly.Finally see if the relative mortality depends on age-at-diagnosisxi: poisson died i.sex i.dxacat , ///

exposure(expected) nolog testparm *xa*

OutputPoisson regression Number of obs = 93

LR chi2(4) = 61.16

Prob > chi2 = 0.0000


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

died | Coef. Std. Err. z P>|z| [95% Conf. Inter]

-----------+--------------------------------------------------

_Isex_1 |-.0495702 .0923718 -0.54 0.592 -.230616 .131475_Idxacat_20|-.7117244 .1615624 -4.41 0.000 -1.02838 -.39507_Idxacat_40|-.7557131 .1461376 -5.17 0.000 -1.04214 -.469289_Idxacat_60|-1.277969 .1599211 -7.99 0.000 -1.59141 -.964530 _cons | 2.776317 .1412925 19.65 0.000 2.49939 3.05325 expected |(exposure)--------------------------------------------------------------

. testparm *dx*

( 1) [died]_Idxacat_20 = 0



chi2( 3) = 64.81

Prob > chi2 = 0.0000

The relative mortality depends clearly on age at diagnosis, but this may well be a time-since-diagnosis effect that is showing up here. Further analysis is needed to uncover this. The model predicts the following SMRs

dxagefemalemale

0-2016.0615.28

20-407.887.50

40-607.547.18

60-4.474.26

Example of obtaining an SMR from the coefficients

Analysis of censored survival data:

Cox regression or Poisson regression?

Analysis of time-to-event data can be analyzed both with Cox regression and Poisson regression models

To use Poisson regression the individual data records must first be aggregated in an event-time table using special software.

This table will often be considerably smaller than the original data set and computations will therefore be faster.

Poisson regression is mainly preferable in large studies with relatively few covariates. Time-dependent covariates can be defined and used when setting up the event-time table. Several time scales are easily accommodated.

Cox regression is mainly preferable in studies with many covariates and if the analyses include more exploratory aspects of working with time-dependent covariate information, e.g. selecting the best way to define a time-dependent covariate. Once a proper representation is found is may be advantageous to continue with Poisson regression.

Age specific mortality rates

EMBED Equation.DSMT4






AGE

33

34

30

31

32

In each cell compute:

Number of events D

Total time at risk S

10

20

25

15

5

1995

2000

1985

1990

1980

Calendar Time

Age

PAGE 1

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Males

Females

age

rate/proportion

2000-01 Danish Life Table

lifetable2000-01

Ddelighedstavle efter kn, Ddelighedstavle, alder og tid.

2000:2001MndKvinderMNDKVINDER

SproprateforholdcheckSproprateforholdcheck

OverlevendeddshyppighedMiddellevetidOverlevendeddshyppighedMiddellevetid

0 r10000055474.530 r10000040179.2010.005540.00555540271.0027802732110.004010.00401806161.00201037621

1 r994463173.951 r995994878.5110.994460.0003117270.00031177561.00015589591.00557086260.995990.00048193250.00048204871.00024104371.0040261448

2 r994152572.972 r995512777.5520.994150.00024141230.00024144141.00012072560.96564904690.995510.00026117270.00026120681.00013060910.9673061677

3 r993912171.993 r995251476.5730.993910.00021128670.00021130911.00010565831.00612731540.995250.00015071590.00015072731.00007536551.0765421466

4 r9937017714 r995101375.5840.99370.00017107780.00017109241.00008554871.00633994160.99510.00013064010.00013064871.00006532581.0049241282

5 r993531770.015 r994971074.5950.993530.00017110710.00017112171.00008556331.00651213350.994970.00010050550.00010051061.00005025611.0050554288

6 r993361769.026 r99487773.660.993360.00017113630.0001711511.00008557791.00668438430.994870.00006030940.00006031121.00003015590.8615626737

7 r993191368.047 r99481872.670.993190.00013089140.00013089991.00006545141.00685669410.994810.00008041740.00008042061.00004021081.0052170766

8 r993061167.058 r99473971.6180.993060.00010069890.00010070391.00005035280.91544409110.994730.00009047680.00009048091.00004524111.00529792

9 r992961466.059 r994641470.6290.992960.00014099260.00014100251.00007050291.0070899130.994640.00014075440.00014076441.00007038381.0053888844

10 r992821465.0610 r994501469.63100.992820.00014101250.00014102241.00007051291.00723192520.99450.00014077430.00014078421.00007039371.0055304173

11 r992681464.0711 r994361068.64110.992680.00014103240.00014104231.00007052281.00737397750.994360.00009051050.00009051461.0000452580.905104791

12 r992541763.0812 r994271167.64120.992540.00017127770.00017129241.00008564861.00751606990.994270.00011063390.00011064011.0000553211.0057630221

13 r992372062.0913 r994161166.65130.992370.00020153770.0002015581.00010078241.00768866450.994160.00011064620.00011065231.00005532721.0058743059

14 r992172461.114 r994051665.66140.992170.0002418940.00024192331.00012096651.00789179270.994050.00016095770.00016097071.00008048751.0059856144

15 r991933560.1215 r993892264.67150.991930.00035284750.00035290971.00017646531.00813565470.993890.00022135250.0002213771.00011069261.0061475616

16 r991585559.1416 r993672063.68160.991580.00054458540.00054473371.00027239160.99015528940.993670.00021133780.00021136011.00010568381.0566888404

17 r991046658.1717 r993462562.7170.991040.00066596710.00066618891.00033313141.00904100740.993460.00024157990.00024160911.00012080940.966319731

18 r990387057.2118 r993223161.71180.990380.00069670230.00069694511.0003485130.99528896560.993220.00031211610.00031216491.00015609061.0068262822

19 r989697256.2519 r992912860.73190.989690.00071739640.00071765381.00035886980.99638382840.992910.00028199940.00028203911.00014102621.007140627

20 r988989255.2920 r992632559.75200.988980.00092013990.00092056351.00046035241.0001521110.992630.00026193040.00026196471.00013098811.047721709

21 r988079254.3421 r992372358.76210.988070.00092098740.00092141171.00046077661.00107323850.992370.00023176840.00023179531.00011590211.0076886645

22 r987167553.3922 r992142257.78220.987160.00075975530.0007600441.00038007011.013007010.992140.00022174290.00022176751.00011088781.007922269

23 r986417652.4323 r991922656.79230.986410.00076033290.00076062211.00038035931.00043805850.991920.00026211790.00026215231.00013108191.0081458182

24 r985667051.4724 r991662755.8240.985660.00070003860.00070028371.00035018271.00005507550.991660.00027227070.00027230781.00013616011.0084101406

25 r984976850.525 r991392854.82250.984970.00068022380.00068045521.00034026621.00032906350.991390.00027234490.0002723821.00013619720.9726603196

26 r984307649.5426 r991123553.83260.98430.00076196280.00076225331.0003811751.00258265290.991120.00035313580.00035319821.00017660951.0089595609

27 r983558348.5827 r990773752.85270.983550.00082354740.00082388671.00041199990.99222572770.990770.00037344690.00037351671.000186771.0093159866

28 r982748447.6228 r990403551.87280.982740.00084457740.00084493431.00042252661.00544929290.99040.00035339260.0003534551.00017673791.0096930533

29 r981918946.6529 r990053550.89290.981910.00089621250.00089661431.00044837421.00698031890.990050.0003434170.0003434761.00017174780.9811914261

30 r981038845.730 r989713549.91300.981030.00087662970.00087701411.00043857120.99617007360.989710.00036374290.00036380911.00018191561.0392654703

31 r980178344.7431 r989354248.92310.980170.00082638730.00082672891.00041342140.99564730040.989350.00041441350.00041449941.0002072640.9866988186

32 r979369043.7732 r988944947.95320.979360.00090875670.00090916991.00045465381.00972971010.988940.000495480.00049560281.00024782191.0111836916

33 r9784710742.8133 r988455146.97330.978470.00107310390.00107368011.00053693611.00290086690.988450.00051595930.00051609251.00025806841.0116849613

34 r9774211741.8634 r987945545.99340.977420.00117656690.00117725961.00058874531.00561271810.987940.00054659190.00054674131.00027339560.9938034514

35 r9762712340.9135 r987406345.02350.976270.00122916820.00122992421.00061508820.99932370770.98740.00062791170.00062810891.00031408730.9966852179

36 r9750713439.9636 r986787444.05360.975070.00133323760.00133412721.00066721190.99495344310.986780.00073977990.00074005371.00037007250.9997025543

37 r9737715339.0137 r986058243.08370.973770.00153013550.00153130731.00076584911.00008853130.986050.00082145940.00082179691.00041095471.0017797049

38 r9722816638.0738 r985249742.11380.972280.00165590160.00165727411.00082886590.99753107960.985240.00096423210.00096469721.00048242620.9940536756

39 r9706718437.1339 r9842910941.15390.970670.00184408710.00184578951.00092317871.00222123580.984290.0010870780.00108766931.00054393320.9973192617

40 r9688821036.240 r9832211840.2400.968880.00210552390.00210774361.0010542421.00263043040.983220.00118996770.00119067621.00059545631.0084471672

41 r9668421135.2741 r9820512639.24410.966840.00210996650.00211219561.00105646960.99998411790.982050.00126266480.00126346271.00063186441.0021149474

42 r9648023634.3542 r9808113438.29420.96480.00236318410.00236598081.00118345691.00134918630.980810.00133563080.00133652351.00066841060.9967393688

43 r9625226233.4343 r9795015637.34430.962520.0026077380.00261114411.00130614020.99531985540.97950.00156202140.00156324271.0007818251.0012957946

44 r9600127732.5144 r9779717436.4440.960010.00277080450.00277465031.00138796671.00029042270.977970.00174851990.00175005031.00087528041.0048964904

45 r9573532131.645 r9762620235.46450.957350.00321721420.00322240051.00161206561.00224740970.976260.00201790510.00201994381.00101031190.9989629041

46 r9542736830.746 r9742922334.53460.954270.00367820430.00368498551.00184362430.99951203310.974290.00222726290.0022297471.00111528780.9987726143

47 r9507641329.8147 r9721225233.61470.950760.00413353530.00414210191.00207248071.00085599930.972120.0025202650.00252344621.00126225371.0001051539

48 r9468343928.9448 r9696728732.69480.946830.00439360810.00440328841.00220325991.00082190090.969670.00286695470.00287107231.00143622310.9989389328

49 r9426745728.0649 r9668930431.79490.942670.00457211960.00458260371.00229305191.00046381130.966890.00304067680.00304530911.00152342741.0002226346

50 r9383651327.1950 r9639533330.88500.938360.00513662130.00514985911.00257713971.00129070940.963950.00333004820.00333560521.00166872981.0000144862

51 r9335454926.3251 r9607436629.98510.933540.00549521180.0055103661.00275771331.00094932140.960740.00365343380.00366012391.00183117830.9982059594

52 r9284156325.4752 r9572339029.09520.928410.0056332870.00564921381.00282726641.00058383970.957230.00389666020.00390427191.00195340620.9991436294

53 r9231860124.6153 r9535041628.2530.923180.00600099660.00601907491.00301255660.99850192270.95350.00416360780.00417229971.00208760051.0008672502

54 r9176466623.7554 r9495345627.32540.917640.00665838460.00668065051.00334404450.99975744020.949530.00456015080.004570581.00228703091.0000330727

55 r9115380222.9155 r9452050026.44550.911530.00801948370.00805181271.0040313090.99993562660.94520.00499365210.00500616211.00250516950.9987304274

56 r9042288522.0956 r9404857225.57560.904220.00884740440.00888677511.00444996880.9997067110.940480.0057311160.00574760191.00287655391.0019433612

57 r8962294221.2857 r9350964524.72570.896220.00941733060.0094619541.00473843770.99971662060.935090.00644857710.0064694591.00323821730.9997794019

58 r88778107420.4858 r9290670523.87580.887780.01074590550.01080405971.00541175721.00054986190.929060.00705013670.00707510631.00354172461.0000193897

59 r87824115919.759 r9225174623.04590.878240.01157997810.01164754821.00583507950.9991353010.922510.00745791370.0074858631.00374760140.9997203399

60 r86807122918.9260 r9156377222.21600.868070.01229163550.01236780241.0061966481.0001330730.915630.00772145950.00775142441.00388071921.0001890598

61 r85740130918.1561 r9085685921.38610.85740.01309773730.01318426911.00660661991.00059108830.908560.00859602010.0086331791.00432280051.0007008237

62 r84617142417.3962 r90075102920.56620.846170.01422881930.01433101951.00718262460.99921483620.900750.01029142380.01034474671.00518129111.0001383687

63 r83413163216.6363 r89148116319.77630.834130.01631640150.01645097991.00824804270.99977950490.891480.01162112440.01168917741.00585597510.9992368377

64 r82052177915.964 r88112125419640.820520.01779359430.01795380361.00900376331.00020204080.881120.01254085710.01262015731.0063233511.0000683486

65 r80592194915.1865 r87007141318.23650.805920.019493250.01968574911.00987516841.00016674960.870070.01412530030.01422601181.00712987080.999667393

66 r79021223814.4766 r85778160917.48660.790210.02237379940.022627891.01135661310.99972293880.857780.01608804120.01621885871.00813135020.9998782612

67 r77253253213.7967 r84398173016.76670.772530.02531940510.02564545661.01287753520.99997650360.843980.01731083670.01746242121.00875662181.0006264017

68 r75297286013.1368 r82937189116.05680.752970.02860671740.02902386431.01458212991.00023487410.829370.01890591650.01908691831.00957381820.9997840581

69 r73143315712.5169 r81369207615.35690.731430.03156829770.03207731771.01612440360.99994607910.813690.02075729090.02097575181.01052454080.9998695019

70 r70834337111.970 r79680226314.66700.708340.03371262390.03429399821.01724500371.00007783690.79680.0226280120.02288795431.01148763160.9999121541

71 r68446371711.371 r77877245513.99710.684460.03716798640.03787632321.019057710.99994582840.778770.02453869560.02484478721.01247383170.9995395371

72 r65902410510.7172 r75966265113.33720.659020.04104579530.04191195841.02110235930.99989757060.759660.02651186060.02686963771.01349498291.0000701837

73 r63197447310.1573 r73952291412.68730.631970.04473313610.04576453891.02305679611.0000701110.739520.02914052360.02957354161.01485965261.0000179678

74 r6037048759.674 r71797317212.04740.60370.04874937880.04997771681.02519699740.99998725810.717970.0317144170.03222821151.01620065910.9998239923

75 r5742752429.0775 r69520344511.42750.574270.05241436950.0538379711.02716051770.99989258960.69520.03445051780.03505792811.01763138351.0000150315

76 r5441758608.5476 r67125382910.81760.544170.05860301010.06039034751.03049907141.00005136670.671250.03828677840.03903897921.01964648990.9999158631

77 r5122866258.0477 r64555422010.22770.512280.06625283050.06854957381.03466634331.00004272430.645550.04219657660.04311271681.02171124580.9999188759

78 r4783474747.5878 r6183145889.65780.478340.07473763430.07767794311.03934174290.99996834790.618310.04588313310.0469691131.02366839121.0000682881

79 r4425982237.1579 r5899450159.09790.442590.08224315960.08582280321.04352512241.00016003380.589940.05014069230.05144140251.02594120970.9998144023

80 r4061987606.7580 r5603654348.54800.406190.08759447550.09167073371.04653556240.99993693480.560360.05434006710.05587225351.02819625561.0000012348

81 r3706194646.3481 r5299160618.01810.370610.09462777580.09940912241.05052794010.9998708350.529910.06061406650.06252887951.03159024021.0000670935

82 r33554103385.9682 r4977968007.49820.335540.10338558740.10912937271.05555692441.00005404730.497790.06800056250.07042306781.03562478381.0000082719

83 r30085112795.5883 r4639475757830.300850.11278045540.11966281361.06102438790.9999153770.463940.07576410740.07878794491.0399112141.0001862367

84 r26692123515.2384 r4287982546.53840.266920.12352015590.13184157071.06736888251.0000822270.428790.0825345740.08614038261.0436884620.9999342626

85 r23395132704.985 r3934092126.08850.233950.13272066680.14239417051.07288618951.00015574080.39340.09211997970.09664304531.04909972470.9999997792

86 r20290145654.5786 r35716105015.64860.20290.14563824540.15740057471.08076401380.99991929590.357160.10502295890.11095721341.05650435491.0001234063

87 r17335160714.2687 r31965114915.25870.173350.16071531580.17520531641.09015942561.00003307720.319650.11490692950.1220624751.06227253330.9999732787

88 r14549171483.9988 r28292126884.86880.145490.17148944940.18812570771.09701038911.00005510520.282920.12685564820.13565438541.06936023190.9998080725

89 r12054185433.7189 r24703143094.5890.120540.18541562970.20507727091.10604090520.99992250270.247030.14310002830.15443408641.07920374421.0000700841

90 r9819205803.4490 r21168156814.16900.098190.20582544050.23045199361.11964776121.00012361750.211680.15679327290.17054312311.08769413390.9998933286

91 r7798229563.291 r17849175613.85910.077980.22954603740.26077537611.13604825860.99993917690.178490.17564009190.19314806271.09968094791.0001713563

92 r600824373392 r14714194893.56920.060080.24367509990.2792842331.14613365580.99977475020.147140.19484844370.21672475051.11227344940.9997867703

93 r4544258832.8193 r11847216233.3930.045440.25880281690.2994885851.15720759390.9998949770.118470.21625728030.24367447621.12678045241.0001261635

94 r3368280882.6194 r9285235673.07940.033680.28087885990.32972545121.17390625770.99999594080.092850.23564889610.26872803531.14037468350.9999104513

95 r2422292092.4495 r7097250362.87950.024220.29232039640.3457638251.18282483621.00078878550.070970.25038748770.28819885621.15101141381.0001097926

96 r1714316982.2496 r5320280302.66960.017140.31680280050.38097173561.20255166620.99944097570.05320.28026315790.32886963091.17343154670.9998685619

97 r1171362872.0597 r3829296602.5970.011710.36293766010.45088776341.24232840221.00018645830.038290.29668320710.35194785871.18627495671.0002805364

98 r746393941.9398 r2693307572.34980.007460.39410187670.50104342041.27135507351.00041091710.026930.3074637950.36739476071.19492039930.9996546966

99 r452418371.8699 r1865336222.15990.004520.01865

lifetable2000-01

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Males

Females

age

rate/proportion

2000-01 Danish Life Table

_1115791368.xlsChart3

1.0000500033

1.0010013353

1.0025083647

1.0050335854

1.0101353659

1.0258658878

1.0536051566

1.08345953

1.1157177566

1.1507282898

1.1889164798

1.3862943611

forhold

proportion

rate/proportion

lifetable2000-01

Ddelighedstavle efter kn, Ddelighedstavle, alder og tid.

2000:2001MndKvinderMNDKVINDER

SproprateforholdcheckSproprateforholdcheck

OverlevendeddshyppighedMiddellevetidOverlevendeddshyppighedMiddellevetid

0 r10000055474.530 r10000040179.2010.005540.00555540271.0027802732110.004010.00401806161.00201037621

1 r994463173.951 r995994878.5110.994460.0003117270.00031177561.00015589591.00557086260.995990.00048193250.00048204871.00024104371.0040261448propforhold

2 r994152572.972 r995512777.5520.994150.00024141230.00024144141.00012072560.96564904690.995510.00026117270.00026120681.00013060910.96730616770.00011.0000500033

3 r993912171.993 r995251476.5730.993910.00021128670.00021130911.00010565831.00612731540.995250.00015071590.00015072731.00007536551.07654214660.0021.0010013353

4 r9937017714 r995101375.5840.99370.00017107780.00017109241.00008554871.00633994160.99510.00013064010.00013064871.00006532581.00492412820.0051.0025083647

5 r993531770.015 r994971074.5950.993530.00017110710.00017112171.00008556331.00651213350.994970.00010050550.00010051061.00005025611.00505542880.011.0050335854

6 r993361769.026 r99487773.660.993360.00017113630.0001711511.00008557791.00668438430.994870.00006030940.00006031121.00003015590.86156267370.021.0101353659

7 r993191368.047 r99481872.670.993190.00013089140.00013089991.00006545141.00685669410.994810.00008041740.00008042061.00004021081.00521707660.051.0258658878

8 r993061167.058 r99473971.6180.993060.00010069890.00010070391.00005035280.91544409110.994730.00009047680.00009048091.00004524111.005297920.11.0536051566

9 r992961466.059 r994641470.6290.992960.00014099260.00014100251.00007050291.0070899130.994640.00014075440.00014076441.00007038381.00538888440.151.08345953

10 r992821465.0610 r994501469.63100.992820.00014101250.00014102241.00007051291.00723192520.99450.00014077430.00014078421.00007039371.00553041730.21.1157177566

11 r992681464.0711 r994361068.64110.992680.00014103240.00014104231.00007052281.00737397750.994360.00009051050.00009051461.0000452580.9051047910.251.1507282898

12 r992541763.0812 r994271167.64120.992540.00017127770.00017129241.00008564861.00751606990.994270.00011063390.00011064011.0000553211.00576302210.31.1889164798

13 r992372062.0913 r994161166.65130.992370.00020153770.0002015581.00010078241.00768866450.994160.00011064620.00011065231.00005532721.00587430590.51.3862943611

14 r992172461.114 r994051665.66140.992170.0002418940.00024192331.00012096651.00789179270.994050.00016095770.00016097071.00008048751.0059856144

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