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
Goodness-of-fit chi2 = 22.27476
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
Goodness-of-fit chi2 = 14.97348
Prob > chi2(5) = 0.0105
. quietly xi:poisson cases i.sex*i.agex i.dose i.dose*i.sex ,
exposure(pyr) irr
. poisgof
Goodness-of-fit chi2 = 9.508152
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
Goodness-of-fit chi2 = 1.890253
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
Log likelihood = -38.115048 Pseudo R2 = 0.9388
--------------------------------------------------------------
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
Log likelihood = -87.668122 Pseudo R2 = 0.2222
--------------------------------------------------------------
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 = .
Log likelihood = -221.45684 Pseudo R2 = 0.0000
--------------------------------------------------------------
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
Log likelihood = -221.13671 Pseudo R2 = 0.0014
--------------------------------------------------------------
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
Log likelihood = -190.87923 Pseudo R2 = 0.1381
--------------------------------------------------------------
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
( 2) [died]_Idxacat_40 = 0
( 3) [died]_Idxacat_60 = 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
EMBED Equation.DSMT4
EMBED Equation.DSMT4
EMBED Equation.DSMT4
EMBED Equation.DSMT4
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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1.00718262461.0051812911
1.00824804271.0058559751
1.00900376331.006323351
1.00987516841.0071298708
1.01135661311.0081313502
1.01287753521.0087566218
1.01458212991.0095738182
1.01612440361.0105245408
1.01724500371.0114876316
1.019057711.0124738317
1.02110235931.0134949829
1.02305679611.0148596526
1.02519699741.0162006591
1.02716051771.0176313835
1.03049907141.0196464899
1.03466634331.0217112458
1.03934174291.0236683912
1.04352512241.0259412097
1.04653556241.0281962556
1.05052794011.0315902402
1.05555692441.0356247838
1.06102438791.039911214
1.06736888251.043688462
1.07288618951.0490997247
1.08076401381.0565043549
1.09015942561.0622725333
1.09701038911.0693602319
1.10604090521.0792037442
1.11964776121.0876941339
1.13604825861.0996809479
1.14613365581.1122734494
1.15720759391.1267804524
1.17390625771.1403746835
1.18282483621.1510114138
1.20255166621.1734315467
1.24232840221.1862749567
1.27135507351.1949203993
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
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 r