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 The LIFEREG Procedure

## Example 36.1: Motorette Failure

This example fits a Weibull model and a lognormal model to the example given in Kalbfleisch and Prentice (1980, p. 5). An output data set called models is specified to contain the parameter estimates. By default, the natural log of the variable time is used by the procedure as the response. After this log transformation, the Weibull model is fit using the extreme value baseline distribution, and the lognormal is fit using the normal baseline distribution.

Since the extreme value and normal distributions do not contain any shape parameters, the variable SHAPE1 is missing in the models data set. An additional output data set, out, is requested that contains the predicted quantiles and their standard errors for values of the covariate corresponding to temp=130 and temp=150. This is done with the control variable, which is set to 1 for only two observations.

Using the standard error estimates obtained from the output data set, approximate 90% confidence limits for the predicted quantities are then created in a subsequent DATA step for the log response. The logs of the predicted values are obtained because the values of the P= variable in the OUT= data set are in the same units as the original response variable, time. The standard errors of the quantiles of the log(time) are approximated (using a Taylor series approximation) by the standard deviation of time divided by the mean value of time. These confidence limits are then converted back to the original scale by the exponential function. The following statements produce Output 36.1.1 through Output 36.1.5:

```   title 'Motorette Failures With Operating Temperature as a Covariate';
data motors;
input time censor temp @@;
if _N_=1 then
do;
temp=130;
time=.;
control=1;
z=1000/(273.2+temp);
output;
temp=150;
time=.;
control=1;
z=1000/(273.2+TEMP);
output;
end;
if temp>150;
control=0;
z=1000/(273.2+temp);
output;
datalines;
8064 0 150 8064 0 150 8064 0 150 8064 0 150 8064 0 150
8064 0 150 8064 0 150 8064 0 150 8064 0 150 8064 0 150
1764 1 170 2772 1 170 3444 1 170 3542 1 170 3780 1 170
4860 1 170 5196 1 170 5448 0 170 5448 0 170 5448 0 170
408 1 190  408 1 190 1344 1 190 1344 1 190 1440 1 190
1680 0 190 1680 0 190 1680 0 190 1680 0 190 1680 0 190
408 1 220  408 1 220  504 1 220  504 1 220  504 1 220
528 0 220  528 0 220  528 0 220  528 0 220  528 0 220
;

proc print data=motors;
run;

proc lifereg data=motors outest=models covout;
a: model time*censor(0)=z;
b: model time*censor(0)=z / dist=lnormal;
output out=out quantiles=.1 .5 .9 std=std p=predtime
control=control;
run;

proc print data=models;
id _model_;
title 'fitted models';
run;

data out1;
set out;
ltime=log(predtime);
stde=std/predtime;
upper=exp(ltime+1.64*stde);
lower=exp(ltime-1.64*stde);
proc print;
id temp;
title 'quantile estimates and confidence limits';
run;
```

Output 36.1.1: Motorette Failure Data

 Motorette Failures With Operating Temperature as a Covariate

 Obs time censor temp control z 1 . 0 130 1 2.48016 2 . 0 150 1 2.36295 3 1764 1 170 0 2.25632 4 2772 1 170 0 2.25632 5 3444 1 170 0 2.25632 6 3542 1 170 0 2.25632 7 3780 1 170 0 2.25632 8 4860 1 170 0 2.25632 9 5196 1 170 0 2.25632 10 5448 0 170 0 2.25632 11 5448 0 170 0 2.25632 12 5448 0 170 0 2.25632 13 408 1 190 0 2.15889 14 408 1 190 0 2.15889 15 1344 1 190 0 2.15889 16 1344 1 190 0 2.15889 17 1440 1 190 0 2.15889 18 1680 0 190 0 2.15889 19 1680 0 190 0 2.15889 20 1680 0 190 0 2.15889 21 1680 0 190 0 2.15889 22 1680 0 190 0 2.15889 23 408 1 220 0 2.02758 24 408 1 220 0 2.02758 25 504 1 220 0 2.02758 26 504 1 220 0 2.02758 27 504 1 220 0 2.02758 28 528 0 220 0 2.02758 29 528 0 220 0 2.02758 30 528 0 220 0 2.02758 31 528 0 220 0 2.02758 32 528 0 220 0 2.02758

Output 36.1.2: Motorette Failure: Model A

 The LIFEREG Procedure

 Model Information Data Set WORK.MOTORS Dependent Variable Log(time) Censoring Variable censor Censoring Value(s) 0 Number of Observations 30 Noncensored Values 17 Right Censored Values 13 Left Censored Values 0 Interval Censored Values 0 Missing Values 2 Name of Distribution WEIBULL Log Likelihood -22.95148315

 Analysis of Parameter Estimates Variable DF Estimate Standard Error Chi-Square Pr > ChiSq Label Intercept 1 -11.89122 1.96551 36.6019 <.0001 Intercept z 1 9.03834 0.90599 99.5239 <.0001 Scale 1 0.36128 0.07950 Extreme value scale

Output 36.1.3: Motorette Failure: Model B

 The LIFEREG Procedure

 Model Information Data Set WORK.MOTORS Dependent Variable Log(time) Censoring Variable censor Censoring Value(s) 0 Number of Observations 30 Noncensored Values 17 Right Censored Values 13 Left Censored Values 0 Interval Censored Values 0 Missing Values 2 Name of Distribution LNORMAL Log Likelihood -24.47381031

 Analysis of Parameter Estimates Variable DF Estimate Standard Error Chi-Square Pr > ChiSq Label Intercept 1 -10.47056 2.77192 14.2685 0.0002 Intercept z 1 8.32208 1.28412 42.0001 <.0001 Scale 1 0.60403 0.11073 Normal scale

Output 36.1.4: Motorette Failure: Fitted Models

 fitted models

 _MODEL_ _NAME_ _TYPE_ _DIST_ _STATUS_ _LNLIKE_ Intercept time z _SCALE_ _SHAPE1_ A time PARMS WEIBULL 0 Converged -22.9515 -11.8912 -1.0000 9.03834 0.36128 . A Intercept COV WEIBULL 0 Converged -22.9515 3.8632 -11.8912 -1.77878 0.03448 . A z COV WEIBULL 0 Converged -22.9515 -1.7788 9.0383 0.82082 -0.01488 . A Scale COV WEIBULL 0 Converged -22.9515 0.0345 0.3613 -0.01488 0.00632 . B time PARMS LNORMAL 0 Converged -24.4738 -10.4706 -1.0000 8.32208 0.60403 . B Intercept COV LNORMAL 0 Converged -24.4738 7.6835 -10.4706 -3.55566 0.03267 . B z COV LNORMAL 0 Converged -24.4738 -3.5557 8.3221 1.64897 -0.01285 . B Scale COV LNORMAL 0 Converged -24.4738 0.0327 0.6040 -0.01285 0.01226 .

Output 36.1.5: Motorette Failure: Quantile Estimates and Confidence Limits

 quantile estimates and confidence limits

 temp time censor control z _PROB_ PREDTIME STD ltime stde upper lower 130 . 0 1 2.48016 0.1 12033.19 5482.34 9.3954 0.45560 25402.68 5700.09 130 . 0 1 2.48016 0.5 26095.68 11359.45 10.1695 0.43530 53285.36 12779.95 130 . 0 1 2.48016 0.9 56592.19 26036.90 10.9436 0.46008 120349.65 26611.42 150 . 0 1 2.36295 0.1 4536.88 1443.07 8.4200 0.31808 7643.71 2692.83 150 . 0 1 2.36295 0.5 9838.86 2901.15 9.1941 0.29487 15957.38 6066.36 150 . 0 1 2.36295 0.9 21336.97 7172.34 9.9682 0.33615 37029.72 12294.62

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