The LIFEREG Procedure

## Example 48.1 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 created 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 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 48.1.1:

```   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
;
run;
```
```   proc print data=motors;
run;
```

Output 48.1.1 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

The following statements produce Output 48.1.2 and Output 48.1.3:

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

Output 48.1.2 Motorette Failure: Model A
 Motorette Failures With Operating Temperature as a Covariate

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
Name of Distribution Weibull
Log Likelihood -22.95148315

Type III Analysis of Effects
Effect DF Wald
Chi-Square
Pr > ChiSq
z 1 99.5239 <.0001

Analysis of Maximum Likelihood Parameter Estimates
Parameter DF Estimate Standard Error 95% Confidence Limits Chi-Square Pr > ChiSq
Intercept 1 -11.8912 1.9655 -15.7435 -8.0389 36.60 <.0001
z 1 9.0383 0.9060 7.2626 10.8141 99.52 <.0001
Scale 1 0.3613 0.0795 0.2347 0.5561
Weibull Shape 1 2.7679 0.6091 1.7982 4.2605

Output 48.1.3 Motorette Failure: Model B
 Motorette Failures With Operating Temperature as a Covariate

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
Name of Distribution Lognormal
Log Likelihood -24.47381031

Type III Analysis of Effects
Effect DF Wald
Chi-Square
Pr > ChiSq
z 1 42.0001 <.0001

Analysis of Maximum Likelihood Parameter Estimates
Parameter DF Estimate Standard Error 95% Confidence Limits Chi-Square Pr > ChiSq
Intercept 1 -10.4706 2.7719 -15.9034 -5.0377 14.27 0.0002
z 1 8.3221 1.2841 5.8052 10.8389 42.00 <.0001
Scale 1 0.6040 0.1107 0.4217 0.8652

The following statements produce Output 48.1.4:

```   data models;
set modela modelb;
run;
```
```   proc print data=models;
id _model_;
title 'fitted models';
run;
```

Output 48.1.4 Motorette Failure: Fitted Models
 fitted models

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

The following statements produce Output 48.1.5:

```   data out;
set outa outb;
run;

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

Output 48.1.5 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 16519.27 5999.85 9.7123 0.36320 29969.51 9105.47
130 . 0 1 2.48016 0.5 32626.65 9874.33 10.3929 0.30265 53595.71 19861.63
130 . 0 1 2.48016 0.9 50343.22 15044.35 10.8266 0.29884 82183.49 30838.80
150 . 0 1 2.36295 0.1 5726.74 1569.34 8.6529 0.27404 8976.12 3653.64
150 . 0 1 2.36295 0.5 11310.68 2299.92 9.3335 0.20334 15787.62 8103.28
150 . 0 1 2.36295 0.9 17452.49 3629.28 9.7672 0.20795 24545.37 12409.24
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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