### Example 51.1 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 51.1.1:

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

Output 51.1.1: 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 51.1.2 and Output 51.1.3:

```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 51.1.2: 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
Number of Parameters 3
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 51.1.3: 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
Number of Parameters 3
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 51.1.4:

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

Output 51.1.4: 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 51.1.5:

```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 51.1.5: 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