### Example 51.4 Analysis of Arbitrarily Censored Data with Interaction Effects

The artificial data in this example are from a study of the natural recovery time of mice after injection of a certain toxin. Twenty mice were grouped by sex (`sex`: 1 = Male, 2 = Female) with equal sizes. Their ages (in days) were recorded at the injection. Their recovery times (in minutes) were also recorded. Toxin density in blood was used to decide whether a mouse recovered. Mice were checked at two times for recovery. If a mouse had recovered at the first time, the observation is left censored, and no further measurement is made. The variable `time1` is set to missing and `time2` is set to the measurement time to indicate left censoring. If a mouse had not recovered at the first time, it was checked later at a second time. If it had recovered by the second measurement time, the observation is interval censored, and the variable `time1` is set to the first measurement time and `time2` is set to the second measurement time. If there was no recovery at the second measurement, the observation is right censored, and `time1` is set to the second measurement time and `time2` is set to missing to indicate right censoring.

The following statements create a SAS data set containing the data from the experiment:

```title 'Natural Recovery Time';
data mice;
input sex age time1 time2;
datalines;
1  57  631   631
1  45  .     170
1  54  227   227
1  43  143   143
1  64  916   .
1  67  691   705
1  44  100   100
1  59  730   .
1  47  365   365
1  74  1916  1916
2  79  1326  .
2  75  837   837
2  84  1200  1235
2  54  .     365
2  74  1255  1255
2  71  1823  .
2  65  537   637
2  33  583   683
2  77  955   .
2  46  577   577
;
```

The following SAS statements create the SAS data sets `xrow1` and `xrow2`:

```data xrow1;
input sex age time1 time2;
datalines;
1  50  .  .
;

data xrow2;
input sex age time1 time2;
datalines;
2  60.6  .  .
;
```

The following SAS statements fit a Weibull model with age, sex, and an age-by-sex interaction term as covariates, and create a plot of predicted probabilities against recovery time for the fixed values of `age` and `sex` specified in the SAS data set `xrow1`:

```ods graphics on;
proc lifereg data=mice xdata=xrow1;
class sex;
model (time1, time2) = age sex age*sex / dist=Weibull;

probplot  / nodata
plower=.5
vref(intersect) = 75
vreflab = '75 Percent';
inset;
run;
```

Standard output is shown in Output 51.4.1. Tables containing general model information, Type III tests for the main effects and interaction terms, and parameter estimates are created.

Output 51.4.1: Parameter Estimates for the Interaction Model

 Natural Recovery Time

The LIFEREG Procedure

Model Information
Data Set WORK.MICE
Dependent Variable Log(time1)
Dependent Variable Log(time2)
Number of Observations 20
Noncensored Values 9
Right Censored Values 5
Left Censored Values 2
Interval Censored Values 4
Number of Parameters 5
Name of Distribution Weibull
Log Likelihood -25.91033295

Type III Analysis of Effects
Effect DF Wald
Chi-Square
Pr > ChiSq
age 1 33.8496 <.0001
sex 1 14.0245 0.0002
age*sex 1 10.7196 0.0011

Analysis of Maximum Likelihood Parameter Estimates
Parameter   DF Estimate Standard Error 95% Confidence Limits Chi-Square Pr > ChiSq
Intercept   1 5.4110 0.5549 4.3234 6.4986 95.08 <.0001
age   1 0.0250 0.0086 0.0081 0.0419 8.42 0.0037
sex 1 1 -3.9808 1.0630 -6.0643 -1.8974 14.02 0.0002
sex 2 0 0.0000 . . . . .
age*sex 1 1 0.0613 0.0187 0.0246 0.0980 10.72 0.0011
age*sex 2 0 0.0000 . . . . .
Scale   1 0.4087 0.0900 0.2654 0.6294
Weibull Shape   1 2.4468 0.5391 1.5887 3.7682

The following two plots display the predicted probability against the recovery time for two different populations. Output 51.4.2 is created with the PROBPLOT statement with the option XDATA= `xrow1`, which specifies the population with `sex` = 1, `age` = 50. Output 51.4.3 is created with the PROBPLOT statement with the option XDATA= `xrow2`, which specifies the population with `sex` = 2, `age` = 60.6. These are the default values that the LIFEREG procedure would use for the probability plot if the XDATA= option had not been specified. Reference lines are used to display specified predicted probability points and their relative locations in the plot.

Output 51.4.2: Probability Plot for Recovery Time with `sex` = 1, `age` = 50

The following SAS statements fit a Weibull model with age, sex, and an age-by-sex interaction term as covariates, and create the plot of predicted probabilities against recovery time shown in Output 51.4.3, for the fixed values of `age` and `sex` specified in the SAS data set `xrow2`:

```proc lifereg data=mice xdata=xrow2;
class sex;
model (time1, time2) = age sex age*sex / dist=Weibull;

probplot  / nodata
plower=.5
vref(intersect) = 75
vreflab = '75 Percent';
inset;
run;
title;
ods graphics off;
```

Output 51.4.3: Probability Plot for Recovery Time with `sex` = 2, `age` = 60.6