The LIFEREG Procedure

## Example 48.7 Bayesian Analysis of Clinical Trial Data

Consider the data on melanoma patients from a clinical trial described in Ibrahim, Chen, and Sinha (2001). A partial listing of the data is shown in Output 48.7.1.

The survival time is modeled by a Weibull regression model with three covariates. An analysis of the right-censored survival data is performed with PROC LIFEREG to obtain Bayesian estimates of the regression coefficients by using the following SAS statements:

```   ods graphics on;
proc lifereg data=e1684;
class Sex;
model Survtime*Survcens(1)=Age Sex Perform / dist=Weibull;
bayes WeibullShapePrior=gamma;
run;
ods graphics off;
```

Output 48.7.1 Clinical Trial Data
Obs survtime survcens age sex perform
1 1.57808 2 35.9945 1 0
2 1.48219 2 41.9014 1 0
3 7.33425 1 70.2164 2 0
4 0.65479 2 58.1753 2 1
5 2.23288 2 33.7096 1 0
6 9.38356 1 47.9726 1 0
7 3.27671 2 31.8219 2 0
8 0.00000 1 72.3644 2 0
9 0.80274 2 40.7151 2 0
10 9.64384 1 32.9479 1 0
11 1.66575 2 35.9205 1 0
12 0.94247 2 40.5068 2 0
13 1.68767 2 57.0384 1 0
14 5.94247 2 63.1452 1 0
15 2.34247 2 62.0630 1 0
16 0.89863 2 56.5342 1 1
17 9.03288 1 22.9945 2 0
18 9.63014 1 18.4712 1 0
19 0.52603 2 41.2521 1 0
20 1.82192 2 29.5178 1 0

Maximum likelihood estimates of the model parameters shown in Output 48.7.2 are displayed by default.

Output 48.7.2 Maximum Likelihood Parameter Estimates
The LIFEREG Procedure

Bayesian Analysis

Analysis of Maximum Likelihood Parameter Estimates
Parameter   DF Estimate Standard Error 95% Confidence Limits
Intercept   1 2.4402 0.3716 1.7119 3.1685
age   1 -0.0115 0.0070 -0.0253 0.0023
sex 1 1 -0.1170 0.1978 -0.5046 0.2707
sex 2 0 0.0000 . . .
perform   1 0.2905 0.3222 -0.3411 0.9220
Scale   1 1.2537 0.0824 1.1021 1.4260
Weibull Shape   1 0.7977 0.0524 0.7012 0.9073

Since no prior distributions for the regression coefficients were specified, the default uniform improper distributions shown in the "Uniform Prior for Regression Coefficients" table in Output 48.7.3 are used. The specified gamma prior for the Weibull shape parameter is also shown in Output 48.7.3.

Output 48.7.3 Model Parameter Priors
The LIFEREG Procedure

Bayesian Analysis

Uniform Prior for Regression
Coefficients
Parameter Prior
Intercept Constant
age Constant
sex1 Constant
perform Constant

Independent Prior Distributions for Model Parameters
Parameter Prior Distribution Hyperparameters
Weibull Shape Gamma Shape 0.001 Inverse Scale 0.001

Fit statistics, descriptive statistics, interval statistics, and the sample parameter correlation matrix for the posterior sample are displayed in the tables in Output 48.7.4. Since noninformative prior distributions for the regression coefficients were used, the mean and standard deviations of the posterior distributions for the model parameters are close to the maximum likelihood estimates and standard errors.

Output 48.7.4 Posterior Sample Statistics
Fit Statistics
DIC (smaller is better) 875.156
pD (effective number of parameters) 4.935

The LIFEREG Procedure

Bayesian Analysis

Posterior Summaries
Parameter N Mean Standard
Deviation
Percentiles
25% 50% 75%
Intercept 10000 2.4762 0.3794 2.2140 2.4697 2.7303
age 10000 -0.0117 0.00717 -0.0165 -0.0117 -0.00693
sex1 10000 -0.1261 0.2024 -0.2622 -0.1253 0.0125
perform 10000 0.3279 0.3352 0.0966 0.3152 0.5407
WeibShape 10000 0.7826 0.0517 0.7467 0.7820 0.8167

Posterior Intervals
Parameter Alpha Equal-Tail Interval HPD Interval
Intercept 0.050 1.7530 3.2344 1.7558 3.2366
age 0.050 -0.0256 0.00231 -0.0255 0.00236
sex1 0.050 -0.5235 0.2654 -0.5217 0.2666
perform 0.050 -0.3079 1.0233 -0.3291 0.9937
WeibShape 0.050 0.6850 0.8864 0.6803 0.8807

Posterior Correlation Matrix
Parameter Intercept age sex1 perform WeibShape
Intercept 1.0000 -.8983 -.2988 -.0759 -.1422
age -.8983 1.0000 -.0467 -.0401 0.0685
sex1 -.2988 -.0467 1.0000 0.0874 0.0485
perform -.0759 -.0401 0.0874 1.0000 -.0320
WeibShape -.1422 0.0685 0.0485 -.0320 1.0000

The default diagnostic statistics are displayed in Output 48.7.5. See the section Assessing Markov Chain Convergence for more details on Bayesian convergence diagnostics.

Output 48.7.5 Convergence Diagnostics
The LIFEREG Procedure

Bayesian Analysis

Posterior Autocorrelations
Parameter Lag 1 Lag 5 Lag 10 Lag 50
Intercept 0.0660 0.0089 0.0194 -0.0107
age 0.0035 -0.0134 0.0180 -0.0175
sex1 0.6230 0.0645 -0.0057 -0.0121
perform 0.6594 0.1132 0.0199 -0.0100
WeibShape 0.0923 0.0322 0.0050 0.0063

Geweke Diagnostics
Parameter z Pr > |z|
Intercept 1.4526 0.1463
age -1.5941 0.1109
sex1 -0.6555 0.5121
perform -0.1008 0.9197
WeibShape 0.0127 0.9899

Effective Sample Sizes
Parameter ESS Correlation
Time
Efficiency
Intercept 7262.3 1.3770 0.7262
age 10000.0 1.0000 1.0000
sex1 2521.7 3.9656 0.2522
perform 2071.8 4.8266 0.2072
WeibShape 6585.9 1.5184 0.6586

Trace, autocorrelation, and density plots for the seven model parameters are shown in Output 48.7.6 through Output 48.7.10. These plots show no indication that the Markov chains have not converged. See the sections Assessing Markov Chain Convergence and Visual Analysis via Trace Plots for more information about assessing the convergence of the chain of posterior samples.

Output 48.7.6 Diagnostic Plots

Output 48.7.7 Diagnostic Plots

Output 48.7.8 Diagnostic Plots

Output 48.7.9 Diagnostic Plots

Output 48.7.10 Diagnostic Plots

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