The LIFEREG Procedure

Example 69.7 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 69.7.1.

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 seed=9999;
run;
ods graphics off;
```

Output 69.7.1: 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 69.7.2 are displayed by default.

Output 69.7.2: 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 69.7.3 are used. The specified gamma prior for the Weibull shape parameter is also shown in Output 69.7.3.

Output 69.7.3: 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 69.7.4. 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 69.7.4: Posterior Sample Statistics

Fit Statistics
DIC (smaller is better) 875.251
pD (effective number of parameters) 4.984

The LIFEREG Procedure

Bayesian Analysis

Posterior Summaries
Parameter N Mean Standard
Deviation
Percentiles
25% 50% 75%
Intercept 10000 2.4668 0.3862 2.1989 2.4621 2.7256
age 10000 -0.0115 0.00733 -0.0163 -0.0115 -0.00652
sex1 10000 -0.1255 0.2004 -0.2584 -0.1247 0.00817
perform 10000 0.3304 0.3317 0.1071 0.3188 0.5470
WeibShape 10000 0.7834 0.0518 0.7481 0.7815 0.8178

Posterior Intervals
Parameter Alpha Equal-Tail Interval HPD Interval
Intercept 0.050 1.7279 3.2368 1.7234 3.2264
age 0.050 -0.0260 0.00263 -0.0261 0.00244
sex1 0.050 -0.5197 0.2676 -0.5260 0.2583
perform 0.050 -0.2898 1.0072 -0.3200 0.9726
WeibShape 0.050 0.6846 0.8905 0.6805 0.8849

Posterior Correlation Matrix
Parameter Intercept age sex1 perform WeibShape
Intercept 1.0000 -.9018 -.3099 -.0888 -.1140
age -.9018 1.0000 -.0259 -.0363 0.0493
sex1 -.3099 -.0259 1.0000 0.1248 0.0371
perform -.0888 -.0363 0.1248 1.0000 -.0355
WeibShape -.1140 0.0493 0.0371 -.0355 1.0000

The default diagnostic statistics are displayed in Output 69.7.5. See the section Assessing Markov Chain Convergence in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details on Bayesian convergence diagnostics.

Output 69.7.5: Convergence Diagnostics

The LIFEREG Procedure

Bayesian Analysis

Posterior Autocorrelations
Parameter Lag 1 Lag 5 Lag 10 Lag 50
Intercept 0.0564 0.0030 0.0082 0.0234
age -0.0079 -0.0184 -0.0015 0.0239
sex1 0.6293 0.0700 0.0055 -0.0199
perform 0.6514 0.0773 0.0397 -0.0123
WeibShape 0.0719 -0.0083 -0.0062 0.0112

Geweke Diagnostics
Parameter z Pr > |z|
Intercept 0.4962 0.6198
age -0.4119 0.6804
sex1 -0.2519 0.8011
perform -0.1049 0.9165
WeibShape -0.6573 0.5110

Effective Sample Sizes
Parameter ESS Autocorrelation
Time
Efficiency
Intercept 7476.1 1.3376 0.7476
age 10000.0 1.0000 1.0000
sex1 2482.1 4.0288 0.2482
perform 2174.0 4.5998 0.2174
WeibShape 8538.8 1.1711 0.8539

Trace, autocorrelation, and density plots for the seven model parameters are shown in Output 69.7.6 through Output 69.7.10. 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 in Chapter 7: Introduction to Bayesian Analysis Procedures, for more information about assessing the convergence of the chain of posterior samples.

Output 69.7.6: Diagnostic Plots

Output 69.7.7: Diagnostic Plots

Output 69.7.8: Diagnostic Plots

Output 69.7.9: Diagnostic Plots

Output 69.7.10: Diagnostic Plots