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|
| 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;
| 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.
| 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.
| 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.
| 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.
| 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.
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