Bayesian Analysis 
PROC PHREG uses the partial likelihood of the Cox model as the likelihood and generates a chain of posterior distribution samples by the Gibbs Sampler. Summary statistics, convergence diagnostics, and diagnostic plots are provided for each parameter. The following statements generate Figure 66.4–Figure 66.10:
ods graphics on; proc phreg data=Rats; model Days*Status(0)=Group; bayes seed=1 outpost=Post; run; ods graphics off;
The BAYES statement invokes the Bayesian analysis. The SEED= option is specified to maintain reproducibility; the OUTPOST= option saves the posterior distribution samples in a SAS data set for postprocessiong; no other options are specified in the BAYES statement. By default, a uniform prior distribution is assumed on the regression coefficient Group. The uniform prior is a flat prior on the real line with a distribution that reflects ignorance of the location of the parameter, placing equal probability on all possible values the regression coefficient can take. Using the uniform prior in the following example, you would expect the Bayesian estimates to resemble the classical results of maximizing the likelihood. If you can elicit an informative prior on the regression coefficients, you should use the COEFFPRIOR= option to specify it.
You should make sure that the posterior distribution samples have achieved convergence before using them for Bayesian inference. PROC PHREG produces three convergence diagnostics by default. If ODS Graphics is enabled before calling PROC PHREG as in the preceding program, diagnostics plots are also displayed.
The results of this analysis are shown in the following figures.
The "Model Information" table in Figure 66.4 summarizes information about the model you fit and the size of the simulation.
Model Information  

Data Set  WORK.RATS  
Dependent Variable  Days  Days from Exposure to Death 
Censoring Variable  Status  
Censoring Value(s)  0  
Model  Cox  
Ties Handling  BRESLOW  
Sampling Algorithm  ARMS  
BurnIn Size  2000  
MC Sample Size  10000  
Thinning  1 
PROC PHREG first fits the Cox model by maximizing the partial likelihood. The only parameter in the model is the regression coefficient of Group. The maximum likelihood estimate (MLE) of the parameter and its 95% confidence interval are shown in Figure 66.5.
Maximum Likelihood Estimates  

Parameter  DF  Estimate  Standard Error 
95% Confidence Limits  
Group  1  0.5959  0.3484  1.2788  0.0870 
Since no prior is specified for the regression coefficient, the default uniform prior is used. This information is displayed in the "Uniform Prior for Regression Coefficients" table in Figure 66.6.
Uniform Prior for Regression Coefficients 


Parameter  Prior 
Group  Constant 
The "Fit Statistics" table in Figure 66.7 lists information about the fitted model. The table displays the DIC (deviance information criterion) and pD (effective number of parameters). See the section Fit Statistics for details.
Fit Statistics  

DIC (smaller is better)  203.444 
pD (Effective Number of Parameters)  1.003 
Summary statistics of the posterior samples are displayed in the "Posterior Summaries" table and "Posterior Intervals" table as shown in Figure 66.8. Note that the mean and standard deviation of the posterior samples are comparable to the MLE and its standard error, respectively, due to the use of the uniform prior.
Posterior Summaries  

Parameter  N  Mean  Standard Deviation 
Percentiles  
25%  50%  75%  
Group  10000  0.5998  0.3511  0.8326  0.5957  0.3670 
Posterior Intervals  

Parameter  Alpha  EqualTail Interval  HPD Interval  
Group  0.050  1.3042  0.0721  1.2984  0.0756 
PROC PHREG provides diagnostics to assess the convergence of the generated Markov chain. Figure 66.9 shows three of these diagnostics: the lag1, lag5, lag10, and lag50 autocorrelations; the Geweke diagnostic; and the effective sample size. There is no indication that the Markov chain has not reached convergence. Refer to the section Statistical Diagnostic Tests for information about interpreting these diagnostics.
Posterior Autocorrelations  

Parameter  Lag 1  Lag 5  Lag 10  Lag 50 
Group  0.0079  0.0091  0.0161  0.0101 
Geweke Diagnostics  

Parameter  z  Pr > z 
Group  0.0149  0.9881 
Effective Sample Sizes  

Parameter  ESS  Autocorrelation Time 
Efficiency 
Group  10000.0  1.0000  1.0000 
You can also assess the convergence of the generated Markov chain by examining the trace plot, the autocorrelation function plot, and the posterior density plot. Figure 66.10 displays a panel of these three plots for the parameter Group. This graphical display is automatically produced when ODS Graphics is enabled. Note that the trace of the samples centers on –0.6 with only small fluctuations, the autocorrelations are quite small, and the posterior density appears bellshaped—all exemplifying the behavior of a converged Markov chain.
The proportional hazards model for comparing the two pretreatment groups is
The probability that the hazard of Group=0 is greater than that of Group=1 is
This probability can be enumerated from the posterior distribution samples by computing the fraction of samples with a coefficient less than 0. The following DATA step and PROC MEANS perform this calculation:
data New; set Post; Indicator=(Group < 0); label Indicator='Group < 0'; run; proc means data=New(keep=Indicator) n mean; run;
Analysis Variable : Indicator Group < 0 


N  Mean 
10000  0.9581000 
The PROC MEANS results are displayed in Figure 66.11. There is a 95.8% chance that the hazard rate of Group=0 is greater than that of Group=1. The result is consistent with the fact that the average survival time of Group=0 is less than that of Group=1.