Sampling from the Posterior Distribution

PROC PHREG uses a Gibbs sampler to generate the posterior samples. See the section Gibbs Sampler for a general discussion.

Let be the parameter vector. For the Cox model, the ’s are the regression coefficients ’s, and for the piecewise constant baseline hazard model, the ’s consist of the baseline hazards ’s (or log baseline hazards ’s) and the regression coefficients ’s. Let be the likelihood function, where is the observed data. Note that for the Cox model, the likelihood contains the infinite-dimensional baseline hazard function and the Gamma process is perhaps the most commonly used prior process (Ibrahim, Chen, and Sinha; 2001); however, Sinha, Ibrahim, and Chen (2003) justify using the partial likelihood as the likelihood function for the Bayesian analysis. Let be the prior distribution. The full conditional distribution of is proportional to the joint distribution; that is,

For instance, the one-dimensional conditional distribution of given , is computed as

Suppose you have a set of arbitrary starting values . Using the ARMS (adaptive rejection Metropolis sampling) algorithm of Gilks, Best, and Tan (1995), you do the following:

• draw from

• draw from

• 

• draw from

This completes one iteration of the Gibbs sampler. After one iteration, you have . After iterations, you have .

You can output these posterior samples into a SAS data set through ODS. The following SAS statement outputs the posterior samples into the SAS data set Post:

    ods output PosteriorSample=Post;

The data set also includes the variable LogPost, representing the log of the posterior log likelihood.