## Bayesian Survival Analysis with SAS/STAT Procedures

Bayesian analysis of survival models can be requested in the LIFEREG and PHREG procedures. In addition to the Cox model, PROC PHREG also allow you to fit a piecewise exponential model. In Bayesian analysis, the model parameters are treated as random variables, and inference about parameters is based on the posterior distribution of the parameters. A posterior distribution is a weighted likelihood function of the data with a prior distribution of the parameters using the Bayes theorem. The prior distribution enables you to incorporate into the analysis knowledge or experience of the likely range of values of the parameters of interest. You can specify normal or uniform prior distributions for the model regression coefficients in both LIFEREG and PHREG procedures. In addition, you can specify a gamma or improper prior distribution for the scale or variance parameter in PROC LIFEREG. For the piecewise exponential model in PROC PHREG, you can specify normal or uniform prior distributions for the log-hazard parameters; alternatively, you can specify gamma or improper prior distributions for the hazards parameters. If you have no prior knowledge of the parameter values, you can use a noninformative prior distribution, and the results of a Bayesian analysis will be very similar to a classical analysis based on maximum likelihood.

A closed form of the posterior distribution is often not feasible, and a Markov chain Monte Carlo method by Gibbs sampling is used to simulate samples from the posterior distribution. You can perform inference by using the simulated samples, for example, to estimate the probability that a function of the parameters of interest lies within a specified range of values.

See Chapter 7: Introduction to Bayesian Analysis Procedures, for an introduction to the basic concepts of Bayesian statistics. Also see Bayesian Analysis: Advantages and Disadvantages for a discussion of the advantages and disadvantages of Bayesian analysis. See Ibrahim, Chen, and Sinha (2001); Gelman et al. (2004); Gilks, Richardson, and Spiegelhalter (1996) for more information about Bayesian analysis, including guidance about choosing prior distributions.