## Bayesian Survival Analysis with SAS/STAT Procedures

You can request Bayesian analysis of survival models in the LIFEREG and PHREG procedures. In addition to enabling you to fit the Cox model, PROC PHREG also enables 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 by using the Bayes theorem. The prior distribution enables you to incorporate into your 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 the 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 hazard 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 are very similar to those of 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 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.

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