#
Bayesian Analysis Using the PHREG Procedure

The PHREG procedure performs
regression analysis of survival data based on the Cox proportional hazards model. Cox’s semiparametric model is
widely used in the analysis of survival data to explain the effect of explanatory variables on hazard rates. The PHREG procedure's
Bayesian analysis capabilities enable you to do the following:

- fit a Cox proportional hazards model
- fit piecewise constant baseline hazard models (also known as piecewise exponential models)
- fit a superset of the Cox model, known as the multiplicative hazards model (also known as the Anderson-Gill model)
- estimate customized hazard ratios
- estimate the survival function
- fit multinomial logit choice models for discrete choice data

To perform Bayesian analyses with PROC PHREG, you specify a model essentially the same way you do for a frequentist
approach, but you add a BAYES statement to request Bayesian estimation methods for fitting the model.
The BAYES statement
requests that the parameters of the model be estimated by Markov chain Monte Carlo sampling techniques and provides
options that enable you to specify prior information, control the sampling, and obtain posterior summary statistics and convergence diagnostics. You can
also save the posterior samples to a SAS data set for further analysis.

### Sampling Algorithms

The PHREG procedure supports the following sampling algorithms:

- The adaptive rejection Metropolis algorithm(Gilks and Wild 1992; Gilks, Best, and Tan 1995) is the default.
- The random walk Metropolis algorithm can be specified for all models.

### Priors

The PHREG procedure supports the following priors:

**Parameter** |
**Prior** |

Regression coefficients |
Normal, uniform, Zellner *g*-prior |

*g* (Zellner) |
Constant, gamma |

Baseline hazards (original scale) |
Improper, uniform, gamma, independent gamma, AR(1) |

Baseline hazards (log scale) |
Uniform, normal |

Log-hazards and regression coefficients |
Joint multivariate normal |

For a Cox model, the model parameters are the regression coefficients. For a piecewise exponential model, the model parameters
consist of the regression coefficients and the hazards or log-hazards.

### Bayesian Anaysis Examples