PROC PHREG: Priors for Model Parameters :: SAS/STAT(R) 9.2 User's Guide, Second Edition

The PHREG Procedure

Priors for Model Parameters

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. The priors for the hazards and the priors for the regression coefficients are assumed to be independent, while you can have a joint multivariate normal prior for the log-hazards and the regression coefficients.

Hazard Parameters

Let $\text{[math]}$ be the constant baseline hazards.

Improper Prior

The joint prior density is given by

$\text{[math]}$

This prior is improper (nonintegrable), but the posterior distribution is proper as long as there is at least one event time in each of the constant hazard intervals.

Uniform Prior

The joint prior density is given by

$\text{[math]}$

This prior is improper (nonintegrable), but the posteriors are proper as long as there is at least one event time in each of the constant hazard intervals.

Gamma Prior

The gamma distribution $\text{[math]}$ has a pdf

$\text{[math]}$

where $\text{[math]}$ is the shape parameter and $\text{[math]}$ is the scale parameter. The mean is $\text{[math]}$ and the variance is $\text{[math]}$ .

Independent Gamma Prior

Suppose for $\text{[math]}$ , $\text{[math]}$ has an independent $\text{[math]}$ prior. The joint prior density is given by

$\text{[math]}$

AR1 Prior

$\text{[math]}$ are correlated as follows:

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

The joint prior density is given by

$\text{[math]}$

Log-Hazard Parameters

Write $\text{[math]}$ .

Uniform Prior

The joint prior density is given by

$\text{[math]}$

Note that the uniform prior for the log-hazards is the same as the improper prior for the hazards.

Normal Prior

Assume $\text{[math]}$ has a multivariate normal prior with mean vector $\text{[math]}$ and covariance matrix $\text{[math]}$ . The joint prior density is given by

$\text{[math]}$

Regression Coefficients

Let $\text{[math]}$ be the vector of regression coefficients.

Uniform Prior

The joint prior density is given by

$\text{[math]}$

This prior is improper, but the posterior distributions for $\text{[math]}$ are proper.

Normal Prior

Assume $\text{[math]}$ has a multivariate normal prior with mean vector $\text{[math]}$ and covariance matrix $\text{[math]}$ . The joint prior density is given by

$\text{[math]}$

Joint Multivariate Normal Prior for Log-Hazards and Regression Coefficients

Assume $\text{[math]}$ has a multivariate normal prior with mean vector $\text{[math]}$ and covariance matrix $\text{[math]}$ . The joint prior density is given by

$\text{[math]}$

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