PROC PHREG: Starting Values of the Markov Chains :: SAS/STAT(R) 9.2 User's Guide, Second Edition

The PHREG Procedure

Starting Values of the Markov Chains

When the BAYES statement is specified, PROC PHREG generates one Markov chain containing the approximate posterior samples of the model parameters. Additional chains are produced when the Gelman-Rubin diagnostics are requested. Starting values (or initial values) can be specified in the INITIAL= data set in the BAYES statement. If INITIAL= option is not specified, PROC PHREG picks its own initial values for the chains. If the prior distribution of the parameter $\text{[math]}$ is proper, the starting values for $\text{[math]}$ are based on the estimated mean ( $\text{[math]}$ ) and standard deviation ( $\text{[math]}$ ) of the posterior distribution given the MLE. If the prior distribution of $\text{[math]}$ is improper, the starting values for $\text{[math]}$ are based on the MLE and its standard error estimate; that is, $\text{[math]}$ is the MLE of $\text{[math]}$ and $\text{[math]}$ is the standard error of maximum likelihood estimator.

Denote $\text{[math]}$ as the integral value of $\text{[math]}$ .

Constant Baseline Hazards $\text{[math]}$ ’s

For the first chain that the summary statistics and diagnostics are based on, the initial values are

$\text{[math]}$

For subsequent chains, the starting values are picked in two different ways according to the total number of chains specified. If the total number of chains specified is less than or equal to 10, initial values of the $\text{[math]}$ th chain ( $\text{[math]}$ ) are given by

$\text{[math]}$

with the plus sign for odd $\text{[math]}$ and minus sign for even $\text{[math]}$ . If the total number of chains is greater than 10, initial values are picked at random over a wide range of values. Let $\text{[math]}$ be a uniform random number between 0 and 1; the initial value for $\text{[math]}$ is given by

$\text{[math]}$

Regression Coefficients and Log-Hazard Parameters $\text{[math]}$ ’s

The $\text{[math]}$ ’s are the regression coefficients $\text{[math]}$ ’s, and in the piecewise exponential model, include the log-hazard parameters $\text{[math]}$ ’s. For the first chain that the summary statistics and regression diagnostics are based on, the initial values are

$\text{[math]}$

If the number of chains requested is less than or equal to 10, initial values for the $\text{[math]}$ th chain ( $\text{[math]}$ ) are given by

$\text{[math]}$

with the plus sign for odd $\text{[math]}$ and minus sign for even $\text{[math]}$ . When there are more than 10 chains, the initial value for the $\text{[math]}$ is picked at random over the range $\text{[math]}$ ; that is,

$\text{[math]}$

where $\text{[math]}$ is a uniform random number between 0 and 1.

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Starting Values of the Markov Chains

Constant Baseline Hazards ’s

Regression Coefficients and Log-Hazard Parameters ’s

Constant Baseline Hazards $\text{[math]}$ ’s

Regression Coefficients and Log-Hazard Parameters $\text{[math]}$ ’s