## Supported Distributions

For each distribution, the baseline survival distribution
function (*S*) and the probability density function(*f*)
are listed for the additive random disturbance.
These distributions apply when the log of the
response is modeled (this is the default analysis).
The corresponding survival distribution function
(*G*) and its density function (*g*) are given
for the untransformed baseline distribution.
For example, for the WEIBULL distribution,
*S*(*w*) and *f*(*w*) are
the baseline survival distribution function and the probability
density function for the extreme value distribution (the log
of the response) while *G*(*t*) and *g*(*t*) are the survival
distribution function and probability distribution function
of a Weibull distribution (using the untransformed response).
The chosen baseline functions define the meaning
of the intercept, scale, and shape parameters.
Only the gamma distribution has a free shape
parameter in the following parameterizations.
Notice that some of the distributions do not have
mean zero and that is not, in general, the
standard deviation of the baseline distribution.

Additionally, it is worth mentioning that, for the Weibull
distribution, the accelerated failure time model is also a
proportional-hazards model. However, the parameterization for the
covariates differs by a multiple of the scale parameter from the
parameterization commonly used for the proportional hazards model.

The distributions supported in the LIFEREG procedure follow.
= Intercept and = Scale in the output.

*Exponential*

where .
*Generalized Gamma*

(with , )

where denotes the complete gamma function,
denotes the incomplete gamma function,
and is a free shape parameter.
The parameter is referred to as Shape by PROC LIFEREG.
Refer to Lawless, 1982, p.240 and Klein and Moeschberger, 1997, p.386
for a description of the generalized gamma distribution.
*Loglogistic*

where and .*Lognormal*

where is the cumulative distribution
function for the normal distribution.
*Weibull*

where and .If your parameterization is different from the ones shown here,
you can still use the procedure to fit your model. For example,
a common parameterization for the Weibull distribution is

so that and .Again note that the expected value of the baseline
log response is, in general, not zero and that
the distributions are not symmetric in all cases.
Thus, for a given set of covariates, **x**, the expected
value of the log response is not always .

Some relations among the distributions are as follows:

- The gamma with Shape=1 is a Weibull distribution.
- The gamma with Shape=0 is a lognormal distribution.
- The Weibull with Scale=1 is an exponential distribution.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.