The LIFEREG Procedure
Supported Distributions
For most distributions, the baseline survival function (S) and the probability density function(f) are listed for the additive random disturbance (
or 
) with location parameter 
and scale parameter 
. See the section Overview: LIFEREG Procedure for more information. These distributions apply when the log of the response is modeled (this is the default analysis). The corresponding survival function (G) and its density function (g) are given for the untransformed baseline distribution (
).
For the normal and logistic distributions, the response is not log transformed by PROC LIFEREG, and the survival functions and probability density functions listed apply to the untransformed response.
For example, for the WEIBULL distribution, 
and 
are the survival function and the probability density function for the extreme-value distribution (distribution of the log of the response), while 
and 
are the survival function and the probability density 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.
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. If there are no covariates in the model, = Intercept in the output; otherwise, 
. = Scale in the output.
Exponential
where 
.
Generalized Gamma
, 
, 
, 
, 
, 
, and
where 
denotes the complete gamma function, 
denotes the incomplete gamma function, and 
is a free shape parameter. The parameter is called Shape by PROC LIFEREG. See Lawless (2003, p. 240), and Klein and Moeschberger (1997, p. 386) for a description of the generalized gamma distribution.
Logistic
Log-Logistic
where 
and 
.
Lognormal
where 
is the cumulative distribution function for the normal distribution.
Normal
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, 
, 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.