Supported Distributions

The LIFEREG Procedure

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 $\sigma$ 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. $\mu$ = Intercept and $\sigma$ = Scale in the output.

Exponential

$S(w) & = & \exp(-\exp(w-\mu)) \ f(w) & = & \exp(w-\mu) \exp(-\exp(w-\mu)) \ G(t) & = & \exp(-\alpha t) \ g(t) & = & \alpha \exp (- \alpha t) \$

where $\exp(-\mu) = \alpha$ .

Generalized Gamma

(with $\mu=0$ , $\sigma=1$ )

$S(w) & = & \{ \frac{ \Gamma ( \delta^{-2}, \delta^{-2} \exp (\delta w) ) } { \... ... } ( \delta^{-2} t^{\delta} )^{\delta^{-2}} \exp ( -t^{\delta} \delta^{-2} ) \$

where $\Gamma(a)$ denotes the complete gamma function, $\Gamma(a,z)$ denotes the incomplete gamma function, and $\delta$ is a free shape parameter. The $\delta$ 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

$S(w) & = & ( 1 + \exp ( \frac{w-\mu}{\sigma} ) )^{-1} \ f(w) & = & \frac{ \... ...(t) & = & \frac{ \alpha \gamma t^{\gamma - 1} } { ( 1 + \alpha t^{\gamma} )^2 }$

where $\gamma = 1 / \sigma$ and $\alpha = \exp(-\mu / \sigma)$ .

Lognormal

$S(w) & = & 1 - \Phi ( \frac{w-\mu}{\sigma} ) \ f(w) & = & \frac{1}{\sqrt{2 \pi}... ...qrt{2 \pi} \sigma t} \exp ( -\frac{1}2 ( \frac{\log(t)-\mu}{\sigma} )^2 ) \$

where $\Phi$ is the cumulative distribution function for the normal distribution.

Weibull

$S(w) & = & \exp ( - exp ( \frac{w-\mu}{\sigma} ) ) \ f(w) & = & \frac{1}{\sig... ...ma} ) \ g(t) & = & \gamma \alpha t^{\gamma - 1} \exp ( -\alpha t^{\gamma} ) \$

where $\sigma = 1/\gamma$ and $\alpha = \exp(-\mu / \sigma)$ .

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

$g(t;\lambda, \beta) & = & (\frac{\beta}{\lambda})^\beta (\frac{t}{\alpha})^{\b... ...\lambda})^\beta) \ G(t;\lambda, \beta) & = & \exp(-(\frac{t}{\lambda})^\beta) \$

so that $\lambda = \exp(\mu)$ and $\beta = 1/\sigma$ .

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 ${x^' \beta}$ .

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.

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