The combination of survreg + weibull + gamma frailty doesn't work as you
(or I) would like.
The algebra for this model all works out when you treat it as a proportional
hazards model, with the given penalty function
(1/theta) \sum exp(omega_j) - omega_j
where theta is the variance of the random effect and omega_j is the frailty
for the jth group. However, survReg is written in accelerated failure time
notation, which means that the `natural' parameter in the code is
omega_j/ sigma, where sigma is the scale parameter. Now for the Gaussian
frailty case the penalty function is (1/ theta) \sum (omega_j)^2: here the
change of notation just factors out into the theta term and everything is
fine. The gamma case doesn't factor, and the code in frailty.gamma to find
the MLE for theta depends on the factorization.
Solutions -
1. Use the AIC or corrected AIC criteria, which doesn't depend on the
pretty algebra.
2. Wait for me to someday get to the task, on my list for 6 months now,
of working through the Weibull/survreg case carefully, and adding the
necessary code modifications (likely minor ones). I'd like to under-
stand the Morris and Christiansen (Lifetime Data Analysis 1995) estimate
of theta while I'm at it.
3. Write your own penalty function.
Terry Therneau
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