Asymptotic and small sample statistical properties of random frailty variance estimates for shared gamma frailty models

This paper concerns maximum likelihood estimation for the semiparametric shared gamma frailty model; that is the Cox proportional hazards model with the hazard function multiplied by a gamma random variable with mean I and variance theta. A hybrid ML-EM algorithm is applied to 26 400 simulated samples of 400 to 8000 observations with Weibull hazards. The hybrid algorithm is much faster than the standard EM algorithm., faster than standard direct maximum likelihood (ML, Newton Raphson) for large samples, and gives almost identical results to the penalised likelihood method in S-PLUS 2000. When the true value theta (0) of theta is zero, the estimates of theta are asymptotically distributed as a 50-50 mixture between a point mass at zero and a normal random variable on the positive axis. When theta (0) > 0, the asymptotic distribution is normal. However, for small samples, simulations suggest that the estimates of 0 are approximately distributed as an x - (100 - x) % mixture, 0 less than or equal to x less than or equal to 50, between a point mass at zero and a normal random variable on the positive axis even for theta (0) > 0. In light of this, p-values and confidence intervals need to be adjusted accordingly. We indicate an approximate method for carrying out the adjustment.