Estimation in Semiparametric Marginal Shared Gamma Frailty Models

Abstract

The semiparametric marginal shared frailty models in survival analysis have the non–parametric hazard functions multiplied by a random frailty in each cluster, and the survival times conditional on frailties are assumed to be independent. In addition, the marginal hazard functions have the same form as in the usual Cox proportional hazard models. In this paper, an approach based on maximum likelihood and expectation–maximization is applied to semiparametric marginal shared gamma frailty models, where the frailties are assumed to be gamma distributed with mean 1 and variance θ. The estimates of the fixed–effect parameters and their standard errors obtained using this approach are compared in terms of both bias and efficiency with those obtained using the extended marginal approach. Similarly, the standard errors of our frailty variance estimates are found to compare favourably with those obtained using other methods. The asymptotic distribution of the frailty variance estimates is shown to be a 50–50 mixture of a point mass at zero and a truncated normal random variable on the positive axis for θ0 = 0. Simulations demonstrate that, for θ0 < 0, it is approximately an x−(100 −x)%, 0 ≤x≤ 50, mixture between a point mass at zero and a truncated normal random variable on the positive axis for small samples and small values of θ0; otherwise, it is approximately normal.