Parametric and Semiparametric Conditional Shared Gamma Frailty Models with Events Before Study Entry

Abstract

It is useful for prevention to detect familial aggregation of age at onset of disease. There are usually two major aspects associated with such a study: dependent survival times and events before study entry. Failure to properly account for events before study entry would seriously underestimate the association between complete survival times in the same family. This article intends to deal with these two aspects together in conditional shared gamma frailty models in which the hazard functions in the same cluster (family) are multiplied by the same (unobserved) random frailties and the survival times conditional on frailties are independent. We describe an algorithm based on maximum likelihood and expectation-maximization for the calculation of maximum likelihood estimates in these models and examine asymptotic and small sample statistical properties for parameter estimates, especially for the frailty variance parameter. Let θ ₀ be the true value of the frailty variance θ. Then the asymptotic distribution of frailty variance estimates is normal for θ ₀ > 0 while it is a 50–50 mixture between a point mass at zero and a normal random variable on the positive axis for θ ₀ = 0. For small samples, simulations suggest that the frailty variance estimates are approximately distributed as an x-(100−x)% mixture, 0 ≤ x ≤ 50, between a point mass at zero and a normal random variable on the positive axis even for θ ₀ > 0.

Keywords:

• Dependent survival times; Events before study entry; Conditional frailty models; Shared gamma frailties; Maximum likelihood estimates; ML-EM; Simulations

Acknowledgment

This research was supported by Grant 139077 from the National Health and Medical Research Council of Australia.