A Bayesian analysis of mixed survival models

Abstract

In proportional hazards models, the hazard of an animal lambda(t),ie, its probability of dying or being culled at time t given it isalive prior to t, is described as lambda(t) = lambda(0)(t)e(w'theta)where lambda(0)(t) is a 'baseline' hazard function and e(w'theta)represents the effect of covariates w on culling rate. A distributioncan be attached to elements s(q) in theta, identifying, for example,genetic effects and leading to mixed survival models, also called'frailty' models. To estimate the parameters tau of the distributionof frailty terms, a Bayesian analysis is proposed. Inferences aredrawn from the marginal posterior density pi(tau) which can be derivedfrom the joint posterior density via Laplacian integration, a powerfultechnique related to saddlepoint approximations. The validity ofthis technique is shown here on simulated examples by comparing theresulting approximate pi(tau) to the one obtained by algebraic integration.This exact calculation is feasible in very specific cases only, whereasthe saddlepoint approximation can be applied to situations whereXo(t) is arbitrary (Cox models) or parametric (eg, Weibull), wherethe frailty terms are correlated through a known relationship matrix,or in more general models with stratification and/or time-dependentcovariates. The influence of the censoring rate and the data structureis also illustrated.