A Bayesian analysis of mixed survival models

Abstract

In proportional hazards models, the hazard of an animal A(t), ie, its probability of dying or being culled at time t given it is alive prior to t, is described as A(t) = '>" o (t)e W ' e where A o (t) is a 'baseline' hazard function and e w 'B represents the effect of covariates w on culling rate. A distribution can be attached to elements sq in 0, identifying, for example, genetic effects and leading to mixed survival models, also called 'frailty' models. To estimate the parameters T of the distribution of frailty terms, a Bayesian analysis is proposed. Inferences are drawn from the marginal posterior density x(T) which can be derived from the joint posterior density via Laplacian integration, a powerful technique related to saddlepoint approximations. The validity of this technique is shown here on simulated examples by comparing the resulting approximate x( T ) to the one obtained by algebraic integration. This exact calculation is feasible in very specific cases only, whereas the saddlepoint approximation can be applied to situations where Ao(t) is arbitrary (Cox models) or parametric (eg, Weibull), where the frailty terms are correlated through a known relationship matrix, or in more general models with stratification and/or time-dependent covariates. The influence of the censoring rate and the data structure is also illustrated.