Analysis of frailty survival models using Poisson variance structures.

Abstract

A key assumption of the popular Cox model is that the observations in the study are statistically independent of each other. However, it is not uncommon in practice to find that observations are correlated. A frailty model approach, where the correlation is induced by latent random variables, can be applied to correlated survival data. In relative risk models, the frailties are usually assumed to follow a parametric distribution and act multiplicatively on the conditional hazard rate. In this dissertation, new methods using Poisson variance structures are introduced to fit multivariate frailty models. The likelihood functions of both parametric (e.g., with piecewise constant baseline hazard) and semi-parametric multivariate frailty models are shown to be proportional to the likelihood functions of a class of mixed Poisson regression models. For multivariate lognormal frailty models, the penalized quasi-likelihood (PQL) or adaptive Gaussian quadrature numerical procedure can be applied for the maximum likelihood based inference of the mixed Poisson regression models. Thus, a rich variety of random effect structures can be modeled for survival analysis. Simulation studies show that mixed Poisson regression models using PQL or adaptive Gaussian quadrature methods perform well in estimating both the fixed and random effect parameters of multivariate lognormal frailty models. The studies also suggest that using penalized full likelihood instead of partial likelihood in computation yields less biased estimates for lognormal frailty parameters and their standard errors (SE) than those from the penalized partial likelihood (PPL) approach. The procedure is applied to the analysis of the national kidney transplantation dataset, the national patient days of waiting to deceased kidney transplant dataset, and a diabetic retinopathy study dataset. Further extensions to marginal relative risk models are also discussed.