Frailty Models for Modelling Heterogeneity

In the analysis of lifetime data, heterogeneity may be present in many situations. This heterogeneity is also referred to as frailty. Analysis that ignore frailty when it is present leads to incorrect inferences. In lifetime analysis, Cox proportional hazards model is commonly used to measure the effects of covariates. The covariates may fail to fully account for the true differences in risk. This may be due to an existence of another variable that is ignored in the model but can be explained by random frailty. Including frailty in the model can avoid underestimation and overestimation of parameters and also correctly measure the effects of the covariates on the response variable. This thesis presents an extension of Cox model to parametric frailty model in which the exponential and Weibull distributions are used as the distributions of baseline hazard, and the gamma and Weibull distributions are used as frailty distributions. We examine the maximum likelihood estimation procedures and propose the use of Monte Carlo integration method or quadrature method in complicated cases where explicit solutions to the likelihood functions can not be obtained. The gamma distribution is one of the most commonly used distributions for frailty. It has a closed form likelihood function that can be readily maximized. In this thesis, we study the performance of the Weibull distribution as a frailty distribution and compare with the gamma frailty model. Through simulation studies, the performance of the parameter estimates are evaluated. The effects of increasing the sample size and cluster size separately are also studied through Monte Carlo simulations. The Akaike Information Criteria (AIC) is used to compare the performance of the gamma and Weibull frailty models. The developed methods are then illustrated with numerical examples.