On the mixture models in survival analysis with competing risks and covariates

Abstract

This thesis consists of two parts, both focus on important issues in mixture models for survival analysis with long-term survivors. In Part I, we study the large-sample properties of a class of parametric mixture models with covariates for competing risks. The models allow general distributions for the survival times and incorporate the idea of long-term survivors. Asymptotic results are obtained under a commonly assumed independent censoring mechanism and some modest regularity conditions on the survival distributions. The existence, consistency, and asymptotic normality of maximum likelihood estimators for the parameters of the model are rigorously derived under general sufficient conditions. Specific conditions for particular models can be derived from the general conditions for ready check. Two most often used models (exponential and Weibull) in survival analysis are selected for illustration. In addition, a likelihood-ratio statistic is proposed to test various hypotheses of practical interest, and its asymptotic distribution is provided. In Part II, we investigate the sufficiency of follow-up in survival analysis with long-term survivors, which is a crucial condition when estimating non-parametrically the proportion of long term-survivors in a mixture model. The concept of the degree of sufficiency of follow-up is introduced and an estimator for such a degree of sufficiency is then derived. This estimator can be used to construct a heuristic and non-parametric test for the sufficiency of follow-up. Guidelines are also provided on how to plan the duration of a follow-up experiment with long-term survivors so that sufficient follow-up can be expected in advance. The whole process is simple to implement.