On competing risks data with covariates and long-term survivors

Abstract

In the study of survival data, one of the important problems is about competing risks combined with the possible existence of long-term survivors (subjects that will never experience the events under consideration, also referred to as "cured" or "immunes"). Under this scenario, it is important to know the failure rates with respect to different risks and the cured proportion. It is also useful to make inferences on the regression co-efficients of the covariates that influence the failure. Further analysis of the significance levels of the parameters plays an important role in the study. In particular, a major attention is paid to the significance level of the cured rate which implies the existence of immunes. In this dissertation, three models are investigated for survival data with competing risks, covariates and immunes: general mixture model, piecewise exponential mixture model and proportional cause-specific hazards model. In the general mixture setting, full maximum likelihood methods are employed to draw statistical inferences on the model attributes and the asymptotic properties of the estimators. Likelihood ratio tests are developed to test the significance levels of the parameters and the relationships among them. Under some regularity conditions and mild assumptions, the estimators are proved to be consistent and asymptotically normally distributed, and the tests are also consistent and follow different distributions according to the underlying hypotheses. The performances of the estimators and tests are assessed by a simulation study. It shows that the approach given in this part provides a satisfactory way to investigate many practical problems. The second part of the dissertation is the piecewise exponential mixture model for competing risks data. The existence, consistency and asymptotic normality of the estimators are rigorously derived under general sufficient conditions. Likelihood ratio tests are investigated for various hypotheses of practical interest. A study of real life data is conducted to illustrate the approach. In addition, a semi-parametric approach is proposed to investigate the competing risks data under the assumptions of independent censoring and proportional cause-specific hazard functions among different risks and covariates. Partial likelihood methods are used to make inferences on the levels of the risks.