Proportional hazards models for survival data with long-term survivors

Abstract

Survival analysis with long-term survivors has received considerable attention in recent years. It is useful in handling situations in which a proportion of subjects under study may never experience the event of interest. A commonly used approach is to formulate the model as a mixture of two populations, one for "long-term survivors" (subjects that will never experience the event of interest) and the other for "susceptibles" (subjects that will "fail" eventualy). This is an attractive approach to analyzing survival data with long-term survivors, in that it contains two parts which can be interpreted separately by adding structure to the standard survival model. Fully parametric approaches have had a long history and recent attention has focused on test for the presence of long-term survivors in the data for mixture model. Recently, Kuk and Chen (1992) proposed a model which is a semiparametric generalization of a parametric model above, which combines a logistic formulation for the probability of occurrence of the event with a proportional hazards specification for the time of occurence of the event. However the model proposed by Kuk and Chen (1992) does not have a proportional hazards structure for the survival function of the entire population, this structure is a desirable property in survival analysis models when doing covariates and is extensively used in survival analysis. In this dissertation, we will investigate an alternative mixture model with covariates, which does have a proportional hazards structure and is proposed by Maller and Zhou (1996) via the different motivation of the model from other cure models and is not further investigated so far. Their idea is to extend Cox model with a parametric or completely unspecified baseline to "improper" Cox model of which the baseline can be modeled as an improper and semiparametric structure of a combination of the probability of occurrence of the event with a proper survival function for the time of occurrence of the event. Partial and full likelihood methods are used to make statistical inference based on counting processes and martingale technique. In addition, we consider the problem of measurement erors for the covariates, and propose corrected partial and full likelihoods to obtain relevant estimators. We show that the resulting estimators are consistent and asymptotically normal. We study this improper proportional hazards model in both interior and boundary cases by maximum likelihood method, and develop a likelihood ratio test for the presence of an immune proportion in a population. Recently, much attention has been attracted to semiparametric transformation model, which provides many interesting statistical models and approaches. In this dissertation, we consider a class of semiparametric transformation models derived from the aforesaid "improper" PH model, which assume a linear relationship between an unknown transformation of the survival time under the proportional hazards model and the covariates. The random errors are modeled by an "improper" extreme value distribution, which is parametricaly specified with unknown parameters and covariates. Estimators for the coefficients of covariates are obtained from pseudo Z-estimator approach with censored observations. The consistency and asymptotic normality of the estimators are proved. This transformation model, coupled with proposed inference procedures, provides many alternatives to Cox proportional hazards models for survival analysis with long-term survivors. Although continuous-time survival analysisis frequently used in many settings, discrete-time survival analysis is often more natural in social and behavioral science applications where the survival data typicaly posses three features: discrete, ties and contain concomitant information. Inspired by the works of Potts (2004) and Linoff (2004), in this dissertation we will review some existing discrete time survival models which have already been proposed to analyze survival data from social and behavioral science by these authors, and then generalize these models to accommodate survival data with long-term survivors. As a natural extension of the continuous cases discussed in Chapters 1-4, in Chapter 5 of this dissertation we will predominantly focus on modeling discrete-time survival data which may accommodate proportional hazards structure and propose an alternative discrete time cure model which does have proportional hazards structure. The maximum likelihood estimation and approximate partial likelihood estimation are proposed to obtain the parameter estimators. The proposed models and approaches can be directly applied to analyze survival data from social and behavioral science such as the economic values for customer retention with long-life customers.