Semiparametric regression models in survival analysis
In survival analysis and clinical trials, it is important to consider the relationship of lifetime to other factors. One way to do this is through regression models, in which the dependence of the lifetime variable on concomitant variables is explicitly recognized. There is a vast literature considering the parametric regression models. For example, Kalbfleisch and Prentice (1980), Lawless (1982) and Fleming and Harrington (1991) discuss parametric regression models for lifetime distribution in detail. But if the relationship between a lifetime and a set of concomitant variables can not be described by a parametric regression model, we may consider nonparametric and semiparametric regression models. Since the semiparametric approach and its asymptotic properties in survival analysis are not fully developed in the literature, this thesis tries to fill the gap to some extent. In the thesis, we propose two classes of semiparametric regression models: semi-parametric proportional hazards model and semiparametric location-scale model for log lifetime log T. We extend the "generalized profile likelihood" method of Severini and Wong (1992) to the survival analysis setting. Local likelihood and generalized profile likelihood are used alternatively to estimate nonparametric and parametric components. Maximum likelihood estimation has a drawback in the sense it has to assume a particular parametric form for the unknown targeting function. We overcome this problem by using a local maximum likelihood in the semiparametric models to estimate the nonparametric components first. Then, "generalized profile likelihood" is used to estimate the parameters of interest. By using a martingale technique and the theory of counting processes, the estimators are proved to be consistent and asymptotically normal under some regularity conditions. For some typical semiparametric regression models, we give the algorithms. A simulation study is developed to explore the efficiency of the estimators. They perform very well in capturing the functional form of the nonparametric component. Furthermore, two real data sets are analysed applying the methodologies we proposed. Compared with traditional parametric regression models, the semiparametric models possess some better properties, and they provide a powerful way to discover the unknown functional forms of the covariates.