Contributions to frailty and copula modelling with applications to clinical trials and dairy cows data

Abstract

In survival analysis the response of interest is the time from a well-defined time origin to the occurrence of a specific event. Some examples are the time from onset of a disease to death, the time from recovery to the time of recurrence of a disease, the time from purchase to breakdown of a machine. This response is called the survival time, the failure time or the event time. A special di±culty that often occurs in the analysis of survival data is the possibility that some responses are not observed for the full event time. Such incomplete observation of the event time is called (right) censoring. A classical model that is frequently used to model univariate survival data subject to right censoring, is the proportional hazards model (Cox, 1972). In many studies there is a natural clustering in the data; event times within the same cluster may be correlated. Such data are known as clustered survival data. Clustered survival data are a particular example of multivariate survival data (see, e.g., Klein and Moeschberger, 2003, p.425). In recent years, extensive research on clustered survival data has been carried out. A lot of attention has been paid to frailty models and copula models. In frailty models the cluster effect is a random effect; therefore frailty models are conditional models. A frailty model is a multiplicative hazard model with three components: a frailty factor that models the random cluster effect, a baseline hazard function (that can be modelled in a parametric way or that can be left unspecified) and a component that models (in a parametric way) the dependence of the hazard on the covariates. Copula models are used to model multivariate survival data with small and equal cluster size. In this thesis we study copula models for four-dimensional survival data. In copula models the joint survival function of the four event times in a cluster is modelled as a function, called the copula, of the marginal survival functions of the four event times. The copula determines the type of dependence. The marginal survival functions can be modelled in a parametric, a semi-parametric or a nonparametric way, possibly taking into account the effect of covariates (Shih and Louis, 1995b; Glidden, 2000; Andersen, 2005). In Section 1.2 we define the basic quantities that are used in survival analysis and we review classical models that can be used to analyse univariate survival data. In Section 1.3 we discuss multivariate survival data, which is the type of data considered in this thesis, in somewhat more detail. The examples, that will be used in future chapters to illustrate the developed methodology, are collected in Section 1.4. Section 1.5 concludes this chapter with a discussion of the thesis objectives.