In survival analysis or time-to-event analysis, one of the primary goals of analysis is to predict the occurrence of an event of interest for subjects within the study. Even though survival analysis methods were originally developed and used in medical re- search, those methods are also commonly used nowadays in other areas as well, such as in predicting the default of a loan and in estimating of the failure of a system. To include covariates in the analysis, the most widely used models are the propor- tional hazard model developed by Cox (1972) and the accelerated failure time model developed by Buckley and James (1979). The proportional hazard (PH) model as- sumes subjects from different groups have their hazard functions proportionally, while the accelerated failure time (AFT) model assumes the effect of covariates is to accel- erate or decelerate the occurrence of event of interest. In some survival analyses, not all subjects in the study will experience the event. Such a group of individuals is referred to `cured' group. To analyze a data set with a cured fraction, Boag (1948) and Berkson and Gage (1952) discussed a mixture cure model. Since then, the cure model and associated inferential methods have been widely stud- ied in the literature. It has also been recognized that subjects in the study are often correlated within clusters or groups; for example, patients in a hospital would have similar conditions and environment. For this reason, Vaupel et al. (1979) proposed a frailty model to model the correlation among subjects within clusters and conse- quently the presence of heterogeneity in the data set. Hougaard (1989), McGilchrist and Aisbett (1991), and Klein (1992) all subsequently developed parametric frailty models. Balakrishnan and Peng (2006) proposed a Generalized Gamma frailty model, which includes many common frailty models, and discussed model fitting and model selection based on it. To combine the key components and distinct features of the mixture cure model and the frailty model, a mixture cure frailty model is discussed here for modelling correlated survival data when not all the subjects under study would experience the occurrence of the event of interest. Longini and Halloran (1996) and Price and Manatunga (2001) developed several parametric survival models and employed the Likelihood Ratio Test (LRT) to perform a model discrimination among cure, frailty and mixture cure frailty models. In this thesis, we first describe the components of a mixture cure frailty model, wherein the flexibility of the frailty distributions and lifetime survival functions are discussed. Both proportional hazard and accelerated failure time models are considered for the distribution of lifetimes of susceptible (or non-cured) individuals. Correlated ran- dom effect is modelled by using a Generalized Gamma frailty term, and an EM-like algorithm is developed for the estimation of model parameters. Some Monte Carlo simulation studies and real-life data sets are used to illustrate the models as well as the associated inferential methods. / Thesis / Doctor of Philosophy (PhD)

Survival analytic methods help to analyze lifetime data arising from medical and reliability experiments. The popular proportional hazards model, proposed by Cox (1972), is widely used in survival analysis to study the effect of risk factors on lifetimes. An important assumption in regression type analysis is that all relative risk factors should be included in the model. However, not all relative risk factors are observed due to measurement difficulty, inaccessibility, cost considerations, and so on. These unobservable risk factors can be modelled by the so-called frailty model, originally introduced by Vaupel et al. (1979). Furthermore, the frailty model is also applicable to clustered data. Cluster data possesses the feature that observations within the same cluster share similar conditions and environment, which are sometimes difficult to observe. For example, patients from the same family share similar genetics, and patients treated in the same hospital share the same group of profes- sionals and same environmental conditions. These factors are indeed hard to quantify or measure. In addition, this type of similarity introduces correlation among subjects within clusters. In this thesis, a semi-parametric frailty model is proposed to address aforementioned issues. The baseline hazards function is approximated by a piecewise constant function and the frailty distribution is assumed to be a Birnbaum-Saunders distribution. Due to the advancement in modern medical sciences, many diseases are curable, which in turn leads to the need of incorporating cure proportion in the survival model. The frailty model discussed here is further extended to a mixture cure rate frailty model by integrating the frailty model into the mixture cure rate model proposed originally by Boag (1949) and Berkson and Gage (1952). By linking the covariates to the cure proportion through logistic/logit link function and linking observable covariates and unobservable covariates to the lifetime of the uncured population through the frailty model, we obtain a flexible model to study the effect of risk factors on lifetimes. The mixture cure frailty model can be reduced to a mixture cure model if the effect of frailty term is negligible (i.e., the variance of the frailty distribution is close to 0). On the other hand, it also reduces to the usual frailty model if the cure proportion is 0. Therefore, we can use a likelihood ratio test to test whether the reduced model is adequate to model the given data. We assume the baseline hazard to be that of Weibull distribution since Weibull distribution possesses increasing, constant or decreasing hazard rate, and the frailty distribution to be Birnbaum-Saunders distribution. D ́ıaz-Garc ́ıa and Leiva-Sa ́nchez (2005) proposed a new family of life distributions, called generalized Birnbaum-Saunders distribution, including Birnbaum-Saunders distribution as a special case. It allows for various degrees of kurtosis and skewness, and also permits unimodality as well as bimodality. Therefore, integration of a generalized Birnbaum-Saunders distribution as the frailty distribution in the mixture cure frailty model results in a very flexible model. For this general model, parameter estimation is carried out using a marginal likelihood approach. One of the difficulties in the parameter estimation is that the likelihood function is intractable. The current technology in computation enables us to develop a numerical method through Monte Carlo simulation, and in this approach, the likelihood function is approximated by the Monte Carlo method and the maximum likelihood estimates and standard errors of the model parameters are then obtained numerically by maximizing this approximate likelihood function. An EM algorithm is also developed for the Birnbaum-Saunders mixture cure frailty model. The performance of this estimation method is then assessed by simulation studies for each proposed model. Model discriminations is also performed between the Birnbaum-Saunders frailty model and the generalized Birnbaum-Saunders mixture cure frailty model. Some illustrative real life examples are presented to illustrate the models and inferential methods developed here. / Thesis / Doctor of Science (PhD)