Univariate and Multivariate Survival Models with Flexible Hazard Functions

Our research focuses on exploring and developing flexible Bayesian methodologies to model both univariate and multivariate survival data. When developing a Bayesian survival model, the most desirable properties are often flexibility of hazard functions, a proper posterior distribution and efficient implementation. The novelty of our work can be classified into three sections: first, we introduce a new distribution to model univariate and bivariate survival data. Although extreme value theory and subsequently the Generalized Extreme Value (GEV) distribution have been explored in the past to model rare events, our work is the first of its kind to extend GEV framework into the foray of survival analysis. We develop a cure rate model and apply it to various types of univariate cancer survival data. Second, we provide a novel method of estimating the copula association parameter for bivariate survival data using an empirical Bayes approach. Lastly we propose a novel Bayesian R-Vine approach to model multivariate survival data. The thesis consists of five chapters. Chapter 1 introduces the problem of survival data analysis and provides a brief overview of both the frequentist and Bayesian methods developed over the past few decades. Chapter 2 briefly introduces the univariate extreme value analysis. In Chapter 3, we use both forms of the GEV distribution, the Maxima and the Minima to develop a Bayesian modeling technique to analyze right-censored log survival data for populations with a surviving fraction. Next in Chapter 4, we consider bivariate survival data and use copula structures to model the association between the survival times. A novel empirical Bayesian method for estimating the copula function has been proposed. Using our model, we enable the user to use different copula functions to model the same data and hence introduce the concept of copula choice using the Bayesian model selection approach. We demonstrate through extensive simulations that the empirical Bayesian approach provides tighter HPD intervals for the copula parameter of association as compared to full Bayesian and two-stage estimation procedures. Lastly, chapter 5 introduces a novel approach to model multivariate survival data using a Bayesian R-vine copula approach. We show that this method provides flexibility and easy computation even for dimensions 3 and higher as compared to direct extension of bivariate copula families to multivariate dimensions