In this dissertation, we are interested in studying processes which generate events repeatedly over the follow-up time of a given subject. Such processes are called recurrent event processes and the data they provide are referred to as recurrent event data. Examples include the cancer recurrences, recurrent infections or disease episodes, hospital readmissions, the filing of warranty claims, and insurance claims for policy holders. In particular, we focus on the multi-type recurrent event times which usually arise when two or more different kinds of events may occur repeatedly over a period of observation. Our main objectives are to describe features of each marginal process simultaneously and study the dependence among different types of events. We present applications to a real dataset collected from the Nutritional Prevention of Cancer Trial. The objective of the clinical trial was to evaluate the efficacy of Selenium in preventing the recurrence of several types of skin cancer among 1312 residents of the Eastern United States.

Four chapters are involved in this dissertation. Chapter 1 introduces a brief background to the statistical techniques used to develop the proposed methodology. We cover some concepts and useful functions related to survival data analysis and present a short introduction to frailty distributions. The Monte Carlo expectation maximization (MCEM) algorithm and copula functions for the multivariate variables are also presented in this chapter.

Chapter 2 develops a multi-type recurrent events model with multivariate Gaussian random effects (frailties) for the intensity functions. In this chapter, we present nonparametric baseline intensity functions and a multivariate Gaussian distribution for the multivariate correlated random effects. An MCEM algorithm with MCMC routines in the E-step is adopted for the partial likelihood to estimate model parameters. Equations for the variances of the estimates are derived and variances of estimates are computed by Louis' formula. Predictions of the individual random effects are obtained because in some applications the magnitude of the random effects is of interest for a better understanding and interpretation of the variability in the data. The performance of the proposed methodology is evaluated by simulation studies, and the developed model is applied to the skin cancer dataset.

Chapter 3 presents copula-based semiparametric multivariate frailty models for multi-type recurrent event data with applications to the skin cancer data. In this chapter, we generalize the multivariate Gaussian assumption of the frailty terms and allow the frailty distributions to have more features than the symmetric, unimodal properties of the Gaussian density. More flexible approaches to modeling the correlated frailty, referred to as copula functions, are introduced. Copula functions provide tremendous flexibility especially in allowing taking the advantages of a variety of choices for the marginal distributions and correlation structures. Semiparametric intensity models for multi-type recurrent events based on a combination of the MCEM with MCMC sampling methods and copula functions are introduced. The combination of the MCEM approach and copula function is flexible and is a generally applicable approach for obtaining inferences of the unknown parameters for high dimension frailty models. Estimation procedures for fixed effects, nonparametric baseline intensity functions, copula parameters, and predictions for the subject-specific multivariate frailties and random effects are obtained. Louis' formula for variance estimates are derived and calculated. We investigate the impact of the specification of the frailty and random effect models on the inference of covariate effects, cumulative baseline intensity functions, prediction of random effects and frailties, and the estimation of the variance-covariance components. Performances of proposed models are evaluated by simulation studies. Applications are illustrated through the dataset collected from the clinical trial of patients with skin cancer. Conclusions and some remarks for future work are presented in Chapter 4.