Statistical Methods for Joint Analysis of Survival Time and Longitudinal Data

In biomedical studies, researchers are often interested in the relationship between patients' characteristics or risk factors and both longitudinal outcomes such as quality of life measured over time and survival time. However, despite the progress in the joint analysis for longitudinal data and survival time, investigation on modeling approach to find which factor or treatment can simultaneously improve the patient's quality of life and reduce the risk of death has been limited. In this dissertation, we investigate joint modeling of longitudinal outcomes and survival time. We consider the generalized linear mixed models for the longitudinal outcomes to incorporate both continuous and categorical data and the stratified multiplicative proportional hazards model for the survival data. We study both Gaussian process and distribution free approaches for the random effect characterizing the joint process of longitudinal data and survival time.;We consider three estimation approaches in this dissertation. First, we consider the maximum likelihood approach with Gaussian process for random effects. The random effects, which are introduced into the simultaneous models to account for dependence between longitudinal outcomes and survival time due to unobserved factors, are assumed to follow a multivariate Gaussian process. The full likelihood, obtained by integrating the complete data likelihood over the random effects, is used for estimation. The Expectation-Maximization (EM) algorithm is used to compute the point estimates for the model parameters, and the observed information matrix is adopted to estimate their asymptotic variances. Second, the normality assumption of random effects in the likelihood approach is relaxed. Assuming the underlying distribution of random effects to be unknown, we propose using a mixture of Gaussian distributions as an approximation in estimation. Weights of the mixture components are estimated with model parameters using the EM algorithm, and the observed information matrix is used for estimation of the asymptotic variances of the proposed estimators. For the two maximum likelihood approaches with and without normality assumption of random effects, asymptotic properties of the proposed estimators are investigated and their finite sample properties are assessed via simulation studies. Third, we consider a penalized likelihood approach. This approach is expected to be computationally less intensive than the maximum likelihood approach. It gives a penalty for regarding the random effect as a fixed effect in the likelihood and avoids the need to integrate the likelihood over random effects. The penalized likelihood is obtained through Laplace approximation. We compare the numerical performances of the penalized likelihood method and the EM algorithm used in maximum likelihood estimation for the simultaneous models with Gaussian process for random effects via simulation studies. All the proposed methods in this dissertation are illustrated with the real data from the Carolina Head and Neck Cancer Study (CHANCE).