Models for Univariate and Multivariate Analysis of Longitudinal and Clustered Data

Abstract

Longitudinal studies of repeated observations on subjects are commonly undertaken in medical and biological sciences. The responses on a given occasion may be either univariate or multivariate. We concentrate on three topics related to longitudinal and clustered data analysis. The first topic is the development of a class of generalized linear latent variable models. The second involves the modelling of count data with excess zeros. The third is the development of a non-Gaussian linear mixed effects model for multiple outcomes. In addressing the first problem, we propose random mean models to account for correlation among repeated measures. We extend random mean models to include mixed outcomes, renaming them random mean joint models. The difficulty in joint modelling of continuous and discrete outcomes is the lack of a natural multivariate distribution. We overcome the difficulty by introducing two cross-correlated latent processes. We apply the Monte Carlo EM (MCEM) algorithm to find the MLEs of regression coefficients and variance components, by treating the latent variables as missing data. This thesis also proposes regression models for count data with excess zeros. We solve the problem from a perspective different from that of mixture model framework. By employing the zero truncated distribution and the zero modified distribution, we establish a broad class of distributions to model data with excess zeros. We consider the zero modified Poisson regression model and zero modified binomial regression model for cross-sectional data. We extend the zero modified regression models to models with random effects. We further extend random mean models to model zero-inflated data, and formulate the corresponding zero modified random mean models. A non-Gaussian linear mixed effects model for multiple outcomes is proposed to the third question. The methodology is motivated by a glaucoma study. The normality assumption for random effects may be unrealistic, raising concerns about the validity of inferences on fixed effects and random effects if it is violated. To accommodate the skewness of the responses and the associations among multiple characteristics, we propose a mixed effects model, in which non-normal random effects are assumed by the log-gamma distribution.