Contributions to mixed effects models for longitudinal data

Mixed effects models not only explicitly distinguish between fixed and random effects so that between-subject and within-subject sources of variability are able to be modeled, but also they allow for prediction of individual response trajectories in addition to estimation of population parameters. Normality is usually assumed for the random effects. However, estimation and inference in mixed effects models would be questionable when the normal assumption is blindly adopted. For example, it is known that the estimation of parameters may be biased if some influential outliers are present in the data set.; We studied mixed effects models with non-normal distributions for random effects in Chapters 2 and 3. In Chapter 2, we investigated how linear mixed models handle outliers arising from different random sources using multivariate t distribution. Models with different distribution assumptions for random effects and error terms were discussed with the development of maximum likelihood estimation; and these models were compared by information criteria. This framework enables researchers to understand which source of outlying observations is more influential, and consequently which model can be suitably selected.; Illustrated with the analysis of Modification of Diet in Renal Disease (MDRD) data whose random slopes had a negatively skewed distribution, Chapter 3 presented a new class of mixed effects models. We proposed to predict the subject-specific trajectories with a linear mixed model using log-gamma distribution for random slopes. We also proposed a lack-of-fit test in order to determine whether the non-normal linear mixed model fitted data significantly better than the traditional linear mixed models. If it was so, predictors of subject-specific random effects would be computed based on the selected non-normal linear mixed model. The MDRD data analysis showed such predictors have a better performance in terms of accuracy than those given by the normal mixed effects models.; In Chapter 4, we presented the comparison of the marginal model with the mixed effects model from a new perspective. We showed that the specification of the marginal model implies a strong assumption of the population, that is, heterogeneity in the study population is prohibited except that explained by the included covariates in the marginal model and random noise unless the link function is linear. We then proposed a new model that relaxes such a restriction by allowing the incorporation of the population heterogeneity in the GEE framework, which essentially takes advantages of both the marginal models and mixed effects models.; Chapter 5 concerns a generalization of mixed effects models. The restriction that random effects have to be a subset of fixed effects is removed by directly assuming the mean vector itself to be random, which is hence termed the random mean model. Clearly mixed effects model is a special case of the random mean model. Although predicting subject-specific effects becomes impossible the random mean model allows much more flexible covariance structures which is desirable in some situations with complex longitudinal data. In addition, this approach facilitates handling of outlying observations via multivariate t distributions. It leads to an alternative methodology for the non-linear regression analysis of longitudinal data.