*Case-control studies with longitudinal covariates.

The case-control design is commonly used for studying rare diseases. In a case-control study, subjects are recruited according to their case or control status, and thus the sampling is outcome-based, retrospective and biased. An emerging problem in case-control studies is the presence of a longitudinal covariate, which is collected longitudinally but retrospectively. We consider case-control studies with longitudinal covariates. We propose a logistic model coupled with a linear mixed model to jointly model the binary outcome and the longitudinal covariate. Specifically, we assume that the longitudinal covariates follow a linear mixed model and the primary binary outcome relates to the longitudinal covariate through latent subject specific random effects, for example, individual baselines and slopes. In Chapter 2, we show that the data can be analyzed prospectively by using plug-in estimation procedures, and we discuss two such approaches: the Two-stage method and the Best Linear Unbiased Predictor (BLUP) method. The proposed BLUP method characterizes the longitudinal profile by using only controls assuming the disease is rare. Compared to the Two-stage approach, the BLUP approach reduces the bias of the parameter estimators. A full likelihood approach is developed in Chapter 3. We derive the true retrospective likelihood for case-control data with longitudinal covariates and obtain estimates via Maximum Likelihood Estimation (MLE). The retrospective likelihood involves un-estimable population information and can be approximated when the disease is rare. The proposed MLE method is demonstrated under normality of random effects. Both the BLUP method and the MLE method require parametric assumptions on the distribution of the random effects. When the distribution of the random effects is misspecified, the parameter estimators are likely to be biased in logistic models. We hence develop a semiparametric approach, known as the sufficiency score method, with the distribution of the random effects completely unspecified in Chapter 4. We apply the proposed methods to analyze a case-control breast-cancer dataset with weight as the longitudinal covariate.Ph.D.Health and Environmental SciencesPublic healthPure SciencesStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/125790/2/3208590.pd