Title: Assessing thought disordered behavior using finite mixture models and comparing approximations for logistic regression
Description: Schizophrenia is a genetically complex mental illness in that multiple genes are thought to be involved and the mode of transmission is not Mendelian. Recent research has focused on studying the clinically well first-degree relatives of schizophrenic patients, some of whom may be non-penetrant carriers of schizophrenia genes. A number of traits that are associated with schizophrenia and that aggregate in relatives of schizophrenics at a rate much higher than that of the clinical disorder have been identified. These traits, provisionally identified endophenotypes, may be alternative manifestations of schizophrenia genes that are more penetrant than schizophrenia itself. The presence of alternative manifestations may help to identify healthy relatives who have inherited one or more of the genes that increases susceptibility for schizophrenia. One of these traits is thought disorder with "schizophrenic" features. In the first chapter of this thesis, we apply finite mixture models to a sample of normal controls and clinically unaffected first-degree relatives of schizophrenic patients as a first step towards using these potential alternative manifestations of schizophrenia genes to uncover more information about the heritability of the disorder.

We note that the model presented in Chapter 1 assumes independent observations. In the second chapter of this thesis, we demonstrate that this model is not appropriate for correlated data and we develop a hierarchical model that can accommodate such data. We demonstrate the efficacy of this model and provide further support for the addition of thought disorder with schizophrenic features to the schizophrenia phenotype. In Chapter 3, we turn to a different topic, namely the issue of comparing approximations. Inference for the parameters of a logistic regression typically requires an approximation to the likelihood function, typically, the asymptotic normal approximation. In this chapter, we explore two other approximations, probit-based and robit-based likelihood approximations to the logistic-based likelihood, and compare those two to the normal approximation. Also compared are the improvements that arise from using the SIR algorithm as an adjunct to improve any such approximation. We also present a new graphical method designed to aid in such comparisons.