PhD, University of Cincinnati, 2018, Arts and Sciences: Mathematical Sciences

In epidemiology and clinical research, the relative risk (RR) is commonly used as a measure of the risk of developing a disease. The log-binomial regression is a preferred statistical model to estimate RR as it provides a convenient form of RR. However, the constrained parameter space of the log-binomial model often causes numerical difficulties in applying this model for data analysis. In this dissertation, we conduct model selections for the log-binomial regression where three objectives are addressed. Firstly, we develop Bayesian variable selection methods for log-binomial model, where the Bayes factor is used as the selection criterion and five Monte Carlo methods are utilized to compute the Bayes factor while dealing with the constrained parameter space. These Monte Carlo methods are then assessed for computational accuracy and efficiency. Secondly, we study the sensitivity of Bayes factor to the prior distributions of the regression parameters by evaluating the performances of five popular priors in the log-binomial variable selection, i.e. independent Gaussian prior, independent t prior, independent Cauchy prior, the g-prior (g=n) and the Zellner-Siow Cauchy prior. Finally, we perform Bayesian model selection between the log-binomial and the logistic regressions where the Bayes factor, the fractional Bayes factor, LPML, DIC and BIC are used as the model selection methods. In addition, four prior densities of the regression parameters are considered: the flat prior, Cauchy prior, Jeffreys prior and an elicited proper prior. The theoretical properties of these priors are also investigated. These methods and priors are compared via simulations on their capabilities of distinguishing between the log-binomial and logit models.

Committee: Siva Sivaganesan Ph.D. (Committee Chair); Emily Kang Ph.D. (Committee Member); Seongho Song Ph.D. (Committee Member); Xia Wang Ph.D. (Committee Member) Subjects: Statistics

Doctor of Philosophy, The Ohio State University, 2005, Statistics

Margin-based classifiers defined by functional margins are generally believed to yield high performance in classification. In this thesis, a general theory that quantifies the size of generalization error of a margin classifier is presented. The trade-off between geometric margins and training errors is captured, in addition to the complexity of a classification problem. The theory permits an investigation of the generalization ability of convex and nonconvex margin classifiers, including support vector machines (SVM), kernel logistic regression (KLR), and ψ-learning. Our theory indicates that the generalization ability of a certain class of nonconvex losses may be substantially faster than those for convex losses. Illustrative examples for both linear and nonlinear classification are provided.

Committee: Xiaotong Shen (Advisor) Subjects: Statistics