Assistant Professor in Department of StatisticsUniversity of Connecticut
Bayesian Modeling Using Latent Structures
This dissertation is devoted to modeling complex data from the Bayesian perspective via constructing priors with latent structures. There are three major contexts in which this is done -- strategies for the analysis of dynamic longitudinal data, estimating shape-constrained functions, and identifying subgroups. The methodology is illustrated in three different interdisciplinary contexts: (1) adaptive measurement testing in education; (2) emulation of computer models for vehicle crashworthiness; and (3) subgroup analyses based on biomarkers. Chapter 1 presents an overview of the utilized latent structured priors and an overview of the remainder of the thesis. Chapter 2 is motivated by the problem of analyzing dichotomous longitudinal data observed at variable and irregular time points for adaptive measurement testing in education. One of its main contributions lies in developing a new class of Dynamic Item Response (DIR) models via specifying a novel dynamic structure on the prior of the latent trait. The Bayesian inference for DIR models is undertaken, which permits borrowing strength from different individuals, allows the retrospective analysis of an individual's changing ability, and allows for online prediction of one's ability changes. Proof of posterior propriety is presented, ensuring that the objective Bayesian analysis is rigorous. Chapter 3 deals with nonparametric function estimation under shape constraints, such as monotonicity, convexity or concavity. A motivating illustration is to generate an emulator to approximate a computer model for vehicle crashworthiness. Although Gaussian processes are very flexible and widely used in function estimation, they are not naturally amenable to incorporation of such constraints. Gaussian processes with the squared exponential correlation function have the interesting property that their derivative processes are also Gaussian processes and are jointly Gaussian processes with the original Gaussian process. This allows one to impose shape constraints through the derivative process. Two alternative ways of incorporating derivative information into Gaussian processes priors are proposed, with one focusing on scenarios (important in emulation of computer models) in which the function may have flat regions. Chapter 4 introduces a Bayesian method to control for multiplicity in subgroup analyses through tree-based models that limit the subgroups under consideration to those that are a priori plausible. Once the prior modeling of the tree is accomplished, each tree will yield a statistical model; Bayesian model selection analyses then complete the statistical computation for any quantity of interest, resulting in multiplicity-controlled inferences. This research is motivated by a problem of biomarker and subgroup identification to develop tailored therapeutics. Chapter 5 presents conclusions and some directions for future research.