This dissertation concerns two essentially independent topics, with the primary link between the two being the use of Bayesian nonparametrics as an inference tool. The first topic concerns inference in the presence of missing data, with emphasis on longitudinal clinical trials with attrition. In this setting, it is well known that many effects of interest are not identified in the absence of untestable assumptions; the best one can do is to conduct a sensitivity analysis to determine the effect that such assumptions have on inferences. The second topic we address is model selection and hyperparameter estimation in hierarchical nonparametric Bayes models, with an emphasis on hierarchical Dirichlet processes. For various hyperparameter values on the boundary of the parameter space, such nonparametric models may reduce to parametric or semiparametric submodels, effectively giving tests of nonparametric models.
Chapter 1 and Chapter 2 provide some necessary background material on nonparametric Bayes and missing data problems. In Chapter 1, we discuss Dirichlet processes and present theoretical results of interest. In Chapter 2, we describe in detail various aspects of the missing data problem, and the generic approach we will take to addressing it via identifying restrictions.
In Chapter 3, we provide a general framework for specifying nonparametric priors in missing data models which allow for fine control over the assumptions made; care is taken to not inadvertently "identify away" the missing data problem. To accomplish this, we place a prior directly on the observed data distribution, leaving the "extrapolation distribution" untouched. This is accomplished essentially by marginalizing over the extrapolation distribution for a given prior on the complete-data distribution. The extrapolation distribution is then identified via a suitable family of identifying restrictions with interpretable sensitivity parameters.
In Chapter 4, we apply this methodology to a longitudinal clinical trial designed to assess the efficacy of a proposed treatment for acute schizophrenia. We construct a Dirichlet process mixture model to model the observed data generating distribution, and identify the distribution of the missing data by introducing a sensitivity parameter representing a location shift. The methodology is additionally validated through simulation.
In Chapter 5, we consider the problem of hierarchical Bayesian modeling. We use hierarchical Dirichlet processes to construct tests of nonparametric models against semiparametric and parametric alternatives. Additionally, we develop empirical Bayes estimators for the hyperparameters of the hierarchical Dirichlet process.
We conclude this dissertation in Chapter 6 with a discussion, and suggest possible avenues for future work.
|Advisor:||Daniels, Michael J., Doss, Hani|
|School:||University of Florida|
|School Location:||United States -- Florida|
|Source:||DAI-B 78/05(E), Dissertation Abstracts International|
|Subjects:||Biostatistics, Mathematics, Statistics|
|Keywords:||Bayesian, Clinical trial, Dirichlet process, Missing data, Model selection, Nonparametric|
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