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Mathematical Sciences Department, PhD Dissertation Defense, Xinyu Chen "Constraint Bayesian Inference for Count Data from Small Areas" (via Zoom)

Friday, December 03, 2021
4:00 pm to 6:00 pm

Xinyu Chen

PhD Candidate - Mathematical Sciences Department

Dissertation Committee: Dr. Balgobin Nandram, WPI (Advisor) Dr. Jai Won Choi, Meho Inc. Dr. Myron Katzoff, National Center for Health and Statistics, CDC (retired) Dr. Buddika Peiris, WPI Dr. Joseph Sedransk, University of Maryland Dr. Fangfang Wang, WPI

Friday, December 3, 2021

4:00PM - 6:00PM

Zoom ID : 5585354913

Title: Constraint Bayesian Inference for Count Data from Small Areas

Abstract: The small area analysis of survey data has received a lot of attention, especially by many US government agencies. Borrowing information from other areas can provide reliable and accurate estimates when the sample size of an area is small. In many cases, it is necessary to consider possible order restrictions of the unknown parameters of interest, and it is reasonable to make such an assumption. With the order restriction assumption, pooling data can provide more accurate estimates. In this dissertation, we develop several Bayesian hierarchical multinomial-Dirichlet models with order restrictions. We incorporate unimodal order restrictions on the cell probabilities of the multinomial Dirichlet model, and we develop three major extensions. First, due to the natural characteristics of the data, we incorporate unimodal order restrictions of the cell probabilities and we present the present models. We develop methods to generate posterior samples for the models with different order restriction assumptions. Using a simulation, we compare these models with order restrictions under different scenarios, where we assume three levels of heterogeneity among areas. Second, we notice the same unimodal order restriction may not hold for all areas. To have a more robust model, we incorporate uncertainty into the unimodal order restriction. We allow the modal position for each area be random, and each area can have different order restrictions. We provide an approximation of log-pseudo marginal likelihood as a model diagnostic procedure. Third, when the area probabilities are clustered into two or more subgroups, shrinking all the areas towards a common weighted average is inappropriate. A useful substitute for exchangeability is partial exchangeability. We present multinomial-Dirichlet exchangeability– nonexchangeability models with order restrictions, which allow borrowing information across similar areas while avoiding too optimistic borrowing from extremely different areas. In an application on body mass index (BMI) data from the National Health and Nutrition Examination Survey, people may have a high chance to have overweight BMI, which is the third category among five categories. We assume the same unimodal order restriction across all counties, where the modal position is at the third. The main issue we focus on here is to borrow information with the unimodal order restrictions on cell probabilities, which can borrow more information among areas than the model without order restrictions. As extensions of our approach, incorporating uncertainty about the order restrictions may solve the problem that the same unimodal order restrictions across areas may not hold. Partial exchangeability of parameters is recommended to allow borrowing across similar areas and to avoid optimistic borrowing from very different areas. We show that there are benefits to incorporate the unimodal order restrictions into the multinomial-Dirichlet model for the BMI data. Our theoretical and methodological work can help provide accurate and efficient small area statistics for numerous national surveys.