Constraint Bayesian Inference for Count Data from Small Areas

Chen, Xinyu

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Constraint Bayesian Inference for Count Data from Small Areas

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The small area analysis of survey data has received a lot of attention. Borrow- ing information from other areas can provide reliable and accurate estimates when the sample size of an area is small. In many applications, it is necessary to take into account 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, where parameters can increase up to a point (mode) and decrease thereafter. In this dissertation, we assume unimodal order restrictions on parameters of interest. First, we describe Bayesian hierarchical multinomial-Dirichlet models with order restrictions for count data from small areas. Second, we incorporate uncertainty into the model with unimodal order restrictions as an extension. Third, we describe the models with exchangeability–nonexchangeability (EXNEX) priors, which allow borrowing information across similar areas while avoiding too optimistic borrowing for extreme areas. First, due to the natural characteristics of the data, making unimodal order restriction assumptions to parameter spaces is relevant. We present the models with order restrictions on different parameters of interest to explore how borrowing information under different order restriction assumptions works differently. In the simulation, we compare these models with order restrictions under different scenarios, where we assume three levels of heterogeneity between areas. In a small heterogeneity scenario, the model with stronger order restrictions on parameters borrow more information among areas and has smaller relative bias, posterior standard deviation, and higher credible interval coverage than other models. We develop methods to generate posterior samples for the models with different order restrictions assumptions. Second, in our application to body mass index (BMI) data from the NHANES III, we assume people may have a high chance to have overweight BMI level. We assume the same unimodal order restriction across all counties, where the mode is at the third position. But we notice the same unimodal order restriction for all areas may not hold. To have a more robust model, we incorporate uncertainty into the unimodal order restriction. We let the modal position for each area be a random variable and have mixture probabilities for the modal position, which means each area can have different order restrictions. We provide an approximation of log-pseudo marginal likelihood as a model diagnostic procedure. In the application to the BMI data and simulated data, we compare the performance of different models with or without order restrictions. We show that the performance of the model, incorporating uncertainty about order restrictions, is consistent and it can provide relatively accurate estimates of parameters in the application. We demonstrate how the model with order restrictions can borrow information among areas differently from the model without order restriction. Third, when population means are clustered into two or more subgroups, shrinking all the means towards a common weighted average is inappropriate. A useful substitute for exchangeability in the above situation is partial exchangeability. We present exchangeability–nonexchangeability (EXNEX) models, which allow borrowing information across similar areas while avoiding too optimistic borrowing for extreme areas. We present a griddy Gibbs sampler to draw samples from the joint posterior distribution of the binomial-Beta EXNEX model. In the simulation, we illustrate the robustness of EXNEX models, which have small relative bias under different scenarios. Then we extend the approach to a multinomial-Dirichlet EXNEX model with order restrictions. In the application to BMI data, we compare the multinomial-Dirichlet EXNEX model with order restrictions and the multinomial-Dirichlet model with order restrictions. We show that the EXNEX model with order restrictions can borrow information across similar areas while avoiding borrowing from very different areas. So the EXNEX model with order restrictions is preferred in some cases. Overall, borrowing information among areas is a key idea in small area estimation. The hierarchical structure of the models with order restrictions is easy to apply to small area estimation problems. 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 are recommended to allow borrowing across similar areas and avoid optimistic borrowing for very different areas. Our theoretical and methodological work can help provide accurate and efficient small area statistics for many national surveys.

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