Bayesian Order-Restricted Inference of Multinomial Counts from Small Areas

Abstract

AbstractBody mass index (BMI) can be a useful indicator of health status, and people can fall in different cells. Estimating BMI cell probabilities for small areas can be difficult, due to a lack of available data from national surveys. We have data from a number of counties in the USA, and it is sensible to assume that BMI may be similar across the counties for each cell. Overall, the cell probabilities for each county follow a unimodal order restriction, and so, it is sensible to assume the same for the individual counties (small areas). Moreover, we assume that the counties are similar with some variations. In this setting, it is convenient to use the Bayesian paradigm to adaptively pool the data over areas. Therefore, we use a hierarchical multinomial Dirichlet model with order restrictions, to model the cell counts and the cell probabilities, thereby permitting a borrowing of strength across areas. We provide efficient Gibbs samplers to make inference about the cell probabilities for multinomial Dirichlet models with and without order restrictions (a model with the same pooling structure). To make inference, we compute the posterior distributions of the cell probabilities for both models. In general for most counties, as expected, the posterior distributions of cell probabilities of the model with order restrictions have significantly less variation, as measured by posterior standard deviations and coefficients of variation, than those of the model without order restrictions.KeywordsBayesian computationBody mass indexMultinomial distributionMonte Carlo methodsUnimodal order restrictions