A Bayesian small area model with order restrictions for contingency tables

Estimating the probabilities of possible outcomes for small areas can be difficult, due to a lack of available data from national surveys. One of the statistical techniques for small area estimation is using multinomial Dirichlet models to borrow information among small areas. We study Bayesian diagnostics for multinomial counts from small areas. Within each area, the probabilities of counts for possible outcomes are ordered (e.g., unimodal ordering). Specifically we consider Bayesian diagnostics for a multinomial Dirichlet model with order restriction which shares a common effect among areas. The log pseudo marginal likelihood (LPML) is a well-known Bayesian criterion for comparing models. Since the order restriction significantly increases the difficulty, we develop an algorithm to compute LPML. We use a special-designed importance function to increase the efficiency of Monte Carlo integration, thereby gaining a higher precision for estimation of LPML. The proposed methodology is applied to a case study of body mass index (BMI) and a simulation study to test different scenarios.