Bayesian Small Area Models for Three-way Contingency Tables with Nonignorability

Abstract

We show how to use Bayesian uncertainty analysis to study several three-way contingency tables, each obtained from a single area, when one, two or three categories are missing. This is an extension of Nandram and Woo (2015) to cover small areas. One approach to analyze these data is to construct several tables (one complete and the others incomplete) with each table corresponding to one or more missing categories. When tables are incomplete and nonignorable nonresponse models are used, there are nonidentifiable parameters. To deal with these parameters, we describe five hierarchical Bayesian models, which are an ignorable nonresponse model and four nonignorable nonresponse models. Rather than performing a sensitivity analysis, we perform the Bayesian uncertainty analysis by placing priors on the nonidentifiable parameters. This is done to reduce the effects of the nonidentifiable parameters that is accomplished by projecting the parameters to a lower dimensional space and allowing the reduced set of parameters to share a common distribution. Also, this procedure allows a "borrowing of strength" from larger areas to improve estimation in smaller areas. We use the griddy Gibbs sampler to fit our models and we use goodness-of-fit procedures to assess model fit. We use an illustrative example and a simulation study to compare our models when inference is made about finite population proportions of the cells of the three-way tables.

Advances in Quasi-Symmetry for Square Contingency Tables

“…Lipsitz and Fitzmaurice (1996), Takai and Kano (2008), and Lin et al (2018) discussed the test of independence (null association) in 2 × 2 contingency tables with nonresponses. Additionally, various topics are discussed in biomedical studies (Tian et al, 2003;Tang and Tang, 2004;Chang, 2009;Tian and Li, 2017). Park et al (2014) and Kim et al (2020) studied boundary solutions for nonresponse log-linear models, while Ng et al (2008) proposed the grouped Dirichlet distributions for statistical analysis of incomplete categorical data.…”