On the occurrence of boundary solutions in two-way incomplete tables

TL;DR

This paper investigates boundary solutions in two-way incomplete contingency tables with nonignorable nonresponse, providing new theoretical results, conditions for occurrence, and practical insights for model selection.

Contribution

It establishes new sufficient and necessary conditions for boundary solutions, connecting different parameterizations and improving computational efficiency.

Findings

Boundary solutions are characterized under various models.

Sufficient conditions are not necessary, disproving previous conjectures.

New simple conditions based on observed counts aid model selection.

Abstract

The analysis of incomplete contingency tables is an important problem, which is also of practical interest. In this paper, we consider boundary solutions under nonignorable nonresponse models in two-way incomplete tables with data on both variables missing. We establish a result similar to Park {\it et al.} (2014) on sufficient conditions for the occurrence of boundary solutions. We also provide a new result, which connects the forms of boundary solutions under various parameterizations of the missing data models. This result helps us to obtain the exact form of boundary solutions in the above tables, which improves a claim made in Baker {\it et al.} (1992) and avoids computational burden. A counterexample is provided to show that the sufficient conditions for the occurrence of boundary solutions are not necessary, thereby disproving a conjecture of Kim and Park (2014). Finally, we establish new necessary conditions for the occurrence of boundary solutions under nonignorable nonresponse models in square two-way incomplete tables, and show that they are not sufficient. These conditions are simple and easy to check as they depend only on the observed cell counts. They are useful and important for model selection also. Some real life data sets are analyzed to illustrate the results.