On the Occurrence of Boundary Solutions in Two-Way Incomplete Tables

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 [9] 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 [2] 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 [7]. 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.

Keywords

• incomplete tables
• boundary solutions
• Log-linear models
• NMAR models