Bayesian Full Rank Marginalization for Two-Way Contingency Tables

A general approach is proposed for modeling the structure of anr ×s contingency table and for drawing marginal inferences about all parameters (e.g., interaction effects) in the model. The main approach is relevant wheneverrs −r −s + 1≥5. The approach may also be used to check the adequacy of Rasch’s multiplicative Poisson model. In general, the posterior estimates of the cell probabilities compromise between the cell frequencies and the fitted values obtained under the reduced model in the spirit ofLeonard (1975). It is also possible to compute reasonable approximations to the full posterior densities of many parameters of interest, followingLeonard (1982) andTierney and Kadane (1984). All prior parameters are evaluated with the assistance of the data via a hierarchical Bayes procedure, thus reducing the subjectivity involved in the analysis. Anr ×s cross classification of 5,648 Marine Corps clerical students by school and test grade is analyzed in detail, and the posterior densities of the 96 possible interactions are used to suggest a simplified structure partitioning and collapsing the table into a meaningful 3 × 2 table.