Bayesian inference about odds ratio structure in ordinal contingency tables

When the goal of a study is to compare two groups on an ordinal categorical scale, a large number of inferential methods are available. Most methods are designed to detect a location effect, such as by focusing a single-degree-of-freedom test on an effect parameter. Often, rather than merely summarizing by a P -value to describe the evidence against a null hypothesis, it is of interest to consider whether a stronger conclusion can be made. For example, can we conclude that the population distributions are stochastically ordered? For parameter space regions described by order restrictions, frequentist methods are not well designed for significance testing. For example, a frequentist P -value for testing identical distributions against an alternative of stochastically ordered distributions can be very small even when the sample distributions give clear evidence that the distributions do not have the ordering property. The Bayesian approach seems better equipped to handle such questions. We discuss this in the context of stochastic ordering and other types of ordinal odds ratio structure, for the two-group comparison and for more general contexts. For Dirichlet priors, simple simulations provide posterior probabilities of particular ordinal odds ratio structures. Copyright © 2013 John Wiley & Sons, Ltd.