Exact Bayes Factors for the Comparison of Multinomial Distributions

Abstract

This article deals with the problem of comparing multinomial distributions with multiple ordered categories. A graphical procedure is proposed for obtaining the posterior probabilities for the hypotheses of a stochastic dominance relationship, positive cumulative odds ratios, and a likelihood ratio ordering. From these expressions we subsequently obtain exact expressions for the Bayes factors related to these hypotheses. Supplemental materials for running the analysis for the examples presented in the article are available online.

Keywords:

• Exact Bayes factor
• Likelihood ratio ordering
• Positive cumulative odds ratios
• Stochastic dominance

Supplementary Materials

Programs for implementing the analyses in R (R Core Team 2017) are available as supplementary materials to this article.

Acknowledgments

The author thanks two referees, the associate editor, and the editor, for their helpful comments and suggestions that substantially improved the article.

Notes

1 Fay and Proschan (2010) provided various perspectives on the Mann–Whitney–Wilcoxon test, one of which takes a stochastic ordering as the alternative hypothesis. Mann and Whitney (1947) showed the test to be consistent with respect to this alternative [albeit for continuous outcomes]. However, the data in Table 1 are too sparse to rely on asymptotic results.