Bayesian P-Values for Testing Independence in 2 × 2 Contingency Tables

Abstract

This article considers Bayesian p-values for testing independence in 2 × 2 contingency tables with cell counts observed from the two independent binomial sampling scheme and the multinomial sampling scheme. From the frequentist perspective, Fisher's p-value (p _F ) is the most commonly used p-value but it can be conservative for small to moderate sample sizes. On the other hand, from the Bayesian perspective, Bayarri and Berger (2000) first proposed the partial posterior predictive p-value (p _PPOST ), which can avoid the double use of the data that occurs in another Bayesian p-value proposed by Guttman (1967) and Rubin (1984), called the posterior predictive p-value (p _POST ). The subjective and objective Bayesian p-values in terms of p _POST and p _PPOST are derived under the beta prior and the (noninformative) Jeffreys prior, respectively. Numerical comparisons among p _F , p _POST , and p _PPOST reveal that p _PPOST performs much better than p _F and p _POST for small to moderate sample sizes from the frequentist perspective.

Keywords:

• Bayesian p-values
• Contingency tables
• Fisher's p-value
• Independence

Mathematics Subject Classification:

• Primary 62F15
• Secondary 62F03

Acknowledgments

The authors are most grateful to an Associate Editor and two anonymous referees for their very careful reading and invaluable comments, which substantially improved the presentation of the article.

Notes

Note: The p-value having the smallest absolute distance has been bolded, and the smaller absolute distance between p _PPOST(U) and p _PPOST(J) is marked by underline.

Note: The p-value having the smallest absolute distance has been bolded, and the smaller absolute distance between p _PPOST(U) and p _PPOST(J) is marked by underline.