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November 1998 The $2\times2$ table: a discussion from a Bayesian viewpoint
J. V. Howard
Statist. Sci. 13(4): 351-367 (November 1998). DOI: 10.1214/ss/1028905830


The $2\times2$ table is used as a vehicle for discussing different approaches to statistical inference. Several of these approaches (both classical and Bayesian) are compared, and difficulties with them are highlighted. More frequent use of one-sided tests is advocated. Given independent samples from two binomial distributions, and taking independent Jeffreys priors, we note that the posterior probability that the proportion of successes in the first population is larger than in the second can be estimated from the standard (uncorrected) chi-square significance level. An exact formula for this probability is derived. However, we argue that usually it will be more appropriate to use dependent priors, and we suggest a particular "standard prior" for the $2\times2$ table. For small numbers of observations this is more conservative than Fisher's exact test, but it is less conservative for larger sample sizes. Several examples are given.


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J. V. Howard. "The $2\times2$ table: a discussion from a Bayesian viewpoint." Statist. Sci. 13 (4) 351 - 367, November 1998.


Published: November 1998
First available in Project Euclid: 9 August 2002

zbMATH: 1059.62526
MathSciNet: MR1705267
Digital Object Identifier: 10.1214/ss/1028905830

Primary: 62F15
Secondary: 62H17

Keywords: ancillarity , Bayesian statistics , chi-square tests , Conditioning , dependent prior distributions , Fisher's exact text , likelihood principles , posterior probability , P-values , significance tests , two by two contingency tables , Yates's correction

Rights: Copyright © 1998 Institute of Mathematical Statistics


Vol.13 • No. 4 • November 1998
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