Statistical Science

The $2\times2$ table: a discussion from a Bayesian viewpoint

J. V. Howard

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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.

Article information

Statist. Sci., Volume 13, Number 4 (1998), 351-367.

First available in Project Euclid: 9 August 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F15: Bayesian inference
Secondary: 62H17: Contingency tables

Bayesian statistics two by two contingency tables Fisher's exact text Yates's correction chi-square tests significance tests p-values likelihood principles conditioning ancillarity dependent prior distributions posterior probability


Howard, J. V. The $2\times2$ table: a discussion from a Bayesian viewpoint. Statist. Sci. 13 (1998), no. 4, 351--367. doi:10.1214/ss/1028905830.

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