The \(2\times2\) table: A discussion from a Bayesian viewpoint. (English) Zbl 1059.62526

Howard, J. V.

Stat. Sci. 13, No. 4, 351-367 (1998).

Summary: 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.

MSC:

62F15	Bayesian inference
62H17	Contingency tables

Keywords:

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

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