The 2 x 2 table: A discussion from a Bayesian viewpoint

The 2 x 2 table is used as a vehicle for discussing different approaches to statistical inference. Several of these approaches (both classical and Bay esian) are compared, and difficulties with them are highlighted. More frequ ent use of one-sided tests is advocated. Given independent samples from two binomial distributions, and taking independent Jeffreys priors, we note th at 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 prob ability is derived. However, we argue that usually it will be more appropri ate to use dependent priors, and we suggest a particular "standard prior" f or the 2 x 2 table. For small numbers of observations this is more conserva tive than Fisher's exact test, but it is less conservative for larger sampl e sizes. Several examples are given.