Exact Bayesian Analysis of a 2 × 2 Contingency Table, and Fisher's “Exact” Significance Test

Abstract

Summary A relationship is derived between the posterior probability of negative association of rows and columns of a 2 times 2 contingency table and Fisher's “exact” probability, as given in existing tables for testing the hypothesis of no association of rows and columns. The result for the 2 times 2 table is generalized to provide the posterior probability that one discrete‐valued random variable is stochastically larger than another.

Bayesian inference for categorical data analysis

“…As in the independent samples case studied by Altham (1969), this is a Bayesian P -value for a prior distribution favoring H 0 . If α 12 = α 21 = γ, with 0 ≤ γ ≤ 1, Altham showed that this is smaller than the frequentist P -value, and the difference between the two is no greater than the null probability of the observed data.…”

“…See Berry (2004) for a recent exposition of advantages of using a Bayesian approach in clinical trials. Weisberg (1972) extended Novick and Grizzle (1965) and Altham (1969) to the comparison of two multinomial distributions with ordered categories. Assuming independent Dirichlet priors, he obtained an expression for the posterior probability that one distribution is stochastically larger than the other.…”

“…From this, Cornfield derived the posterior probability that µ 1 = µ 2 and showed connections with stopping-rule theory. Altham (1969) discussed Bayesian testing for 2×2 tables in a multinomial context. She treated the cell probabilities {π ij } as multinomial parameters having a Dirichlet prior with parameters {α ij }.…”

“…These approaches primarily used conjugate beta and Dirichlet priors. Altham (1969Altham ( , 1971 presented Bayesian analogs of small-sample frequentist tests for 2×2 tables using such priors. An alternative approach using normal priors for logits received considerable attention in the 1970s by Leonard and others (e.g., Leonard 1972).…”

Bayesian point null hypothesis testing via the posterior likelihood ratio

“…Exact results for the posterior distribution of the attributable risk or any of the other measures of difference are complex and involve sums of hypergeometric probabilities (Altham 1969, Hashemi et al 1997. However the marginal posterior distribution of any parametric function of p 1 and p 2 can be simulated directly, by generating N realizations from the posterior distributions of p 1 and p 2 , and calculating the appropriate function (Tanner 1996).…”