Exact Bayesian p-values for a test of independence in a 2 × 2 contingency table with missing data

Altham (Altham PME. Exact Bayesian analysis of a 2 × 2 contingency table, and Fisher's "exact" significance test. J R Stat Soc B 1969; 31: 261-269) showed that a one-sided p-value from Fisher's exact test of independence in a 2 × 2 contingency table is equal to the posterior probability of negative association in the 2 × 2 contingency table under a Bayesian analysis using an improper prior. We derive an extension of Fisher's exact test p-value in the presence of missing data, assuming the missing data mechanism is ignorable (i.e., missing at random or completely at random). Further, we propose Bayesian p-values for a test of independence in a 2 × 2 contingency table with missing data using alternative priors; we also present results from a simulation study exploring the Type I error rate and power of the proposed exact test p-values. An example, using data on the association between blood pressure and a cardiac enzyme, is presented to illustrate the methods.