Bayesian Procedures for Prediction Analysis of Implication Hypotheses in 2 × 2 Contingency Tables

Procedures for prediction analysis in 2 x 2 contingency tables are illustrated by the analysis of successes to six types of problems associated with the acquisition of fractions. According to Hildebrand, Laing, and Rosenthal (1977), hypotheses such as "success to problem type A implies in most cases success to problem type B" can be evaluated from a numerical index. This index has been considered in various other frameworks and can be interpreted in terms of a measure of predictive efficiency of implication hypotheses. Confidence interval procedures previously proposed for this index are reviewed and extended. Then, under a multinomial model with a conjugate Dirichlet prior distribution, the Bayesian posterior distribution of this index is characterized, leading to straightforward numerical methods. 1 The choices of "noninformative" priors for discrete data are shown to be no more arbitrary or subjective than the choices involved in the frequentist approach. Moreover, a simulation study of frequentist coverage probabilities favorably compares Bayesian credibility intervals with conditional confidence intervals.