Determination of exact sample sizes in the Bayesian estimation of the difference of two proportions

Abstract

Using generalized hypergeometric functions in several variables in a Bayesian context, we compute the exact minimum double-sample size (n 1 , n 2 ) required in the Bernoulli sampling of two independent populations, so that the expected length (or the maximum length) of the highest posterior density credible interval of P = P 1 - P 2 is less than a preset quantity, where P 1 and P 2 are two independent proportions. This precise and computer-intensive approach permits the treatment of this Bayesian sample size determination problem under very general hypotheses and also provides a relationship between the minimal values of n 1 and n 2 . Similar results are derived in an applied Bayesian decision theory context, with a quadratic loss function, and the criteria used are now the posterior risk, the Bayes risk and the expected value of sample information.