Bayesian sample size determination for binomial proportions

Abstract

This paper presents several new results on Bayesian sample size determination for estimating binomial proportions, and provides a comprehensive comparative overview of the subject. We investigate the binomial sample size problem using generalized versions of the Average Length and Average Coverage Criteria, the Median Length and Median Coverage Criteria, as well as the Worst Outcome Criterion and its modified version. We compare sample sizes derived from highest posterior density and equal-tailed credible intervals. In some cases, we derive, for the first time, closed form sample size formulae, and where this is not possible, we describe various numerical approaches. These range in complexity from Monte Carlo simulations to more sophisticated curve fitting techniques, third order analytic approximations, and exact, but more computationally-intensive, methods. We compare the accuracy and efficiency of the different computational methods for each of the criteria and make recommendations about which methods are preferred. Finally, we consider, again for the first time, issues surrounding prior robustness on the choice of sample size. Examples are given throughout the text.