Bayes Models of Clinical Trials with Dichotomous Outcomes and Sample Size Determination

Bayesian methods are used in the clinical trial environment to reduce sample sizes and/or increase power. The Beta distribution is a natural prior for binomial models. Under the empirical Bayes approach, the parameters of this distribution are the maximum likelihood estimator of the marginal beta-binomial distribution. This distribution is defined by a combination of gamma functions. Because of factorial growth of these functions, the straightforward numerical search for the maximum likelihood solution is frequently impractical given available software. In this article, we consider some simplifications to the marginal beta-binomial distribution, which are easily computationally tractable and very precise. Using empirical Bayes priors is restricted to the case of complete exchangeability of historical trials as opposed to the current trial. In order to reflect some difference between the historical studies and the current studies, we introduce an adjustment to the maximum likelihood estimate. The exchangeability is measured by the confidence interval for the historical rate of events. With this prior, the formula for the sample size calculation is completely defined.