Bayesian Power and Sample Size Calculations for Bayes Factors in the Binomial Setting

Bayesian design of experiments and sample size calculations usually rely on complex Monte Carlo simulations in practice. Obtaining bounds on Bayesian notions of the false-positive rate and power therefore often lack closed-form or approximate numerical solutions. In this paper, we focus on the sample size calculation in the binomial setting via Bayes factors, the predictive updating factor from prior to posterior odds. We discuss the drawbacks of sample size calculations via Monte Carlo simulations and propose a numerical root-finding approach which allows to determine the necessary sample size to obtain prespecified bounds of Bayesian power and type-I-error rate almost instantaneously. Real-world examples and applications in clinical trials illustrate the advantage of the proposed method. We focus on point-null versus composite and directional hypothesis tests, derive the corresponding Bayes factors, and discuss relevant aspects to consider when pursuing Bayesian design of experiments with the introduced approach. In summary, our approach allows for a Bayes-frequentist compromise by providing a Bayesian analogue to a frequentist power analysis for the Bayes factor in binomial settings. A case study from a Phase II trial illustrates the utility of our approach. The methods are implemented in our R package bfpwr.