Sample size determination for a binary response in a superiority clinical trial using a hybrid classical and Bayesian procedure

When designing studies that have a binary outcome as the primary endpoint, the hypothesized proportion of patients in each population experiencing the endpoint of interest (i.e., π ₁,π ₂) plays an important role in sample size and power calculations. Point estimates for π ₁ and π ₂ are often calculated using historical data. However, the uncertainty in these estimates is rarely addressed.

This paper presents a hybrid classical and Bayesian procedure that formally integrates prior information on the distributions of π ₁ and π ₂ into the study’s power calculation. Conditional expected power (CEP), which averages the traditional power curve using the prior distributions of π ₁ and π ₂ as the averaging weight conditional on the presence of a positive treatment effect (i.e., π ₂>π ₁), is used, and the sample size is found that equates the pre-specified frequentist power (1−β) and the conditional expected power of the trial.

Notional scenarios are evaluated to compare the probability of achieving a target value of power with a trial design based on traditional power and a design based on CEP. We show that if there is uncertainty in the study parameters and a distribution of plausible values for π ₁ and π ₂, the performance of the CEP design is more consistent and robust than traditional designs based on point estimates for the study parameters. Traditional sample size calculations based on point estimates for the hypothesized study parameters tend to underestimate the required sample size needed to account for the uncertainty in the parameters. The greatest marginal benefit of the proposed method is achieved when the uncertainty in the parameters is not large.

Through this procedure, we are able to formally integrate prior information on the uncertainty and variability of the study parameters into the design of the study while maintaining a frequentist framework for the final analysis. Solving for the sample size that is necessary to achieve a high level of CEP given the available prior information helps protect against misspecification of hypothesized treatment effect and provides a substantiated estimate that forms the basis for discussion about the study’s feasibility during the design phase.