Selection of the effect size for sample size determination for a continuous response in a superiority clinical trial using a hybrid classical and Bayesian procedure

Abstract

Background:

When designing studies that have a continuous outcome as the primary endpoint, the hypothesized effect size ( ${\Delta }_A$ ), that is, the hypothesized difference in means ( ${\theta }_A$ ) relative to the assumed variability of the endpoint ( ${\sigma }_A$ ), plays an important role in sample size and power calculations. Point estimates for $\theta$ and $\sigma$ are often calculated using historical data. However, the uncertainty in these estimates is rarely addressed.

Methods:

This article presents a hybrid classical and Bayesian procedure that formally integrates prior information on the distributions of $\theta$ and $\sigma$ into the study’s power calculation. Conditional expected power, which averages the traditional power curve using the prior distributions of $\theta$ and $\sigma$ as the averaging weight, is used, and the value of ${\Delta }_A$ is found that equates the prespecified frequentist power ( $1-\beta$ ) and the conditional expected power of the trial. This hypothesized effect size is then used in traditional sample size calculations when determining sample size for the study.

Results:

The value of ${\Delta }_A$ found using this method may be expressed as a function of the prior means of $\theta$ and $\sigma$ , ${\mu }_{\theta }/{\mu }_{\sigma }$ , and their prior standard deviations, ${\tau }_{\theta }/{\tau }_{\sigma }$ . We show that the “naïve” estimate of the effect size, that is, the ratio of prior means, should be down-weighted to account for the variability in the parameters. An example is presented for designing a placebo-controlled clinical trial testing the antidepressant effect of alprazolam as monotherapy for major depression.

Conclusion:

Through this method, we are able to formally integrate prior information on the uncertainty and variability of both the treatment effect and the common standard deviation into the design of the study while maintaining a frequentist framework for the final analysis. Solving for the effect size which the study has a high probability of correctly detecting based on the available prior information on the difference $\theta$ and the standard deviation $\sigma$ provides a valuable, substantiated estimate that can form the basis for discussion about the study’s feasibility during the design phase.