Loss of power from an optimistic alternative hypothesis

In the planning of a clinical trial to compare the proportion of responses to two treatments one determines the sample size to yield the desired power of achieving a significant difference at a pre-selected type I error under the assumption that the expected treatment difference exceeds a prescribed minimum. To achieve a practicable sample size, an investigator may be tempted to require a large treatment difference and thereby risk the chance of missing a somewhat smaller but clinically important difference. We obtain in this paper an expression for this loss of power if the true treatment difference is smaller than the minimum used to plan the study. The expression does not explicitly depend on the sample size and for most practical purposes does not depend on the actual response rates but rather varies only as a function of the fractional difference between the required minimum and true treatment difference. With provision of a prior distribution for this fractional difference, we can use the expression to calculate an expected power for the study. An illustration considers the case where the prior follows a beta distribution.