Sample size determination for a three-arm equivalence trial of normally distributed responses

The equivalence assessment is often conducted through a three-arm clinical trial (namely, test, reference, and placebo) and it usually consists of three tests. The first two tests are to demonstrate the superiority of the test and the reference treatment to the placebo, and they are followed by an equivalence test between the test treatment and the reference treatment. When the response variable is continuous, equivalence is commonly defined in terms of mean difference, mean ratio, or ratio of mean differences, that is, the mean difference of the test and the placebo to the mean difference of the reference and the placebo. These equivalence tests can be performed with both a hypothesis-testing approach and a confidence-interval approach. The advantage of applying the equivalence test by ratio of mean differences is that it can test both superiority of the test treatment over placebo and equivalence between the test and the reference simultaneously through a single hypothesis. In this article, we derive the test statistics and the power function for the ratio of mean differences hypothesis and solve the required sample size for a three-arm clinical trial. Examples of required sample size are given in this article, and are compared with the required sample size by the traditional mean difference equivalence test. After a careful examination, we suggest increasing the power of the ratio of mean differences approach by appropriately adjusting the lower limit of the equivalence interval.