Power for Testing Multiple Instances of the Two One-Sided Tests Procedure

The two one-sided tests procedure is used to test the equivalence of two measurements taken under different conditions. For example, two formulations of a drug are said to be bioequivalent if the average blood levels of the drug over time (AUC) are similar for the two formulations. In some studies there may be more than one parameter to test, such as a drug's AUC and maximum concentration, Cmax, or AUCs from a parent drug and a metabolite. The power of testing two or more equivalence hypotheses simultaneously is less than the power to test any one hypothesis separately, and depends on the correlations of the measurements. This paper develops an exact mathematical formula for the power for two or more simultaneous comparisons for normally distributed variables when several comparisons are evaluated separately. The formula requires numerical integration with respect to the variance-covariance terms. These terms are distributed according to the Wishart distribution, and are integrated over a subset of positive-definite matrices defined by the equivalence criteria. An R program for the case of two comparisons is included.