Ratio of means vs. difference of means as measures of superiority, noninferiority, and average bioequivalence.

Ratio of means (ROM) and difference of means (DOM) are often used in a superiority, noninferiority (NI), or average bioequivalence (ABE) test to evaluate whether the test mean is superior, NI, or equivalent to the reference (placebo or active control) mean. The literature provides recommendations regarding how to choose between ROM and DOM, mainly for superiority testing. In this article, we evaluated these two measures from other perspectives and cautioned the potential impact of different scoring systems/transformation for the same outcome (which is not rarely seen in practice) on the power of a ROM or DOM test for superiority, NI, or ABE. 1) For superiority, with the same margin, power remains the same for a location, scale, or combined shift (no other transformations) to scoring systems for both measures; however, for NI and ABE, different shifts can change the power of the test significantly. 2) Direction of scores (larger or smaller value indicating desirable effects) does not change the power for a DOM superiority, NI, or ABE test, but it does change the power tremendously for a ROM, NI, or ABE test. Caution should be taken when defining scoring systems. Data transformation is not encouraged in general, and if needed, should be statistically justified.