A unified approach to permutation testing for equivalence

The notion of testing for equivalence of two treatments is widely used in clinical trials, pharmaceutical experiments, bioequivalence and quality control. It is traditionally operated within the intersection–union principle (IU). According to this principle the null hypothesis is stated as the set of effects the differences δ of which lie outside a suitable equivalence interval and the alternative as the set of δ that lie inside it. In the literature related solutions are essentially based on likelihood techniques, which in turn are rather difficult to deal with. A recently published paper goes beyond most of likelihood limitations by using the IU approach within the permutation theory. One more paper, based on Roy’s union–intersection principle (UI) within the permutation theory, goes beyond some limitations of traditional two-sided tests. Such UI approach, effectively a mirror image of IU, assumes a null hypothesis where δ lies inside the equivalence interval and an alternative where it lies outside. Since testing for equivalence can rationally be analyzed by both principles but, as the two differ in terms of the mirror-like roles assigned to the hypotheses under study, they are not strictly comparable. The present paper’s main goal is to look into these problems and provide a sort of comparative analysis of both by highlighting the related requirements, properties, limitations, difficulties, and pitfalls so as to get practitioners properly acquainted with their correct use in practical contexts.