Equivalence testing with data-dependent and post-hoc equivalence margins

Equivalence testing compares the hypothesis that an effect μ is large against the alternative that it is negligible. Here, `large' is classically expressed as being larger than some `equivalence margin' Delta. A longstanding problem is that this margin must be specified but can rarely be objectively justified in practice. We lay the foundation for an alternative paradigm, arguing to instead report a data-dependent margin widehatDeltaα that bounds the true effect μ with probability 1 - α. Our key argument is that widehatDeltaα is more useful than a test outcome at a fixed margin Delta, as measured by the guarantees it offers to decision makers. We generalize this to a curve of margins α mapsto widehatDeltaα, uniformly valid under the post-hoc selection of the margin. These ideas rely on e-values, which we derive for models that are strictly totally positive of order 3, nesting the classical z-test and t-test settings.