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

Abstract: 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 someequivalence margin' $Δ$. 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 $\widehatΔα$ that bounds the true effect $μ$ with probability $1 - α$. Our key argument is that $\widehatΔα$ is more useful than a test outcome at a fixed margin $Δ$, as measured by the guarantees it offers to decision makers. We generalize this to a curve of margins $α\mapsto \widehatΔ_α$, 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.