🤖 AI Summary
Traditional equivalence testing requires pre-specifying an equivalence margin, which is often difficult to determine objectively in practice. This work proposes a data-driven paradigm that leverages e-values to construct an equivalence margin guaranteed to cover the true effect with probability at least \(1 - \alpha\), and further generalizes this into a unified post-hoc selectable boundary curve. By abandoning fixed margins, the method applies to third-order strictly totally positive models—encompassing classical z- and t-tests—and yields boundaries with posteriorly valid coverage, offering stronger guarantees for decision-making. Compared to conventional fixed-margin approaches, the proposed framework enhances practical applicability and provides more informative guidance for inference.
📝 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 some `equivalence 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 $\widehatΔ_α$ that bounds the true effect $μ$ with probability $1 - α$. Our key argument is that $\widehatΔ_α$ 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.