Confirming the Null: Remarks on Equivalence Testing and the Topology of Confirmation

TL;DR

This paper explores the conditions under which null hypotheses can be confirmed in statistical testing, revealing that equivalence hypotheses are confirmable while two-sided hypotheses are not, based on topological properties.

Contribution

It introduces a modal logic framework linking hypothesis confirmation to topological conditions, clarifying when null hypotheses can be statistically confirmed.

Findings

Equivalence hypotheses are confirmable based on topological criteria.

Two-sided hypotheses are not confirmable due to topological defects.

Confirmation depends on hypotheses having nonempty interior in a topological space.

Abstract

Null Hypothesis Statistical Testing is a dominant framework for conducting statistical analysis across the sciences. There remains considerable debate as to whether, and under what circumstances, evidence can be said to be confirmatory of a null hypothesis. This paper presents a modal logic of short-run frequentist confirmation developed by leveraging the duality between hypothesis testing and statistical estimation. It is shown that a hypothesis is confirmable if and only if it satisfies the topological condition of having nonempty interior. Consequently, two-sided hypotheses are not statistically confirmable owing to defects in their topological structure. Equivalence hypotheses are, by contrast, confirmable.