Familial inference: tests for hypotheses on a family of centres

TL;DR

This paper introduces a Bayesian nonparametric method to test hypotheses on a family of centres, addressing the gap between scientific hypotheses and statistical tests that assume a single centre, with theoretical and practical validation.

Contribution

It proposes a novel approach for testing a family of plausible centres, such as the Huber family, using a new pathwise optimization routine and Bayesian methods.

Findings

The new test has favorable theoretical properties.

Experimental validation demonstrates effectiveness.

Case studies show practical applicability.

Abstract

Statistical hypotheses are translations of scientific hypotheses into statements about one or more distributions, often concerning their centre. Tests that assess statistical hypotheses of centre implicitly assume a specific centre, e.g., the mean or median. Yet, scientific hypotheses do not always specify a particular centre. This ambiguity leaves the possibility for a gap between scientific theory and statistical practice that can lead to rejection of a true null. In the face of replicability crises in many scientific disciplines, significant results of this kind are concerning. Rather than testing a single centre, this paper proposes testing a family of plausible centres, such as that induced by the Huber loss function (the Huber family). Each centre in the family generates a testing problem, and the resulting family of hypotheses constitutes a familial hypothesis. A Bayesian nonparametric procedure is devised to test familial hypotheses, enabled by a novel pathwise optimization routine to fit the Huber family. The favourable properties of the new test are demonstrated theoretically and experimentally. Two examples from psychology serve as real-world case studies.