On statistical deficiency: Why the test statistic of the matching method is hopelessly underpowered and uniquely informative

TL;DR

This paper reveals that the matching method's test statistic is fundamentally underpowered due to its statistical deficiency, limiting its effectiveness in null hypothesis testing of association.

Contribution

It introduces the concept of a deficient statistic, demonstrating the inherent limitations of the matching method's test statistic through theoretical reinterpretation and derivation.

Findings

Matching method's test statistic has low statistical power.

The concept of deficient statistic is introduced and exemplified.

Application shows how deficiency affects sampling error estimation.

Abstract

The random variate m is, in combinatorics, a basis for comparing permutations, as well as the solution to a centuries-old riddle involving the mishandling of hats. In statistics, m is the test statistic for a disused null hypothesis statistical test (NHST) of association, the matching method. In this paper, I show that the matching method has an absolute and relatively low limit on its statistical power. I do so first by reinterpreting Rae's theorem, which describes the joint distributions of m with several rank correlation statistics under a true null. I then derive this property solely from m's unconditional sampling distribution, on which basis I develop the concept of a deficient statistic: a statistic that is insufficient and inconsistent and inefficient with respect to its parameter. Finally, I demonstrate an application for m that makes use of its deficiency to qualify the sampling error in a jointly estimated sample correlation.