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arXiv · 2609.26211

An Honest Effect Size for Contingency Tables: Why Nothing Can Be Unbiased, Where to Put the Error Instead, and How to Route the Report

William J. Dwyer
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Abstract

Cramer's V is the effect size reported beside almost every chi-square test, read against Cohen's labels, yet a large fraction of such numbers describe sampling noise and the standard bias correction (Bergsma 2013) does not fix it. We assemble three results and claim only the consequence of each. First, the squared effect size phi^2 admits no unbiased estimator at any sample size: under fixed-N multinomial sampling the expectation of any estimator is a polynomial in the cell probabilities, while phi^2 is not. Second, unbiasedness transfers across a rescaling of the effect size only if the rescaling is affine, and among affine choices V^2 = phi^2/k is the one bounded in [0,1] with value 1 at perfect association. Third, we give an interval with conservative, asymptotically valid coverage, obtained by projecting a likelihood-ratio confidence set for the cell probabilities through the effect-size map; its lower endpoint is zero in closed form exactly when the test of independence fails to reject. Because nothing is unbiased, the only question is where the irreducible error is placed. Bergsma's correction puts zero error at the null but several percent under the alternative; a delete-one jackknife on the V^2 scale spreads it thin everywhere (absolute bias at most 0.008 over a 180-design grid). Pooling does not remove bias: across 3,000 simulated meta-analyses, pooling 200 studies drives the naive estimator's chance of landing within 0.01 of the truth to zero, while the jackknife's rises to 0.99. The projected interval covers 0.997-1.000 across the tested designs, while a noncentral inversion undercovers (0.936 at phi^2 = 0.18) at two to three times the width. The point estimate and the interval are different problems with different answers; conflating them is why the literature has neither. We give a routing rule and a browser tool that implements it.

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William J. Dwyer. 2026-08-12. An Honest Effect Size for Contingency Tables: Why Nothing Can Be Unbiased, Where to Put the Error Instead, and How to Route the Report. https://arxiv.org/abs/2609.26211

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