Searcharxiv⌕ Search
Home/Archive/arXiv

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
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

stat.MEstat.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

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

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Bayesian Framework for Multivariate Differential Analysis

Differential analysis is a routine procedure in the statistical analysis toolbox across many applied fields, including quantitative proteomics, the main illustration of the present paper. The state-of-the-art limma approach uses a hierarchical formulation with moderated-variance estimators for each analyte directly injected into the t-statistic. While standard hypothesis testing strategies are recognised for their low computational cost, allowing for quick extraction of the most differential among thousands of elements, they generally overlook key aspects such as handling missing values, inter-element correlations, and uncertainty quantification. The present paper proposes a fully Bayesian framework for differential analysis, leveraging a conjugate hierarchical formulation for both the mean and the variance. Inference is performed by computing the posterior distribution of compared experimental conditions and sampling from the distribution of differences. This approach provides well-calibrated uncertainty quantification at a similar computational cost as hypothesis testing by leveraging closed-form equations. Furthermore, a natural extension enables multivariate differential analysis that accounts for possible inter-element correlations. We also demonstrate that, in this Bayesian treatment, missing data should generally be ignored in univariate settings, and further derive a tailored approximation that handles multiple imputation for the multivariate setting. We argue that probabilistic statements in terms of effect size and associated uncertainty are better suited to practical decision-making. Therefore, we finally propose simple and intuitive inference criteria, such as the overlap coefficient, which express group similarity as a probability rather than traditional, and often misleading, p-values.

stat.ME↗
arXiv2026

Interpretable Deep Neural Network for Modeling Functional Surrogates

Developing surrogates for computer models has become increasingly important for addressing complex problems in science and engineering. This article introduces an artificial intelligent (AI) surrogate, referred to as the DeepSurrogate, for analyzing functional outputs with vector-valued inputs. The relationship between the functional output and vector-valued input is modeled as an infinite sequence of unknown functions, each representing the relationship at a specific location within the functional domain. These spatially indexed functions are expressed through a combination of basis functions and their corresponding coefficient functions, both of which are modeled using deep neural networks (DNN). The proposed framework accounts for spatial dependencies across locations, while capturing the relationship between the functional output and scalar predictors. It also integrates a Monte Carlo (MC) dropout strategy to quantify prediction uncertainty, enhancing explainability in the deep neural network architecture. The proposed method enables efficient inference on datasets with approximately 50,000 spatial locations and 20 simulations, achieving results in under 10 minutes using standard hardware. The approach is validated on extensive synthetic datasets and a large-scale simulation from the Sea Lake and Overland Surge from Hurricanes (SLOSH) simulator. An open-source Python package implementing the method is made available.

stat.ME↗
arXiv2026

Bayesian inference for the learning rate in Generalised Bayesian inference

In Generalised Bayesian Inference (GBI), the learning rate and hyperparameters of the loss must be estimated. These inference-hyperparameters can't be estimated jointly with the other parameters, from the data, by giving them a prior. However, in some settings there exist unknown ``true'' hyperparameter-values about which it is meaningful to have prior belief. It is then possible to use Bayesian inference with held-out data to get hyperparameter-posteriors. We define two hyperparameter posteriors, one based on an Expected Log Pointwise Predictive Density (ELPPD)-utility and one aiming to cover the pseudo-true parameter. The new framework supports estimation and uncertainty quantification for multiple hyperparameters jointly. Experiments show that the resulting GBI-posteriors outperform Bayesian inference on simulated test data and select optimal or near-optimal hyperparameter values in a large real problem of text analysis. Generalised Bayesian inference is particularly useful for combining multiple data sets and most of our examples belong to that setting. We also give asymptotic results for some of the special ``multi-modular'' Generalised Bayes posteriors which we use in our examples.

stat.ME↗