Association measures for two-way contingency tables based on multi-categorical proportional reduction in error

Kouji Tahata; Sadao Tomizawa; Wataru Urasaki

Association measures for two-way contingency tables based on multi-categorical proportional reduction in error

T0 review · 2 major / 0 minor · reviewed 2026-05-23 · grok-4.3

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Pith's one-line read New association measures extend proportional reduction in error to multiple categories in contingency tables.

desk verdict A modest extension of classic PRE measures that fixes their tendency to hit zero under dependence for multi-category responses. read the letter →

arxiv 2503.06538 v2 submitted 2025-03-09 stat.ME

Wataru Urasaki , Kouji Tahata , Sadao Tomizawa This is my paper

classification stat.ME

keywords contingencytablesassociationmeasuresproportionalreductioninerrorGoodman-Kruskallambdacategoricaldataasymmetricprediction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper seeks to extend measures of proportional reduction in error so they quantify how much an explanatory variable improves prediction of a response variable that has any number of categories. Existing PRE measures such as Goodman-Kruskal lambda can return zero even when the variables are dependent, because they rely only on the single largest joint and marginal probabilities. The authors construct versions that draw on the full set of category probabilities to compute the error reduction. They derive the basic properties of these measures and illustrate their behavior on numerical examples. If the extension works as intended, analysts obtain a value strictly between zero and one that reflects dependence whenever it is present in the table.

What carries the argument

Multi-categorical proportional reduction in error measures, which extend the error-reduction calculation from the single most probable outcome to all categories of the response variable.

What would settle it

A contingency table with three or more response categories in which the variables are dependent yet any of the new measures equals zero would show the extension fails to correct the original limitation.

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Extended reading notes

Core claim

The proposed measures are extensions of PRE designed for the proportional reduction in error with multiple categories. These measures address the limitation of returning zero despite dependence by using more of the information in the contingency table. The properties of the proposed measures are examined, and their utility is demonstrated through numerical experiments.

Load-bearing premise

That extending PRE to multiple categories will systematically use more table information and avoid returning zero when dependence exists, without new limitations or the need for post-hoc adjustments.

Editorial extensions

If this is right

The measures take values in [0,1] and equal zero only under independence of the row and column variables.
They apply directly to asymmetric tables that distinguish explanatory and response variables.
They remain simple to interpret as the proportional reduction in prediction error.
Numerical checks confirm they produce positive values on tables where standard lambda equals zero.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Analysts could replace lambda with these measures in software routines for categorical data without changing downstream workflows.
The approach might generalize to three-way tables by conditioning the multi-category reduction on a third variable.
Direct comparison on the same data sets would reveal how much additional association is captured beyond single-maximum-probability versions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The paper extends Goodman-Kruskal lambda and related PRE association measures to cases where the response variable has more than two categories. The standard versions rely only on the maximum probabilities and can return zero even when dependence is present; the new versions are constructed to pull in additional table information to reduce that problem. They lay out the definitions, check basic properties such as the 0-1 range and behavior under independence, and run numerical experiments to show the measures pick up association where the older ones do not. That is the main new piece. The motivation is clear and the limitation they target is genuine in applied work with contingency tables. The experiments are described at a level that supports the claim without overstatement. The soft spot is that the extension stays close to the original PRE logic rather than introducing a different framework, so the advance is incremental. It would be useful to see direct comparisons against other existing multi-category association measures beyond the PRE family. The paper is aimed at applied statisticians who already use lambda or similar tools in social science or health data analysis and want a version that avoids the zero artifact. It is coherent on its own terms and the argument does not rely on hidden fitting or circular definitions. A serious editor should send it to peer review so the formulas and experiments can be checked in detail.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes an extension of proportional reduction in error (PRE) measures, such as Goodman-Kruskal's lambda, for two-way contingency tables in asymmetric settings (explanatory row variable, response column variable). The extension targets multi-categorical cases to address the known limitation that standard PRE measures can return zero despite dependence, by incorporating more information from the full table rather than only maximum probabilities. Properties of the new measures are examined and their utility is illustrated via numerical experiments, with the conclusion that they have potential as practical tools in applied statistics.

Significance. If the extension indeed uses additional table information to avoid the zero-value problem without introducing new limitations or requiring post-hoc adjustments, the measures could provide a simple, interpretable addition to existing association measures for categorical data analysis.

major comments (2)

[Abstract] Abstract: the central claim that the extension 'uses more table information' and avoids returning zero under dependence cannot be evaluated, as no explicit definition, derivation, or verification of the proposed measures is provided.
[Abstract] Abstract: the numerical experiments are invoked to demonstrate utility, but no results, comparisons to existing measures, or verification steps appear, preventing assessment of whether the extension fulfills the stated motivation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their review. We address the major comments on the abstract point by point below, noting that abstracts are summaries and the full details appear in the manuscript body.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the extension 'uses more table information' and avoids returning zero under dependence cannot be evaluated, as no explicit definition, derivation, or verification of the proposed measures is provided.

Authors: The abstract is a concise summary of the motivation and contribution. The explicit definitions of the multi-categorical PRE measures, their derivation extending the standard PRE framework to incorporate full table probabilities rather than only maxima, and the verification that they avoid zero under dependence are provided in Sections 2 and 3 of the manuscript. revision: no
Referee: [Abstract] Abstract: the numerical experiments are invoked to demonstrate utility, but no results, comparisons to existing measures, or verification steps appear, preventing assessment of whether the extension fulfills the stated motivation.

Authors: The abstract summarizes the use of numerical experiments. The actual results, direct comparisons to measures such as Goodman-Kruskal lambda, and verification steps confirming the measures address the zero-value limitation are reported in Section 4, including tables and figures. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The paper proposes new multi-categorical PRE association measures as an explicit extension of existing PRE concepts (e.g., Goodman-Kruskal lambda) to address a known limitation of returning zero under dependence. The abstract and description frame this as a definitional construction of new quantities that incorporate more table information, with properties examined and utility shown via numerical experiments. No equations or steps are visible that reduce a claimed prediction or result to a fitted parameter by construction, nor do any load-bearing premises rest on self-citations whose content is unverified or imported as uniqueness theorems. The central claim remains an independent definitional proposal rather than a renaming or tautological fit, making the derivation self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are described.

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Cite this review

Pith. "Pith review of Association measures for two-way contingency tables based on multi-categorical proportional reduction in error." pith.science (2026). https://pith.science/paper/2503.06538

@misc{pith2026250306538,
  author       = {Pith},
  title        = {Pith review of: Association measures for two-way contingency tables based on multi-categorical proportional reduction in error},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2503.06538}},
  note         = {Machine review of arXiv:2503.06538}
}

abstract

In two-way contingency tables under an asymmetric situation, where the row and column variables are defined as explanatory and response variables, respectively, quantifying the extent to which the explanatory variable contributes to predicting the response variable is important. One quantification method is the association measure, which indicates the degree of association in a range from $0$ to $1$. Among various measures that have been proposed, those based on proportional reduction in error (PRE) are particularly notable for their simplicity and intuitive interpretation. These measures, including Goodman-Kruskal's lambda proposed in 1954, are widely implemented in statistical software such as R and SAS and remain extensively used. However, a well-known limitation of PRE measures is their potential to return a value of $0$ despite no independence. This issue arises because the measures are constructed based solely on the maximum joint and marginal probabilities, failing to make full use of the information available in the contingency table. To address this problem, we propose an extension of PRE measures designed for the proportional reduction in error with multiple categories. The properties of the proposed measures are examined, and their utility is demonstrated through numerical experiments. The results suggest their potential as practical tools in applied statistics.

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Reviewed May 23, 2026 · model on record in the stance chip above.