A family of divergence-based correlation measures for contingency tables under bivariate normality

Wataru Urasaki

arxiv: 2603.13848 · v2 · submitted 2026-03-14 · 📊 stat.ME

A family of divergence-based correlation measures for contingency tables under bivariate normality

Wataru Urasaki This is my paper

Pith reviewed 2026-05-15 11:43 UTC · model grok-4.3

classification 📊 stat.ME

keywords divergence-based correlationcontingency tablesbivariate normalitylatent correlationpower-divergenceassociation measuresasymptotic distributions

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The pith

A family of measures indexed by a parameter approximates the latent correlation coefficient from contingency tables by inverting a closed-form power-divergence approximation under bivariate normality.

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

This paper develops a family of association measures for two-way contingency tables that assume an underlying bivariate normal distribution for the latent variables. Under that assumption the power-divergence from independence admits a closed-form expression in terms of the latent correlation. Inverting the expression produces the indexed family ρ_{(λ)} that estimates the latent correlation directly for any choice of the parameter λ between negative one and one. The construction recovers Linfoot’s informational measure of correlation when λ equals zero and Pearson’s contingency coefficient when λ equals one. Asymptotic theory supplies confidence intervals, and simulations show the new measures track the true correlation more closely than prior divergence-based statistics while running thousands of times faster than the polychoric correlation.

Core claim

When the latent variables follow a bivariate normal distribution, the power-divergence measuring departure from independence can be approximated in closed form as a function of the latent correlation coefficient. Inverting this relationship yields a family of measures ρ_{(λ)} indexed by the scalar parameter λ in the interval from −1 to 1 that directly approximate the latent correlation. Special cases include the informational measure of correlation at λ = 0 and Pearson’s contingency coefficient at λ = 1. Asymptotic distributions are obtained via the delta method, enabling the construction of confidence intervals.

What carries the argument

The closed-form approximation of the power-divergence statistic under bivariate normality, inverted to express the latent correlation as a function of the observed divergence value.

If this is right

The proposed measures approximate the true latent correlation more faithfully than conventional divergence-based measures.
They successfully distinguish between weak and moderate associations where existing measures give indistinguishable values.
Computation requires several thousand times less time than the polychoric correlation coefficient.
The measures remain numerically stable even when the latent correlation is close to one.
Asymptotic distributions derived via the delta method support two families of confidence intervals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Adoption in statistical software could enable routine analysis of large collections of contingency tables that are currently limited by polychoric computation time.
The parameter λ provides a tunable knob that might be chosen by cross-validation or information criteria in applied work.
Similar inversion techniques could be investigated for other divergence families or for non-normal latent distributions.

Load-bearing premise

The latent continuous variables are jointly normally distributed.

What would settle it

Generating many contingency tables from a bivariate normal distribution with a fixed known correlation ρ and verifying whether the sample averages of the proposed measures ρ_{(λ)} converge to that known ρ.

read the original abstract

We propose a family of association measures for two-way contingency tables whose latent distribution can be assumed to be bivariate normal. When this assumption holds, the power-divergence measuring departure from independence can be approximated in closed form as a function of the latent correlation coefficient. By inverting this relationship, we obtain a family of measures $\rho_{(\lambda)}$, indexed by a scalar parameter $-1 \leq \lambda \leq 1$, that directly approximates the latent correlation. Special cases include the informational measure of correlation proposed by Linfoot (1957) at $\lambda = 0$ and Pearson's contingency coefficient $C$ at $\lambda = 1$. Additionally, we derive asymptotic distributions via the delta method and construct two families of confidence intervals. Simulation studies confirm that the proposed measures approximate the true latent correlation more faithfully than conventional divergence-based measures, and that they successfully distinguish between weak and moderate associations where existing measures tend to give indistinguishable values. Compared with the polychoric correlation coefficient, the proposed measures are computed several thousand times faster and remain numerically stable even when the latent correlation is close to one.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a clean, fast family of correlation measures for tables under bivariate normality that generalizes two older ones and adds usable asymptotics.

read the letter

The main point is a parameterized family ρ_λ derived by approximating power divergences in closed form under the bivariate normal latent model and inverting the expression. It recovers the latent correlation directly, covers Linfoot at λ=0 and Pearson's C at λ=1, supplies delta-method standard errors, and includes two confidence-interval constructions. Simulations reportedly show tighter recovery of the true value and better distinction between weak and moderate associations than plain divergence measures, plus several-thousand-fold speed gains and stability near ρ=1 compared with polychoric correlation. That combination is the practical contribution. The construction is internally consistent: the normality assumption is used only for the approximation step, the inversion is explicit, and the asymptotics follow standard delta-method arguments. No circularity or hidden fitting appears. The main soft spot is the scope. Everything stands on the latent bivariate-normal assumption, which the paper states plainly but which will not hold for every contingency table in practice. If the approximation error grows for coarse tables or extreme probabilities, the advantage over polychoric could shrink, though the reported simulations suggest it holds up under the conditions they tested. Without the exact Monte Carlo design in front of me I cannot judge whether the replications cover small samples or highly unbalanced margins thoroughly enough. This is for applied statisticians who routinely assume normal latents for ordinal data and want something quicker and more stable than full maximum-likelihood estimation. It is not a general-purpose replacement for polychoric or rank correlations. I would send it to peer review. The core derivation is straightforward, the comparisons address a real computational pain point, and the work is scoped tightly enough that referees can evaluate it on its own terms.

Referee Report

0 major / 3 minor

Summary. The paper proposes a family of association measures ρ_{(λ)} (-1 ≤ λ ≤ 1) for two-way contingency tables under the assumption that the latent variables follow a bivariate normal distribution. It derives a closed-form approximation to the power-divergence family as a function of the latent correlation ρ, inverts the relationship to obtain ρ_{(λ)}, recovers special cases such as Linfoot's informational measure (λ=0) and Pearson's contingency coefficient (λ=1), obtains delta-method asymptotic distributions, constructs confidence intervals, and reports simulation results indicating that the new measures recover the true latent correlation more accurately than conventional divergence-based statistics while remaining numerically stable and orders of magnitude faster than polychoric correlation.

Significance. If the approximation and inversion hold under the stated normality assumption, the work supplies a parameterized family of measures that directly target the latent correlation, unifies several classical coefficients, and supplies ready-to-use asymptotic inference. The computational speed and stability advantages relative to polychoric correlation, together with the explicit delta-method justification, would make the family practically useful for large tables or repeated analyses where polychoric methods become unstable or slow.

minor comments (3)

[Simulation studies] The simulation section should explicitly state the number of Monte Carlo replications, the grid of table dimensions, and the range of ρ values examined so that the reported fidelity gains can be reproduced.
[Derivation of the approximation] The precise algebraic form of the closed-form approximation to the power-divergence integral (prior to inversion) should be displayed as an equation rather than described only in prose.
[Asymptotic results] Notation for the two families of confidence intervals should be introduced once and used consistently in both the theoretical and numerical sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. We are pleased that the significance of the proposed family of measures, their computational advantages, and the delta-method inference are recognized.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives a closed-form approximation of the power-divergence statistic as an explicit function of the latent correlation ρ under the bivariate normality assumption, then algebraically inverts the resulting expression to define the family ρ_{(λ)}. This is a direct model-based construction rather than a fit to data or a renaming of an input quantity. No self-citations appear in the load-bearing steps, no uniqueness theorems are invoked from prior author work, and the delta-method asymptotics follow standard statistical arguments. Simulations are presented only as numerical confirmation of recovery accuracy, not as part of the derivation itself. The central claim remains scoped to the stated assumption and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on a single domain assumption that enables the closed-form approximation; no free parameters are fitted inside the measure itself and no new entities are postulated.

axioms (1)

domain assumption The latent variables follow a bivariate normal distribution
This assumption is invoked to obtain the closed-form expression for the power-divergence as a function of the latent correlation coefficient.

pith-pipeline@v0.9.0 · 5487 in / 1290 out tokens · 48029 ms · 2026-05-15T11:43:30.649839+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

I_λ({p_ij};{p_i·p·j}) ≈ 1/[λ(λ+1)] [(1-ρ²)^{-λ/2}(1-λ²ρ²)^{-1/2}-1] ... invert to obtain ρ_{(λ)}
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Special cases ... λ=0 informational measure ... λ=1 Pearson's C

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.