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arxiv: 2606.26949 · v1 · pith:25TBF52Enew · submitted 2026-06-25 · 📊 stat.ME

Exact Comparison of Explanatory Strength of Two Dependent Predictors

Tom\'a\v{s} Mrkvi\v{c}ka , Jan Radimsk\'y This is my paper

Pith reviewed 2026-06-26 02:47 UTC · model grok-4.3

classification 📊 stat.ME
keywords permutation testexplanatory strengthdependent predictorsexact testnonparametric inferenceMonte Carlo simulationcategorical datacontinuous data
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The pith

A Paired Swap Permutation Test delivers exact comparison of which of two dependent predictors has greater explanatory strength.

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

The paper presents a non-parametric test to decide which of two dependent predictors better explains a shared target variable. Standard asymptotic tests lose control of error rates or lose power with heavy tails and sparse categories, while ordinary permutation or bootstrap methods either break the dependence structure or introduce artificial ties. The new procedure rests on functional exchangeability under the null: it swaps the two predictors within each subject for categorical data and maps continuous observations through their empirical distribution functions before swapping. Monte Carlo experiments confirm that the test keeps the nominal significance level exactly and attains higher power than alternatives precisely when those alternatives become liberal or conservative. The method is then used on a large set of Italian noun-noun compounds to obtain reliable rankings.

Core claim

The Paired Swap Permutation Test, grounded in the principle of functional exchangeability under the null hypothesis, utilizes a symmetric within-subject swapping mechanism for categorical data together with an Empirical Cumulative Distribution Function mapping step for continuous domains; this copula-based transposition generates the exact null distribution while perfectly preserving marginal densities and empirical support, thereby furnishing an exact non-parametric procedure for comparing the explanatory strength of two dependent predictors.

What carries the argument

The Paired Swap Permutation Test, which implements symmetric within-subject swapping and ECDF mapping under functional exchangeability to produce the exact null distribution.

If this is right

  • The test strictly maintains the nominal significance level across the simulated conditions examined.
  • The test attains higher statistical power than Vuong, Hotelling-Williams, naive permutation, and paired bootstrap procedures under heavy-tailed and sparse-categorical regimes.
  • The procedure produces usable p-values on a high-dimensional linguistic corpus of noun-noun compounds where classical methods are unreliable.

Where Pith is reading between the lines

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

  • The same swapping-plus-ECDF construction could be reused to obtain exact tests for other symmetric functionals of paired predictors.
  • Because the method preserves the joint empirical distribution, it may be directly applicable to paired observations arising in longitudinal or matched-pair designs outside linguistics.

Load-bearing premise

Functional exchangeability under the null hypothesis is sufficient for the symmetric swapping and ECDF mapping steps to generate the exact null distribution while preserving marginal densities.

What would settle it

A Monte Carlo experiment under the null hypothesis in which the empirical rejection rate at nominal level alpha deviates systematically from alpha, or in which power falls below that of a correctly calibrated competitor, would falsify the exactness and optimality claims.

Figures

Figures reproduced from arXiv: 2606.26949 by Jan Radimsk\'y, Tom\'a\v{s} Mrkvi\v{c}ka.

Figure 1
Figure 1. Figure 1: Continuous domain performance for Pearson’s [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Continuous domain performance for Kendall’s [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Categorical domain performance using Mutual Information. The top panels show Type [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

Comparing the relative explanatory power of two dependent predictors regarding a common target variable is a fundamental challenge across scientific disciplines. Classical asymptotic procedures, such as Vuong's closeness test or the Hotelling-Williams test, frequently collapse under pathological data conditions, including heavy-tailed distributions and extreme categorical sparsity. To bypass these limitations, practitioners often turn to non-parametric resampling. However, naive permutation tests destroy the natural covariance structure of dependent predictors, while the paired bootstrap evaluating variance around the alternative hypothesis and introducing artificial ties suffers from metric space compression and categorical omission, rendering it highly unreliable in finite samples. In this paper, we introduce the Paired Swap Permutation Test, a novel and exact non-parametric methodology. Grounded in the principle of functional exchangeability under the null hypothesis, our algorithm utilizes a symmetric within-subject swapping mechanism for categorical data, and introduces an Empirical Cumulative Distribution Function (ECDF) mapping step for continuous domains. This copula-based transposition perfectly preserves marginal densities and the empirical support without introducing resampling ties. Through extensive Monte Carlo simulations, we demonstrate that the proposed test strictly maintains the nominal significance level and maximizes statistical power under conditions where standard methods become catastrophically liberal or pathologically conservative. Finally, we apply the framework to a high-dimensional linguistic dataset of Italian noun-noun compounds, proving its capacity to deliver robust, exact inference in environments where conventional analytical methods inherently fail.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces the Paired Swap Permutation Test for comparing the explanatory strength of two dependent predictors X1 and X2 for a response Y. The method relies on a symmetric within-subject swap (for categorical data) combined with ECDF mapping (for continuous data) to generate an exact null distribution under the principle of functional exchangeability when the predictors have equal explanatory power. It claims that Monte Carlo simulations show strict type-I error control at the nominal level and power advantages over Vuong, Hotelling-Williams, naive permutation, and paired bootstrap procedures, especially under heavy tails or categorical sparsity; an application to Italian noun-noun compound data is also presented.

Significance. A genuinely exact, non-asymptotic procedure that preserves marginals while respecting dependence would be valuable in statistical methodology, particularly for the pathological regimes where classical tests are known to fail. The copula-style transposition idea is technically interesting and could, if valid, address a real gap; however, the paper provides no machine-checked proofs or closed-form derivations, and the simulation evidence is asserted rather than quantified in the abstract.

major comments (1)
  1. [Abstract (method description) and the section defining the Paired Swap Permutation Test] The central exactness claim rests on the assertion that functional exchangeability holds under the null of equal explanatory strength (equal association or R² with Y). The swapping-plus-ECDF construction generates a reference distribution only if the joint law of (X1,X2,Y) is invariant under within-subject transposition once marginals are matched. Nothing in the null forces this invariance when dependence structures are asymmetric; equal marginal associations can arise without exchangeability. Consequently the generated null need not coincide with the true sampling distribution of the test statistic, so type-I error control is not guaranteed to be exact. This issue is load-bearing for the title claim of 'Exact Comparison' and for the simulation-based assertion of strict level control.
minor comments (2)
  1. [Abstract] The abstract states that 'extensive Monte Carlo simulations' demonstrate control and power advantages, yet supplies no design parameters (sample sizes, dependence strengths, tail indices, sparsity levels) or quantitative results (empirical rejection rates, power curves). These details belong in the main text or a dedicated simulation section with tables.
  2. [Introduction / Method] Notation for the test statistic and the precise definition of 'explanatory strength' (e.g., whether it is R², mutual information, or another measure) should be introduced earlier and used consistently.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. The central issue raised concerns the justification for functional exchangeability under the null of equal explanatory strength. We respond point-by-point below and propose revisions to clarify the null hypothesis.

read point-by-point responses

  1. Referee: [Abstract (method description) and the section defining the Paired Swap Permutation Test] The central exactness claim rests on the assertion that functional exchangeability holds under the null of equal explanatory strength (equal association or R² with Y). The swapping-plus-ECDF construction generates a reference distribution only if the joint law of (X1,X2,Y) is invariant under within-subject transposition once marginals are matched. Nothing in the null forces this invariance when dependence structures are asymmetric; equal marginal associations can arise without exchangeability. Consequently the generated null need not coincide with the true sampling distribution of the test statistic, so type-I error control is not guaranteed to be exact. This issue is load-bearing for the title claim of 'Exact Comparison' and for the simulation-based assertion of strict level control.

    Authors: We agree that equal marginal associations (such as identical R² values) do not automatically entail functional exchangeability when the dependence structure between (X1, X2, Y) is asymmetric. Our procedure is constructed to be exact under the null that the predictors are functionally exchangeable with respect to Y after marginal matching via ECDF mapping. We view this exchangeability null as the appropriate operationalization of 'equal explanatory strength' for a symmetric comparison test, because it directly encodes the idea that the two predictors play interchangeable roles in their joint relationship with Y. Under this null the within-subject swaps generate the exact reference distribution by construction. We will revise the abstract, the definition of the test, and the discussion of the null hypothesis to make this distinction explicit, including a note on the distinction between marginal association equality and exchangeability. We will also add a brief remark on the scope of the exactness claim. This addresses the load-bearing concern for the title while preserving the method's validity under the stated null. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation rests on explicit exchangeability assumption without reduction to inputs

full rationale

The paper defines the Paired Swap Permutation Test by direct construction from the stated principle of functional exchangeability under the null, using within-subject swapping for categorical data and ECDF mapping for continuous data to produce the reference distribution. No equations, test statistic, or null distribution is shown to equal a fitted quantity or prior self-citation by construction. Monte Carlo results are presented only as empirical confirmation of type-I error control, not as the source of the exactness claim. The chain is therefore self-contained against the external exchangeability premise.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on the domain assumption of functional exchangeability under the null; no free parameters or invented entities are described in the abstract.

axioms (1)
  • domain assumption functional exchangeability under the null hypothesis
    Invoked to justify that the swapping mechanism produces the exact null distribution without destroying covariance.

pith-pipeline@v0.9.1-grok · 5785 in / 1222 out tokens · 85742 ms · 2026-06-26T02:47:52.697035+00:00 · methodology

discussion (0)

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