Exact Comparison of Explanatory Strength of Two Dependent Predictors

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.