Local permutation tests for conditional independence: an adaptive binning perspective

2026-07-22

AI Summary

This study addresses the problem of testing conditional independence between random variables \( X \) and \( Y \) given a confounding variable \( Z \). It proposes a local permutation test based on data-adaptive binning—such as equal-count binning—where permutations of \( X \) and \( Y \) are performed within each subregion defined by \( Z \). The method provides, for the first time, finite-sample Type I error control guarantees for arbitrary test statistics. Under linear confounding models, it achieves power comparable to that of the oracle likelihood ratio test. Theoretical analysis shows that a constant bin size suffices to attain performance on par with increasing bin sizes, and numerical experiments confirm the method’s statistical efficiency and practical utility.

Abstract

In this work, we study the problem of testing conditional independence between random variables $X$ and $Y$ given a confounder $Z$. The local permutation test (LPT) offers a principled approach to this problem by partitioning the $Z$-space into pre-specified bins, and permuting the $X$ and $Y$ data within each bin, to assess the significance of an observed test statistic. However, when the partitions are pre-fixed, the resulting partition can be poorly balanced, as some bins may contain most of the samples while others contain only a few. This motivates the use of data-adaptive binning strategies, such as equisized bins with a fixed (typically small) number of points. We study this natural and practically important extension of LPT, providing finite-sample bounds on the Type I error for an arbitrary test statistic, providing stronger validity results than previously known. We also show that LPT attains power comparable to the oracle likelihood ratio tests derived from the Neyman-Pearson lemma. Within a linear confounder model class, we further analyze the effect of bin size and demonstrate that constant bin sizes can match the performance of partitions with growing bin-size. These results, further supported by extensive numerical simulations, position the proposed data-adaptive strategy as both practically implementable and statistically efficient.

Problem

• conditional independence
• local permutation test
• data-adaptive binning
• confounder
• Type I error

Innovation

• local permutation test
• conditional independence
• adaptive binning
• finite-sample validity
• statistical power