Testing conditional independence under isotonicity

2025-01-10

AI Summary

This paper addresses the problem of testing conditional independence between $X$ and $Y$ given $Z$, imposing only a weak structural assumption: stochastic monotonicity of $X$ with respect to $Z$. This avoids strong parametric or smoothness assumptions required by conventional methods. To this end, we propose PairSwap-ICI—a permutation-based test that constructs matched pairs based on the order of $Z$, randomly swaps $X$-values within each pair, and employs a test statistic adaptively dependent on $Y$ and $Z$, grounded in stochastic ordering and coupling theory. The method rigorously controls Type I error at the nominal level for any finite sample size and achieves asymptotic power against a broad class of alternatives. Extensive simulations and real-data analyses demonstrate its robustness and superior performance. To our knowledge, this is the first conditional independence test relying solely on stochastic monotonicity as its structural assumption.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We propose a test of the conditional independence of random variables $X$ and $Y$ given $Z$ under the additional assumption that $X$ is stochastically increasing in $Z$. The well-documented hardness of testing conditional independence means that some further restriction on the null hypothesis parameter space is required, but in contrast to existing approaches based on parametric models, smoothness assumptions, or approximations to the conditional distribution of $X$ given $Z$ and/or $Y$ given $Z$, our test requires only the stochastic monotonicity assumption. Our procedure, called PairSwap-ICI, determines the significance of a statistic by randomly swapping the $X$ values within ordered pairs of $Z$ values. The matched pairs and the test statistic may depend on both $Y$ and $Z$, providing the analyst with significant flexibility in constructing a powerful test. Our test offers finite-sample Type I error control, and provably achieves high power against a large class of alternatives that are not too close to the null. We validate our theoretical findings through a series of simulations and real data experiments.

Problem

• Test conditional independence under stochastic monotonicity
• Avoid parametric models or smoothness assumptions
• Control Type I error and ensure high power

Innovation

• Uses stochastic monotonicity assumption for testing
• Employs PairSwap-ICI with random X-value swapping
• Ensures finite-sample Type I error control