Multiscale Cochran-Mantel-Haenszel Scanning for Conditional Dependency

TL;DR

This paper introduces a nonparametric multiscale scanning method for testing and estimating conditional independence and association, extending the CMH test to continuous spaces with efficient computation and reliable error control.

Contribution

It generalizes the CMH test to continuous sample spaces, enabling consistent, scalable, and easy-to-interpret conditional dependency analysis without large sample size constraints.

Findings

Method achieves asymptotic null distribution with linear scaling

Reliable Type I error control demonstrated in simulations

Case study shows ability to test and identify local associations

Abstract

We propose a nonparametric approach to testing conditional independence and estimating conditional association, generalizing the Cochran-Mantel-Haenszel (CMH) test and odds-ratio estimator to continuous sample spaces. It leverages a multiscale scanning approach to decompose the sample space into a cascade of $2\times 2 \times T$ tables. Following the CMH test, we condition on the marginal order statistics, which are "almost ancillary" regarding conditional dependency. This strategy helps overcome a key challenge faced by other methods that discretize the sample space: we achieve consistency without requiring stratum sample sizes to grow to infinity, a constraint often difficult to satisfy in practice. Our method produces easy-to-compute test statistics with a known asymptotic null distribution under the conditional sampling model, scaling almost linearly with the sample size. Our simulation results demonstrate reliable Type I error control, even with small samples and high-dimensional conditioning, and competitive power compared to state-of-the-art tests. Finally, a case study on Uber ride-share data highlights the method's unique dual capability, inherited from the CMH, to both test and identify the nature of the inferred conditional association. By providing summary statistics that capture the strength and direction of local associations, our method offers practitioners a useful tool for learning conditional dependencies.