🤖 AI Summary
This study addresses the challenge of achieving consistent conditional independence testing and association estimation in continuous sample spaces under small-sample, high-dimensional settings. The authors propose a nonparametric multiscale approach that decomposes the continuous space via cascaded 2×2×T contingency tables and conditions on marginal order statistics. This framework extends the Cochran–Mantel–Haenszel (CMH) test and odds ratio estimation to continuous variables for the first time, ensuring statistical consistency without requiring asymptotic layer-wise sample sizes. The method simultaneously supports hypothesis testing and identification of local association strength and direction, with near-linear computational complexity. Empirical evaluations demonstrate its superior or competitive statistical power while properly controlling Type I error, and it successfully uncovers local conditional dependence structures in real-world Uber mobility data.
📝 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.