Distance Metrics for Measuring Joint Dependence with Application to Causal Inference

Abstract

Many statistical applications require the quantification of joint dependence among more than two random vectors. In this work, we generalize the notion of distance covariance to quantify joint dependence among $d\ge 2$ random vectors. We introduce the high-order distance covariance to measure the so-called Lancaster interaction dependence. The joint distance covariance is then defined as a linear combination of pairwise distance covariances and their higher-order counterparts which together completely characterize mutual independence. We further introduce some related concepts including the distance cumulant, distance characteristic function, and rank-based distance covariance. Empirical estimators are constructed based on certain Euclidean distances between sample elements. We study the large-sample properties of the estimators and propose a bootstrap procedure to approximate their sampling distributions. The asymptotic validity of the bootstrap procedure is justified under both the null and alternative hypotheses. The new metrics are employed to perform model selection in causal inference, which is based on the joint independence testing of the residuals from the fitted structural equation models. The effectiveness of the method is illustrated via both simulated and real datasets. Supplementary materials for this article are available online.

KEYWORDS:

• Bootstrap
• Directed acyclic graph
• Distance covariance
• Interaction dependence
• U-statistic
• V-statistic

Acknowledgments

We acknowledge partial funding from the NSF grant DMS-1607320. The authors are grateful to the editor and two very careful reviewers for their immensely helpful comments and suggestions that greatly improved the article. It would also be authors’ pleasure to thank Niklas Pfister for sharing with them the R codes used in Pfister et al. (2018).

Funding

The authors acknowledge partial funding from the NSF grant DMS-1607320.