Testing Mediation Effects Using Logic of Boolean Matrices

Abstract

A central question in high-dimensional mediation analysis is to infer the significance of individual mediators. The main challenge is that the total number of potential paths that go through any mediator is super-exponential in the number of mediators. Most existing mediation inference solutions either explicitly impose that the mediators are conditionally independent given the exposure, or ignore any potential directed paths among the mediators. In this article, we propose a novel hypothesis testing procedure to evaluate individual mediation effects, while taking into account potential interactions among the mediators. Our proposal thus fills a crucial gap, and greatly extends the scope of existing mediation tests. Our key idea is to construct the test statistic using the logic of Boolean matrices, which enables us to establish the proper limiting distribution under the null hypothesis. We further employ screening, data splitting, and decorrelated estimation to reduce the bias and increase the power of the test. We show that our test can control both the size and false discovery rate asymptotically, and the power of the test approaches one, while allowing the number of mediators to diverge to infinity with the sample size. We demonstrate the efficacy of the method through simulations and a neuroimaging study of Alzheimer’s disease. A Python implementation of the proposed procedure is available at https://github.com/callmespring/LOGAN.

KEYWORDS:

• Boolean matrix
• Directed acyclic graph
• Gaussian graphical model
• High-dimensional inference
• Mediation analysis
• Neuroimaging analysis

Supplementary Materials

The supplementary material consists of a multi-split version of the proposed test, a discussion of regularity conditions, technical proofs and some additional numerical results.

Acknowledgments

The authors thank the AE, and the reviewers for their constructive comments, which have led to a significant improvement of the earlier version of this article.

Additional information

Funding

Li’s research was partially supported by NIH grants R01AG061303, R01AG062542, and R01AG034570.