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.

On Learning Necessary and Sufficient Causal Graphs

“…A feasible approach to handle MEC is to list all DAGs in the MEC from a given CPDAG (Chakrabortty et al, 2018). It is common practice to summarize a set of possible effects or probabilities by its average or the minimum absolute value (Chakrabortty et al, 2018;Shi and Li, 2021). Yet, such an approach is typically computationally infeasible for large graphs with thousands of nodes, and provided computational shortcuts to obtain the causal effects or probabilities of causation without listing all DAGs in the MEC of the estimated CPDAG.…”

On optimal two‐stage testing of multiple mediators

“…In addition, the authors provide high‐dimensional consistency and distributional results for the proposed method, which can be employed to obtain asymptotic confidence intervals for the individual mediation effects. Shi and Li (2021) also assume a directed acyclic graphical structure, but introduce a slightly different definition of the individual mediation effect which circumvents the problem of disjunctive effects cancelling each other out and resulting in a zero mediation effect. The authors propose a novel method for testing mediation effects based on the logic of Boolean matrices, which allows taking into account directed paths among mediators, and still obtaining a tractable, limiting distribution of the test statistic under the null hypothesis.…”

Sequential pathway inference for multimodal neuroimaging analysis

“…The latter means that all the zero entries of the DAG estimator have to match those of the true DAG. For the method of Zheng et al (2018), Shi and Li (2021) established its estimation consistency.…”

“…This condition is often imposed; see, for example, Peters and Bühlmann (2014) and Yuan et al (2019). Meanwhile, it is also possible to relax this condition; see Shi and Li (2021) for more discussion.…”

“…Though proven useful in some applications, ignoring potential paths and interactions among the mediators seems not sensible in plenty of other scientific applications; for example, different brain regions are generally conceived to influence each other, and different genes are expected to interact with each other. More recently, Chakrabortty et al (2018) proposed a mediation test based on interventional calculus, and Shi and Li (2021) proposed a test based on logic of Boolean matrices, while both allowed interactions among the mediators. Nevertheless, none of the existing mediation inference solutions tackle multimodality of mediators.…”