Optimal tests of the composite null hypothesis arising in mediation analysis

TL;DR

This paper introduces new hypothesis tests for mediation analysis that address the limitations of traditional tests under the composite null hypothesis, improving power especially when effects are small.

Contribution

It proposes two novel tests for the composite null in mediation analysis, one minimax optimal and one based on sparse linear programming, with implementation in an R package.

Findings

The tests maintain correct type I error rates.

They show improved power near the null when effects are small.

The methods are adaptable for large-scale testing.

Abstract

The indirect effect of an exposure on an outcome through an intermediate variable can be identified by a product of two regression coefficients under certain causal and regression modeling assumptions. In this context, the null hypothesis of no indirect effect is a composite null hypothesis, as the null holds if either regression coefficient is zero. A consequence is that traditional hypothesis tests are severely underpowered near the origin (i.e., when both coefficients are small with respect to standard errors). We propose hypothesis tests that (i) preserve level alpha type~1 error, (ii) meaningfully improve power when both true underlying effects are small relative to sample size, and (iii) preserve power when at least one is not. One approach gives a closed-form test that is minimax optimal with respect to local power over the alternative parameter space. Another uses sparse linear programming to produce an approximately optimal test for a Bayes risk criterion. We discuss adaptations for performing large-scale hypothesis testing as well as modifications that yield improved interpretability. We provide an R package that implements our proposed methodology.