Power of testing for exposure effects under incomplete mediation

Mediation analysis studies situations where an exposure may affect an outcome both directly and indirectly through intervening variables called mediators. It is frequently of interest to test for the effect of the exposure on the outcome, and the standard approach is simply to regress the latter on the former. However, it seems plausible that a more powerful test statistic could be achieved by also incorporating the mediators. This would be useful in cases where the exposure effect size might be small, which for example is common in genomics applications. Previous work has shown that this is indeed possible under complete mediation, where there is no direct effect. In most applications, however, the direct effect is likely nonzero. In this paper we study linear mediation models and find that under certain conditions, power gain is still possible under this incomplete mediation setting for testing the null hypothesis that there is neither a direct nor an indirect effect. We study a class of procedures that can achieve this performance and develop their application to both low- and high-dimensional mediators. We then illustrate their performances in simulations as well as in an analysis using DNA methylation mediators to study the effect of cigarette smoking on gene expression.