Model Selection for Exposure-Mediator Interaction.

In mediation analysis, the exposure often influences the mediating effect, i.e., there is an interaction between exposure and mediator on the dependent variable. When the mediator is high-dimensional, it is necessary to identify non-zero mediators $\left(M\right)$ and exposure-by-mediator ( $X$ -by- $M$ ) interactions. Although several high-dimensional mediation methods can naturally handle $X$ -by- $M$ interactions, research is scarce in preserving the underlying hierarchical structure between the main effects and the interactions. To fill the knowledge gap, we develop the XMInt procedure to select $M$ and $X$ -by- $M$ interactions in the high-dimensional mediators setting while preserving the hierarchical structure. Our proposed method employs a sequential regularization-based forward-selection approach to identify the mediators and their hierarchically preserved interaction with exposure. Our numerical experiments showed promising selection results. Further, we applied our method to ADNI morphological data and examined the role of cortical thickness and subcortical volumes on the effect of amyloid-beta accumulation on cognitive performance, which could be helpful in understanding the brain compensation mechanism.