  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 (M) 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.
