A Dimension Reduction Approach to Multivariate Mediation Analysis

Abstract

Mediation analysis is an important tool for understanding possible mechanistic relationships among three variables representing an exposure (X), a mediator (M), and an outcome (Y ). Here we propose a method for conducting multivariate mediation analysis via dimension reduction. Specifically, we aim to project X, M, and Y to subspaces of minimal dimension that capture all correlations that could reflect mediation. In particular, the reduced M must simultaneously correlate with the reduced X and the reduced Y . We identify these mediation subspaces by minimizing a loss function based on determinants of covariance matrices. To aid in the interpretation, we rotate the reduced spaces to approximate “parallel mediation†, in which the multivariate mediation approximately splits into separate univariate mediations on projected one-dimensional subspaces. We also develop a type of biplot to aid in the visualization of the findings. We present simulation studies showing that our method effectively identifies mediation structures, and we illustrate the approach by analyzing data from a mouse model of kidney disease, considering gene expression as a mediator between exposures and disease outcomes.