Learning the Optimal Composite Mediator: Closed-Form Solution and Inference

TL;DR

This paper introduces MaxIE, a closed-form solution for identifying the optimal composite mediator in high-dimensional settings, enabling efficient inference of indirect effects in observational studies.

Contribution

The paper derives a geometric, closed-form solution for the optimal composite mediator, significantly reducing computational cost compared to numerical optimization.

Findings

MaxIE efficiently finds the optimal mediator with linear solves.

The method accurately tests for the existence of a composite mediator.

Applied to UK Biobank data, it identified a significant proteomic mediator.

Abstract

Understanding how an exposure transmits its effect through high-dimensional intermediaries is a central problem in observational research. We study the problem of finding a composite mediator that maximises the indirect effect of an exposure on an outcome in a linear structural equation model. Although the objective is non-convex in the weight vector, a geometric argument yields a closed-form global solution: the optimal weight bisects the angle between two computable path vectors in a weighted inner product space, recovered via two linear solves. The resulting algorithm, MaxIE, runs at the same cost as ordinary least squares -- orders of magnitude lower than numerical optimisation -- with a dual formulation for settings where mediators outnumber observations. The same path vectors yield a test for the global null that no composite mediator exists, with t(p-1) in the classical and t(n-2) in the dual regime. Power is characterised analytically as a function of the population path angle; simulations confirm size control and the power characterisation. Applied to a UK Biobank proteomics dataset (n=38,383, p=2,916), the method rejects the global null (p-value = 6.4e-9) and identifies the optimal proteomic composite mediating age's effect on dementia.