Multilevel Regression Modeling of Covariance Matrix Outcomes

TL;DR

This paper introduces a multilevel regression framework for covariance matrix outcomes in neuroimaging, accommodating hierarchical data and improving estimation over existing single-level methods.

Contribution

The paper presents MCAP, a multilevel covariance regression model that accounts for nested data structures and borrows information across clusters using a von Mises-Fisher distribution.

Findings

MCAP outperforms single-level methods in simulations.

Identifies age and sex effects on brain connectivity.

Reveals neural reorganization in late adulthood.

Abstract

Covariance matrix outcomes arise naturally in neuroimaging experiments to study brain functional connectivity. It is also of interest to understand how brain network organization varies with subject-level covariates. Existing covariance regression methods operate in a single-level framework and do not accommodate the hierarchically nested data structure in which subjects are grouped into clusters, such as age cohorts in lifespan studies. A Multilevel Covariate-Assisted Principal Regression (MCAP) framework is introduced, which identifies, for each cluster, a linear projection such that a generalized linear mixed effects model can be formulated with the covariates. The cluster-specific projections are modeled on the unit sphere via a von Mises-Fisher distribution, enabling principled borrowing of information across clusters. Model parameters are estimated by maximizing a hierarchical likelihood. For inference, a two-stage bootstrap procedure is proposed. Asymptotic properties of the estimators are established. Simulation studies demonstrate that MCAP substantially outperforms single-level competitors in estimating regression coefficients. Applied to the Human Connectome Project Lifespan Study spanning ages from five to ninety, MCAP identifies a dominant spectral brain network capturing age and sex effects on functional connectivity, and reveals findings including the convergence of neural reorganization patterns in late adulthood and the coordinated lifespan modulation of cross-network regions linked to language and executive function.