Multilevel Regression Modeling of Covariance Matrix Outcomes

2026-05-06

AI Summary

Existing covariance regression methods struggle to accommodate the multilevel structure inherent in neuroimaging data, where subjects are nested within groups such as age cohorts, thereby failing to accurately characterize the relationship between functional brain connectivity and individual covariates. This work proposes the Multilevel Covariance Analysis via Projections (MCAP) framework, which introduces multilevel modeling into covariance matrix regression for the first time. MCAP constructs group-specific linear projections and jointly models them using a generalized linear mixed-effects model, while regularizing these projections via the von Mises–Fisher distribution on the unit sphere to enable information sharing across groups. Theoretically, the method establishes asymptotic properties of the estimators and a two-stage bootstrap inference procedure. Simulations demonstrate its marked superiority over single-level approaches. Applied to the Human Connectome Project Lifespan dataset, MCAP reveals associations between dominant spectral brain networks and age or sex, uncovers convergent neural reorganization in later life, and identifies coordinated modulation between language and executive function networks.

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.

Problem

• covariance matrix
• multilevel modeling
• functional connectivity
• hierarchical data
• neuroimaging

Innovation

• Multilevel regression
• Covariance matrix outcomes
• Functional connectivity
• von Mises-Fisher distribution
• Hierarchical likelihood