Longitudinal regression of covariance matrix outcomes

TL;DR

This paper introduces a novel longitudinal regression model for covariance matrices that identifies covariate effects, estimates parameters, and accounts for within-subject variation, with proven efficiency and application to Alzheimer's disease data.

Contribution

It develops a multilevel generalized linear model for covariance matrices, providing optimal estimators for both low- and high-dimensional data, and demonstrates improved power in neuroimaging analysis.

Findings

Identifies brain networks differing by gender and disease stage in ADNI data.

Proposes estimators that are asymptotically consistent and efficient.

Shows improved statistical power over cross-sectional analysis.

Abstract

In this study, a longitudinal regression model for covariance matrix outcomes is introduced. The proposal considers a multilevel generalized linear model for regressing covariance matrices on (time-varying) predictors. This model simultaneously identifies covariate associated components from covariance matrices, estimates regression coefficients, and estimates the within-subject variation in the covariance matrices. Optimal estimators are proposed for both low-dimensional and high-dimensional cases by maximizing the (approximated) hierarchical likelihood function and are proved to be asymptotically consistent, where the proposed estimator is the most efficient under the low-dimensional case and achieves the uniformly minimum quadratic loss among all linear combinations of the identity matrix and the sample covariance matrix under the high-dimensional case. Through extensive simulation studies, the proposed approach achieves good performance in identifying the covariate related components and estimating the model parameters. Applying to a longitudinal resting-state fMRI dataset from the Alzheimer's Disease Neuroimaging Initiative (ADNI), the proposed approach identifies brain networks that demonstrate the difference between males and females at different disease stages. The findings are in line with existing knowledge of AD and the method improves the statistical power over the analysis of cross-sectional data.