Longitudinal regression of covariance matrix outcomes

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 captures 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. These estimators are proved to be asymptotically consistent, where the proposed covariance matrix 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 functional magnetic resonance imaging data set from the Alzheimer's Disease (AD) Neuroimaging Initiative, 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.

Keywords: Covariance regression; Hierarchical likelihood; Multilevel model; Shrinkage estimator.

Fig. 1.

Estimation performance in estimating the parameters in the simulation study. For , (a) bias, (b) mean squared error (MSE), and (c) coverage probability (CP) from bootstrap samples. For , (e) similarity to . For , (e) bias and (f) MSE. Data dimension . Sample sizes vary from , , and .

Fig. 2.

The brain map of the five identified components in the ADNI analysis. (a), (c), (d), and (e) primarily show the cortical regions; (b) shows two subcortical regions, the left and right caudate nucleus.