  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.
