A random covariance model for bi-level graphical modeling with application to resting-state fMRI data

This paper considers a novel problem, bi-level graphical modeling, in which multiple individual graphical models can be considered as variants of a common group-level graphical model and inference of both the group- and individual-level graphical models are of interest. Such problem arises from many applications including multi-subject neuroimaging and genomics data analysis. We propose a novel and efficient statistical method, the random covariance model, to learn the group- and individual-level graphical models simultaneously. The proposed method can be nicely interpreted as a random covariance model that mimics the random effects model for mean structures in linear regression. It accounts for similarity between individual graphical models, identifies group-level connections that are shared by individuals in the group, and at the same time infers multiple individual-level networks. Compared to existing multiple graphical modeling methods that only focus on individual-level networks, our model learns the group-level structure underlying the multiple individual networks and enjoys computational efficiency that is particularly attractive for practical use. We further define a measure of degrees-of-freedom for the complexity of the model that can be used for model selection. We demonstrate the asymptotic properties of the method and show its finite sample performance through simulation studies. Finally, we apply the proposed method to our motivating clinical data, a multi-subject resting-state functional magnetic resonance imaging (fMRI) dataset collected from schizophrenia patients.