Joint Estimation of the Two-Level Gaussian Graphical Models Across Multiple Classes

Abstract

The Gaussian graphical model has been a popular tool for investigating the conditional dependency structure between random variables by estimating sparse precision matrices. The estimated precision matrices can be mapped into networks for visualization. However, the ability to investigate the conditional dependency structure when a two-level structure exists among the variables is still limited. Some variables are considered as higher-level variables while others are nested in these higher-level variables—the latter are called lower-level variables. For instance, genes are grouped into pathways for particular functions, so that pathways are the higher-level variables and genes within the pathways are the lower-level variables. Higher-level variables are not isolated; instead, they work together to accomplish certain tasks. Therefore, our main interest is to simultaneously explore conditional dependency structures among higher-level variables and among lower-level variables. Given two-level data from heterogeneous classes, we propose a method to jointly estimate the two-level Gaussian graphical models across multiple classes, so that common structures in terms of the two-level conditional dependency are shared during the estimation procedure, yet unique structures for each class are retained as well. Our proposed approach is achieved by first introducing higher-level variable factors within classes and then introducing common factors across classes. The performance of our approach is evaluated on several simulated networks. We also demonstrate the advantages of our approach using breast cancer patient data. Supplementary materials for this article are available online.