Multilevel Gaussian Graphical Model for Multilevel Networks

Gaussian graphical models have become a popular tool to represent networks among variables such as genes. They use the conditional correlations from the joint distribution to describe the dependencies between gene pairs, and employ the precision matrix of the genes. Because of the sparse nature of the gene networks and small sample sizes in high dimensional genetic data, regularization approaches attracted much attention in aim at obtaining the shrinkage estimates of the precision matrix. However, existing methods have been focused on the Gaussian graphical model among genes; that is, they are only applicable to a single level Gaussian graphical model. It is known that pathways are not independent of each other because of shared genes and interactions among pathways. Developing multipathway analysis has been a challenging problem because of the complex dependence structure among pathways. By considering the dependency among pathways as well as the genes within each pathway, we propose a multilevel Gaussian graphical model (MGGM) in which one level describes the networks for genes and the other for pathways. We have developed a multilevel L1 penalized likelihood approach to achieve the sparseness on both levels. In addition, we have developed an iterative weighted graphical LASSO algorithm for MGGM. Our simulation results supported the advantages of our approach; our method estimated the network more accurately on the pathway level and sparser on the gene level. We also demonstrated the usefulness of our approach using a canine genes-pathways data set.