Learning directed acyclic graphical structures with genetical genomics data

Motivation: Large amount of research efforts have been focused on estimating gene networks based on gene expression data to understand the functional basis of a living organism. Such networks are often obtained by considering pairwise correlations between genes, thus may not reflect the true connectivity between genes. By treating gene expressions as quantitative traits while considering genetic markers, genetical genomics analysis has shown its power in enhancing the understanding of gene regulations. Previous works have shown the improved performance on estimating the undirected network graphical structure by incorporating genetic markers as covariates. Knowing that gene expressions are often due to directed regulations, it is more meaningful to estimate the directed graphical network. Results: In this article, we introduce a covariate-adjusted Gaussian graphical model to estimate the Markov equivalence class of the directed acyclic graphs (DAGs) in a genetical genomics analysis framework. We develop a two-stage estimation procedure to first estimate the regression coefficient matrix by l(1) penalization. The estimated coefficient matrix is then used to estimate the mean values in our multi-response Gaussian model to estimate the regulatory networks of gene expressions using PC-algorithm. The estimation consistency for high dimensional sparse DAGs is established. Simulations are conducted to demonstrate our theoretical results. The method is applied to a human Alzheimer's disease dataset in which differential DAGs are identified between cases and controls. R code for implementing the method can be downloaded at http://www.stt.msu.edu/similar to cui.