Variable selection and regression analysis for graph-structured covariates with an application to genomics

Graphs and networks are common ways of depicting biological information. In biology, many different biological processes are represented by graphs, such as regulatory networks, metabolic pathways and protein--protein interaction networks. This kind of a priori use of graphs is a useful supplement to the standard numerical data such as microarray gene expression data. In this paper we consider the problem of regression analysis and variable selection when the covariates are linked on a graph. We study a graph-constrained regularization procedure and its theoretical properties for regression analysis to take into account the neighborhood information of the variables measured on a graph. This procedure involves a smoothness penalty on the coefficients that is defined as a quadratic form of the Laplacian matrix associated with the graph. We establish estimation and model selection consistency results and provide estimation bounds for both fixed and diverging numbers of parameters in regression models. We demonstrate by simulations and a real data set that the proposed procedure can lead to better variable selection and prediction than existing methods that ignore the graph information associated with the covariates.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS332 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Network-constrained Regularization and Variable Selection for Analysis of Genomic Data

Graphs or networks are common ways of depicting information. In biology in particular, many different biological processes are represented by graphs, such as regulatory networks or metabolic pathways. This kind of {\it a priori} information gathered over many years of biomedical research is a useful supplement to the standard numerical genomic data such as microarray gene expression data. How to incorporate information encoded by the known biological networks or graphs into analysis of numerical data raises interesting statistical challenges. In this paper, we introduce a network-constrained regularization procedure for linear regression analysis in order to incorporate the information from these graphs into an analysis of the numerical data, where the network is represented as a graph and its corresponding Laplacian matrix. We define a network-constrained penalty function that penalizes the $L_1$-norm of the coefficients but encourages smoothness of the coefficients on the network. An efficient algorithm is also proposed for computing the network-constrained regularization paths, much like the Lars algorithm does for the lasso. We illustrate the methods using simulated data and analysis of a microarray gene expression data set of glioblastoma

A Network-constrained Empirical Bayes Method for Analysis of Genomic Data

Empirical Bayes methods are widely used in the analysis of microarray gene expression data in order to identify the differentially expressed genes or genes that are associated with other general phenotypes. Available methods often assume that genes are independent. However, genes are expected to function interactively and to form molecular modules to affect the phenotypes. In order to account for regulatory dependency among genes, we propose in this paper a network-constrained empirical Bayes method for analyzing genomic data in the framework of general linear models, where the dependency of genes is modeled by a discrete Markov random field model defined on a pre-defined biological network. This method provides a statistical framework for integrating the known biological network information into the analysis of genomic data. We present an iterated conditional mode algorithm for parameter estimation and for estimating the posterior probabilities using Gibbs sampling. We demonstrate the application of the proposed methods using simulations and analysis of a human brain aging microarray gene expression data set