Gaussian graphical models have become popular tools for identifying relationships between genes when analyzing microarray expression data. In the classical undirected Gaussian graphical model setting, conditional independence relationships can be inferred from partial correlations obtained from the concentration matrix (= inverse covariance matrix) when the sample size n exceeds the number of parameters p which need to estimated. In situations where n < p, another approach to graphical model estimation may rely on calculating unconditional (zero-order) and first-order partial correlations. In these settings, the goal is to identify a lower-order conditional independence graph, sometimes referred to as a ‘0-1 graphs’. For either choice of graph, model selection may involve a multiple testing problem, in which edges in a graph are drawn only after rejecting hypotheses involving (saturated or lower-order) partial correlation parameters. Most multiple testing procedures applied in previously proposed graphical model selection algorithms rely on standard, marginal testing methods which do not take into account the joint distribution of the test statistics derived from (partial) correlations. We propose and implement a multiple testing framework useful when testing for edge inclusion during graphical model selection. Two features of our methodology include (i) a computationally efficient and asymptotically valid test statistics joint null distribution derived from influence curves for correlation-based parameters, and (ii) the application of empirical Bayes joint multiple testing procedures which can effectively control a variety of popular Type I error rates by incorpo- rating joint null distributions such as those described here (Dudoit and van der Laan, 2008). Using a dataset from Arabidopsis thaliana, we observe that the use of more sophisticated, modular approaches to multiple testing allows one to identify greater numbers of edges when approximating an undirected graphical model using a 0-1 graph. Our framework may also be extended to edge testing algorithms for other types of graphical models (e.g., for classical undirected, bidirected, and directed acyclic graphs).
Bioinformatics | Biostatistics | Computational Biology | Statistical Methodology | Statistical Theory
Gilbert, Houston N.; van der Laan, Mark J.; and Dudoit, Sandrine, "Joint Multiple Testing Procedures for Graphical Model Selection with Applications to Biological Networks" (April 2009). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 245.