Abstract
Gaussian Graphical Models (GGMs) are extensively used in many research areas, such as genomics, proteomics, neuroimaging, and psychology, to study the partial correlation structure of a set of variables. This structure is visualized by drawing an undirected network, in which the variables constitute the nodes and the partial correlations the edges. In many applications, it makes sense to impose sparsity (i.e., some of the partial correlations are forced to zero) as sparsity is theoretically meaningful and/or because it improves the predictive accuracy of the fitted model. However, as we will show by means of extensive simulations, state-of-the-art estimation approaches for imposing sparsity on GGMs, such as the Graphical lasso, ℓ1 regularized nodewise regression, and joint sparse regression, fall short because they often yield too many false positives (i.e., partial correlations that are not properly set to zero). In this paper we present a new estimation approach that allows to control the false positive rate better. Our approach consists of two steps: First, we estimate an undirected network using one of the three state-of-the-art estimation approaches. Second, we try to detect the false positives, by flagging the partial correlations that are smaller in absolute value than a given threshold, which is determined through cross-validation; the flagged correlations are set to zero. Applying this new approach to the same simulated data, shows that it indeed performs better. We also illustrate our approach by using it to estimate (1) a gene regulatory network for breast cancer data, (2) a symptom network of patients with a diagnosis within the nonaffective psychotic spectrum and (3) a symptom network of patients with PTSD.
Highlights
Often, a sparse Gaussian Graphical Models (GGMs) is fitted, which implies that many of the partial correlations are forced to zero and that the corresponding edges in the network can be dropped
Recent studies on the use of 1 penalization in standard regression analysis have shown that it tends to yield too many non-zero regression weights[14,15,16]. Translating these results to the estimation of sparse GGMs, we expect regularized nodewise regression, SPACE and the Graphical lasso (Glasso) to often yield false positives, implying that some of the drawn edges should have been dropped. We will test this hypothesis in extensive simulations, in which we will evaluate the effect of the tuning approach
The results for Glasso are affected by the penalty tuning approach: whereas using cross-validation tends to introduce a large number of false positives across all different conditions, applying Extended Bayesian Information Criterion (EBIC) yields many false negatives
Summary
A sparse GGM is fitted, which implies that many of the partial correlations are forced to zero and that the corresponding edges in the network can be dropped. Recent studies on the use of 1 penalization in standard regression analysis have shown that it tends to yield too many non-zero regression weights[14,15,16] Translating these results to the estimation of sparse GGMs, we expect regularized nodewise regression, SPACE and the Glasso to often yield false positives, implying that some of the drawn edges should have been dropped. We will test this hypothesis in extensive simulations, in which we will evaluate the effect of the tuning approach (i.e., information criteria, k-fold cross validation or finite sample derivations). The Methods section presents a detailed description of the evaluated tuning approaches for each of the state-of-the-art estimation approaches and of the PCS procedure
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