Network Inference With the Lasso

Abstract

Calculating confidence intervals and p-values of edges in networks is useful to decide their presence or absence and it is a natural way to quantify uncertainty. Since lasso estimation is often used to obtain edges in a network, and the underlying distribution of lasso estimates is discontinuous and has probability one at zero when the estimate is zero, obtaining p-values and confidence intervals is problematic. It is also not always desirable to use the lasso to select the edges because there are assumptions required for correct identification of network edges that may not be warranted for the data at hand. Here, we review three methods that either use a modified lasso estimate (desparsified or debiased lasso) or a method that uses the lasso for selection and then determines p-values without the lasso. We compare these three methods with popular methods to estimate Gaussian Graphical Models in simulations and conclude that the desparsified lasso and its bootstrapped version appear to be the best choices for selection and quantifying uncertainty with confidence intervals and p-values.

Keywords: bootstrap; debiased lasso; desparsified lasso; multisplit method; p-values for lasso.

Figure 1.

(a) Example of a network with four nodes 1, 2, 3 and 4 with associated variables $X1,$ $X2,$ $X3,$ and $X4.$ The edges $X1-X2,$ $X1-X3$ and $X2-X4$ correspond to the conditional dependence statements, and the absence of edges correspond to conditional independence statements: X2 ⊥⊥X3|X1, $X1\perp \perp X4|X2$ and $X3\perp \perp X4|X1,X2.$ (b) The variable $X4$ is not part of the neighborhood of $X1,$ while $X2$ and $X3$ are part of the neighborhood of $X1.$

Figure 2.

Top panel: bootstrapped sampling distributions of lasso estimates of partial correlations $\rho =0,0.2,0.5.$ Bottom panel: the corresponding bootstrapped sampling distributions based on unbiased least squares estimates. The dashed vertical line indicates the true parameter.

Figure 3.

Sensitivity (first row), precision (second row), and coverage (third row) for edge probabilities 0.2 (left column) and 0.4 (right column), as a function of sample size (x-axis). The dashed horizontal line at 0.95 in the last row indicates the desired coverage based on the chosen threshold $\alpha =0.05.$

Figure 4.

Significant partial correlations in the PTSD data; blue/red edges correspond to positive/negative partial correlations; the width of edges is a function of the absolute value of the associated parameter.

Figure 5.

The point estimates and confidence intervals of the 50 partial correlations with the largest point estimate. Hollow points represent estimates whose CIs overlap with zero.

Figure I.1.

Specificity (first row), coverage for absent edges (second row), and coverage for present edges (third row) for edge probabilities 0.2 (left column) and 0.4 (right column), as a function of sample size (x-axis).