Network Inference With the Lasso

Figures & data

Figure 1. (a) Example of a network with four nodes 1, 2, 3 and 4 with associated variables $X_1,$ $X_2,$ $X_3,$ and $X_4.$ The edges $X_1-X_2,$ $X_1-X_3$ and $X_2-X_4$ correspond to the conditional dependence statements, and the absence of edges correspond to conditional independence statements: $X_2\perp \perp X_3|X_1,$ $X_1\perp \perp X_4|X_2$ and $X_3\perp \perp X_4|X_1,X_2.$ (b) The variable $X_4$ is not part of the neighborhood of $X_1,$ while $X_2$ and $X_3$ are part of the neighborhood of $X_1.$

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).

Table 1. The table provides an overview of the assumptions that are required for each of the three methods of obtaining p-values, the lasso, the graphical lasso (glasso) and OLS.

	Method
Assumptions	OLS	lasso	Multisplit	Desp lasso	Boot desp	Glasso
(a) sparsity	no	yes	yes	yes	yes	yes
(b) no collinearity	yes	yes	yes	yes	yes	yes
(c) incoherence	no	yes	yes	no	no	yes
(d) beta-min	no	yes	yes	no	no	yes

Table 2. Sparsity (for nodewise estimation) and beta-min violations for each of the settings in the simulation with probability of an edge $p_e=0.2$ or 0.4. Sparsity is in the second column for each number of observations (from 50 to 500) and $p=20$ nodes. In the third and fourth columns, the number of nodes in the network that violate the sparsity assumption. In the fifth column, is the heuristic beta-min value for each of the number of observations. Sixth and seventh columns are the percentage of edges that violate the beta-min assumption (i.e., are smaller than the beta-min value).

	$\#>$ Sparsity $p_e$			$\%<$ Beta-min $p_e$
n	Sparsity	0.2	0.4	Beta-min	0.2	0.4
50	4	9	20	0.25	22	29
100	5	4	18	0.17	17	17
200	8	1	9	0.12	2	6
500	12	0	0	0.08	0	0
800	16	0	0	0.06	0	0
1000	18	0	0	0.05	0	0