Moderated Network Models

Abstract

Pairwise network models such as the Gaussian Graphical Model (GGM) are a powerful and intuitive way to analyze dependencies in multivariate data. A key assumption of the GGM is that each pairwise interaction is independent of the values of all other variables. However, in psychological research, this is often implausible. In this article, we extend the GGM by allowing each pairwise interaction between two variables to be moderated by (a subset of) all other variables in the model, and thereby introduce a Moderated Network Model (MNM). We show how to construct MNMs and propose an ${\ell }_1$-regularized nodewise regression approach to estimate them. We provide performance results in a simulation study and show that MNMs outperform the split-sample based methods Network Comparison Test (NCT) and Fused Graphical Lasso (FGL) in detecting moderation effects. Finally, we provide a fully reproducible tutorial on how to estimate MNMs with the R-package mgm and discuss possible issues with model misspecification.

Keywords:

• Network models
• moderation
• higher-order interactions

Notes

1 $(\begin{array}{c}n\\ k\end{array})=\frac{n!}{k!(n-k)!},$ the $k^{\text{th}}$ binomial coefficient of the polynomial expansion of ${(1+x)}^n.$

2 We do not allow the initial six edges to be connected to the moderator, because otherwise for some graph configurations (1) moderation effects can turn into quadratic effects and (2) unmoderated pairwise interactions can turn into moderated pairwise interactions.

3 We use the implementation of the algorithm of Higham (2002) in the R-package Matrix (Bates & Maechler, 2017).