Relations between networks, regression, partial correlation, and latent variable model

TL;DR

This paper clarifies the relationship between Gaussian graphical models, regression, and latent variables, demonstrating that partial correlations do not eliminate shared variance and that ULVMs produce fully connected networks.

Contribution

It establishes a theoretical connection between ULVMs and GGMs, proving that ULVMs correspond to fully connected networks, countering previous claims that partial correlations remove shared variance.

Findings

ULVMs are associated with fully connected networks.

Partial correlations do not remove shared variance.

Theoretical link between GGMs and linear regression.

Abstract

The Gaussian graphical model (GGM) has become a popular tool for analyzing networks of psychological variables. In a recent paper in this journal, Forbes, Wright, Markon, and Krueger (FWMK) voiced the concern that GGMs that are estimated from partial correlations wrongfully remove the variance that is shared by its constituents. If true, this concern has grave consequences for the application of GGMs. Indeed, if partial correlations only capture the unique covariances, then the data that come from a unidimensional latent variable model ULVM should be associated with an empty network (no edges), as there are no unique covariances in a ULVM. We know that this cannot be true, which suggests that FWMK are missing something with their claim. We introduce a connection between the ULVM and the GGM and use that connection to prove that we find a fully-connected and not an empty network associated with a ULVM. We then use the relation between GGMs and linear regression to show that the partial correlation indeed does not remove the common variance.