Parts and Wholes: A Note on Interpretation of Partial Covariance Matrices and Latent Variable Models

In a discussion of partial covariance matrices, Gaussian graphical models (GGMs), and unidimensional latent variable models (ULVMs), Waldorp and Marsman (2021) make a claim that “for the ULVM, observed partial correlations would all be positive... [proving] that the GGM applied to data coming from a ULVM will be fully-connected and not empty.” In a note, we show that although this is technically true, it is misleading, as with a ULVM parts of the partial covariance matrix corresponding to less informative indicators will in fact become approximately empty as other indicator variables become increasingly informative about the latent variable. We note that in this way, even though in their entirety covariance and concentration matrices are statistically equivalent, interpretations of their elements are not. We discuss interpretation of partial covariance matrices under random and incomplete samplings of observed variables, which is the norm in the behavioral sciences.