Unidimensional factor models imply weaker partial correlations than zero-order correlations
Riet van Bork, Raoul P. P. P. Grasman, Lourens J. Waldorp
TL;DR
The paper addresses how, in a unidimensional factor model, partial correlations among indicators are constrained relative to zero-order correlations. It formalizes the model, derives the implications for covariance structure, and then proves that for any set of Gaussian indicators loading on a single factor, the partial correlation between two indicators, conditioned on all other indicators loading on the same factor, is always closer to zero than their zero-order correlation. The authors provide both a theoretical proof (via the partial-covariance formula and a Sherman–Morrison-based argument) and a practical formula for computing model-implied partial correlations from the inverse covariance (precision) matrix. These results yield a simple, model-based way to assess partial relationships and aid in model diagnostics, including interpreting vanishing tetrads and evaluating local independence assumptions in unidimensional factor models.
Abstract
In a unidimensional factor model it is assumed that the set of indicators that loads on this factor are conditionally independent given the latent factor. Two indicators are, however, never conditionally independent given (a set of) other indicators that load on this factor, as this would require one of the indicators that is conditioned on to correlate one with the latent factor. Although partial correlations between two indicators given the other indicators can thus never equal zero (Holland and Rosenbaum, 1986), we show in this paper that the partial correlations do need to be always weaker than the zero-order correlations. More precisely, we prove that the partial correlation between two observed variables that load on one factor given all other observed variables that load on this factor, is always closer to zero than the zero-order correlation between these two variables.