Unidimensional factor models imply weaker partial correlations than zero-order correlations

TL;DR

This paper demonstrates that in unidimensional factor models, the partial correlations between indicators are always weaker (closer to zero) than their zero-order correlations, clarifying the relationship between observed variables and latent factors.

Contribution

It provides a formal proof that partial correlations in unidimensional factor models are always weaker than zero-order correlations, clarifying their theoretical relationship.

Findings

Partial correlations are always closer to zero than zero-order correlations.

Conditional independence assumptions imply weaker partial correlations.

Theoretical proof of the relationship between partial and zero-order correlations.

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.