A note on testing perfect correlations in SEM

This article is concerned with the seemingly simple problem of testing whether latent factors are perfectly correlated (i.e., statistically indistinct). In recent literature, researchers have used different approaches, which are not always correct or complete. We discuss the parameter constraints required to obtain such perfectly correlated latent factors in the context of 4 commonly used models: (a) the oblique factor model, (b) the hierarchical factor model, (c) models in which the factors are predicted by a covariate, and (d) models in which the factors are predictors of a dependent variable. It is shown that the necessary constraints depend on the choice of scaling. We illustrate testing the indistinctiveness of factors with 2 real data examples.