Diagnostic procedures for detecting nonlinear relationships between latent variables

Abstract

Structural equation models are commonly used to estimate relationships between latent variables. Almost universally, the fitted models specify that these relationships are linear in form. This assumption is rarely checked empirically, largely for lack of appropriate diagnostic techniques. This article presents and evaluates two procedures that can be used to visualize and detect nonlinear relationships between latent variables. The first procedure involves fitting a linear structural equation model and then inspecting plots of factor score estimates for evidence of nonlinearity. The second procedure is to use a mixture of linear structural equation models to approximate the underlying, potentially nonlinear function. Targeted simulations indicate that the first procedure is more efficient, but that the second procedure is less biased. The mixture modeling approach is recommended, particularly with medium to large samples.