Assessing linearity in structural equation models through graphics

Abstract

While Structural Equation Models (SEMs) generally assume linear linkages between variables, it is a well-worn issue that this may not adequately describe the complexity and richness of social phenomena. For this reason, nonlinear SEMs that include interaction effects between latent factors have been developed. However, while a large literature is available on methods for their estimation, few efforts have been devoted to the development of adequate diagnostic tools. In particular, the use of graphics has been rather limited so far, probably because of the partial information provided by the SEM residuals. Hence, with this paper we introduce a graphical device which aims to evaluate the SEM linearity assumption, without any previous estimation of nonlinear models. Specifically, we define a series of plots based on the individual latent variable scores in order to investigate nonlinear effects involving latent variables. In doing so, we also highlight the potential for graphical tools within SEM when factor scores for each individual in the sample are visualized. We call our graphical device the latent joint effect plot, as it displays the joint effect of two latent variables on some other response variable. The idea is presented through both simulated data and an illustrative example regarding the determinants that lead high school students to drop out of the Italian education system.