# Seminar in Psychometrics

**Old and new approaches for including interaction and quadratic terms in structural equation modeling**

**Date and time: **February 27, 2023 (4:00 PM CET)

**Place** ICS CAS room 318, Pod Vodárenskou věží 2, Prague 8.

**Abstract.** In a linear regression model with multiple predictors, it is trivial to extend
the model by adding interaction terms or quadratic terms. But when the
predictors are latent variables, this is far from trivial. In this
presentation, I will give an overview of several approaches that have been
proposed in the literature to extend linear structural equation models (SEMs)
to so-called nonlinear structural equation models (NSEMs) that include
quadratic or interaction terms involving latent variables.

I will discuss the latent moderated structural equations approach (LMS; Klein &
Moosbrugger, 2000), the nonlinear structural equation mixture approach (NSEMM;
Kelava & Brandt, 2014), and several variants of the product indicator (PI)
approach (Kenny & Judd, 1984; Marsh, Wen, & Hau, 2004). All these methods use a
system-wide estimation approach and estimate the free parameters of the model
simultaneously. This is in contrast to the 2-stage method of moments estimator
(2SMM; Wall & Amemiya, 2003) where factor scores are computed in a first stage,
and an errors-in-variable regression approach is used in the second stage. In
this presentation, I will also propose an alternative approach that is similar
in spirit to 2SMM, but where we avoid the explicit calculation of factor
scores. The approach builds on the (local) structural-after-measurement (SAM)
approach that was recently proposed by Rosseel & Loh (2022).

**References.**

Klein, A.G. & Moosbrugger, H. (2000). Maximum likelihood estimation of latent
interaction effects with the LMS method. Psychometrika, 65, 457-474.

Kelava, A. & Brandt, H. (2014). A general nonlinear multilevel structural
equation mixture model. Frontiers in Psychology, 5, 748.

Kenny, D. Judd, C.M. (1984). Estimating the nonlinear and interactive
effects of latent variables. Psychological Bulletin 96, 201-210.

Marsh, H.W., Wen, Z., & Hau, K.-T. (2004). Structural equation models of
latent interactions: evaluation of alternative estimation strategies and
indicator construction. Psychological Methods 9, 275-300.

Wall, M.M. & Amemiya, Y. (2000). Estimation for polynomial structural
equation models. Journal of the American Statistical Association, 95,
929-940.

Rosseel, Y. & Loh, W.W. (2022). A structural after measurement (SAM) approach
to structural equation modeling. Psychological Methods. Advance online
publication. https://dx.doi.org/10.1037/met0000503

**Yves Rosseel**

Ghent University

Yves Rosseel obtained his PhD from Ghent University, Belgium. He is now a full professor at the Department of Data Analysis, Faculty of Psychology and Educational Sciences, Ghent University. He is the developer of an open-source software package for structural equation modeling: the R package `lavaan' (https://lavaan.org). His main research interest today is structural equation modeling.