Advanced nonlinear latent variable modeling: Distribution analytic LMS and QML estimators of interaction and quadratic effects

Augustin Kelava, Christina S. Werner, Karin Schermelleh-Engel, Helfried Moosbrugger, Dieter Zapf, Yue Ma, Heining Cham, Leona S. Aiken, Stephen West

Research output: Contribution to journalArticlepeer-review

122 Scopus citations


Interaction and quadratic effects in latent variable models have to date only rarely been tested in practice. Traditional product indicator approaches need to create product indicators (e.g., x21, x1x4) to serve as indicators of each nonlinear latent construct. These approaches require the use of complex nonlinear constraints and additional model specifications and do not directly address the nonnormal distribution of the product terms. In contrast, recently developed, easy-to-use distribution analytic approaches do not use product indicators, but rather directly model the nonlinear multivariate distribution of the measured indicators. This article outlines the theoretical properties of the distribution analytic Latent Moderated Structural Equations (LMS; Klein & Moosbrugger, 2000) and Quasi-Maximum Likelihood (QML; Klein & Muthén, 2007) estimators. It compares the properties of LMS and QML to those of the product indicator approaches.A small simulation study compares the two approaches and illustrates the advantages of the distribution analytic approaches as multicollinearity increases, particularly in complex models with multiple nonlinear terms. An empirical example from the field of work stress applies LMS and QML to a model with an interaction and 2 quadratic effects. Example syntax for the analyses with both approaches is provided.

Original languageEnglish (US)
Pages (from-to)465-491
Number of pages27
JournalStructural Equation Modeling
Issue number3
StatePublished - Jul 2011


  • Estimators
  • Interaction
  • Multicollinearity
  • Nonlinear structural equation models
  • Quadratic

ASJC Scopus subject areas

  • Decision Sciences(all)
  • Modeling and Simulation
  • Sociology and Political Science
  • Economics, Econometrics and Finance(all)


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