Products of Variables in Structural Equation Models

Abstract

A general method is introduced in which variables that are products of other variables in the context of a structural equation model (SEM) can be decomposed into the sources of variance due to the multiplicands. The result is a new category of SEM which we call a Products of Variables Model (PoV). Some useful and practical features of PoV models include the estimation of interactions between latent variables, latent variable moderators, manifest moderators with missing values, and manifest or latent squared terms. Expected means and covariances are analytically derived for a simple product of two variables and it is shown that the method reproduces previously published results for this special case. It is shown algebraically that using centered multiplicands results in an unidentified model, but if the multiplicands have non-zero means, the result is identified. The method has been implemented in OpenMx and Ωnyx and is applied in five extensive simulations.

Keywords:

• Latent interaction
• latent moderation
• missing data
• non-linear SEM

Acknowledgments

The authors would also like to acknowledge Robert M. Kirkpatrick for his helpful advice on the Monte Carlo simulations reported in this manuscript and the OpenMx development team for their support in user interface design.

Notes

1 Operators often take one or more elements from a set back onto that set. A unary operator maps one member of a set onto the set; whereas a binary operator maps two elements of a set onto that set, and a n-ary operator maps n elements onto that set.

Additional information

Funding

Funding for this work was provided in part by NIH Grant R01DA018673, the Max Planck Institute for Human Development, and a fellowship from the University of Zurich Research Priority Program on Dynamics of Healthy Aging. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Institutes of Health.