Structural Equation Models of Latent Interactions: An Appropriate Standardized Solution and Its Scale-Free Properties

Standardized parameter estimates are routinely used to summarize the results of multiple regression models of manifest variables and structural equation models of latent variables, because they facilitate interpretation. Although the typical standardization of interaction terms is not appropriate for multiple regression models, straightforward alternatives are well known (Aiken & West, 1991 Aiken, L. S. and West, S. G. 1991. Multiple regression: Testing and interpreting interactions, Newbury Park, CA: Sage. [Crossref] , [Google Scholar]; Friedrich, 1982 Friedrich, R. J. 1982. In defense of multiplicative terms in multiple regression equations. American Journal of Political Science, 26: 797–833. [Crossref], [Web of Science ®] , [Google Scholar]). Whereas the analogous problem exists for the estimation of latent interactions in structural equation modeling (SEM), the problem is more complex and apparently has not been resolved. Here we demonstrate that the appropriate “standardized” parameter estimates are easily formulated from parameter estimates routinely available from existing SEM software packages. Some properties of the appropriate “standardized” solution are mathematically derived, including the demonstration that the main and interaction effects are scale-free, as are the factor loadings. These desirable properties of the standardized solution are illustrated with a simulation data set using the unconstrained approach (Marsh, Wen, & Hau, 2004 Marsh, H. W., Wen, Z. and Hau, K. T. 2004. Structural equation models of latent interactions: Evaluation of alternative estimation strategies and indicator construction. Psychological Methods, 9: 275–300. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to estimating latent interactions. These results support the use of the appropriate “standardized” solutions in interpreting and comparing SEM estimates of latent interactions.