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

Abstract

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; Friedrich, 1982). 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) to estimating latent interactions. These results support the use of the appropriate “standardized” solutions in interpreting and comparing SEM estimates of latent interactions.

Notes

¹As in many other statistical models, there are Bayesian approaches and non-Bayesian approaches in estimating latent interaction models. Although well developed (e.g., Arminger & Muthén, 1998; Lee et al., 2004), Bayesian approaches and their calculation algorithms are relatively difficult for applied researchers and thus, although available in the special WinBUGS software, these approaches have not been adopted for the most popular commercial SEM software.

²This linear constraint comes from κ₃ = E(ξ₁ξ₂) = cov(ξ₁, ξ₂) = Φ₂₁ and could help model convergence, but is not absolutely necessary for the unconstrained approach (see Marsh et al., 2004).

³For general bootstrap sampling, different sizes of the resamples are sometimes used. For our purpose of standard error calculation, each bootstrap sample has the same size as the original sample.