A method of moments technique for fitting interaction effects in structural equation models

Abstract

The desire to fit structural equation models containing an interaction term has received much methodological attention in the social science literature. This paper presents a technique for the cross-product structural model that utilizes factor score estimates and results in closed-form moments-type estimators. The technique, which does not require normality for the underlying factors, was originally introduced in a very general form by Wall and Amemiya (2000) for any polynomial structural model. In this paper, the practical implementation of this method, including standard error estimation, is presented specifically for the cross-product model. The procedure is applied to an example from social/behavioural epidemiology where the flexibility of the cross-product model provides a useful description of the underlying theory. A simulation study is also presented comparing the method of moments for the cross-product model with three other procedures.

On Insensitivity of the Chi-Square Model Test to Nonlinear Misspecification in Structural Equation Models

“…4 of the book of Lee (2007) that is devoted to this problem; see also the paper of Arminger and Muthén (1998). To mention a few other methods, Wall and Amemiya (2003) propose a two-stage method in which at the first stage the measurement parameters are estimated and the estimates are used for estimating the factor scores. On the basis of these factor scores, errors are estimated and then estimation of model parameters is carried out by using higher-order moments of these errors.…”

A Heterogeneous Growth Curve Model for Nonnormal Data

“…In contrast to predictor variables that are conventionally used to model interaction effects (e.g., in product indicator approaches or moment-based approaches; Jöreskog & Yang, 1996;Kelava & Brandt, 2009;Kenny & Judd, 1984;Marsh et al, 2004;Wall & Amemiya, 2003), there is no measurement model for the residual variable. As a consequence, most of the approaches for nonlinear structural equation modeling are not applicable.…”

A general non-linear multilevel structural equation mixture model

“…Numerous parametric approaches for the estimation of non-linear effects have been developed (for a review, see Schumacker and Marcoulides, 1998; Algina and Moulder, 2001; Marsh et al, 2004, 2006), including product indicator approaches (e.g., Kenny and Judd, 1984; Bollen, 1995; Jaccard and Wan, 1995; Ping, 1995; Jöreskog and Yang, 1996; Algina and Moulder, 2001; Marsh et al, 2004, 2006; Little et al, 2006; Kelava and Brandt, 2009), distribution analytic approaches (Klein and Moosbrugger, 2000; Klein and Muthén, 2007), Bayesian approaches (e.g., Arminger and Muthén, 1998; Lee et al, 2007), and method of moments based approaches (Wall and Amemiya, 2003; Mooijaart and Bentler, 2010). Whereas most product indicator approaches have been ad-hoc methods for the specification of non-linear interaction effects and have thus suffered from requiring complicated measurement models, recent distribution analytic and Bayesian approaches have tried to overcome the need for non-linear measurement models.…”

“…Whereas most product indicator approaches have been ad-hoc methods for the specification of non-linear interaction effects and have thus suffered from requiring complicated measurement models, recent distribution analytic and Bayesian approaches have tried to overcome the need for non-linear measurement models. Method-of-moments-based approaches (Wall and Amemiya, 2003; Mooijaart and Bentler, 2010) and some indicator approaches (Bollen, 1995; Jöreskog and Yang, 1996) have been proposed as methods that do not rely as heavily on the normality assumption of the latent variables as other approaches (e.g., the distribution analytic approaches). The relaxation of distributional assumptions may lead to a reduction in the threat of biased estimates for non-linear effects in situations in which data are non-normally distributed, but for most of these approaches, relaxing these assumptions is associated with a low power for detecting the effects (Schermelleh-Engel et al, 1998; Brandt et al, 2014).…”