Academic journal article Canadian Journal of Behavioural Science

Analysis of Interaction Terms in Structural Equation Models: A Non-Technical Demonstration Using the Deviation Score Approach

Academic journal article Canadian Journal of Behavioural Science

Analysis of Interaction Terms in Structural Equation Models: A Non-Technical Demonstration Using the Deviation Score Approach

Article excerpt


One of the major shortcomings in using Structural Equation Modeling data analytic techniques has been the difficulty in handling interaction terms in the modeling process. The issue is that interaction terms that are created by cross-multiplying raw scores result in the matrix of covariances or correlations being singular (there is at least one linear dependency in the matrix). The data analyses will not proceed as the matrix is not positive definite. This paper demonstrates a valid and easy way to cope with the problem of interaction terms by using deviation scores or "centred" variables as the interaction terms.


Une des grandes lacunes de l'utilisation des techniques d'analyse des donnees bases sur la modelisation de l'equation structurelle est la difficulte de manier les termes d'interaction dans la modelisation. Le probleme est que les termes d'interaction crees par une multiplication croisee des notes brutes entrainent la singularite de la matrice des covariances ou correlations (il y a au moms une dependance lineaire dans la matrice). Les analyses de donnees ri auront pas lieu, car la matrice nest pas definie positive. Ce document expose un moyen valide et facile de regler le probleme des termes d'interaction, en utilisant des notes-ecarts ou des variables << centrees >> comme termes d'interaction. Within the context of traditional path analysis that uses a least squares regression approach, interaction effects (i.e., moderating variables) in models are not difficult to incorporate. This is due in large part to Aiken and West's (1991) readable exposition on centring the main effect variables prior to creating the interaction term. Once the centring occurs, the regression proceeds as usual. Extending the work of Aiken and West (1991) into the realm of Structural Equation Modeling (SEM) has not occurred and is thus not available for wide consumption and use by sEM program users. The purpose of this paper is to fill this gap. It is a pedagogical demonstration, using both an actual data set and a simulated set, to show how incorporating moderating variables into sENt programs can be easily and thus, hopefully, more frequently used. Given the large number of models in psychology that propose interactive effects, this tutorial should be useful to many readers.

The problem of dealing with interaction terms in structural equation models has been a source of aggravation for users of sENt programs (e.g., AMOS, cosAN, EQs, LISREL). The problem occurs if one creates an interaction variable by cross-multiplying raw scores of two original variables to create a third variable. First, the interactions are highly correlated with the raw score terms resulting in collinearity problems (e.g., Pedhazur, 1997), and the resulting matrix of covariances or correlations among the variables is now singular (there is a linear dependency in the variables). The programs will not analyze data where the matrix is singular as it is not positive definite. Jaccard and Wan (1995) also note that measurement error is exacerbated when product terms are created.

The literature suggests that the interaction term problem in SEM has been around for a while and is not readily tractable. For example, Ping (1995) suggested a procedure whereby single indicators are used to specify the interaction terms. Hayduk (1987, 1996) provides another alternative using an approach advocated by Kenny and Judd (1984). This technique entails constraining the estimation of a number of parameters. It is very cumbersome and easily prone to error in that convenience variables are created that estimate the covariances between observed variables. In addition, model fit is degraded due to the nonnormality introduced by specifying many product indicators. Ping (1996) noted these problems and provided a much more simplified approach to estimating interaction and quadratic effects using a two-step estimation procedure. …

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