Multiple Group Structural Equation Modeling

Multiple group structural equation modeling (MG-SEM) is a structural equation model (SEM) analysis extended to more than one set of data. The groupings may be preexisting (e.g., gender or race), or they may be imposed (e.g., assignment to treatment or control group, or groupings based on other measured variables). Recall, SEM is a form of latent variable analysis. A path analysis is an examination of regression equations in a (potentially complex) network of interconnected relationships. An SEM analysis is different from a path analysis in that the latter uses only manifest (or directly observable) variables, whereas the former allows for latent (or unobserved) variables to be modeled. Consequently, many SEM analyses involve both a structural model (the path model, if you will) and a measurement model. The measurement model is the means by which the influence of the latent variables on the overall relationship among all the data is parsed out and identified. This entry first reviews a basic example of model parameter equivalence (multiple regression). Next, the concept of measurement model equivalence is addressed. Finally, the complete stages of an MG-SEM analysis are presented.

To better understand MG-SEM, it is useful to review how group comparisons can be made in the multiple regression framework. As a quick review, consider a multiple regression model predicting a scalar dependent variable (e.g., math performance). If the multiple regression model includes a dichotomous independent variable (a grouping variable such as gender), a scalar independent variable (such as math anxiety), and the interaction of these two variables, then one can assess if the two separate bivariate regression models for each group are the same or not. That is, are the slopes and intercepts for each bivariate model the same for each group. In particular, the bivariate model would predict [Page 975]math performance from math anxiety. Obtaining the bivariate regression for each group would result in two different slopes. However, by running the multiple regression model with the three predictors (one dichotomous variable, one scalar variable, and the interaction), the interaction term can serve as a test as to whether or not the slopes are the same between the two groups. If they are the same (or if they are at least not statistically significantly different), this suggests that the slope part of the model is the same for the two groups. This is what might be referred to as model equivalence between the two groups.

With this idea of model equivalence in mind, the overarching importance of MG-SEM can be explained. As an SEM analysis can involve a structural part (the path model), it is reasonable to ask if the same model was run for two (or more) different groups, would any part (if not all) of the model be the same for the two groups. For each part of the model that is the same, it is possible to constrain the parameters being estimated in the model to be equivalent across the groups. (This is not dissimilar to the aforementioned multiple regression example when an analysis of covariance [ANCOVA] model is being run; in the ANCOVA model, the slopes for the different groups are constrained to be the same.)

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