Discovering Interaction in Structural Equation Models through a Diagnostic Plot
Giovanni C. Porzio1, Maria P. Vitale2
1Department of Economics, University of Cassino, Cassino, Frosinone, Italy; 2Department of Economics and Statistics, University of Salerno, Fisciano, Salerno, Italy

Structural Equation Models are increasingly applied in many research areas in order to analyze causal relationships between theoretical latent concepts. Within these models, linear linkages between variables are generally assumed. Nevertheless, this assumption may not adequately describe the complexity and richness of social phenomena.

For this reason, models including interaction effects between latent variables have been developed, with a large literature on methods for their estimation (see e.g.: Schumacker and Marcoulides, 2005). However, few efforts have been devoted to the development of adequate diagnostic tools.

Aim of this work is to introduce thus a graphical device, the latent joint effect plot, that allows users to evaluate the presence of interaction effects between latent variables. The plot is in line with the graphical diagnostic methods used in the framework of multiple linear regression (Cook and Weisberg, 1999). In particular, it belongs to the framework of the conditional plot, and mimics the joint effect plot introduced in path analysis by Porzio and Vitale (2007).

However, plots for diagnostics in multiple regression are based on the visualization of observed variables. Here, given that we are dealing with latent variables, factor scores are plotted. It is worth noting that, in spite of the well known factor score indeterminacy issue (Grice, 2001), the plot seems to be effective, as shown by a proper simulation study. By varying sample size, reliability, and interaction effect size, we evaluate how the choice of a different factor score estimation method affects the plot efficacy. The study shows that the proposed plot has some efficacy beyond the factor score estimation method one may select. On the other hand, the same study suggests that the factor scores estimated according to the Krijnen proposal (Krijnen et al., 1996) should be generally preferred over the others.


Cook, R.D., Weisberg, S. (1999). Applied Regression Including Computing and Graphics. New York: John Wiley & Sons.

Grice, J.W. (2001). Computing and evaluating factor scores. Psychological Methods 6: 430-450.

Krijnen, W.P., Wansbeek, T.J., Ten Berge J.M.F. (1996). Best linear predictors for factor scores. Communications in Statistics: Theory and Methods 25: 3013-3025.

Porzio, G.C., Vitale, M.P. (2007). Exploring nonlinearities in path models. Quality & Quantity 41: 937-954.

Schumacker, R.E., Marcoulides, G.A. (eds) (2005). Interaction and Nonlinear Effects in Structural Equation Modeling, edited by Mahwah, NJ: Lawrence Erlbaum Associates.

Keywords: graphical device; interaction; simulation study; structural equation models

Biography: Since 2005 Assistant Professor of Social Statistics, Faculty of Humanities, University of Salerno (Italy).

B.A. in Sociology, University of Salerno (1998). Ph.D. in Statistics, University of Chieti “G. D'Annunzio” (Italy) (2002). Research Fellow (2002-2004), University of Salerno.

International courses and summer schools attended on: Lisrel Model (2000- ICPSR University of Michigan), Logit and Probit Models, SEM and Longitudinal Data (2005- ECPR University of Essex), Network Analysis (2007- ECPR University of Ljubljana) and Advanced Methods for Network Analysis (2008- University of Lisbona). Visiting researcher at the University of Ljubljana (November-December 2008).

Teaching experience: Statistics, Indicators for the analysis of socio-demographic phenomena and Statistical Methods for evaluation of public policies.

Member of National Research Program “G. D'Annunzio” (2010-2012).

The research interests are: Analysis of survey data questionnaire, Structural Equations Models, Social Network Analysis.