Abstract
The model chi-square that is used in linear structural equation modeling compares the fitted covariance matrix of a target model to an unstructured covariance matrix to assess global fit. For models with nonlinear terms, i.e., interaction or quadratic terms, this comparison is very problematic because these models are not nested within the saturated model that is represented by the unstructured covariance matrix. We propose a novel measure that quantifies the heteroscedasticity of residuals in structural equation models. It is based on a comparison of the likelihood for the residuals under the assumption of heteroscedasticity with the likelihood under the assumption of homoscedasticity. The measure is designed to respond to omitted nonlinear terms in the structural part of the model that result in heteroscedastic residual scores. In a small Monte Carlo study, we demonstrate that the measure appears to detect omitted nonlinear terms reliably when falsely a linear model is analyzed and the omitted nonlinear terms account for substantial nonlinear effects. The results also indicate that the measure did not respond when the correct model or an overparameterized model were used.
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References
Arbuckle, J.L.: AMOS Users’ Guide Version 3.6. Small Waters Corporation, Chicago (1997)
Barendse, M.T., Oort, F.J., Garst, G.J.A.: Using restricted factor analysis with latent moderated structures to detect uniform and nonuniform measurement bias: A simulation study. Adv. Stat. Anal. (2010). doi:10.1007/s10182-010-0126-1
Bentler, P.M.: EQS Structural Equations Program Manual, Multivariate Software, Encino (1995)
Bentler, P.M., Wu, E.J.C.: EQS/Windows User’s Guide, BMDP Statistical Software, Los Angeles (1993)
Ganzach, Y.: Misleading interaction and curvilinear terms. Psychol. Methods 2, 235–247 (1997)
Geiser, C., Eid, M., Nussbeck, F.W., Courvoisier, D.S., Cole, D.A.: Multitrait-multimethod change modeling. Adv. Stat. Anal. (2010). doi:10.1007/s10182-010-0127-0
Jak, S., Oort, F.J., Garst, G.J.A.: Measurement bias and multidimensionality: An illustration of bias detection in multidimensional measurement models. Adv. Stat. Anal. (2010). doi:10.1007/s10182-010-0128-z
Jöreskog, K.G., Sörbom, D.: New Features in LISREL 8. Scientific Software, Chicago (1993)
Jöreskog, K.G., Sörbom, D.: LISREL 8: User’s Reference Guide. Scientific Software, Chicago (1996)
Jöreskog, K.G., Yang, F.: Nonlinear structural equation models: The Kenny-Judd model with interaction effects. In: Marcoulides, G.A., Schumacker, R.E. (eds.) Advanced Structural Equation Modeling, pp. 57–87. Erlbaum, Mahwah (1996)
King-Kallimanis, B.L., Oort, F.J., Garst, G.J.A.: Using structural equation modelling to detect measurement bias and response shift in longitudinal data. Adv. Stat. Anal. (2010). doi:10.1007/s10182-010-0129-y
Klein, A., Moosbrugger, H.: Maximum likelihood estimation of latent interaction effects with the LMS method. Psychometrika 65, 457–474 (2000)
Muthén, L.K., Muthén, B.O.: Mplus User’s Guide, 5th edn. Muthén & Muthén, Los Angeles (2007)
Schermelleh-Engel, K., Werner, C.S., Klein, A.G., Moosbrugger, H.: Nonlinear structural equation modeling: Is partial least squares an alternative? Stat. Anal. (2010). doi:10.1007/s10182-010-0132-3
Stuart, A., Ord, J.K., Arnold, S.: Kendall’s Advanced Theory of Statistics, 6th edn, vol. 5A. Oxford University Press, New York (1999)
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Klein, A.G., Schermelleh-Engel, K. Introduction of a new measure for detecting poor fit due to omitted nonlinear terms in SEM. AStA Adv Stat Anal 94, 157–166 (2010). https://doi.org/10.1007/s10182-010-0130-5
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DOI: https://doi.org/10.1007/s10182-010-0130-5