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Equivalent Models in Covariance Structure Analysis

Published online by Cambridge University Press:  01 January 2025

Thom C. W. Luijben
Thom C. W. Luijben*
Affiliation:
University of Groningen
*
Requests for reprints can be sent to Thom C. W. Luijben, PTT Post, Department of Quantitative Support, Room H691, PO Box 30250, 2500 GG The Hague, THE NETHERLANDS.
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Abstract

Defining equivalent models as those that reproduce the same set of covariance matrices, necessary and sufficient conditions are stated for the local equivalence of two expanded identified models M1 and M2 when fitting the more restricted model M0. Assuming several regularity conditions, the rank deficiency of the Jacobian matrix, composed of derivatives of the covariance elements with respect to the union of the free parameters of M1 and M2 (which characterizes model M12), is a necessary and sufficient condition for the local equivalence of M1 and M2. This condition is satisfied, in practice, when the analysis dealing with the fitting of M0, predicts that the decreases in the chi-square goodness-of-fit statistic for the fitting of M1 or M2, or M12 are all equal for any set of sample data, except on differences due to rounding errors.

Keywords

equivalent modelscovariance structure analysisLISRELEQSmodel modificationmodification indexLagrange multiplier statisticidentificationregular point

Information

Type
Original Paper
Information
Psychometrika , Volume 56 , Issue 4 , December 1991 , pp. 653 - 665
Copyright
Copyright © 1991 The Psychometric Society

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Footnotes

This research was supported by the Foundation of Social-Cultural Sciences which is subsidized by the Dutch Scientific Organization (N.W.O.) under project number 500-278-003. The author wishes to thank Anne Boomsma, Ivo Molenaar, Albert Satorra, and Tom Snijders for their stimulating and crucial comments during the research, and the Editor, Paul Bekker, Henk Broer, and anonymous reviewers for their helpful suggestions.

References

Apostol, T. M. (1957). Mathematical analysis, Reading, MA: Addison-Wesley.Google Scholar
Bekker, P. A., Pollock, D. S. (1986). Identification of linear stochastic models with covariance restrictions. Journal of Econometrics, 31, 179–208.CrossRefGoogle Scholar
Bentler, P. M. (1986). Lagrange multiplier and Wald statistic for EQS and EQS/PC, Los Angeles: BMDP Statistical Software.Google Scholar
Bentler, P. M. (1989). EQS structural equations program manual, Los Angeles: BMDP Statistical Software.Google Scholar
Bröcker, T., Jänich, K. (1973). Einführung in der differentialtopologie [Introduction to differential topology], Berlin: Springer-Verlag.CrossRefGoogle Scholar
Brouwer, L. E. J. (1911). Beweis der invarianz der dimensionenzahl [Proof of the invariance of the number of dimensions]. Mathematische Annalen, 70, 161–165.CrossRefGoogle Scholar
Burkill, F. R. S., Burkill, H. (1970). A second course in mathematical analysis, Cambridge: Cambridge University Press.Google Scholar
Cox, D. R., Hinkley, D. V. (1974). Theoretical statistics, London: Chapman & Hall.CrossRefGoogle Scholar
Dijkhuizen, A. A. (1978). LISREL-procedure. Rotterdamse mededelingen nr.34, Rotterdam: University of Rotterdam.Google Scholar
Fisher, F. M. (1966). The identification problem in econometrics, New York: McGraw-Hill.Google Scholar
Hayduk, L. A. (1987). Structural equation modeling with LISREL, Baltimore and London: University Press.Google Scholar
Hurwicz, L. (1950). Generalization of the concept of identification. In Koopmans, T. C. (Eds.), Statistical inference in dynamic economic models (pp. 245–257). New York: Wiley.Google Scholar
Jöreskog, K. G., Sörbom, D. (1988). LISREL 7: A guide to the program and applications, Chicago: SPSS.Google Scholar
Lee, S.-Y., Bentler, P. M. (1980). Some asymptotic properties of constrained generalized least squares estimation in covariance structure models. South African Statistical Journal, 14, 121–136.Google Scholar
Lipschutz, S. (1965). General topology, New York: McGraw-Hill.Google Scholar
Luijben, T. C. W. (1989). Statistical guidance for model modification in covariance structure analysis, Amsterdam: Sociometric Research Foundation.Google Scholar
Luijben, T. C., Boomsma, A., Molenaar, I. W. (1988). Modification of a factor analysis model in covariance structure analysis: A Monte Carlo study. In Dijkstra, T. K. (Eds.), On model uncertainty and its statistical implications, Berlin: Springer-Verlag.Google Scholar
MacCallum, R. (1986). Specification searches in covariance structure modeling. Psychological Bulletin, 100, 107–120.CrossRefGoogle Scholar
Magnus, J. R., Neudecker, H. (1988). Matrix differential calculus, Chichester: Wiley.Google Scholar
McDonald, R. P. (1978). A simple comprehensive model for the analysis of covariance structures. The British Journal of Mathematical and Statistical Psychology, 31, 59–72.CrossRefGoogle Scholar
Rudin, W. (1964). Principles of mathematical analysis, New York: McGraw-Hill.Google Scholar
Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified Approach. Psychometrika, 54, 131–151.CrossRefGoogle Scholar
Seidel, G., Eicheler, C. (1990). Identification structure of linear structural models. Quality and Quantity, 24, 345–366.CrossRefGoogle Scholar
Sen, P. K. (1979). Asymptotic properties of maximum likelihood estimators based on conditional specification. The Annals of Statistics, 7, 1019–1033.CrossRefGoogle Scholar
Shapiro, A. (1986). Asymptotic theory of overparameterized structural models. Journal of American Statistical Association, 81, 142–149.CrossRefGoogle Scholar
Shapiro, A., Browne, M. W. (1983). On the investigations of local identifiability: A counterexample. Psychometrika, 48, 303–304.CrossRefGoogle Scholar
Silvey, S. D. (1959). The Lagrangian multiplier test. The Annals of Mathematical Statistics, 30, 389–407.CrossRefGoogle Scholar
Sörbom, D. (1989). Model modification. Psychometrika, 54, 371–384.CrossRefGoogle Scholar
Stelzl, I. (1986). Changing a causal hypothesis without changing the fit; some rules for generating equivalent path models. Multivariate Behavioral Research, 21, 309–331.CrossRefGoogle ScholarPubMed
Wald, A. (1950). Note on the identification of economic relations. In Koopmans, T. C. (Eds.), Statistical inference in dynamic economic models (pp. 238–244). New York: Wiley.Google Scholar