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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
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.

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