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Constructing a Covariance Matrix that Yields a Specified Minimizer and a Specified Minimum Discrepancy Function Value

Published online by Cambridge University Press:  01 January 2025

Robert Cudeck  and
Michael W. Browne
Robert Cudeck*
Affiliation:
University of Minnesota
Michael W. Browne*
Affiliation:
Ohio State University
*
Requests for reprints or a copy of the computer program that implements this method should be sent to Robert Cudeck, Department of Psychology, University of Minnesota, 75 East River Road, Minneapolis, MN 55455,
Michael W. Browne, Departments of Psychology and Statistics, Ohio State University, Columbus, OH 43210.
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Abstract

A method is presented for constructing a covariance matrix Σ*0 that is the sum of a matrix Σ(γ0) that satisfies a specified model and a perturbation matrix,E, such that Σ*0=Σ(γ0) +E. The perturbation matrix is chosen in such a manner that a class of discrepancy functions F(Σ*0, Σ(γ0)), which includes normal theory maximum likelihood as a special case, has the prespecified parameter value γ0 as minimizer and a prespecified minimum δ A matrix constructed in this way seems particularly valuable for Monte Carlo experiments as the covariance matrix for a population in which the model does not hold exactly. This may be a more realistic conceptualization in many instances. An example is presented in which this procedure is employed to generate a covariance matrix among nonnormal, ordered categorical variables which is then used to study the performance of a factor analysis estimator.

Keywords

Monte Carlo experimentscovariance structure analysisfactor analysismodel misspecification
Type
Original Paper
Information
Psychometrika , Volume 57 , Issue 3 , September 1992 , pp. 357 - 369
Copyright
Copyright © 1992 The Psychometric Society

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Footnotes

We are grateful to Alexander Shapiro for suggesting the proof of the solution in section 2.

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