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Circumplex Models for Correlation Matrices

Published online by Cambridge University Press:  01 January 2025

Michael W. Browne
Michael W. Browne*
Affiliation:
The Ohio State University
*
Requests for reprints should be sent to Michael W. Browne, Department of Psychology, The Ohio State University, 1885 Neil Avenue Mall, Columbus OH 43210, U.S.A.
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Abstract

Structural models that yield circumplex inequality patterns for the elements of correlation matrices are reviewed. Particular attention is given to a stochastic process defined on the circle proposed by T. W. Anderson. It is shown that the Anderson circumplex contains the Markov Process model for a simplex as a limiting case when a parameter tends to infinity.

Anderson's model is intended for correlation matrices with positive elements. A replacement for Anderson's correlation function that permits negative correlations is suggested. It is shown that the resulting model may be reparametrzed as a factor analysis model with nonlinear constraints on the factor loadings. An unrestricted factor analysis, followed by an appropriate rotation, is employed to obtain parameter estimates. These estimates may be used as initial approximations in an iterative procedure to obtain minimum discrepancy estimates.

Practical applications are reported.

Keywords

circumplexcovariance structurescorrelation structuresfactor analysissimplex
Type
Original Paper
Information
Psychometrika , Volume 57 , Issue 4 , December 1992 , pp. 469 - 497
Copyright
Copyright © 1992 The Psychometric Society

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Footnotes

Presented as the 1992 Psychometric Society Presidential Address. I am greatly indebted to Stephen Du Toit for help in the development of the computer program employed here. Part of this research was carried out at the University of South Africa and at the Institute for Statistical Research of the South African Human Sciences Research Council.

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