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On Equivariance and Invariance of Standard Errors in Three Exploratory Factor Models

Published online by Cambridge University Press:  02 January 2025

Ke-Hai Yuan  and
Peter M. Bentler
Ke-Hai Yuan*
Affiliation:
University of North Texas
Peter M. Bentler
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to Ke-Hai Yuan, Department of Psychology, University of North Texas, PO Box 311280, Denton TX 76203-1280.
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Abstract

Current practice in factor analysis typically involves analysis of correlation rather than covariance matrices. We study whether the standard z-statistic that evaluates whether a factor loading is statistically necessary is correctly applied in such situations and more generally when the variables being analyzed are arbitrarily rescaled. Effects of rescaling on estimated standard errors of factor loading estimates, and the consequent effect on z-statistics, are studied in three variants of the classical exploratory factor model under canonical, raw varimax, and normal varimax solutions. For models with analytical solutions we find that some of the standard errors as well as their estimates are scale equivariant, while others are invariant. For a model in which an analytical solution does not exist, we use an example to illustrate that neither the factor loading estimates nor the standard error estimates possess scale equivariance or invariance, implying that different conclusions could be obtained with different scalings. Together with the prior findings on parameter estimates, these results provide new guidance for a key statistical aspect of factor analysis.

Keywords

maximum likelihoodaugmented information matrixcanonical rotationvarimax rotationstandard errorscale equivariance
Type
Original Paper
Information
Psychometrika , Volume 65 , Issue 1 , March 2000 , pp. 121 - 133
Copyright
Copyright © 2000 The Psychometric Society

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Footnotes

We gratefully acknowledge the help of the Associate Editor and three referees whose constructive comments lead to an improved version of the paper. This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017 and by the University of North Texas Faculty Research Grant Program.

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