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Standard Errors for Rotated Factor Loadings

Published online by Cambridge University Press:  01 January 2025

Claude O. Archer  and
Robert I. Jennrich
Claude O. Archer
Affiliation:
Naval Missile Center
Robert I. Jennrich
Affiliation:
University of California at Los Angeles and Educational Testing Service
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Abstract

Beginning with the results of Girshick on the asymptotic distribution of principal component loadings and those of Lawley on the distribution of unrotated maximum likelihood factor loadings, the asymptotic distribution of the corresponding analytically rotated loadings is obtained. The principal difficulty is the fact that the transformation matrix which produces the rotation is usually itself a function of the data. The approach is to use implicit differentiation to find the partial derivatives of an arbitrary orthogonal rotation algorithm. Specific details are given for the orthomax algorithms and an example involving maximum likelihood estimation and varimax rotation is presented.

Type
Original Paper
Information
Psychometrika , Volume 38 , Issue 4 , December 1973 , pp. 581 - 592
Copyright
Copyright © 1973 The Psychometric Society

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Footnotes

*

This research was supported in part by NIH Grant RR-3. The authors are grateful to Dorothy T. Thayer who implemented the algorithms discussed here as well as those of Lawley and Maxwell. We are particularly indebted to Michael Browne for convincing us of the significance of this work and for helping to guide its development and to Harry H. Harman who many years ago pointed out the need for standard errors of estimate.

References

Anderson, T. W. and Rubin, H. Statistical inference in factor analysis. Proc. Third Berkeley Symp. Math. Statist. Probab., 1956, 5, 111–150Google Scholar
Girshick, M. A. On the sampling theory of roots of determinantal equations. Annals of Mathematical Statistics, 1939, 10, 203–224CrossRefGoogle Scholar
Harman, H. H. Modern factor analysis, 1967, Chicago: University of Chicago PressGoogle Scholar
Jöreskog, K. G. A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 1969, 34, 183–202CrossRefGoogle Scholar
Lawley, D. N. A modified method of estimation in factor analysis and some large sample results. Uppsala Symposium on Psychological Factor Analysis. Stockholm: Almqvist and Wiksell. 1953, 35–42Google Scholar
Lawley, D. N. Some new results in maximum likelihood factor analysis. Proceedings of the Royal Society of Edinburgh, 1967, 67A, 256–264Google Scholar
Lawley, D. N. and Maxwell, A. E. Factor analysis as a statistical method, 1971, New York: American ElsevierGoogle Scholar
Pennell, R. Routinely computable confidence intervals for factor loadings using the “jack-knife”. British Journal of Mathematical and Statistical Psychology, 1972, 25, 107–114CrossRefGoogle Scholar
Rao, C. R. Linear statistical inference, 1965, New York: WileyGoogle Scholar
Wexler, N. An evaluation of an asymptotic formula for factor loading variance by random methods. Unpublished doctoral dissertation. Rutgers University, 1968.Google Scholar