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A Study of Algorithms for Covariance Structure Analysis with Specific Comparisons using Factor Analysis

Published online by Cambridge University Press:  01 January 2025

S. Y. Lee  and
R. I. Jennrich
S. Y. Lee*
Affiliation:
The Chinese University of Hong Kong
R. I. Jennrich
Affiliation:
University of California at Los Angeles
*
Requests for reprints should be sent to Dr. Sik-Yum Lee, Department of Mathematics, University of Science Centre, The Chinese University of Hong Kong, Shatin, N.T., HONG KONG.
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Abstract

Several algorithms for covariance structure analysis are considered in addition to the Fletcher-Powell algorithm. These include the Gauss-Newton, Newton-Raphson, Fisher Scoring, and Fletcher-Reeves algorithms. Two methods of estimation are considered, maximum likelihood and weighted least squares. It is shown that the Gauss-Newton algorithm which in standard form produces weighted least squares estimates can, in iteratively reweighted form, produce maximum likelihood estimates as well. Previously unavailable standard error estimates to be used in conjunction with the Fletcher-Reeves algorithm are derived. Finally all the algorithms are applied to a number of maximum likelihood and weighted least squares factor analysis problems to compare the estimates and the standard errors produced. The algorithms appear to give satisfactory estimates but there are serious discrepancies in the standard errors. Because it is robust to poor starting values, converges rapidly and conveniently produces consistent standard errors for both maximum likelihood and weighted least squares problems, the Gauss-Newton algorithm represents an attractive alternative for at least some covariance structure analyses.

Keywords

covariance structure analysisfactor analysisGauss-NewtonNewton-RaphsonFisher ScoringFletcher-PowellFletcher-Reevesmaximum likelihoodweighted least squaresHessian matrixFisher information matrix
Type
Original Paper
Information
Psychometrika , Volume 44 , Issue 1 , March 1979 , pp. 99 - 113
Copyright
Copyright © 1979 The Psychometric Society

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Footnotes

Work by the first author has been supported in part by Grant No. Da01070 from the U. S. Public Health Service. Work by the second author has been supported in part by Grant No. MCS 77-02121 from the National Science Foundation.

References

Reference Note

Bentler, P. M. & Lee, S. Y.Maximum likelihood factor analysis with unique solution. Unpublished manuscript, 1975.Google Scholar

References

Bentler, P. M. Multistructure statistical model applied to factor analysis. Multivariate Behavioral Research, 1976, 11, 3–25.CrossRefGoogle ScholarPubMed
Browne, M. W. Generalized least square estimators in the analysis of covariance structure. South African Statistical Journal, 1974, 8, 1–24.Google Scholar
Clarke, M. R. B. A rapidly convergent method for maximum likelihood factor analysis. British Journal of Mathematical and Statistical Psychology, 1970, 23, 43–52.CrossRefGoogle Scholar
Emmett, W. G. Factor analysis by Lawley's method of maximum likelihood. British Journal of Mathematical and Statistical Psychology, 1949, 2, 90–97.CrossRefGoogle Scholar
Fletcher, R. & Powell, M. J. D. A rapidly convergent descent method for minimization. Computer Journal, 1963, 6, 163–168.CrossRefGoogle Scholar
Fletcher, R. & Reeves, C. M. Function minimization by conjugate gradients. Computer Journal, 1964, 7, 149–154.CrossRefGoogle Scholar
Holzinger, K. J. & Swineford, F. A. A study in factor analysis: The stability of a bi-factor solution. Supplementary Educational Monographs, No. 48. University of Chicago: University of Chicago Press, 1939.Google Scholar
Jennrich, R. I. Simplified formulae for standard errors in maximum-likelihood factor analysis. British Journal of Mathematical and Statistical Psychology, 1974, 27, 122–131.CrossRefGoogle Scholar
Jennrich, R. I. & Moore, R. A. Maximum likelihood estimation by means of nonlinear least squares. Statistical Computing Section Proceedings of the American Statistical Association, 1975, 57–65.Google Scholar
Jennrich, R. I. Rotational equivalence of factor loading matrices with specified values. Psychometrika, 1978, 43, 421–426.CrossRefGoogle Scholar
Jöreskog, K. G. Some contributions to maximum likelihood factor analysis. Psychometrika, 1967, 32, 443–482.CrossRefGoogle Scholar
Jöreskog, K. G. A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 1969, 34, 183–202.CrossRefGoogle Scholar
Jöreskog, K. G. Analysis of covariance structures. In Krishnaiah, P. R.(Eds.), Multivariate analysis—III. New York: Academic Press. 1973, 263–285.CrossRefGoogle Scholar
Jöreskog, K. G. & Goldberger, A. S. Factor analysis by generalized least squares. Psychometrika, 1972, 37, 243–260.CrossRefGoogle Scholar
Thurstone, L. L. & Thurstone, T. G. Factorial studies of intelligence, 1941, Chicago: University of Chicago Press.Google Scholar