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.
Similar content being viewed by others
Reference note
Bentler, P. M. & Lee, S. Y.Maximum likelihood factor analysis with unique solution. Unpublished manuscript, 1975.
References
Bentler, P. M. Multistructure statistical model applied to factor analysis.Multivariate Behavioral Research, 1976,11, 3–25.
Browne, M. W. Generalized least square estimators in the analysis of covariance structure.South African Statistical Journal, 1974,8, 1–24.
Clarke, M. R. B. A rapidly convergent method for maximum likelihood factor analysis.British Journal of Mathematical and Statistical Psychology, 1970,23, 43–52.
Emmett, W. G. Factor analysis by Lawley's method of maximum likelihood.British Journal of Mathematical and Statistical Psychology, 1949,2, 90–97.
Fletcher, R. & Powell, M. J. D. A rapidly convergent descent method for minimization.Computer Journal, 1963,6, 163–168.
Fletcher, R. & Reeves, C. M. Function minimization by conjugate gradients.Computer Journal, 1964,7, 149–154.
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.
Jennrich, R. I. Simplified formulae for standard errors in maximum-likelihood factor analysis.British Journal of Mathematical and Statistical Psychology, 1974,27, 122–131.
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.
Jennrich, R. I. Rotational equivalence of factor loading matrices with specified values.Psychometrika, 1978,43, 421–426.
Jöreskog, K. G. Some contributions to maximum likelihood factor analysis.Psychometrika, 1967,32, 443–482.
Jöreskog, K. G. A general approach to confirmatory maximum likelihood factor analysis.Psychometrika, 1969,34, 183–202.
Jöreskog, K. G. Analysis of covariance structures. In P. R. Krishnaiah (Ed.),Multivariate analysis—III. New York: Academic Press, 1973, pp. 263–285.
Jöreskog, K. G. & Goldberger, A. S. Factor analysis by generalized least squares.Psychometrika, 1972,37, 243–260.
Thurstone, L. L. & Thurstone, T. G. Factorial studies of intelligence.Psychometric Monographs, No. 2, Chicago: University of Chicago Press, 1941.
Author information
Authors and Affiliations
Additional information
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.
Rights and permissions
About this article
Cite this article
Lee, S.Y., Jennrich, R.I. A study of algorithms for covariance structure analysis with specific comparisons using factor analysis. Psychometrika 44, 99–113 (1979). https://doi.org/10.1007/BF02293789
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02293789