Hostname: page-component-5f745c7db-szhh2 Total loading time: 0 Render date: 2025-01-06T07:39:08.845Z Has data issue: true hasContentIssue false
  • English
  • Français

Factor Analysis by Generalized Least Squares

Published online by Cambridge University Press:  01 January 2025

Karl G. Jöreskog  and
Arthur S. Goldberger
Karl G. Jöreskog
Affiliation:
University Institute of Statistics, Uppsala, Sweden
Arthur S. Goldberger
Affiliation:
University of Wisconsin
Get access Rights & Permissions [Opens in a new window]

Abstract

Aitken’s generalized least squares (GLS) principle, with the inverse of the observed variance-covariance matrix as a weight matrix, is applied to estimate the factor analysis model in the exploratory (unrestricted) case. It is shown that the GLS estimates are scale free and asymptotically efficient. The estimates are computed by a rapidly converging Newton-Raphson procedure. A new technique is used to deal with Heywood cases effectively.

Keywords

Public PolicyStatistical TheoryWeight MatrixGeneralize Little SquareFactor Analysis Model
Type
Original Paper
Information
Psychometrika , Volume 37 , Issue 3 , September 1972 , pp. 243 - 260
Copyright
Copyright © 1972 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

The work on this project was done when the first author was Research Statistician at Educational Testing Service, Princeton, N. J. The second author was in part supported by a grant from the Research Committee of the University of Wisconsin Graduate School. The authors wish to thank Michael Browne for many helpful comments and Marielle van Thillo for valuable assistance in the numerical computations.

References

Aitken, A. C. On least squares and the linear combination of observations. Proceedings of the Royal Society of Edinburgh, 1934, 55, 42–48CrossRefGoogle Scholar
Anderson, T. W. Some scaling models and estimation procedures in the latent class model. In Grenander, U. (Eds.), Probability and statistics: The Harald Cramér volume. New York: Wiley. 1959, 9–38Google Scholar
Anderson, T. W. & Rubin, H. Statistical inference in factor analysis. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Berkeley: University of California Press. 1956, 111–149Google Scholar
Browne, M. W. Analysis of covariance structures. Paper presented at the annual conference of the South African Statistical Association, October 1970.Google Scholar
Clarke, M. R. B. A rapidly convergent method for maximum-likelihood factor analysis. The British Journal of Mathematical and Statistical Psychology, 1970, 23, 43–52CrossRefGoogle Scholar
Ferguson, T. S. A method of generating best asymptotically normal estimates with application to the estimation of bacterial densities. Annals of Mathematical Statistics, 1958, 29, 1049–1062CrossRefGoogle Scholar
Harman, H. H. Modern factor analysis, 2nd ed., Chicago: University of Chicago Press, 1967Google Scholar
Jennrich, R. I. & Robinson, S. M. A Newton-Raphson algorithm for maximum likelihood factor analysis. Psychometrika, 1969, 34, 111–123CrossRefGoogle Scholar
Jöreskog, K. G. Statistical estimation in factor analysis, 1963, Stockholm: Almqvist & WiksellGoogle Scholar
Jöreskog, K. G. Some contributions to maximum likelihood factor analysis. Psychometrika, 1967, 32, 443–482CrossRefGoogle Scholar
Lawley, D. N. A modified method of estimation in factor analysis and some large sample results. Stockholm: Almqvist & Wiksell. 1953, 35–42Google Scholar
Lawley, D. N. Some new results in maximum likelihood factor analysis. Proceedings of the Royal Society of Edinburgh, 1967, 67, 256–264Google Scholar
Malinvaud, E. Statistical methods of econometrics, 1966, Chicago: Rand-McNallyGoogle Scholar
Neyman, J. Contribution to the theory of x2 test. Proceedings of the First Berkeley Symposium on Mathematical Statistics and Probability. Berkeley: University of California Press. 1949, 239–273Google Scholar
Rothenberg, T. Structural restrictions and estimation efficiency in linear econometric models, 1966, New Haven, Conn.: Yale UniversityGoogle Scholar
Taylor, W. F. Distance functions and regular best asymptotically normal estimates. Annals of Mathematical Statistics, 1953, 24, 85–92CrossRefGoogle Scholar
Wilkinson, J. H. The algebraic eigenvalue problem, 1965, Oxford: Oxford University PressGoogle Scholar
Zellner, A. An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. Journal of the American Statistical Association, 1962, 57, 348–368CrossRefGoogle Scholar