Hostname: page-component-89b8bd64d-4ws75 Total loading time: 0 Render date: 2026-05-09T00:05:08.555Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Contributions to Maximum Likelihood Factor Analysis

Published online by Cambridge University Press:  01 January 2025

K. G. Jöreskog
K. G. Jöreskog*
Affiliation:
Educational Testing Service
Get access Rights & Permissions [Opens in a new window]

Abstract

A new computational method for the maximum likelihood solution in factor analysis is presented. This method takes into account the fact that the likelihood function may not have a maximum in a point of the parameter space where all unique variances are positive. Instead, the maximum may be attained on the boundary of the parameter space where one or more of the unique variances are zero. It is demonstrated that such improper (Heywood) solutions occur more often than is usually expected. A general procedure to deal with such improper solutions is proposed. The proposed methods are illustrated using two small sets of empirical data, and results obtained from the analyses of many other sets of data are reported. These analyses verify that the new computational method converges rapidly and that the maximum likelihood solution can be determined very accurately. A by-product obtained by the method is a large sample estimate of the variance-covariance matrix of the estimated unique variances. This can be used to set up approximate confidence intervals for communalities and unique variances.

Information

Type
Original Paper
Information
Psychometrika , Volume 32 , Issue 4 , December 1967 , pp. 443 - 482
Copyright
Copyright © 1967 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.)

Article purchase

Temporarily unavailable

Footnotes

*

The first part of this work was done at the University of Uppsala, Sweden and supported by the Swedish Council for Social Science Research. The second part was done at Educational Testing Service and supported by a grant (NSF-GB-1985) from National Science Foundation to Educational Testing Service.

The author is deeply indebted to Dr. D. N. Lawley whose contributions amounted nearly to coauthorship. The author also wishes to thank Mr. G. Gruvaeus for much valuable assistance in constructing the computer program.

References

Aitchison, J. and Silvey, S. D. Maximum likelihood estimation procedures and associated tests of significance. Journal of the Royal Statistical Society, 1960, 22, 154–171.CrossRefGoogle Scholar
Anderson, T. W. An introduction to multivariate statistical analysis, New York: Wiley, 1958.Google Scholar
Bargmann, R. A study of independence and dependence in multivariate normal analysis. University of North Carolina, Institute of Statistics Mimeo Series No. 186, 1957.Google Scholar
Bechtoldt, H. P. An empirical study of the factor analysis stability hypothesis. Psychometrika, 1961, 26, 405–432.CrossRefGoogle Scholar
Bellman, R. Introduction to matrix analysis, New York: McGraw-Hill, 1960.Google Scholar
Box, G. E. P. A general distribution theory for a class of likelihood criteria. Biometrika, 1949, 36, 317–346.CrossRefGoogle ScholarPubMed
Browne, M. W. A comparison of factor analytic techniques, South Africa: University of Witwatersrand, 1965.Google Scholar
Davis, F. B. Fundamental factors of comprehension in reading. Psychometrika, 1944, 9, 185–197.CrossRefGoogle Scholar
Emmett, W. G. Factor analysis by Lawley's method of maximum likelihood. British Journal of Psychology, 1949, 2, 90–97.Google Scholar
Fletcher, R. and Powell, M. J. D. A rapidly convergent descent method for minimization. Computer Journal, 1963, 2, 163–168.CrossRefGoogle Scholar
Harman, H. H. Modern factor analysis, Chicago: University of Chicago Press, 1960.Google Scholar
Hemmerle, W. J. Obtaining maximum-likelihood estimates of factor loadings and communalities using an easily implemented iterative procedure. Psychometrika, 1965, 30, 291–302.CrossRefGoogle Scholar
Howe, W. G. Some contributions to factor analysis. Report No. ORNL-1919, Oak Ridge, Tennessee: Oak Ridge National Laboratory, 1955.Google Scholar
Jöreskog, K. G. Statistical estimation in factor analysis, Stockholm: Almqvist & Wiksell, 1963.Google Scholar
Jöreslkog, K. G. Testing a simple structure hypothesis in factor analysis. Psychometrika, 1966, 31, 165–178.CrossRefGoogle Scholar
Jöreskog, K. G. UMLFA-a computer program for unrestricted maximum likelihood factor analysis. Research Memorandum 66-20. Princeton (Revised Edition), New Jersey: Educational Testing Service, 1967.Google Scholar
Kendall, M. G. and Stuart, A. The advanced theory of statistics, London: Charles Griffin and Co., 1961.CrossRefGoogle Scholar
Lawley, D. N. The estimation of factor loadings by the method of maximum likelihood. Proceedings of the Royal Society of Edinburgh, 1940, 60, 64–82.CrossRefGoogle Scholar
Lawley, D. N. Further investigations in factor estimation. Proceedings of the Royal Society of Edinburgh, 1942, 61, 176–185.Google Scholar
Lawley, D. N. The application of the maximum likelihood method to factor analysis. British Journal of Psychology, 1943, 33, 172–175.Google Scholar
Lawley, D. N. A statistical examination of the centroid method. Proceedings of the Royal Society of Edinburgh, 1955, 64, 175–189.Google Scholar
Lawley, D. N. Some new results in maximum likelihood factor analysis. Proceedings of the Royal Society of Edinburgh, 1967, 67, 256–264.Google Scholar
Lawley, D. N. and Maxwell, A. E. Factor analysis as a statistical method, London: Butterworths, 1963.Google Scholar
Lord, F. M. A study of speed factors in tests and academic grades. Psychometrika, 1956, 21, 31–50.CrossRefGoogle Scholar
Mattsson, A., Olsson, U., and Rosen, M. The maximum likelihood method in factor analysis with special consideration to the problem of improper solutions, Sweden: Institute of Statistics, University of Uppsala, 1966.Google Scholar
Maxwell, E. A. Recent trends in factor analysis. Journal of the Royal Statistical Society, 1961, 124, 49–59.CrossRefGoogle Scholar
Neyman, J. and Pearson, E. S. On the use and interpretation of certain test criteria for purposes of statistical inference. Biometrika, 1928, 20A, 175–240.Google Scholar
Ortega, J. M. On sturm-sequences for tri-diagonal matrices. Journal of the Association of Computing Machinery, 1960, 7, 260–263.CrossRefGoogle Scholar
Ortega, J. M. and Kaiser, H. F. The LLT and QR methods for symmetric tri-diagonal matrices. Computer Journal, 1963, 6, 99–101.CrossRefGoogle Scholar
Rao, C. R. Estimation and tests of significance in factor analysis. Psychometrika, 1955, 20, 93–111.CrossRefGoogle Scholar
Rippe, D. D. Application of a large sampling criterion to some sampling problems in factor analysis. Psychometrika, 1953, 18, 191–205.CrossRefGoogle Scholar
Wilkinson, J. H. The calculation of eigenvectors of codiagonal matrices. Computer Journal, 1958, 1, 90–96.CrossRefGoogle Scholar
Wilkinson, J. H. Householder's method for the solution of the algebraic eigenproblem. Computer Journal, 1960, 3, 23–27.CrossRefGoogle Scholar
Wilks, S. S. The large-sample distribution of the likelihood ratio for testing composite hypotheses. Annals of Mathematical Statistics, 1938, 9, 60–62.CrossRefGoogle Scholar