A general approach to confirmatory maximum likelihood factor analysis

Psychometrika volume 34, pages183–202(1969)Cite this article

Abstract

We describe a general procedure by which any number of parameters of the factor analytic model can be held fixed at any values and the remaining free parameters estimated by the maximum likelihood method. The generality of the approach makes it possible to deal with all kinds of solutions: orthogonal, oblique and various mixtures of these. By choosing the fixed parameters appropriately, factors can be defined to have desired properties and make subsequent rotation unnecessary. The goodness of fit of the maximum likelihood solution under the hypothesis represented by the fixed parameters is tested by a large samplex 2 test based on the likelihood ratio technique. A by-product of the procedure is an estimate of the variance-covariance matrix of the estimated parameters. From this, approximate confidence intervals for the parameters can be obtained. Several examples illustrating the usefulness of the procedure are given.

This is a preview of subscription content, log in to check access.

References

  1. Anderson, T. W.An introduction to multivariate statistical analysis. New York: Wiley, 1958.

    Google Scholar 

  2. Anderson, T. W., & Rubin, H. Statistical inference in factor analysis. In J. Neyman (Ed.),Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. V. Berkeley: University of California Press, 1956. pp. 111–150.

    Google Scholar 

  3. Fletcher, R., & Powell, M. J. D. A rapidly convergent descent method for minimization.Computer Journal, 1963,2, 163–168.

    Google Scholar 

  4. Holzinger, K. J., & Swineford, F.A study in factor analysis: The stability of a bi-factor solution. University of Chicago: Supplementary Educational Monographs, No. 48, 1939.

  5. Howe, W. G. Some contributions to factor analysis. Report No. ORNL-1919, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 1955.

    Google Scholar 

  6. Jöreskog, K. G. Testing a simple structure hypothesis in factor analysis.Psychometrika, 1966,31, 165–178.

    Google Scholar 

  7. Jöreskog, K. G. UMLFA — A computer program for unrestricted maximum likelihood factor analysis. Research Memorandum 66-20. Princeton, N. J.: Educational Testing Service. Revised Edition, 1967. (a)

    Google Scholar 

  8. Jöreskog, K. G. Some contributions to maximum likelihood factor analysis.Psychometrika, 1967,32, 443–482. (b)

    Google Scholar 

  9. Jöreskog, K. G., & Gruvaeus, G. RMLFA — A computer program for restricted maximum likelihood factor analysis. Research Memorandum 67-21. Princeton, N. J.: Educational Testing Service, 1967.

    Google Scholar 

  10. Jöreskog, K. G., & Lawley, D. N. New methods in maximum likelihood factor analysis.British Journal of Mathematical and Statistical Psychology, 1968,21, in press.

  11. Kendall, M. G., & Stuart, A.The advanced theory of statistics, vol. 2. London: Charles Griffin and Co., 1961.

    Google Scholar 

  12. Lawley, D. N. Estimation in factor analysis under various initial assumptions.British Journal of Statistical Psychology, 1958,11, 1–12.

    Google Scholar 

  13. Lawley, D. N. Some new results in maximum likelihood factor analysis.Proceedings of the Royal Society of Edinburgh, Section A, 1967,67, 256–264.

    Google Scholar 

  14. Lawley, D. N., & Maxwell, A. E.Factor analysis as a statistical method. London: Butterworth, 1963.

    Google Scholar 

  15. Lawley, D. N., & Maxwell, A. E. Factor transformation methods.British Journal of Statistical Psychology, 1964,17, 97–103.

    Google Scholar 

  16. Lockhart, R. S. Asymptotic sampling variances for factor analytic models identified by specified zero parameters.Psychometrika, 1967,32, 265–277.

    Google Scholar 

  17. Reiersøl, O. On the identifiability of parameters in Thurstone's multiple factor analysis.Psychometrika, 1950,15, 121–149.

    Google Scholar 

  18. Thurstone, L. L.Multiple-factor analysis. Chicago: University of Chicago Press, 1947.

    Google Scholar 

  19. Thurstone, L. L., & Thurstone, T. G. Factorial studies of intelligence.Psychometric Monographs No. 2, 1941. Chicago: University of Chicago Press.

    Google Scholar 

  20. Tucker, L. R. An inter-battery method of factor analysis.Psychometrika, 1958,23, 111–136.

    Google Scholar 

  21. Tucker, L. R, Koopman, R. F., & Linn, R. L. Evaluation of factor analytic research procedures by means of simulated correlation matrices.Psychometrika, 1968, in press.

  22. Vernon, P. E.The structure of human abilities. London: Methuen, 1951.

    Google Scholar 

Download references

Author information

Affiliations

  1. Educational Testing Service, USA

    K. G. Jöreskog

Authors
  1. K. G. Jöreskog
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by a grant (NSF-GB 1985) from the National Science Foundation to Educational Testing Service.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Jöreskog, K.G. A general approach to confirmatory maximum likelihood factor analysis. Psychometrika 34, 183–202 (1969). https://doi.org/10.1007/BF02289343

Download citation

Keywords

  • Confidence Interval
  • Likelihood Ratio
  • Public Policy
  • General Procedure
  • Free Parameter