Hostname: page-component-669899f699-cf6xr Total loading time: 0 Render date: 2025-05-02T01:36:14.964Z Has data issue: false hasContentIssue false
  • English
  • Français

A Comparison of Factor Analytic Techniques

Published online by Cambridge University Press:  01 January 2025

Michael W. Browne
Michael W. Browne*
Affiliation:
National Institute for Personnel Research, South Africa
Get access Rights & Permissions [Opens in a new window]

Abstract

Statistical properties of several methods for obtaining estimates of factor loadings and procedures for estimating the number of factors are compared by means of random sampling experiments. The effect of increasing the ratio of the number of observed variables to the number of factors, and of increasing sample size, is examined. A description is given of a procedure which makes use of the Bartlett decomposition of a Wishart matrix to generate random correlation matrices.

Type
Original Paper
Information
Psychometrika , Volume 33 , Issue 3 , September 1968 , pp. 267 - 334
Copyright
Copyright © 1968 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

References

Anderson, T. W. An introduction to multivariate statistical analysis, New York: Wiley, 1958.Google Scholar
Anderson, T. W., and Rubin, H. Statistical inference in factor analysis. In Neyman, J. (Eds.), Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume V. Berkeley: Univer. California Press, 1956, 111–150.Google Scholar
Appel, K. Solution of eigenvalue problems with approximately known eigenvectors. Communications of the Association for Computing Machinery, 1962, 5, 381–381.CrossRefGoogle Scholar
Bargmann, R. E. A study of independence and dependence in multivariate normal analysis, Chapel Hill: Univer. North Carolina, Institute of Statistics, 1957.Google Scholar
Bartlett, M. S. On the theory of statistical regression. Proceedings of the Royal Society of Edinburgh, 1933, 53, 260–283.CrossRefGoogle Scholar
Bartlett, M. S. Tests of significance in factor analysis. British Journal of Psychology, Statistical Section, 1950, 3, 77–85.CrossRefGoogle Scholar
Bartlett, M. S. A note on the multiplying factors for various approximations. Journal of the Royal Statistical Society, 1954, 16, 296–298.CrossRefGoogle Scholar
Bechtoldt, H. P. An empirical study of the factor analysis stability hypothesis. Psychometrika, 1961, 26, 405–432.CrossRefGoogle Scholar
Box, G. E. P., andMuller, M. E. A note on the generation of random normal deviates. Annals of Mathematical Statistics, 1958, 29, 610–611.CrossRefGoogle Scholar
Brown, R. H. A comparison of the maximum determinant solution in factor analysis with various approximations. In Bargmann, R. E. (Eds.), Representative ordering and selection of variables, Part B, Blacksburg: Virginia Polytechnic Institute, 1962.Google Scholar
Browne, M. W. On oblique Procrustes rotation, Research Bulletin 66-8, Princeton, N. J.: Educational Testing Service, 1966.Google Scholar
Dwyer, P. S. The contribution of an orthogonal multiple factor solution to multiple correlation. Psychometrika, 1939, 4, 163–171.CrossRefGoogle Scholar
Dwyer, P. S. The square root method and its use in correlation and regression. Journal of the American Statistical Association, 1945, 40, 493–503.CrossRefGoogle Scholar
Greenstadt, J. The determination of the characteristic roots of a matrix by the Jacobi method. In Ralston, A., and Wilf, H. S. (Eds.), Mathematical methods for digital computers. New York: Wiley, 1960, 84–92.Google Scholar
Guttman, L. Multiple rectilinear prediction and the resolution into components. Psychometrika, 1940, 5, 75–99.CrossRefGoogle Scholar
Guttman, L. Some necessary conditions for common-factor analysis. Psychometrika, 1954, 19, 149–161.CrossRefGoogle Scholar
Guttman, L. “Best possible” systematic estimates of communalities. Psychometrika, 1956, 21, 273–285.CrossRefGoogle Scholar
Harman, H. H. Modern factor analysis, Chicago: Univer. Chicago Press, 1960.Google Scholar
Harman, H. H., & Fukuda, Y. Resolution of the Heywood case in the minres solution. Psychometrika, 1966, 31, in press.CrossRefGoogle Scholar
Harman, H. H., and Jones, W. H. Factor analysis by minimizing residuals (minres). Psychometrika, 1966, 31, 351–368.CrossRefGoogle ScholarPubMed
Harris, C. W. Some Rao-Guttman relationships. Psychometrika, 1962, 27, 247–263.CrossRefGoogle Scholar
Henrysson, S. The significance of factor loadings—Lawley's test examined by artificial samples. British Journal of Psychology, Statistical Section, 1950, 3, 159–165.CrossRefGoogle Scholar
Hotelling, H. Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology, 1933, 24, 417–441.CrossRefGoogle Scholar
Howe, W. G. Some contributions to factor analysis, Oak Ridge, Tenn.: Oak Ridge National Laboratory, 1955.CrossRefGoogle Scholar
Hull, T. E., and Dobell, A. R. Random number generators. Society for Industrial and Applied Mathematics Review, 1962, 4, 230–254.Google Scholar
Jöreskog, K. G. On the statistical treatment of residuals in factor analysis. Psychometrika, 1962, 27, 335–354.CrossRefGoogle Scholar
Jöreskog, K. G. Statistical estimation in factor analysis, Stockholm: Almqvist and Wiksell, 1963.Google Scholar
Jöreskog, K. G. Some contributions to maximum likelihood factor analysis, Princeton, N.J.: Educational Testing Service, 1966.CrossRefGoogle Scholar
Kaiser, H. F. The application of electronic computers to factor analysis. Educational and Psychological Measurement, 1960, 20, 141–151.CrossRefGoogle Scholar
Kaiser, H. F., and Dickman, K. Sample and population score matrices and sample correlation matrices from an arbitrary population correlation matrix. Psychometrika, 1962, 27, 179–182.CrossRefGoogle Scholar
Kendall, M. G. The advanced theory of statistics, Volume II, London: Charles Griffin, & Co., Ltd., 1946.Google Scholar
Kendall, M. G., and Stuart, A. The advanced theory of statistics, Volume II, London: Charles Griffin & Co., Ltd., 1961.Google Scholar
Kshirsagar, A. M. Bartlett decomposition and Wishart distribution. Annals of Mathematical Statistics, 1959, 30, 239–241.CrossRefGoogle Scholar
Lawley, D. N. The estimation of factor loadings by the method of maximum likelihood. Proceedings of the Royal Society of Edinburgh, Section A, 1940, 60, 64–82.CrossRefGoogle Scholar
Lawley, D. N. Further investigations in factor estimation. Proceedings of the Royal Society of Edinburgh, Section A, 1941, 61, 176–185.Google Scholar
Lawley, D. N. Tests of significance for the latent roots of covariance and correlation matrices. Biometrika, 1956, 43, 128–136.CrossRefGoogle Scholar
Lawley, D. N., and Swanson, Z. Tests of significance in a factor analysis of artificial data. British Journal of Statistical Psychology, 1954, 7, 75–79.CrossRefGoogle Scholar
Linn, R. L. A Monte Carlo approach to the number of factors problem. Unpublished doctoral dissertation, Univer. of Illinois, 1965.Google Scholar
Lord, F. M. A study of speed factors in tests and academic grades. Psychometrika, 1956, 21, 31–50.CrossRefGoogle Scholar
Maxwell, A. E. A comparative study of some factor analytic techniques. Unpublished doctoral dissertation, Univer. of Edinburgh, 1957.Google Scholar
McNemar, Q. On the sampling error of factor loadings. Psychometrika, 1941, 6, 141–152.CrossRefGoogle Scholar
McNemar, Q. On the number of factors. Psychometrika, 1942, 7, 9–18.CrossRefGoogle Scholar
Mosier, C. I. Influence of chance error on simple structure. Psychometrika, 1939, 4, 33–44.CrossRefGoogle Scholar
Mosier, C. I. Determining a simple structure when loadings for certain tests are known. Psychometrika, 1939, 4, 149–162.CrossRefGoogle Scholar
Muller, M. E. A comparison of methods for generating normal deviates on digital computers. Journal of the Association for Computing Machinery, 1959, 6, 376–383.CrossRefGoogle Scholar
Odell, P. L., and Feiveson, A. H. A numerical procedure to generate a sample covariance matrix. Journal of the American Statistical Association, 1966, 61, 199–203.CrossRefGoogle Scholar
Rao, C. R. Estimation and tests of significance in factor analysis. Psychometrika, 1955, 20, 93–111.CrossRefGoogle Scholar
Saunders, D. R. Further implications of Mundy-Castle's correlations between EEG and Wechsler-Bellevue variables. Journal of the National Institute for Personnel Research, 1960, 8, 91–101.Google Scholar
Sokal, R. R. A comparison of five tests for completeness of factor extraction. Transactions of the Kansas Academy of Science, 1959, 62, 141–152.CrossRefGoogle Scholar
Taussky, O., and Todd, J. Generation and testing of pseudo random numbers. In Meycr, H. A. (Eds.), Symposium on Monte Carlo methods. New York: Wiley, 1956, 15–28.Google Scholar
Teichroew, D., and Sitgreaves, R. Computation of an empirical sampling distribution for the W classification statistic. In Solomon, H. (Eds.), Studies in item analysis and prediction. Stanford: Stanford Univer. Press, 1961, 252–275.Google Scholar
Thomson, G. H. Hotelling's method modified to give Spearman's g. Journal of Educational Psychology, 1934, 25, 366–374.CrossRefGoogle Scholar
Thurstone, L. L. Multiple factor analysis, Chicago: Univer. Chicago Press, 1947.Google Scholar
Whittle, P. On principal components and least square methods of factor analysis. Skandinavisk Aktuarietidskrift, 1952, 35, 223–239.Google Scholar
Wijsman, R. A. Applications of a certain representation of the Wishart matrix. Annals of Mathematical Statistics, 1959, 30, 597–601.CrossRefGoogle Scholar
Wold, H. Some artificial experiments in factor analysis. Uppsala Symposium on psychological factor analysis. Uppsala: Almqvist and Wiksell, 1953, 43–64.Google Scholar
Wrigley, C. The effect upon the communalities of changing the estimate of the number of factors. British Journal of Statistical Psychology, 1959, 12, 34–54.CrossRefGoogle Scholar
Wrigley, C., and Neuhaus, J. O. The use of an electronic computer in principal axes factor analysis. Journal of Educational Psychology, 1955, 46, 31–41.CrossRefGoogle Scholar
Young, G. Maximum likelihood estimation and factor analysis. Psychometrika, 1941, 6, 49–53.CrossRefGoogle Scholar