Hostname: page-component-74656f9884-54qdv Total loading time: 0 Render date: 2026-11-10T08:20:51.387Z Has data issue: false hasContentIssue false
  • English
  • Français

Conditions for Factor (in)Determinacy in Factor Analysis

Published online by Cambridge University Press:  01 January 2025

Wim P. Krijnen ,
Theo K. Dijkstra  and
Richard D. Gill
Wim P. Krijnen*
Affiliation:
University of Groningen
Theo K. Dijkstra
Affiliation:
University of Groningen
Richard D. Gill
Affiliation:
University of Utrecht
*
Requests for reprints should be sent to Wim P. Krijnen, Lisdodde I, 9679 MC Scheemda, THE NETHERLANDS.
Get access Rights & Permissions [Opens in a new window]

Abstract

The subject of factor indeterminacy has a vast history in factor analysis (Guttman, 1955; Lederman, 1938; Wilson, 1928). It has lead to strong differences in opinion (Steiger, 1979). The current paper gives necessary and sufficient conditions for observability of factors in terms of the parameter matrices and a finite number of variables. Five conditions are given which rigorously define indeterminacy. It is shown that (un)observable factors are (in)determinate. Specifically, the indeterminacy proof by Guttman is extended to Heywood cases. The results are illustrated by two examples and implications for indeterminacy are discussed.

Keywords

indeterminacyHeywood casesmean squared errorfactor score prediction

Information

Type
Original Paper
Information
Psychometrika , Volume 63 , Issue 4 , December 1998 , pp. 359 - 367
Copyright
Copyright © 1998 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., & Rubin, H. (1956). Statistical inference in factor analysis. Proceedings of the Third Berkeley Symposium, 5, 111–150.Google Scholar
Anderson, T. W. (1984). An introduction to multivariate statistical analysis, New York: Wiley.Google Scholar
Bargmann, R. E. (1957). A study of independence and dependence in multivariate normal analysis, Chapel Hill: University of North Carolina, Institute of Statistics.Google Scholar
Bartlett, M. S. (1937). The statistical conception of mental factors. British Journal of Psychology, 28, 97–104.Google Scholar
Boomsma, A. (1985). Nonconvergence, improper solutions, and starting values in Lisrel Maximum Likelihood estimation. Psychometrika, 50, 229–242.CrossRefGoogle Scholar
Browne, M. W. (1968). A comparison of factor analytic techniques. Psychometrika, 33, 267–334.CrossRefGoogle ScholarPubMed
Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.CrossRefGoogle ScholarPubMed
Cramér, H. (1946). Mathematical methods of statistics, Princeton: University Press.Google Scholar
Dijkstra, T. K. (1992). On statistical inference with parameter estimates on the boundary of the parameter space. British Journal of Statistical and Mathematical Psychology, 45, 289–309.CrossRefGoogle Scholar
Elffers, H., Bethlehem, J., & Gill, R. D. (1978). Indeterminacy problems and the interpretation of factor analysis results. Statistica Neerlandica, 32, 181–199.CrossRefGoogle Scholar
Ferguson, T. S. (1958). A method of generating best asymptotically normal estimates with application to the estimation of bacterial densities. Annals of Mathematical Statistics, 29, 1046–1062.CrossRefGoogle Scholar
Guttman, L. (1955). The determinacy of factor score matrices with implications for five other basic problems of common-factor theory. The British Journal of Statistical Psychology, 8, 65–81.CrossRefGoogle Scholar
Heywood, H. B. (1931). On finite sequences of real numbers. Proceedings of the Royal Society London, 134, 486–501.Google Scholar
Howe, W. G. (1955). Some contributions to factor analysis (Report No. ORNL-1919), Oak Ridge: Oak Ridge National Laboratory.CrossRefGoogle Scholar
Jöreskog, K. G. (1967). Some contributions to maximum likelihood factor analysis. Psychometrika, 32, 443–482.CrossRefGoogle Scholar
Jöreskog, K. G. (1971). Statistical analysis of sets of congeneric tests. Psychometrika, 36, 109–133.CrossRefGoogle Scholar
Kano, Y. (1986). A condition for the regression predictor to be consistent in a single common factor model. British Journal of Mathematical and Statistical Psychology, 39, 221–227.CrossRefGoogle Scholar
Krijnen, W. P. (1996). Algorithms for unweighted least squares factor analysis. Computational Statistics and Data Analysis, 21(2), 133–147.CrossRefGoogle Scholar
Krijnen, W. P. (1997). Using single factor-factor analysis as a measurement model. Submitted for publication.Google Scholar
Krijnen, W. P. (in press). A note on the parameter set for factor analysis models. Linear Algebra and its Applications.Google Scholar
Krijnen, W. P., Wansbeek, T. J., & ten Berge, J. M. F. (1996). Best linear estimators for factor scores. Communications in statistics: Theory and methods, 25, 3013–3025.CrossRefGoogle Scholar
Lawley, D. N., & Maxwell, A. E. (1971). Factor analysis as a statistical method, Durban: Lawrence Erlbaum.Google Scholar
Lederman, W. (1938). The orthogonal transformations of a factorial matrix into itself. Psychometrika, 3, 181–187.CrossRefGoogle Scholar
Lee, S. Y. (1980). Estimation of covariance structure models with parameters subject to functional restraints. Psychometrika, 45, 309–324.CrossRefGoogle Scholar
Lord, M., & Novick, M. R. (1968). Statistical theories of mental test scores, Reading, MA: Addison-Wesley.Google Scholar
Löwner, K. (1934). Über monotone Matrixfunctionen. Mathematisches Zeitschrift, 38, 177–216 [On monotone matrix functions]CrossRefGoogle Scholar
Luenberger, D. G. (1969). Optimization by vector space methods, New York, NY: John Wiley.Google Scholar
Magnus, J. R., & Neudecker, H. (1991). Matrix differential calculus with applications in statistics and economics, Chichester: John Wiley and Sons.Google Scholar
McDonald, R. P. (1974). The measurement of factor indeterminacy. Psychometrika, 39, 203–222.CrossRefGoogle Scholar
McDonald, R. P. (1977). The indeterminacy of components and the definition of common factors. British Journal of Mathematical and Statistical Psychology, 30, 165–176.CrossRefGoogle Scholar
McDonald, R. P., & Burr, E. J. (1967). A comparison of four methods of constructing factor scores. Psychometrika, 32, 381–401.CrossRefGoogle Scholar
Muirhead, R. J. (1982). Aspects of multivariate statistical theory, New York: John Wiley & Sons.CrossRefGoogle Scholar
Penrose, R. (1955). A generalized inverse for matrices. Proceedings of the Cambridge Philosophical Society, 51, 406–413.CrossRefGoogle Scholar
Schneeweiss, H., & Mathes, H. (1995). Factor analysis and principal components. Journal of Multivariate Analysis, 55, 105–124.CrossRefGoogle Scholar
Schönemann, P. H., & Wang, M.-M. (1972). Some new results on factor indeterminacy. Psychometrika, 37, 61–91.CrossRefGoogle Scholar
Sen, P. K., & Singer, J. M. (1993). Large sample methods in statistics, New York: Chapman & Hall.CrossRefGoogle Scholar
Serfling, R. J. (1980). Approximation theorems of mathematical statistics, New York, NY: John Wiley.CrossRefGoogle Scholar
Shapiro, A. (1986). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika, 72, 133–144.CrossRefGoogle Scholar
Spearman, C. (1904). “General Intelligence”, objectively determined and measured. American Journal of Psychology, 15, 201–293.CrossRefGoogle Scholar
Spearman, C. (1933). The uniqueness and exactness of g. British Journal of Psychology, 24, 106–108.Google Scholar
Steiger, J. H. (1979). Factor indeterminacy in the 1930's and the 1970's some interesting parallels. Psychometrika, 44, 157–167.CrossRefGoogle Scholar
ten Berge, J. M. F., Krijnen, W. P., Wansbeek, T. J. & Shapiro, A. (in press). Some new results on correlation preserving factor scores prediction methods. Linear Algebra and its Applications.Google Scholar
ten Berge, J. M. F., & Nevels, K. (1977). A general solution to Mosier's oblique Procrustes problem. Psychometrika, 42, 593–600.CrossRefGoogle Scholar
Thurstone, L. L. (1935). The vectors of mind, Chicago, IL: University of Chicago Press.Google Scholar
Thomson, G. H. (1950). The factorial analysis of human ability, London: University Press.Google Scholar
van Driel, O. P. (1978). On various causes of improper solutions in maximum likelihood factor analysis. Psychometrika, 43, 225–243.CrossRefGoogle Scholar
Williams, J. S. (1978). A definition for the common-factor analysis model and the elimination of problems of factor score indeterminacy. Psychometrika, 43, 293–306.CrossRefGoogle Scholar
Wilson, E. B. (1928). On hierarchical correlation systems. Proceedings, National Academy of Science, 14, 283–291.CrossRefGoogle ScholarPubMed