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.
Similar content being viewed by others
References
Anderson, T. W., & Rubin, H. (1956). Statistical inference in factor analysis.Proceedings of the Third Berkeley Symposium, 5, 111–150.
Anderson, T. W. (1984).An introduction to multivariate statistical analysis. New York: Wiley.
Bargmann, R. E. (1957).A study of independence and dependence in multivariate normal analysis (Mimeo Series No. 186). Chapel Hill: University of North Carolina, Institute of Statistics.
Bartlett, M. S. (1937). The statistical conception of mental factors.British Journal of Psychology, 28, 97–104.
Boomsma, A. (1985). Nonconvergence, improper solutions, and starting values in Lisrel Maximum Likelihood estimation.Psychometrika, 50, 229–242.
Browne, M. W. (1968). A comparison of factor analytic techniques.Psychometrika, 33, 267–334.
Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures.British Journal of Mathematical and Statistical Psychology, 37, 62–83.
Cramér, H. (1946).Mathematical methods of statistics. Princeton: University Press.
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.
Elffers, H., Bethlehem, J., & Gill, R. D. (1978). Indeterminacy problems and the interpretation of factor analysis results.Statistica Neerlandica, 32, 181–199.
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.
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.
Heywood, H. B. (1931). On finite sequences of real numbers.Proceedings of the Royal Society London, 134, 486–501.
Howe, W. G. (1955).Some contributions to factor analysis (Report No. ORNL-1919). Oak Ridge: Oak Ridge National Laboratory.
Jöreskog, K. G. (1967). Some contributions to maximum likelihood factor analysis.Psychometrika, 32, 443–482.
Jöreskog, K. G. (1971). Statistical analysis of sets of congeneric tests.Psychometrika, 36, 109–133.
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.
Krijnen, W. P. (1996). Algorithms for unweighted least squares factor analysis.Computational Statistics and Data Analysis, 21(2), 133–147.
Krijnen, W. P. (1997).Using single factor-factor analysis as a measurement model. Submitted for publication.
Krijnen, W. P. (in press). A note on the parameter set for factor analysis models.Linear Algebra and its Applications.
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.
Lawley, D. N., & Maxwell, A. E. (1971).Factor analysis as a statistical method. Durban: Lawrence Erlbaum.
Lederman, W. (1938). The orthogonal transformations of a factorial matrix into itself.Psychometrika, 3, 181–187.
Lee, S. Y. (1980). Estimation of covariance structure models with parameters subject to functional restraints.Psychometrika, 45, 309–324.
Lord, M., & Novick, M. R. (1968).Statistical theories of mental test scores. Reading, MA: Addison-Wesley.
Löwner, K. (1934). Über monotone Matrixfunctionen [On monotone matrix functions].Mathematisches Zeitschrift, 38, 177–216.
Luenberger, D. G. (1969).Optimization by vector space methods. New York, NY: John Wiley.
Magnus, J. R., & Neudecker, H. (1991).Matrix differential calculus with applications in statistics and economics. Chichester: John Wiley and Sons.
McDonald, R. P. (1974). The measurement of factor indeterminacy.Psychometrika, 39, 203–222.
McDonald, R. P. (1977). The indeterminacy of components and the definition of common factors.British Journal of Mathematical and Statistical Psychology, 30, 165–176.
McDonald, R. P., & Burr, E. J. (1967). A comparison of four methods of constructing factor scores.Psychometrika, 32, 381–401.
Muirhead, R. J. (1982).Aspects of multivariate statistical theory. New York: John Wiley & Sons.
Penrose, R. (1955). A generalized inverse for matrices.Proceedings of the Cambridge Philosophical Society, 51, 406–413.
Schneeweiss, H., & Mathes, H. (1995). Factor analysis and principal components.Journal of Multivariate Analysis, 55, 105–124.
Schönemann, P. H., & Wang, M.-M. (1972). Some new results on factor indeterminacy.Psychometrika, 37, 61–91.
Sen, P. K., & Singer, J. M. (1993).Large sample methods in statistics. New York: Chapman & Hall.
Serfling, R. J. (1980).Approximation theorems of mathematical statistics. New York, NY: John Wiley.
Shapiro, A. (1986). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints.Biometrika, 72, 133–144.
Spearman, C. (1904). “General Intelligence”, objectively determined and measured.American Journal of Psychology, 15, 201–293.
Spearman, C. (1933). The uniqueness and exactness ofg.British Journal of Psychology, 24, 106–108.
Steiger, J. H. (1979). Factor indeterminacy in the 1930's and the 1970's some interesting parallels.Psychometrika, 44, 157–167.
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.
ten Berge, J. M. F. & Nevels, K. (1977). A general solution to Mosier's oblique Procrustes problem.Psychometrika, 42, 593–600.
Thurstone, L. L. (1935).The vectors of mind. Chicago, IL: University of Chicago Press.
Thomson, G. H. (1950).The factorial analysis of human ability. London: University Press.
van Driel, O. P. (1978). On various causes of improper solutions in maximum likelihood factor analysis.Psychometrika, 43, 225–243.
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.
Wilson, E. B. (1928). On hierarchical correlation systems.Proceedings, National Academy of Science, 14, 283–291.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Krijnen, W.P., Dijkstra, T.K. & Gill, R.D. Conditions for factor (in)determinacy in factor analysis. Psychometrika 63, 359–367 (1998). https://doi.org/10.1007/BF02294860
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02294860