, Volume 69, Issue 4, pp 655–660 | Cite as

Positive loadings and factor correlations from positive covariance matrices

  • Wim P. KrijnenEmail author
Note And Comments


In many instances it is reasonable to assume that the population covariance matrix has positive elements. This assumption implies for the single factor analysis model that the loadings and regression weights for best linear factor prediction are positive. For the multiple factor analysis model where each variable loads on a single factor and a hierarchical factor model, it implies that the loadings and the factor correlations are positive. For the latter model it also implies that the regression weights for first- and second-order factor prediction are positive.

Key words

Classical test theory congeneric tests structural equation models hierarchical factor analysis regression weights best linear factor prediction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bekker, P.A. & Ten Berge, J.M.F. (1997). Generic global identification in factor analysis.Linear Algebra And Its Applications, 264, 255–263.CrossRefGoogle Scholar
  2. Bock, R.D. & Liebermann, M. (1970). Fitting a response model forn dichotomously scored items.Psychometrika, 35, 179–197.Google Scholar
  3. Bollen, K.A. (1989).Structural equations with latent variables. New York: Wiley.Google Scholar
  4. Browne, M.W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures.British Journal of Mathematical and Statistical Psychology, 37, 62–83.PubMedGoogle Scholar
  5. Carroll, J.B. (1993).Human cognitive abilities: A survey of factor analytic studies. Cambridge: Cambridge University Press.Google Scholar
  6. Fisher, F.M. (1966).The identification problem in econometrics. New York: McGraw-Hill.Google Scholar
  7. Gustafsson, J.E. (1988). Hierarchical models of individual differences in cognitive abilities. In R.J. Sternberg (Ed.),Advances in the psychology of human intelligence. Vol. 4 (pp. 35–71). Hillsdale: Erlbaum.Google Scholar
  8. Henderson, H.V. & Searle, S.R. (1981). On deriving the inverse of a sum of matrices.SIAM Review, 23, 53–60.CrossRefGoogle Scholar
  9. Jöreskog, K.G. (1971a). Statistical analysis of sets of congeneric tests.Psychometrika, 36, 109–133.Google Scholar
  10. Jöreskog, K.G. (1971b). Simultaneous factor analysis in several populations.Psychometrika, 36, 409–426.Google Scholar
  11. 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.Google Scholar
  12. Lawley, D.N. & Maxwell, A.E. (1971).Factor analysis as a statistical method (2nd ed.). London: Butterworth.Google Scholar
  13. Lord, M. & Novick, M.R. (1968).Statistical theories of mental test scores. Reading, MA: Addison-Wesley.Google Scholar
  14. Luenberger, D.G. (1969).Optimization by vector space methods. New York: Wiley.Google Scholar
  15. Magnus, J.R. & Neudecker, H. (1988).Matrix differential calculus with applications in statistics and economics. Chichester: Wiley.Google Scholar
  16. Rao, C.R. & Rao, M.B. (1998).Matrix algebra and its applications to statistics and econometrics. Singapore: World Scientific.Google Scholar
  17. Schmid, J. & Leiman, J.M. (1957). The developments of hierarchical factor solutions.Psychometrika, 22, 53–61.CrossRefGoogle Scholar
  18. Schneeweiss, H. & Mathes, H. (1995). Factor analysis and principal components.Journal of Multivariate Analysis, 55, 105–124.CrossRefGoogle Scholar
  19. Serfling, R.J. (1980).Approximation theorems of mathematical statistics. New York: Wiley.Google Scholar
  20. Shapiro, A. (1983). Asymptotic distribution theory in the analysis of covariance structures (A unified approach).South African Statistical Journal, 17, 33–81.Google Scholar
  21. Shapiro, A. (1985). Identifiability of factor analysis: Some results and open problems.Linear Algebra And Its Applications, 70, 1–7.CrossRefGoogle Scholar
  22. Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis.International Statistical Review, 56, 49–62.Google Scholar
  23. Shapiro, A. (1989). Erratum.Linear Algebra And Its Applications, 125, 149.CrossRefGoogle Scholar
  24. Spearman, C. (1927).The abilities of man. London: Macmillan.Google Scholar
  25. Spearman, C. (1933). The uniqueness and exactness of g.British Journal Psychology, 24, 106–108.Google Scholar
  26. Takane, Y. & De Leeuw, J. (1987). On the relationship between item response theory and factor analysis of discretized variables.Psychometrika, 52, 393–408.Google Scholar
  27. Thurstone, L.L. (1947).Multiple-factor analysis: A development and expansion of the vectors of mind. Chicago: University of Chicago Press.Google Scholar
  28. Yung Y.F., Thissen, D., & Cleoid, L.D. (1999). On the relationship between the higher-order factor model and the hierarchical factor model.Psychometrika, 64, 113–128.CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2004

Authors and Affiliations

  1. 1.Department of Psychological MethodsUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations