Hostname: page-component-5f745c7db-szhh2 Total loading time: 0 Render date: 2025-01-06T22:05:02.119Z Has data issue: true hasContentIssue false
  • English
  • Français

Positive Loadings and Factor Correlations from Positive Covariance Matrices

Published online by Cambridge University Press:  01 January 2025

Wim P. Krijnen
Wim P. Krijnen*
Affiliation:
University of Amsterdam
*
Requests for reprints should be sent to Wim P. Krijnen, Department of Psychological Methods, University of Amsterdam, Roetersstraat 15, 1018 WB Amsterdam, The Netherlands. E-mail: Wim.Krijnen@hetnet.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Classical test theorycongeneric testsstructural equation modelshierarchical factor analysisregression weightsbest linear factor prediction
Type
Note and Comments
Information
Psychometrika , Volume 69 , Issue 4 , December 2004 , pp. 655 - 660
Copyright
Copyright © 2004 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.)

Footnotes

I would like to thank Conor Dolan for fruitful discussions on factor analysis, and the associate editor as well as three reviewers for making useful remarks on earlier versions of the manuscript.

References

Bekker, P.A., Ten Berge, J.M.F. (1997). Generic global identification in factor analysis. Linear Algebra And Its Applications, 264, 255–263CrossRefGoogle Scholar
Bock, R.D., Liebermann, M. (1970). Fitting a response model forn dichotomously scored items. Psychometrika, 35, 179–197CrossRefGoogle Scholar
Bollen, K.A. (1989). Structural equations with latent variables, New York: WileyCrossRefGoogle Scholar
Browne, M.W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83CrossRefGoogle ScholarPubMed
Carroll, J.B. (2003). Human cognitive abilities: A survey of factor analytic studies, Cambridge: Cambridge University PressGoogle Scholar
Fisher, F.M. (1966). The identification problem in econometrics, New York: McGraw-HillGoogle Scholar
Gustafsson, J.E. (1988). Hierarchical models of individual differences in cognitive abilities. In Sternberg, R.J. (Eds.), Advances in the psychology of human intelligence. Vol. 4 (pp. 35–71). Hillsdale: ErlbaumGoogle Scholar
Henderson, H.V., Searle, S.R. (1981). On deriving the inverse of a sum of matrices. SIAM Review, 23, 53–60CrossRefGoogle Scholar
Jöreskog, K.G. (1971). Statistical analysis of sets of congeneric tests. Psychometrika, 36, 109–133CrossRefGoogle Scholar
Jöreskog, K.G. (1971). Simultaneous factor analysis in several populations. Psychometrika, 36, 409–426CrossRefGoogle 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, 221–227CrossRefGoogle Scholar
Lawley, D.N., Maxwell, A.E. (1971). Factor analysis as a statistical method 2nd ed.,, London: ButterworthGoogle Scholar
Lord, M., Novick, M.R. (1968). Statistical theories of mental test scores, Reading, MA: Addison-WesleyGoogle Scholar
Luenberger, D.G. (1969). Optimization by vector space methods, New York: WileyGoogle Scholar
Magnus, J.R., Neudecker, H. (1988). Matrix differential calculus with applications in statistics and economics, Chichester: WileyGoogle Scholar
Rao, C.R., Rao, M.B. (1998). Matrix algebra and its applications to statistics and econometrics, Singapore: World ScientificCrossRefGoogle Scholar
Schmid, J., Leiman, J.M. (1957). The developments of hierarchical factor solutions. Psychometrika, 22, 53–61CrossRefGoogle Scholar
Schneeweiss, H., Mathes, H. (1995). Factor analysis and principal components. Journal of Multivariate Analysis, 55, 105–124CrossRefGoogle Scholar
Serfling, R.J. (1980). Approximation theorems of mathematical statistics, New York: WileyCrossRefGoogle Scholar
Shapiro, A. (1983). Asymptotic distribution theory in the analysis of covariance structures (A unified approach). South African Statistical Journal, 17, 33–81Google Scholar
Shapiro, A. (1985). Identifiability of factor analysis: Some results and open problems. Linear Algebra And Its Applications, 70, 1–7CrossRefGoogle Scholar
Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis. International Statistical Review, 56, 49–62CrossRefGoogle Scholar
Shapiro, A. (1989). Erratum. Linear Algebra And Its Applications, 125, 149–149CrossRefGoogle Scholar
Spearman, C. (1927). The abilities of man, London: MacmillanGoogle Scholar
Spearman, C. (1933). The uniqueness and exactness of g. British Journal Psychology, 24, 106–108Google Scholar
Takane, Y., De Leeuw, J. (1987). On the relationship between item response theory and factor analysis of discretized variables. Psychometrika, 52, 393–408CrossRefGoogle Scholar
Thurstone, L.L. (1947). Multiple-factor analysis: A development and expansion of the vectors of mind, Chicago: University of Chicago PressGoogle Scholar
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, 113–128CrossRefGoogle Scholar