, Volume 52, Issue 1, pp 125–135 | Cite as

The rank of reduced dispersion matrices

  • Paul A. Bekker
  • Jan de Leeuw


Psychometricians working in factor analysis and econometricians working in regression with measurement error in all variables are both interested in the rank of dispersion matrices under variation of the diagonal elements. Psychometricians concentrate on cases in which low rank can be attained, preferably rank one, the Spearman case. Econometricians cocentrate on cases in which the rank cannot be reduced below the number of variables minus one, the Frisch case. In this paper we give an extensive historial discussion of both fields, we prove the two key results in a more satisfactory and uniform way, we point out various small errors and misunderstandings, and we present a methodological comparison of factor analysis and regression on the basis of our results.

Key words

factor analysis errors of measurement structural regression functional models communalities errors in variables 


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  1. Albert, A. A. (1944a). The matrices of factor analysis.Proceedings of the National Academy of Sciences, 30, 90–95.Google Scholar
  2. Albert, A. A. (1944b). The minimum rank of a correlation matrix.Proceedings of the National Academy of Sciences, 30, 144–148.Google Scholar
  3. Bekker, P. A. (1986). Comment on identification in the linear errors in variables model.Econometrica, 54, 215–217.Google Scholar
  4. Bekker, P. A., Kapteyn, A., & Wansbeek, T. J. (1984). Measurement error and endogeneity in regression: Bounds for ML and VI estimates. In T. K. Dijkstra (ed.),Misspecification analysis. Berlin: Springer Verlag.Google Scholar
  5. Burt, C. (1909). Experimental tests of general intelligence.British Journal of Psychology, 3, 94–177.Google Scholar
  6. Camp, B. H. (1932). The converse of Spearman's two-factor theorem.Biometrika, 24, 418–428.Google Scholar
  7. Dhondt, A. (1960). Sur une généralisation d'un theorème de R. Frisch en analyse de confluence [On a generalization of a theorem by R. Frisch about confluence analysis].Cahiers du Centre d'Etude de Recherche Operationelle, 2, 37–46.Google Scholar
  8. Fiedler, M. (1969). A characterization of tridiagonal matrices.Linear Algebra and its Applications, 2, 191–197.Google Scholar
  9. Frisch, R. (1934).Statistical confluence analysis by means of complete regression systems (Publication No. 5). Oslo: University of Oslo, Economic Institute.Google Scholar
  10. Garnett, J. C. M. (1919a). On certain independent factors in mental measurement.Proceedings of the Royal Society of London, 96, 91–111.Google Scholar
  11. Garnett, J. C. M. (1919b). General ability, cleverness, and purpose.British Journal of Psychology, 9, 345–366.Google Scholar
  12. Garnett, J. C. M. (1920). The single general factor in dissimilar mental measurements.British Journal of Psychology, 10, 242–258.Google Scholar
  13. Guttman, L. (1954). Some necessary conditions for common factor analysis.Psychometrika, 19, 149–161.Google Scholar
  14. Guttman, L. (1956). “Best possible” systematic estimates of communalities.Psychometrika, 21, 273–285.Google Scholar
  15. Guttman, L. (1958). To what extent can communalities reduce rank?Psychometrika, 23, 297–308.Google Scholar
  16. Hakim, M., Kochard, E. O., Olivier, J. P., & Terouanne, E. (1976).Sur les traces de Spearman [On the traces of Spearman] (Série Recherche N 25). Paris: Cahiers de Bureau Universitaire de Recherche Operationelle, Université Pierre et Marie Curie.Google Scholar
  17. Hart, B., & Spearman, C. E. (1912). General ability, its existence and nature.British Journal of Psychology, 5, 51–84.Google Scholar
  18. Hearnshaw, L. S. (1981).Cyril Burt, psychologist. New York: Random House.Google Scholar
  19. Kalman, R. E. (1982a). System identification from noisy data. In A. R. Bednarek & L. Cesari (Eds.),Dynamical Systems II. New York: Academic Press.Google Scholar
  20. Kalman, R. E. (1982b). Identification from real data. In M. Hazewinkel & A. H. G. Rinnooy Kan (Eds.),Current developments in the interface economics, econometrics, mathematics. Dordrecht: D. Reidel.Google Scholar
  21. Kalman, R. E. (1983). Identifiability and modelling in econometrics. In P. R. Krishnaiah (Ed.),Developments in Statistics 4. New York: Academic Press.Google Scholar
  22. Kalman, R. E. (1984). We can do something about multicollinearity!Communications in Statistics—Theory and Methods, 13, 115–125.Google Scholar
  23. Kelley, T. L. (1928).Crossroads in the mind of man: A study of differentiable mental abilities. Stanford University Press.Google Scholar
  24. Klepper, S. & Leamer, E. E. (1984). Consistent sets of estimates for regressions with errors in all variables.Econometrica, 52, 162–183.Google Scholar
  25. Koopmans, T. C. (1937).Linear regression analysis of economic time series. Haarlem: De Erven F. Bohn NV.Google Scholar
  26. Krueger, F. & Spearman, C. E. (1907). Die Korrelation zwischen verscheidenen geistigen Leistungsfähigkeiten [The correlation between several mental abilities].Zeitschrift für Psychologie, 54, 50–114.Google Scholar
  27. Ledermann, W. (1937). On the rank of the reduced correlational matrix in multiple factor analysis.Psychometrika, 2, 85–93.Google Scholar
  28. Marsaglia, G., & Styan, G. P. H. (1974). Equalities and inequalities for ranks of matrices.Linear and Multilinear Algebra, 2, 269–292.Google Scholar
  29. Patefield, W. M. (1981). Multivariate linear relationships: Maximum likelihood estimation and regression bounds.Journal of the Royal Statistical Society (Series B), 43, 342–352.Google Scholar
  30. Pearson, K. (1901). On lines and planes of closest fit to points in space.Philosophical Magazine 2, 559–572.Google Scholar
  31. Reiersøl, O. (1941). Confluence analysis by means of lag moments and other methods of confluence analysis.Econometrica, 9, 1–24.Google Scholar
  32. Reiersøl, O. (1945). Confluence analysis by means of instrumental sets of variables.Arkiv för Mathematik, Astronomi och Fysik, 32(A), 1–119.Google Scholar
  33. Rheinboldt, W. C., & Sheperd, R. A. (1974). On a characterization of tridiagonal matrices by M. Fiedler.Linear Algebra and its Applications, 8, 87–90.Google Scholar
  34. Shapiro, A. (1982a). Rank reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis.Psychometrika, 47, 187–199.Google Scholar
  35. Shapiro, A. (1982b). Weighted minimum trace factor analysis.Psychometrika, 47, 243–264.Google Scholar
  36. Spearman, C. E. (1904). General intelligence objectively measured and defined.American Journal of Psychology, 15, 201–299.Google Scholar
  37. Spearman, C. E. (1927).The abilities of man. London: McMillan.Google Scholar
  38. Spearman, C. E., & Holzinger, K. J. (1924). The sampling error in the theory of two factors.British Journal of Psychology, 15, 17–19.Google Scholar
  39. Spearman, C. E., & Holzinger, K. J. (1925). Note on the sampling error of tetrad differences.British Journal of Psychology, 15, 86–89.Google Scholar
  40. Thurstone, L. L. (1935).The vectors of mind. University of Chicago Press.Google Scholar
  41. Tumura, Y., & Fukutomi, K. (1968). On the identification in factor analysis.Report of Statistical Application Research, Union of Japanese Scientists and Engineers, 15, 6–11.Google Scholar
  42. Wilson, E. B. (1928). On hierarchical correlation systems.Proceedings of the National Academy of Sciences, 14, 283–291.Google Scholar
  43. Wilson, E. B. (1929). Review of Kelley, Crossroads in the mind of man.Journal of General Psychology, 2, 153–169.Google Scholar
  44. Wilson, E. B., & Worcester, J. (1934). The resolution of four tests.Proceedings of the National Academy of Sciences, 20, 189–192.Google Scholar
  45. Wilson, E. B., & Worcester, J. (1939). The resolution of six tests into three general factors.Proceedings of the National Academy of Sciences, 25, 73–79.Google Scholar

Copyright information

© The Psychometric Society 1987

Authors and Affiliations

  • Paul A. Bekker
    • 1
  • Jan de Leeuw
    • 2
  1. 1.Department of EconometricsTilburg UniversityThe Netherlands
  2. 2.Department of Data Theory FSW/RULLeidenThe Netherlands

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