, Volume 64, Issue 1, pp 83–90 | Cite as

Coefficients alpha and reliabilities of unrotated and rotated components

  • Jos M. F. ten Berge
  • Willem K. B. Hofstee


It has been shown by Kaiser that the sum of coefficients alpha of a set of principal components does not change when the components are transformed by an orthogonal rotation. In this paper, Kaiser's result is generalized. First, the invariance property is shown to hold for any set of orthogonal components. Next, a similar invariance property is derived for the reliability of any set of components. Both generalizations are established by considering simultaneously optimal weights for components with maximum alpha and with maximum reliability, respectively. A short-cut formula is offered to evaluate the coefficients alpha for orthogonally rotated principal components from rotation weights and eigenvalues of the correlation matrix. Finally, the greatest lower bound to reliability and a weighted version are discussed.

Key words

reliability coefficient alpha components factors rotation greatest lower bound to reliability 


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  1. Barchard, K.A. & Hakstian, R.A. (1997). The robustness of confidence intervals for coefficient alpha under violation of the assumption of essential parallellism.Multivariate Behavioral Research, 32, 169–191.Google Scholar
  2. Bentler, P.M. (1968). Alpha-maximized factor analysis (Alphamax): Its relation to alpha and canonical factor analysis.Psychometrika, 33, 335–345.Google Scholar
  3. Bentler, P.M., & Woodward, J.A. (1980). Inequalities among lower-bounds to reliability: With applications to test construction and factor analysis.Psychometrika, 45, 249–267.Google Scholar
  4. Cliff, N. (1992). Derivations of the reliability of components.Psychological Bulletin, 71, 667–670.Google Scholar
  5. Cronbach, L.J. (1951). Coefficient alpha and the internal structure of tests.Psychometrika, 16, 297–334.Google Scholar
  6. Cronbach, L.J. (1988). Internal consistency of tests: Analyses old and new.Psychometrika, 53, 63–70.Google Scholar
  7. Dressel, P.L. (1940). Some remarks on the Kuder-Richardson reliability coefficient.Psychometrika, 5, 305–310.Google Scholar
  8. Green, B.F., Jr. (1950). A note on the calculation of weights for maximum battery reliability.Psychometrika, 15, 57–61.Google Scholar
  9. Guttman, L. (1945). A basis for analyzing test-retest reliability.Psychometrika, 10, 225–282.Google Scholar
  10. Kaiser, H.F. (1992). On the invariance of the sum of coefficients alpha for factors under orthogonal rotation.Psychological Reports, 70, 545–546.Google Scholar
  11. Kaiser, H.F., & Caffrey, J. (1965). Alpha factor analysis.Psychometrika, 30, 1–14.Google Scholar
  12. Li, H. (1997). A unifying expression for the maximal reliability of a composite.Psychometrika, 62, 245–249.Google Scholar
  13. Li, H., Rosenthal, R., & Rubin, D.B. (1996). Reliability of measurement in Psychology: From Spearman-Brown to maximal reliability.Psychological Methods, 1, 98–107.Google Scholar
  14. Lord, F.M. (1958). Some relations between Guttman's principal components of scale analysis and other psychometric theory.Psychometrika, 23, 291–296.Google Scholar
  15. Magnus, J.R., & Neudecker, H. (1988).Matrix differential calculus with applications in statistics and econometrics. New York, Wiley.Google Scholar
  16. Mulaik, S.A. (1972).The foundations of factor analysis. New York: McGraw-Hill.Google Scholar
  17. Murakami, T. (1987). Hierarchical component analysis of second order composites I.Japanese Journal of Administrative Behavior, 2, 37–47. (In Japanese)Google Scholar
  18. Novick, M.R., & Lewis, C. (1967). Coefficient alpha and the reliability of composite measurements.Psychometrika, 32, 1–13.Google Scholar
  19. Nunnally, J.C., & Bernstein, I.H. (1994).Psychometric theory (3rd ed.). New York: McGraw-Hill.Google Scholar
  20. Peel, E.A. (1948). Prediction of a complex criterion and battery reliability.British Journal of Psychology, 1, 84–94.Google Scholar
  21. Shapiro, A. (1982). Weighted minimum trace factor analysis.Psychometrika, 47, 243–263.Google Scholar
  22. ten Berge, J.M.F. (1986). Some relationships between descriptive comparisons of components from different studies.Multivariate Behavioral Research, 21, 29–40.Google Scholar
  23. ten Berge, J.M.F., Snijders, T.A.B., & Zegers, F.E. (1981). Computational aspects of the greatest lower bound to reliability and constrained minimum trace factor analysis.Psychometrika, 46, 201–213.Google Scholar
  24. ten Berge, J.M.F., & Zegers, F.E. (1978). A series of lower bounds to the reliability of a test.Psychometrika, 43, 575–579.Google Scholar
  25. Thomson, G.H. (1940). Weighting for maximum battery reliability and prediction.British Journal of Psychology, 30, 357–366.Google Scholar
  26. Van Zijl, J.M., Neudecker, H., & Nel, D.G. (1997, July).On the distribution of the estimator of Cronbach's alpha. Paper presented at the 10th European Meeting of the Psychometric Society, Santiago de Compostela.Google Scholar
  27. Woodhouse, B., & Jackson, P.H. (1977). Lower bounds for the reliability of a total score on a test composed of nonhomogeneous items: II. A search procedure to locate the greatest lower bound.Psychometrika, 42, 579–591.Google Scholar
  28. Woodward, J.A., & Bentler, P.M. (1978). A statistical lower bound to population reliability.Psychological Bulletin, 85, 1323–1326.Google Scholar

Copyright information

© The Psychometric Society 1999

Authors and Affiliations

  • Jos M. F. ten Berge
    • 1
  • Willem K. B. Hofstee
    • 1
  1. 1.Heijmans Institute of Psychological ResearchUniversity of GroningenGroningenThe Netherlands

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