Abstract
The asymptotic normal distribution of the maximum likelihood estimator of Cronbach's alpha (under normality) is derived for the case when no assumptions are made about the covariances among items. The asymptotic distribution is also considered for the special case of compound symmetry and compared to the exact distribution.
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References
Browne, M.W. (1974). Generalized least squares estimation in the analysis of covariance structures.South African Statistical Journal, 8, 1–24.
Cronbach, L.J. (1951). Coefficient alpha and the internal structure of tests.Psychometrika, 16, 297–334.
Feldt, L.S. (1965). The approximate sampling distribution of Kuder-Richardson reliability coefficient twenty.Psychometrika, 30, 357–370.
Feldt, L.S., & Brennan, R.L. (1989).Reliability. In R.L. Linn (Ed.),Educational measurement (3rd ed.). New York: Macmillan.
Fisher, R.A. (1991). Statistical methods for research workers. In J.H. Bennett (Ed.),Statistical methods, experimental design, and scientific inference. Oxford: Oxford University Press.
Kristof, W. (1963). The statistical theory of stepped-up reliability when a test has been divided into several equivalent parts.Psychometrika, 28, 221–238.
Lord, F.M., & Novick, M.R. (1968).Statistical theories of mental test scores. Reading, MA: Addison-Wesley.
Magnus, J.R., & Neudecker, H. (1999).Matrix differential calculus with applications in statistics and econometrics (rev. ed.) Chichester: John Wiley and Sons.
Mood, A., Graybill, F.A., & Boes, D.C. (1974).Introduction to the theory of statistics (3rd ed.). New York: McGraw Hill.
Muirhead R.J. (1982).Aspects of multivariate statistical theory. New York: Wiley.
Nel, D.G. (1980). On matrix differentiation in statistics.South African Statistical Journal, 14, 137–193.
Nel, D.G. (1985). A matrix derivation of the asymptotic covariance matrix of sample correlation coefficients.Linear algebra and its applications, 67, 137–145.
Roth, W.E. (1934). On direct product matrices.Bulletin of the American Mathematical Society, 40, 461–468.
Siotani, M., Hayakawa, T., & Fujikoshi, Y. (1985).Modern multivariate statistical analysis: A graduate course and handbook. Columbus, Ohio: American Sciences Press.
Traub, R.E. (1994).Reliability for the social sciences (Measurement methods for the social sciences, Volume 3). London: Sage Publications.
Winer, B.J. (1991).Statistical principles in experimental design, (3rd ed.) New York: McGraw Hill.
Zacks, S. (1971).The theory of statistical inference. New York: Wiley.
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The authors would like to thank Willem J. Heiser, an associate editor and the reviewers for valuable and helpful comments to improve the quality of this work.
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van Zyl, J.M., Neudecker, H. & Nel, D.G. On the distribution of the maximum likelihood estimator of Cronbach's alpha. Psychometrika 65, 271–280 (2000). https://doi.org/10.1007/BF02296146
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DOI: https://doi.org/10.1007/BF02296146