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Coefficient Alpha and the Reliability of Composite Measurements

Published online by Cambridge University Press:  01 January 2025

Melvin R. Novick  and
Charles Lewis
Melvin R. Novick
Affiliation:
Educational Testing service*
Charles Lewis
Affiliation:
Princeton University
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Abstract

Following a general approach due to Guttman, coefficientα is rederived as a lower bound on the reliability of a test. A necessary and sufficient condition under which equality is attained in this inequality and hence that α is equal to the reliability of the test is derived and shown to be closely related to the recent redefinition of the concept of parallel measurements due to Novick. This condition is then also shown to be closely related to the unit rank assumption originally adopted by Kuder and Richardson in the derivation of their formula 20. The assumption later adopted by Jackson and Ferguson and the one adopted by Gulliksen are shown to be related to the necessary and sufficient condition derived here. It is then pointed out that the statement that “coefficient α is equal to the mean of the split-half reliabilities” is true only under the restricted condition assumed by Cronbach in the body of his derivation of this result. Finally some limitations on the uses of any function of α as a measure of internal consistency are noted.

Type
Original Paper
Information
Psychometrika , Volume 32 , Issue 1 , March 1967 , pp. 1 - 13
Copyright
Copyright © 1967 The Psychometric Society

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Footnotes

*

Research reported herein was supported in part by the Logistics and Mathematical Statistics Branch of the Office of Naval Research under contract Nonr-4866(00), NR 042-249, Melvin R. Novick, principal investigator. Reproduction, translation, publication, use and disposal in whole or in part by or for the United States Government is permitted. The authors are indebted to Frederic M. Lord, Michael Browne and an unknown referee for constructive criticism of earlier drafts of this manuscript.

References

Cronbach, L. J. Coefficient alpha and the internal structure of tests. Psychometrika, 1951, 16, 297–334.CrossRefGoogle Scholar
Cronbach, L. J., Rajaratnam, Nageswari and Gleser, Goldine C.. Theory of generalizability: A liberalization of reliability theory. Brit. J. Statist. Psychol., 1963, 16, 137–163.CrossRefGoogle Scholar
Cureton, E. E. The definition and estimation of test reliability. Educ. Psychol. Measmt., 1958, 18, 715–738.CrossRefGoogle Scholar
Dressel, P. L. Some remarks on the Kuder-Richardson reliability coefficient. Psychometrika, 1940, 5, 305–310.CrossRefGoogle Scholar
Gulliksen, H. Theory of mental tests, New York: Wiley, 1950.CrossRefGoogle Scholar
Guttman, L. Reliability formulas that do not assume experimental independence. Psychometrika, 1953, 18, 225–239.CrossRefGoogle Scholar
Hoyt, C. Test reliability estimated by analysis of variance. Psychometrika, 1941, 6, 153–160.CrossRefGoogle Scholar
Jackson, R. W. B., & Ferguson, G. A. Studies on the reliability of tests. Bulletin 12, Department of Educational Research, University of Toronto, 1941.Google Scholar
Kuder, G. F., andRichardson, M. W. The theory of the estimation of test reliability. Psychometrika, 1937, 2, 151–160.CrossRefGoogle Scholar
Lord, F. M. Statistical inferences about true scores. Psychometrika, 1959, 24, 1–17.CrossRefGoogle Scholar
Lyerly, S.B. The Kuder-Richardson formula (21) as a split-half coefficient and some remarks on its basic assumption. Psychometrika, 1958, 23, 267–270.Google Scholar
Novick, M. R. The axioms and principal results of classical test theory. J. Math. Psychol., 1966, 3, 1–18.CrossRefGoogle Scholar
Rulon, P. J. A simplified procedure for determining the reliability of a test by splithalves. Harvard Educ. Rev., 1939, 9, 99–103.Google Scholar