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The Statistical Theory of Stepped-Up Reliability Coefficients when a Test has been divided into Several Equivalent Parts

Published online by Cambridge University Press:  01 January 2025

Walter Kristof
Walter Kristof*
Affiliation:
Educational Testing Service and University of Marburg/Lahn, Germany
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Abstract

This paper is a presentation of the statistical sampling theory of stepped-up reliability coefficients when a test has been divided into any number of equivalent parts. Maximum-likelihood estimators of the reliability are obtained and shown to be biased. Their sampling distributions are derived and form the basis of the definition of new unbiased estimators with known sampling distributions. These unbiased estimators have a smaller sampling variance than the maximum-likelihood estimators and are, because of this and some other favorable properties, recommended for general use. On the basis of the variances of the unbiased estimators the gain in accuracy in estimating reliability connected with further division of a test can be expressed explicitly. The limits of these variances and thus the limits of accuracy of estimation are derived. Finally, statistical small sample tests of the reliability coefficient are outlined. This paper also covers the sampling distribution of Cronbach's coefficient alpha.

Type
Original Paper
Information
Psychometrika , Volume 28 , Issue 3 , September 1963 , pp. 221 - 238
Copyright
Copyright © 1963 The Psychometric Society

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References

Cronbach, L. J. Coefficient alpha and the internal structure of tests. Psychometrika, 1951, 16, 297–334.CrossRefGoogle Scholar
Gulliksen, H. Theory of mental tests, New York: Wiley, 1950.CrossRefGoogle Scholar
Jackson, R. and Ferguson, G. Studies on the reliability of tests. Toronto: Dep. Educ. Res. Bull. No. 12, 1941.Google Scholar
Kristof, W. Small sample frequency distribution of stepped-up reliability coefficients based on split-halves, Princeton, N. J.: Educational Testing Service, 1962.CrossRefGoogle Scholar
Lewis, D. Quantitative methods in psychology, New York: McGraw-Hill, 1960.CrossRefGoogle Scholar
Wilks, S. S. Sample criteria for testing equality of means, equality of variances, and equality of covariances in a normal multivariate distribution. Ann. math. Statist., 1946, 17, 257–281.CrossRefGoogle Scholar
Wilks, S. S. Mathematical statistics, New York: Wiley, 1962.Google Scholar