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Statistical Inferences about the Error Variance

Published online by Cambridge University Press:  01 January 2025

Walter Kristof
Walter Kristof*
Affiliation:
Educational Testing Service, Princeton, N. J. University of Marburg/Lahn, Germany
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Abstract

This paper is a presentation of an essential part of the sampling theory of the error variance and the standard error of measurement. An experimental assumption is that several equivalent tests with equal variances are available. These may be either final forms of the same test or obtained by dividing one test into several parts. The simple model of independent and normally distributed errors of measurement with zero mean is employed. No assumption is made about the form of the distributions of true and observed scores. This implies unrestricted freedom in defining the population. First, maximum-likelihood estimators of the error variance and the standard error of measurement are obtained, their sampling distributions given, and their properties investigated. Then unbiased estimators are defined and their distributions derived. The accuracy of estimation is given special consideration from various points of view. Next, rigorous statistical tests are developed to test hypotheses about error variances on the basis of one and two samples. Also the construction of confidence intervals is treated. Finally, Bartlett's test of homogeneity of variances is used to provide a multi-sample test of equality of error variances.

Type
Original Paper
Information
Psychometrika , Volume 28 , Issue 2 , June 1963 , pp. 129 - 143
Copyright
Copyright © 1963 The Psychometric Society

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References

Alexander, H. W. The estimation of reliability when several trials are available. Psychometrika, 1947, 12, 79–100.CrossRefGoogle ScholarPubMed
Bartlett, M. S. Properties of sufficiency and statistical tests. Proceedings of the Royal Society (London), 1937, 50, 1–273.Google Scholar
Ebel, R. L. Estimation of the reliability of ratings. Psychometrika, 1951, 16, 407–424.CrossRefGoogle Scholar
Green, B. F. Jr. A test of the equality of standard errors of measurement. Psychometrika, 1950, 15, 251–257.CrossRefGoogle ScholarPubMed
Gulliksen, H. Theory of mental tests, New York: Wiley, 1950.CrossRefGoogle Scholar
Hoyt, C. Test reliability estimated by analyses of variance. Psychometrika, 1941, 6, 153–160.CrossRefGoogle Scholar
Johnson, P. O. Statistical methods in research, New York: Prentice-Hall, 1950.Google Scholar
Kendall, M. G. and Stuart, A. The advanced theory of statistics. Vol. 1, New York: Hafner, 1958.Google Scholar
Kristof, W. The statistical theory of stepped-up reliability coefficients when a test has been divided into several equivalent parts. Princeton, N. J.: Educational Testing Service Research Bulletin 62-20, 1962.CrossRefGoogle Scholar
Moonan, W. J. Simultaneous examination and method analysis by variance algebra. J. exp. Educ., 1955, 23, 253–257.CrossRefGoogle Scholar
Walker, Helen M. and Lev, J. Statistical inference, New York: Holt, 1953.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