Abstract
The seven analysis-of-variance mean squares for an unreplicated three-way classification may be written as linear combinations of a mean variance and three mean covariances. Formulas are presented for computing the mean variances and mean covariances from linear combinations of mean squares. The relevance of these formulas for assessing rater biases and trait independence is discussed, a numerical example is provided, and proposed extensions are briefly noted.
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Abelson, R. P. A discriminant approach to factoring three-way data tables.Amer. Psychologist, 1958,13, 375. (Abstract) (More extensive reports privately circulated.)
Campbell, D. T. and Fiske, D. W. Convergent and discriminant validation by the multitrait-multimethod matrix.Psychol. Bull., 1959,56, 81–105.
Chi, P.-L. Statistical analysis of personality rating.J. exp. Educ., 1937,5, 229–245.
Cornfield, J. and Tukey, J. W. Average values of mean squares in factorials.Ann. math. Statist., 1956,27, 907–949.
Cronbach, L. J. Coefficient alpha and the internal structure of tests.Psychometrika, 1951,16, 297–334.
Cronbach, L. J. The two disciplines of scientific psychology.Amer. Psychologist, 1957,12, 671–684.
Grant, D. A. Analysis-of-variance tests in the analysis and comparison of curves.Psychol. Bull., 1956,53, 141–154.
Guilford, J. P.Psychometric methods (2nd ed.) New York: McGraw-Hill, 1954.
Gulliksen, H.Theory of mental tests. New York: Wiley, 1950.
Hoyt, C. J. Test reliability estimated by analysis of variance.Psychometrika, 1941,6, 153–160.
Hoyt, C. J. and Stunkard, C. L. Estimation of test reliability for unrestricted item scoring methods.Educ. psychol. Measmt, 1951,12, 756–758.
Humphreys, L. G. Note on the multitrait-multimethod matrix.Psychol. Bull., 1960,57, 86–88.
Johnson, D. M. and Vidulich, R. N. Experimental manipulation of the halo effect.J. appl. Psychol., 1956,40, 130–134.
Stanley, J. C. Fixed, random, and mixed models in the analysis of variance as special cases of a finite model.Psychol. Rep., 1956,2, 369.
Stanley, J. C. K-R 20 as the stepped-up mean item intercorrelation.14th Yrbk Natl Coun. Meas. used in Educ., 1957. Pp. 78–92.
Wilk, M. B. and Kempthorne, O. Fixed, mixed, and random models.J. Amer. statist. Ass., 1955,50, 1144–1167.
Wilk, M. B. and Kempthorne, O. Derived linear models and their use in the analysis of randomized experiments.WADC Tech. Rep. 55–244, Vol. II, Mar. 1956.
Wilk, M. B. and Kempthorne, O. Some aspects of the analysis of factorial experiments in a completely randomized design.Ann. math. Statist., 1956,27, 950–985.
Willingham, W. W. and Jones, M. B. On the identification of halo through analysis of variance.Educ. psychol. Measmt, 1958,18, 403–407.
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The research reported herein was performed pursuant to a contract with the United States Office of Education, Department of Health, Education, and Welfare. The assistance of Sister M. Jacinta Mann, S. C., at one stage of this investigation is gratefully acknowledged.
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Stanley, J.C. Analysis of unreplicated three-way classifications, with applications to rater bias and trait independence. Psychometrika 26, 205–219 (1961). https://doi.org/10.1007/BF02289715
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DOI: https://doi.org/10.1007/BF02289715