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Testing a Linear Relation between True Scores of Two Measures

Published online by Cambridge University Press:  01 January 2025

Walter Kristof
Walter Kristof*
Affiliation:
Educational Testing Service
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Abstract

We concern ourselves with the hypothesis that two variables have a perfect disattenuated correlation, hence measure the same trait except for errors of measurement. This hypothesis is equivalent to saying, within the adopted model, that true scores of two psychological tests satisfy a perfect linear relation. Statistical tests of this hypothesis are derived when the relation is specified with the exception of the additive constant. Two approaches are presented and various assumptions concerning the error parameters are used. Then the results are reinterpreted in terms of the possible existence of an unspecified perfect linear relation between true scores of two psychological tests. A numerical example is appended by way of illustration.

Type
Original Paper
Information
Psychometrika , Volume 38 , Issue 1 , March 1973 , pp. 101 - 111
Copyright
Copyright © 1973 The Psychometric Society

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Footnotes

*

Research reported in this paper has been supported by grant GB-18230 from National Science Foundation.

†

Throughout this paper the term “linear relation” indicates a perfect linear relationship between variables, not just linearity of regression.

References

Cacoullos, T. A relation between t and F-distributions. Journal of the American Statistical Association, 1965, 60, 528–531Google Scholar
Forsyth, R. A., Feldt, L. S. An investigation of empirical sampling distributions of correlation coefficients corrected for attenuation. Educational and Psychological Measurement, 1969, 29, 61–72CrossRefGoogle Scholar
Forsyth, R. A., Feldt, L. S. Some theoretical and empirical results related to McNemar's test that the population correlation coefficient corrected for attenuation equals 1.0. American Educational Research Journal, 1970, 7, 197–207CrossRefGoogle Scholar
Jöreskog, K. G. Statistical analysis of sets of congeneric tests. Psychometrika, 1971, 36, 109–133CrossRefGoogle Scholar
Kristof, W. On a statistic arising in testing correlation. Psychometrika, 1972, 37, 377–384CrossRefGoogle Scholar
Lord, F. M. A significance test for the hypothesis that two variables measure the same trait except for errors of measurement. Psychometrika, 1957, 22, 207–220CrossRefGoogle Scholar
Lord, F. M. Testing if two measuring procedures measure the same dimension. Psychological Bulletin, 1973, 79, 71–72CrossRefGoogle Scholar
McNemar, Q. Attenuation and interaction. Psychometrika, 1958, 23, 259–266CrossRefGoogle Scholar
Scheffé, H. The analysis of variance, 1959, New York: WileyGoogle Scholar
Villegas, G. Confidence region for a linear relation. The Annals of Mathematical Statistics, 1964, 35, 780–788CrossRefGoogle Scholar