Reactions to the Attenuation Paradox

"Perhaps the most paradoxical aspect of the attenuation paradox is that Gulliksen, who appears to deserve credit for discovering it, failed to include any reference to it in his comprehensive summary of mental test theory" (Loevinger 1954 p. 501).

In 1945, Gulliksen discovered that, under certain conditions, increasing the reliability of test scores decreases their validity. Professional reaction to the "attenuation paradox" of classical true-score theory (CTT) illustrates five typical reactions to challenges of familiar theories:

1) Gulliksen (1950) ignores the paradox because he decided that true-score theory provides useful results concerning test reliability anyway. All current psychometric texts follow Gulliksen's footsteps.

2) Lord (1952 p. 501) implies that the paradox is due to lack of skill on the part of psychometricians, rather than a deficient theory. He suggests a curvilinear index to produce a different summary of the relationship between reliability and validity.

3) Tucker (1946 p. 11) accepts the paradox in theory, but resists its implications for practice. "A result which seemed amazing was the low values of the item reliabilities which yielded best measurement... It is safer for the reliabilities to be too high."

4) Davis (1952 p.105) introduces an untestable hypothesis, that of "common sense" (Loevinger 1954 p.105) to save the theory. "It is not proper to deduce from Tucker's data that to obtain high test validity one should make items of low reliability. There is no inconsistency between high item reliability and efficient measurement."

5) Humphreys (1956) accepts the paradox and rejects true-score theory and its assumption that test scores are interval level data. He proposes an alternative theory based on the normal distribution in which test scores are ordinal. His theory explains and overcomes the attenuation paradox, but has its own anomalies.

Though hardly a reaction to the attenuation paradox, Rasch theory does provide a useful perspective for understanding it, as will be shown in my next column.

George Engelhard, Jr.

Davis F B (1952) Item analysis in relation to educational and psychological testing. Psychological Bulletin 49(2) 97-121

Gulliksen H (1945) The relation of item difficulty and inter-item correlation to test variance and reliability. Psychometrika 10(2) 79-91

Gulliksen H (1950) Theory of mental tests. Wiley

Humphreys L G (1956) The normal curve and the attenuation paradox in test theory. Psychological Bulletin 53(6) 472-3

Loevinger J (1954) The attenuation paradox in test theory. Psychological Bulletin 51 493-504

Lord F (1952) A theory of test scores. Psychometrika Monograph No. 7.

Tucker L R (1946) Maximum validity of a test with equivalent items. Psychometrika 11(1) 1-13

Reactions to the attenuation paradox. Engelhard G Jr. … 1993, 7:2 p.294

Reactions to the attenuation paradox. Engelhard G Jr. … Rasch Measurement Transactions, 1993, 1993, 7:2 p.294

Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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