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 Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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