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The Greatest Lower Bound to the Reliability of a Test and the Hypothesis of Unidimensionality

Published online by Cambridge University Press:  01 January 2025

Jos M. F. Ten Berge  and
Gregor Sočan
Jos M. F. Ten Berge*
Affiliation:
University of Groningen
Gregor Sočan
Affiliation:
University of Groningen
*
Request for reprints should be sent to Jos Ten Berge, Department of Psychology, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, Netherlands. E-mail: tberge@ppsw.rug.nl.
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Abstract

To assess the reliability of congeneric tests, specifically designed reliability measures have been proposed. This paper emphasizes that such measures rely on a unidimensionality hypothesis, which can neither be confirmed nor rejected when there are only three test parts, and will invariably be rejected when there are more than three test parts. Jackson and Agunwamba's (1977) greatest lower bound to reliability is proposed instead. Although this bound has a reputation for overestimating the population value when the sample size is small, this is no reason to prefer the unidimensionality-based reliability. Firstly, the sampling bias problem of the glb does not play a role when the number of test parts is small, as is often the case with congeneric measures. Secondly, glb and unidimensionality based reliability are often equal when there are three test parts, and when there are more test parts, their numerical values are still very similar. To the extent that the bias problem of the greatest lower bound does play a role, unidimensionality-based reliability is equally affected. Although unidimensionality and reliability are often thought of as unrelated, this paper shows that, from at least two perspectives, they act as antagonistic concepts. A measure, based on the same framework that led to the greatest lower bound, is discussed for assessing how close is a set of variables to unidimensionality. It is the percentage of common variance that can be explained by a single factor. An empirical example is given to demonstrate the main points of the paper.

Keywords

Reliabilitycongeneric testunidimensionality of a test
Type
Theory And Methods
Information
Psychometrika , Volume 69 , Issue 4 , December 2004 , pp. 613 - 625
Copyright
Copyright © 2004 The Psychometric Society

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Footnotes

The authors are obliged to Henk Kiers for commenting on a previous version. Gregor Sočan is now at the University of Ljubljana.

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