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Lower Bounds for the Reliability of the Total Score on a Test Composed of Non-Homogeneous Items: I: Algebraic Lower Bounds

Published online by Cambridge University Press:  01 January 2025

Paul H. Jackson  and
Christian C. Agunwamba
Paul H. Jackson*
Affiliation:
University College of Wales
Christian C. Agunwamba
Affiliation:
University College of Wales
*
Requests for reprints should be sent to P. H. Jackson, Department of Statistics, University College of Wales, Aberystwyth, Dyfed, Wales.
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Abstract

Let ∑x be the (population) dispersion matrix, assumed well-estimated, of a set of non-homogeneous item scores. Finding the greatest lower bound for the reliability of the total of these scores is shown to be equivalent to minimizing the trace of ∑x by reducing the diagonal elements while keeping the matrix non-negative definite. Using this approach, Guttman's bounds are reviewed, a method is established to determine whether his λ4 (maximum split-half coefficient alpha) is the greatest lower bound in any instance, and three new bounds are discussed. A geometric representation, which sheds light on many of the bounds, is described.

Keywords

reliability boundscoefficient alphanon-homogeneous composites
Type
Original Paper
Information
Psychometrika , Volume 42 , Issue 4 , December 1977 , pp. 567 - 578
Copyright
Copyright © 1977 The Psychometric Society

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Footnotes

Present affiliation of the second author: Department of Statistics, University of Nigeria (Nsukka Campus). Work on this paper was carried out while on study leave in Aberystwyth.

References

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