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Lower Bounds for the Reliability of the Total Score on a Test Composed of Non-Homogeneous Items: II: A Search Procedure to Locate the Greatest Lower Bound

Published online by Cambridge University Press:  01 January 2025

Brian Woodhouse  and
Paul H. Jackson
Brian Woodhouse
Affiliation:
University College of Wales
Paul H. Jackson*
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

Finding the greatest lower bound for the reliability of the total score on a test comprising n non-homogenous items with dispersion matrix Σx is equivalent to maximizing the trace of a diagonal matrix ΣE with elements θI, subject to ΣE and ΣT=Σx − ΣE being non-negative definite. The cases n=2 and n=3 are solved explicity. A computer search in the space of the θi is developed for the general case. When Guttman's λ4 (maximum split-half coefficient alpha) is not the g.l.b., the maximizing set of θi makes the rank of ΣT less than n − 1. Numerical examples of various bounds are given.

Keywords

reliability boundsnon-homogeneous compositessearch proceduretrace minimization with constraints
Type
Original Paper
Information
Psychometrika , Volume 42 , Issue 4 , December 1977 , pp. 579 - 591
Copyright
Copyright © 1977 The Psychometric Society

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Footnotes

Present affiliation of the first author: St. Hild's College of Education, Durham City, England.

References

Reference Note

Woodhouse, B. Lower bounds for the reliability of a test. University of Wales M.Sc. thesis, 1976.Google Scholar

References

Bentler, P. M. A lower-bound method for the dimension-free measurement of internal consistency. Social Science Research, 1972, 1, 343–357.CrossRefGoogle Scholar
Guttman, L. A basis for analyzing test-retest reliability. Psychometrika, 1945, 10, 255–282.CrossRefGoogle ScholarPubMed