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Inequalities Among Lower Bounds to Reliability: With Applications to Test Construction and Factor Analysis

Published online by Cambridge University Press:  01 January 2025

P. M. Bentler  and
J. Arthur Woodward
P. M. Bentler*
Affiliation:
University of California, Los Angeles
J. Arthur Woodward
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to P. M. Bentler, Department of Psychology, University of California, Los Angeles, California 90024.
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Abstract

A chain of lower-bound inequalities leading to the greatest lower bound to reliability is established for the internal consistency of a composite of unit-weighted components. The chain includes the maximum split-half coefficient, the lowest coefficient consistent with nonimaginary common factors, and the lowest coefficient consistent with nonimaginary common and unique factors. Optimization theory is utilized to determine the conditions that are requisite for the inequalities. Convergence proofs demonstrate that the coefficients can be attained. Rapid algorithms obtain estimates of the coefficients with sample data. The theory yields methods for splitting items into maximally similar sets and for exploratory factor analysis based on a theoretical solution to the communality problem.

Keywords

reliabilityinternal consistencytest theoryfactor analysis
Type
Original Paper
Information
Psychometrika , Volume 45 , Issue 2 , June 1980 , pp. 249 - 267
Copyright
Copyright © 1980 The Psychometric Society

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Footnotes

This investigation was supported in part by a Research Scientist Development Award (K02-DA00017) and a research grant (DA01070) from the U.S. Public Health Service. The manuscript profited substantially from the critiques of several anonymous reviewers, whose assistance is gratefully acknowledged.

References

Reference Notes

Bentler, P. M. & Woodward, J. A. Inequalities among lower bounds to reliability. Paper presented at the annual meeting of the Psychometric Society, Chapel Hill, June, 1977.Google Scholar
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Rozeboom, W. W. Personal communication, March, 1979.Google Scholar
Bentler, P. M. Fast, legitimate, rank-independent factor analytic procedures. Paper presented at the annual meeting of the Society of Multivariate Experimental Psychology, Fort Worth, October, 1972.Google Scholar

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