Hostname: page-component-5d84bcc8dc-7kmtv Total loading time: 0 Render date: 2026-08-15T12:32:16.252Z Has data issue: false hasContentIssue false
  • English
  • Français

Coefficients Alpha, Beta, Omega, and the glb: Comments on Sijtsma

Published online by Cambridge University Press:  01 January 2025

William Revelle  and
Richard E. Zinbarg
William Revelle*
Affiliation:
Department of Psychology, Northwestern University
Richard E. Zinbarg
Affiliation:
Department of Psychology, The Family Institute at Northwestern University, Northwestern University
*
Requests for reprints should be sent to William Revelle, Department of Psychology, Northwestern University, Evanston, IL, USA. E-mail: revelle@northwestern.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

There are three fundamental problems in Sijtsma (Psychometrika, 2008): (1) contrary to the name, the glb is not the greatest lower bound of reliability but rather is systematically less than ωt (McDonald, Test theory: A unified treatment, Erlbaum, Hillsdale, 1999), (2) we agree with Sijtsma that when considering how well a test measures one concept, α is not appropriate, but recommend ωt rather than the glb, and (3) the end user needs procedures that are readily available in open source software.

Keywords

reliabilityinternal consistencyhomogeneitytest theorycoefficient alphacoefficient omegacoefficient beta

Information

Type
Theory and Methods
Information
Psychometrika , Volume 74 , Issue 1 , March 2009 , pp. 145 - 154
Copyright
Copyright © 2008 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bentler, P., Woodward, J. (1980). Inequalities among lower bounds to reliability: With applications to test construction and factor analysis. Psychometrika, 45(2), 249–267.10.1007/BF02294079CrossRefGoogle Scholar
Brown, W. (1910). Some experimental results in the correlation of mental abilities. British Journal of Psychology, 3(3), 296–322.Google Scholar
Cronbach, L.J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297–334.10.1007/BF02310555CrossRefGoogle Scholar
Cronbach, L.J. (1988). Internal consistency of tests: Analyses old and new. Psychometrika, 53(1), 63–70.10.1007/BF02294194CrossRefGoogle Scholar
Cronbach, L.J., Shavelson, R.J. (2004). My current thoughts on coefficient alpha and successor procedures. Educational and Psychological Measurement, 64(3), 391–418.10.1177/0013164404266386CrossRefGoogle Scholar
De Leeuw, J. (1983). Models and methods for the analysis of correlation coefficients. Journal of Econometrics, 22 1–2113–137.10.1016/0304-4076(83)90096-9CrossRefGoogle Scholar
Guttman, L. (1945). A basis for analyzing test-retest reliability. Psychometrika, 10(4), 255–282.10.1007/BF02288892CrossRefGoogle ScholarPubMed
Jackson, P., Agunwamba, C. (1977). Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: I: Algebraic lower bounds. Psychometrika, 42(4), 567–578.10.1007/BF02295979CrossRefGoogle Scholar
Lord, F.M. (1955). Sampling fluctuations resulting from the sampling of test items. Psychometrika, 20(1), 1–22.10.1007/BF02288956CrossRefGoogle Scholar
Lord, F.M., Novick, M.R. (1968). Statistical theories of mental test scores, Reading: Addison-Wesley.Google Scholar
McDonald, R.P. (1978). Generalizability in factorable domains: “domain validity and generalizability” 1. Educational and Psychological Measurement, 38(1), 75–79.10.1177/001316447803800111CrossRefGoogle Scholar
McDonald, R.P. (1999). Test theory: A unified treatment, Hillsdale: Erlbaum.Google Scholar
McNemar, Q. (1946). Opinion-attitude methodology. Psychological Bulletin, 43(4), 289–374.10.1037/h0060985CrossRefGoogle ScholarPubMed
R Development Core Team. (2008). R: A language and environment for statistical computing. Vienna, Austria (ISBN 3-900051-07-0).Google Scholar
Rafaeli, E., Revelle, W. (2006). A premature consensus: Are happiness and sadness truly opposite affects?. Motivation and Emotion, 30(1), 1–12.10.1007/s11031-006-9004-2CrossRefGoogle Scholar
Raykov, T., Shrout, P.E. (2002). Reliability of scales with general structure: Point and interval estimation using a structural equation modeling approach. Structural Equation Modeling, 9(2), 195–212.10.1207/S15328007SEM0902_3CrossRefGoogle Scholar
Revelle, W. (1979). Hierarchical cluster-analysis and the internal structure of tests. Multivariate Behavioral Research, 14(1), 57–74.10.1207/s15327906mbr1401_4CrossRefGoogle ScholarPubMed
Revelle, W. (2008). psych: Procedures for personality and psychological research (R package version 1.0-51).Google Scholar
Schmid, J.J., Leiman, J.M. (1957). The development of hierarchical factor solutions. Psychometrika, 22(1), 83–90.10.1007/BF02289209CrossRefGoogle Scholar
Sijtsma, K. (2008). On the use, the misuse, and the very limited usefulness of Cronbach’s alpha. Psychometrika.Google Scholar
Spearman, C. (1904). The proof and measurement of association between two things. The American Journal of Psychology, 15(1), 72–101.10.2307/1412159CrossRefGoogle Scholar
Spearman, C. (1910). Correlation calculated from faulty data. British Journal of Psychology, 3(3), 271–295.Google Scholar
Ten Berge, J.M.F., Hofstee, W.K.B. (1999). Coefficients alpha and reliabilities of unrotated and rotated components. Psychometrika, 64(1), 83–90.10.1007/BF02294321CrossRefGoogle Scholar
Ten Berge, J.M.F., Socan, G. (2004). The greatest lower bound to the reliability of a test and the hypothesis of unidimensionality. Psychometrika, 69(4), 613–625.10.1007/BF02289858CrossRefGoogle Scholar
Ten Berge, J.M.F., Zegers, F.E. (1978). A series of lower bounds to the reliability of a test. Psychometrika, 43(4), 575–579.10.1007/BF02293815CrossRefGoogle Scholar
Warner, W.L. (1960). Social class in America: a manual of procedure for the measurement of social status, New York: Harper.Google Scholar
Woodhouse, B., Jackson, P. (1977). 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. Psychometrika, 42(4), 579–591.10.1007/BF02295980CrossRefGoogle Scholar
Zinbarg, R.E., Revelle, W., Yovel, I., Li, W. (2005). Cronbach’s α, Revelle’s β, and McDonald’s ω H: Their relations with each other and two alternative conceptualizations of reliability. Psychometrika, 70(1), 123–133.10.1007/s11336-003-0974-7CrossRefGoogle Scholar
Zinbarg, R.E., Yovel, I., Revelle, W., McDonald, R.P. (2006). Estimating generalizability to a latent variable common to all of a scale’s indicators: A comparison of estimators for ω h. Applied Psychological Measurement, 30(2), 121–144.10.1177/0146621605278814CrossRefGoogle Scholar
Zinbarg, R.E., Revelle, W., Yovel, I. (2007). Estimating ω h for structures containing two group factors: Perils and prospects. Applied Psychological Measurement, 31(2), 135–157.10.1177/0146621606291558CrossRefGoogle Scholar