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On Robusiness of the Normal-Theory Based Asymptotic Distributions of Three Reliability Coefficient Estimates

Published online by Cambridge University Press:  01 January 2025

Ke-Hai Yuan  and
Peter M. Bentler
Ke-Hai Yuan*
Affiliation:
University of Notre Dame
Peter M. Bentler
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to Ke-Hai Yuan, Department of Psychology, University of Notre Dame, Notre Dame, IN 46556. E-Mail: kyuan@nd.edu
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Abstract

This paper studies the asymptotic distributions of three reliability coefficient estimates: Sample coefficient alpha, the reliability estimate of a composite score following a factor analysis, and the estimate of the maximal reliability of a linear combination of item scores following a factor analysis. Results indicate that the asymptotic distribution for each of the coefficient estimates, obtained based on a normal sampling distribution, is still valid within a large class of nonnormal distributions. Therefore, a formula for calculating the standard error of the sample coefficient alpha, recently obtained by van Zyl, Neudecker and Nel, applies to other reliability coefficients and can still be used even with skewed and kurtotic data such as are typical in the social and behavioral sciences.

Keywords

asymptotic distributionCronbach's alphakurtosismaximal reliabilitypseudo normal distributionsskewness
Type
Articles
Information
Psychometrika , Volume 67 , Issue 2 , June 2002 , pp. 251 - 259
Copyright
Copyright © 2002 The Psychometric Society

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Footnotes

This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of North Texas faculty research grant. We would like to thank the Associate Editor and two reviewers for suggestions that helped to improve the paper.

References

Allen, M.J., & Yen, W.M. (1979). Introduction to measurement theory. Monterey, CA: Brooks-Cole.Google Scholar
Azzalini, A., & Valle, A.D. (1996). The multivariate skew-normal distribution. Biometrika, 83, 715–726.CrossRefGoogle Scholar
Bentler, P.M. (1968). Alpha-maximized factor analysis (Alphamax): Its relation to alpha and canonical factor analysis. Psychometrika, 33, 335–345.CrossRefGoogle ScholarPubMed
Bentler, P.M. (1972). A lower-bound method for the dimension-free measurement of internal consistency. Social Science Research, 1, 343–357.CrossRefGoogle Scholar
Bentler, P.M., & Woodward, J.A. (1983). The greatest lower bound to reliability. In Wainer, H., & Messick, S. (Eds.), Principals of modern psychological measurement: A Festschrift for Frederic M. Lord (pp. 237–253). Hillsdale, NJ: Erlbaum.Google Scholar
Crocker, L., Algina, J. (1986). Introduction to classical and modern test theory. New York, NY: Hott, Rinehart, and Winston.Google Scholar
Cronbach, L.J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297–334.CrossRefGoogle Scholar
Drewes, D.W. (2000). Beyond the Spearman-Brown: A structural approach to maximal reliability. Psychological Methods, 5, 214–227.CrossRefGoogle ScholarPubMed
Feldt, L.S. (1965). The approximate sampling distribution of Kuder-Richardson reliability coefficient twenty. Psychometrika, 30, 357–370.CrossRefGoogle ScholarPubMed
Ferguson, T. (1996). A course in large sample theory. London, England: Chapman & Hall.CrossRefGoogle Scholar
Fleishman, A. (1978). A method for simulating non-normal distributions. Psychometrika, 43, 521–532.CrossRefGoogle Scholar
Guttman, L. (1945). A basis for analyzing test-retest reliability. Psychometrika, 10, 255–282.CrossRefGoogle ScholarPubMed
Hancock, G.R., Mueller, R.O. (2001). Rethinking construct reliability. In Cudeck, R., du Toit, S., & Sörbom, D. (Eds.), Structural equation modeling: Present and future (pp. 195–216). Lincolnwood, IL: Scientific Software International.Google Scholar
Headrick, T.C., & Sawilowsky, S.S. (1999). Simulating correlated multivariate nonnormal distributions: Extending the Fleishman power method. Psychometrika, 64, 25–35.CrossRefGoogle Scholar
Kristof, W. (1963). The statistical theory of stepped-up reliability when a test has been divided into several equivalent parts. Psychometrika, 28, 221–238.CrossRefGoogle Scholar
Li, H. (1997). A unifying expression for the maximal reliability of a linear composite. Psychometrika, 62, 245–249.CrossRefGoogle Scholar
Li, H., Rosenthal, R., & Rubin, D.B. (1996). Reliability of measurement in psychology: From Spearman-Brown to maximal reliability. Psychological Methods, 1, 98–107.CrossRefGoogle Scholar
Li, H., & Wainer, H. (1997). Toward a coherent view of reliability in test theory. Journal of Educational and Behavioral Statistics, 22, 478–484.Google Scholar
Lord, F.M., & Novick, M.R. (1968). Statistical theories of mental test scores. Reading, MA: Addison-Wesley.Google Scholar
Magnus, J.R., & Neudecker, H. (1999). Matrix differential calculus with applications in statistics and econometrics rev. ed., New York, NY: Wiley.Google Scholar
McDonald, R.P. (1999). Test theory: A unified treatment. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
Mendoza, J.L., & Stafford, K.L., Stauffer, J.M. (2000). Large-sample confidence intervals for validity and reliability coefficients. Psychological Methods, 5, 356–369.CrossRefGoogle ScholarPubMed
Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105, 156–166.CrossRefGoogle Scholar
Nunnally, J.C., & Bernstein, (1994). Psychometric theory 3rd ed., New York, NY: McGraw-Hill.Google Scholar
Osburn, H.G. (2000). Coefficient alpha and related internal consistency reliability coefficients. Psychological Methods, 5, 343–355.CrossRefGoogle ScholarPubMed
Raykov, T. (1997). Estimation of composite reliability for congeneric measures. Applied Psychological Measurement, 21, 173–184.CrossRefGoogle Scholar
Satorra, A., & Bentler, P.M. (1986). Some robustness properties of goodness of fit statistics in covariance structure analysis. American Statistical Association 1986 Proceedings of Business and Economics Sections (pp. 549–554). Alexander, VA: American Statistical Association.Google Scholar
Shapiro, A., & ten Berge, J.M.F. (2000). The asymptotic bias of minimum trace factor analysis with applications to the greatest lower bound to reliability. Psychometrika, 65, 413–425.CrossRefGoogle Scholar
Tadikamalla, P.R. (1980). On simulating non-normal distributions. Psychometrika, 45, 273–279.CrossRefGoogle Scholar
ten Berge, J.M.F. (2000). Linking reliability and factor analysis: Recent developments in some classical psychometric problems. In Hampson, S.E. (Eds.), Advances in personality psychology, Vol. 1 (pp. 138–156). Philadelphia, PA: Psychology Press/Taylor & Francis.Google Scholar
ten Berge, J.M.F., & Hofstee, W.K.B. (1999). Coefficients alpha and reliabilities of unrotated and rotated components. Psychometrika, 64, 83–90.CrossRefGoogle Scholar
ten Berge, J.M.F., Kiers, H.A.L. (1991). A numerical approach to the exact and the approximate minimum rank of a covariance matrix. Psychometrika, 56, 309–315.CrossRefGoogle Scholar
Thomson, G.H. (1940). Weighting for battery reliability and prediction. British Journal of Psychology, 30, 357–366.Google Scholar
Vale, D., Maurelli, V.A. (1983). Simulating multivariate nonnormal distributions. Psychometrika, 48, 465–471.CrossRefGoogle Scholar
van Zyl, J.M., Neudecker, H., & Nel, D.G. (2000). On the distribution of the maximum likelihood estimator of Cronbach's alpha. Psychometrika, 65, 271–280.CrossRefGoogle Scholar
Woodhouse, B., & Jackson, P.H. (1977). Lower bounds for the reliability of a total score on a test composed of nonhomogeneous items: II. A search procedure to locate the greatest lower bound. Psychometrika, 42, 579–591.CrossRefGoogle Scholar
Yuan, K.-H., & Bentler, P.M. (1997). Generating multivariate distributions with specified marginal skewness and kurtosis. In Bandilla, W., & Faulbaum, F. (Eds.), SoftStat'97—Advances in statistical software 6 (pp. 385–391). Stuttgart, Germany: Lucius & Lucius.Google Scholar
Yuan, K.-H., & Bentler, P.M. (1999). On normal theory and associated test statistics in covariance structure analysis under two classes of nonnormal distributions. Statistica Sinica, 9, 831–853.Google Scholar
Yuan, K.-H., & Bentler, P.M. (2000). Inferences on correlation coefficients in some classes of nonnormal distributions. Journal of Multivariate Analysis, 72, 230–248.CrossRefGoogle Scholar