The Greatest Lower Bound to Reliability: Corrected And Resampling Estimators*


In spite of being the best possible lower bound to reliability, the greatest lower bound (g.l.b.) has rarely been used in practice, no doubt due to its serious upward sampling bias. A pioneering attempt to estimate this bias was made by Shapiro and ten Berge (2000), but their bias formula has both theoretical and empirical limitations. We propose a new Wherry-like adjusted g.l.b. estimator that has less sampling bias than the classical estimator. Resampling methods are further used to correct the bias of this new estimator in both direct and indirect ways. Numerical simulations confirm the effectiveness of the bootstrap bias-correction to the adjusted g.l.b.

Общая информация

* This study was supported, in part, by grants DA01070 and DA00017 from the US National Institute on Drug Abuse and by grant 5R44CA137841 from the US National Cancer Institute. This study was presented at the 82th Symposium of the Behaviormetric Society of Japan on Recent Developments in Latent Variables Modeling, Tokyo University, Tokyo, Japan, August, 2004. Requests for reprints should be sent to Peter M. Bentler, Department of Psychology, UCLA, Box 951563, Los Angeles, CA 90095-1563. E-mail:

Ключевые слова: greatest lower bound, reliability, Cronbachs alpha, constrained minimum trace factor analysis, bootstrap

Рубрика издания: Анализ данных

Тип материала: научная статья

Для цитаты: Ли Л., Бентлер П. The Greatest Lower Bound to Reliability: Corrected And Resampling Estimators // Моделирование и анализ данных. 2011. Том 1. № 1. С. 87–104.


  1. Bentler, P. M. (1972). A lower-bound method for the dimension-free measurement of internal consistency.  - Social Science Research,  1, 343-357.
  2. Bentler, P. M., & Woodward, J. A. (1980). Inequalities among lower bounds to reliability: With applications to test construction and factor analysis. - Psychometrika,  45, 249-267.
  3. Bentler, P. M., & Woodward, J. A. (1983). The greatest lower bound to reliability. - In: H. Wainer & S. Messick (Eds.),  Principals of modern psychological measurement: A Festschrift for Frederic M. Lord (pp. 237-253). Hillsdale, NJ: Erlbaum.
  4. Browne, M. W. (1974). Generalized least squares estimators in the analysis of covariance structures. - South African Statistical Journal, 8, 1-24.
  5. Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003).  Applied multiple regression/correlation analysis for the behavioral sciences. - Mahwah, NJ: Erlbaum.
  6. Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. - Psychometrika,  16, 297-334.
  7. Cronbach, L. J. (1988). Internal consistency of tests: Analyses old and new. - Psychometrika,  53, 63-70.
  8. Davison, A. C., & Hinkley, S. V. (1997). Bootstrap methods and their application. - Cambridge: Cambridge University Press.
  9. Efron, B. & Tibshirani, R. J. (1994).  An introduction to the bootstrap. - New York: Chapman & Hall.
  10. Guttman, L. (1945). A basis for analyzing test-retest reliability. - Psychometrika,  10, 255-282.
  11. Jackson, P. H., & Agunwamba, C. C. (1977). Lower bounds for the reliability of total scores on a test composed of nonhomogeneous items: I. Algebraic lower bounds. - Psychometrika,  42, 567-578.
  12. Lord, F. M., & Novich, M. R. (1968).  Statistical theories of mental test scores. - Reading, MA: Addison-Wesley.
  13. Shapiro, A. (1982a). Rank reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis. - Psychometrika,  47, 187-199.
  14. Shapiro, A. (1982b). Weighted minmum trace factor analysis. - Psychometrika,  47, 243-264.
  15. 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.
  16. Sijtsma, K. (2009). On the use, the misuse, and the very limited usefulness of Cronbach's alpha. -  Psychometrika, 74, 107-120.
  17. 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, 613-625.
  18. Ten Berge, J. M. F., Snijders, T. A. B., & Zegers, F. E. (1981). Computational aspects of the greatest lower bound to reliability and constrained minimum trace factor analysis. - Psychometrika,  46, 201-213.
  19. 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.
  20. Wherry, R. J. (1931). A new formula for predicting the shrinkage of the multiple correlation coefficient. - Annals of Mathematical Statistics, 2, 440-457.
  21. Woodhouse, B., & Jackson, P. M. (1977). Lower bounds for the reliability of the total score on a test composed of nonhomogeneous items: II. A search procedure to locate the greatest lower bound. - Psychometrika,  42, 579-591.
  22. Yung, Y. F., & Bentler, P. M. (1994). Bootstrap-corrected ADF test statistics in covariance structure analysis. - British Journal of Mathematical and Statistical Psychology,  47, 63-84.   

Информация об авторах

Ли Либо, PhD, старший статистик кафедры психиатрии и наук о поведении факультета медицины, Д. Геффена Калифорнийский университет в Лос-Анджелесе, Лос-Анджелес, США, e-mail:

Бентлер Питер, кандидат психологических наук, профессор, Калифорнийский университет, Лос-Анджелес, США, e-mail:



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