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.
* 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: firstname.lastname@example.org.
Ссылка для цитирования
Bentler, P. M. (1972). A lower-bound method for the dimension-free
measurement of internal consistency. - Social Science Research, 1,
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.
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.
Browne, M. W. (1974). Generalized least squares estimators in the analysis
of covariance structures. - South African Statistical Journal, 8, 1-24.
Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied
multiple regression/correlation analysis for the behavioral sciences. - Mahwah,
Cronbach, L. J. (1951). Coefficient alpha and the internal structure of
tests. - Psychometrika, 16, 297-334.
Cronbach, L. J. (1988). Internal consistency of tests: Analyses old and
new. - Psychometrika, 53, 63-70.
Davison, A. C., & Hinkley, S. V. (1997). Bootstrap methods and their
application. - Cambridge: Cambridge University Press.
Efron, B. & Tibshirani, R. J. (1994). An introduction to the
bootstrap. - New York: Chapman & Hall.
Guttman, L. (1945). A basis for analyzing test-retest reliability. -
Psychometrika, 10, 255-282.
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.
Lord, F. M., & Novich, M. R. (1968). Statistical theories of
mental test scores. - Reading, MA: Addison-Wesley.
Shapiro, A. (1982a). Rank reducibility of a symmetric matrix and sampling
theory of minimum trace factor analysis. - Psychometrika, 47,
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.
Sijtsma, K. (2009). On the use, the misuse, and the very limited usefulness
of Cronbach's alpha. - Psychometrika, 74, 107-120.
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.
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,
Van Zyl, J. M., Neudecker, H. & Nel, D. G. (2000). On the distribution
of the maximum likelihood estimator of Cronbach's alpha. - Psychometrika,
Wherry, R. J. (1931). A new formula for predicting the shrinkage of the
multiple correlation coefficient. - Annals of Mathematical Statistics, 2,
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,
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.