The greatest lower bound to the reliability of a test and the hypothesis of unidimensionality Theory And Methods

Received: 13 June 2002 Revised: 08 September 2003 DOI :
10.1007/BF02289858

Cite this article as: Ten Berge, J.M.F. & Sočan, G. Psychometrika (2004) 69: 613. doi:10.1007/BF02289858
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Abstract To assess the reliability of congeneric tests, specifically designed reliability measures have been proposed. This paper emphasizes that such measures rely on a unidimensionality hypothesis, which can neither be confirmed nor rejected when there are only three test parts, and will invariably be rejected when there are more than three test parts. Jackson and Agunwamba's (1977) greatest lower bound to reliability is proposed instead. Although this bound has a reputation for overestimating the population value when the sample size is small, this is no reason to prefer the unidimensionality-based reliability. Firstly, the sampling bias problem of the glb does not play a role when the number of test parts is small, as is often the case with congeneric measures. Secondly, glb and unidimensionality based reliability are often equal when there are three test parts, and when there are more test parts, their numerical values are still very similar. To the extent that the bias problem of the greatest lower bound does play a role, unidimensionality-based reliability is equally affected. Although unidimensionality and reliability are often thought of as unrelated, this paper shows that, from at least two perspectives, they act as antagonistic concepts. A measure, based on the same framework that led to the greatest lower bound, is discussed for assessing how close is a set of variables to unidimensionality. It is the percentage of common variance that can be explained by a single factor. An empirical example is given to demonstrate the main points of the paper.

Key words Reliability congeneric test unidimensionality of a test The authors are obliged to Henk Kiers for commenting on a previous version. Gregor Sočan is now at the University of Ljubljana.

References Bekker, P.A., & De Leeuw, J. (1987). The rank of reduced dispersion matrices.

Psychometrika, 52 , 125–135.

CrossRef Google Scholar Bentler, P.M. (1972). A lower-bound method for the dimension-free measurement of reliability.

Social Science Research, 1 , 343–357.

CrossRef Google Scholar 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.

Google Scholar Cortina, J.M. (1993). What is coefficient alpha? An examination of theory and applications.

Journal of Applied Psychology, 78 , 98–104.

Google Scholar Cronbach, L.J. (1951). Coefficient alpha and the internal structure of tests.

Psychometrika, 16 , 297–334.

CrossRef Google Scholar Cronbach, L.J. (1988). Internal consistency of tests.

Psychometrika, 53 , 63–70.

CrossRef Google Scholar De Leeuw, J. (1983). Models and methods for the analysis of correlation coefficients.

Journal of Econometrics, 22 , 113–137.

Google Scholar Feldt, L.S., Woodruff, D.J., & Salih, F.A. (1987). Statistical inference for coefficient alpha.

Applied Psychological Measurement, 11 , 93–103.

Google Scholar Guttman, L. (1945). A basis for analyzing test-retest reliability.

Psychometrika, 10 , 255–282.

CrossRef Google Scholar Guttman, L. (1958). To what extent can communalities reduce rank.

Psychometrika, 23 , 297–308.

CrossRef Google Scholar Jackson, P.H., & Agunwamba, C.C. (1977). Lower bounds for the reliability of the total score on a test composed of nonhomogeneous items: I. Algebraic lower bounds.

Psychometrika, 42 , 567–578.

CrossRef Google Scholar Kristof, W. (1974). Estimation of reliability and true score variance from a split of the test into three arbitrary parts.

Psychometrika, 39 , 245–249.

Google Scholar Ledermann, W. (1937). On the rank of reduced correlation matrices in multiple factor analysis.

Psychometrika, 2 , 85–93.

CrossRef Google Scholar McDonald, R.P. (1970). The theoretical foundations of principal factor analysis, canonical factor analysis, and alpha factor analysis.

British Journal of Mathematical and Statistical Psychology, 23 , 1–21.

Google Scholar Nicewander, W.A. (1990). A latent-trait based reliability estimate and upper bound.

Psychometrika, 55 , 65–74.

CrossRef Google Scholar Novick, M.R., & Lewis, C. (1967). Coefficient alpha and the reliability of composite measurements.

Psychometrika, 32 , 1–13.

CrossRef PubMed Google Scholar Osburn, H.G. (2000). Coefficient alpha and related internal consistency reliability coefficients.

Psychological Methods, 5 , 343–355.

CrossRef PubMed Google Scholar Schmitt, N. (1996). Uses and abuses of coefficient alpha.

Psychological Assessment, 8 , 350–353.

Google Scholar Shapiro, A. (1982a). Rank reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis.

Psychometrika, 47 , 187–199.

Google Scholar Shapiro, A. (1982b). Weighted minimum trace factor analysis.

Psychometrika, 47 , 243–264.

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.

CrossRef Google Scholar Shapiro, A., & Ten Berge, J.M.F. (2002). Statistical inference of minimum rank factor analysis.

Psychometrika, 67 , 79–94.

CrossRef Google Scholar Spearman, C.E. (1927).

The abilities of man . London: McMillan.

Google Scholar Ten Berge, J.M.F. (1998). Some recent developments in factor analysis and the search for proper communalities. In A. Rizzi, M. Vichi, and H.-H. Bock (Eds.):

Advances in data science and classification (pp. 325–334). Berlin: Springer.

Google 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.

Google Scholar 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 , 357–366.

CrossRef Google Scholar Van Zijl, J.M., Neudecker, H., & Nel, D.G. (2000). On the distribution of the maximum likelihood estimator of Cronbach's alpha.

Psychometrika, 65 , 271–280.

Google Scholar Verhelst, N.D. (1998).

Estimating the reliability of a test from a single test administration (Measurement and Research Department Report No. 98-2). Arnhem: CITO.

Google Scholar Wilson, E.B., & Worcester, J. (1939). The resolution of six tests into three general factors.

Proceedings of the National Academy of Sciences, 25 , 73–79.

Google Scholar Woodhouse, B., & Jackson, P.H. (1977). Lower bounds for the reliability of a test composed of nonhomogeneous items II: A search procedure to locate the greatest lower bound.

Psychometrika, 42 , 579–591.

CrossRef Google Scholar Yuan, K.-H., & Bentler, P.M. (2002). On robustness of the normal-theory based asymptotic distributions of three reliability coefficient estimates.

Psychometrika, 67 , 251–259.

Google Scholar © The Psychometric Society 2004

Authors and Affiliations 1. Department of Psychology University of Groningen Groningen Netherlands