Abstract
In the last decade several algorithms for computing the greatest lower bound to reliability or the constrained minimum-trace communality solution in factor analysis have been developed. In this paper convergence properties of these methods are examined. Instead of using Lagrange multipliers a new theorem is applied that gives a sufficient condition for a symmetric matrix to be Gramian. Whereas computational pitfalls for two methods suggested by Woodhouse and Jackson can be constructed it is shown that a slightly modified version of one method suggested by Bentler and Woodward can safely be applied to any set of data. A uniqueness proof for the solution desired is offered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Reference Notes
Della Riccia, G. & Shapiro, A.Minimum rank and minimum trace of covariance matrices. Beer-Sheva, Israel: Dept. of Mathematics, Ben Gurion University of the Negev, 1980.
Hakim, M., Lochard, E. O., Olivier, J. P. & Térouanne, E.Sur les traces de Spearman (I) (Cahiers du bureau universitaire du recherche opérationelle, Serie recherche no. 25). Paris: Université Pierre et Marie Curie, 1976.
References
Bentler, P. M. A lower-bound method for the dimension-free measurement of internal consistency.Social Science Research, 1972,1, 343–357.
Bentler, P. M. & Woodward, J. A. Inequalities among lower-bounds to reliability: With applications to test construction and factor analysis.Psychometrika, 1980,45, 249–267.
Guttman, L. A basis for analyzing test-retest reliability.Psychometrika, 1945,10, 255–282.
Jackson, P. H. & Agunwamba, C. C. Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: I. Algebraic lower bounds.Psychometrika, 1977,42, 567–578.
Ledermann, W. On a problem concerning matrices with variable diagonal elements.Proceedings of the Royal Society of Edinburgh, 1939,60, 1–17.
Mulaik, S. A.The foundations of factor analysis. New York: McGraw-Hill Inc., 1972.
Nicewander, W. A. A relationship between Harris factors and Guttman's sixth lower bound to reliability.Psychometrika, 1975,40, 197–203.
Woodhouse, B. & Jackson, P. H. 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, 1977,42, 579–591.
Author information
Authors and Affiliations
Additional information
The authors are obliged to Charles Lewis and Dirk Knol for helpful comments, and to Frank Brokken and Henk Camstra for developing computer programs.
Rights and permissions
About this article
Cite this article
ten Berge, J.M.F., Snijders, T.A.B. & Zegers, F.E. Computational aspects of the greatest lower bound to the reliability and constrained minimum trace factor analysis. Psychometrika 46, 201–213 (1981). https://doi.org/10.1007/BF02293900
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02293900