Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-02-11T16:19:29.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Computational Aspects of the Greatest Lower Bound to the Reliability and Constrained Minimum Trace Factor Analysis

Published online by Cambridge University Press:  01 January 2025

Jos M. F. ten Berge ,
Tom A. B. Snijders  and
Frits E. Zegers
Jos M. F. ten Berge*
Affiliation:
University of Groningen
Tom A. B. Snijders
Affiliation:
University of Groningen
Frits E. Zegers
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Jos M. F. ten Berge, Subfaculteit voor de Psychologie der Rijksuniversiteit, Oude Boteringestraat 34, 9712 GK Groningen. THE NETHERLANDS.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

communalityinternal consistencyHeywood casepositive semidefinite
Type
Original Paper
Information
Psychometrika , Volume 46 , Issue 2 , June 1981 , pp. 201 - 213
Copyright
Copyright © 1981 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The authors are obliged to Charles Lewis and Dirk Knol for helpful comments, and to Frank Brokken and Henk Camstra for developing computer programs.

References

Reference Notes

Della Riccia, G. & Shapiro, A. Minimum rank and minimum trace of covariance matrices, 1980, Beer-Sheva, Israel: Dept. of Mathematics, Ben Gurion University of the Negev.Google Scholar
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), 1976, Paris: Université Pierre et Marie Curie.Google Scholar

References

Bentler, P. M. A lower-bound method for the dimension-free measurement of internal consistency. Social Science Research, 1972, 1, 343–357.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Guttman, L. A basis for analyzing test-retest reliability. Psychometrika, 1945, 10, 255–282.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Ledermann, W. On a problem concerning matrices with variable diagonal elements. Proceedings of the Royal Society of Edinburgh, 1939, 60, 1–17.CrossRefGoogle Scholar
Mulaik, S. A. The foundations of factor analysis, 1972, New York: McGraw-Hill Inc..Google Scholar
Nicewander, W. A. A relationship between Harris factors and Guttman's sixth lower bound to reliability. Psychometrika, 1975, 40, 197–203.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar