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Statistical Inference of Minimum Rank Factor Analysis

Published online by Cambridge University Press:  01 January 2025

Alexander Shapiro  and
Jos M. F. Ten Berge
Alexander Shapiro*
Affiliation:
School of Industrial and Systems Engineering, Georgia Institute of Technology
Jos M. F. Ten Berge
Affiliation:
Heymans Institute of Psychological Research, University of Groningen
*
Requests for reprints should be sent to Alexander Shapiro, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0205. E-Mail: ashapiro@isye.gatech.edu
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Abstract

For any given number of factors, Minimum Rank Factor Analysis yields optimal communalities for an observed covariance matrix in the sense that the unexplained common variance with that number of factors is minimized, subject to the constraint that both the diagonal matrix of unique variances and the observed covariance matrix minus that diagonal matrix are positive semidefinite. As a result, it becomes possible to distinguish the explained common variance from the total common variance. The percentage of explained common variance is similar in meaning to the percentage of explained observed variance in Principal Component Analysis, but typically the former is much closer to 100 than the latter. So far, no statistical theory of MRFA has been developed. The present paper is a first start. It yields closed-form expressions for the asymptotic bias of the explained common variance, or, more precisely, of the unexplained common variance, under the assumption of multivariate normality. Also, the asymptotic variance of this bias is derived, and also the asymptotic covariance matrix of the unique variances that define a MRFA solution. The presented asymptotic statistical inference is based on a recently developed perturbation theory of semidefinite programming. A numerical example is also offered to demonstrate the accuracy of the expressions.

Keywords

factor analysiscommunalitiesproper solutionsexplained common variancesemidefinite programminglarge samples asymptoticsasymptotic normalityasymptotic bias
Type
Articles
Information
Psychometrika , Volume 67 , Issue 1 , March 2002 , pp. 79 - 94
Copyright
Copyright © 2002 The Psychometric Society

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Footnotes

This work was supported, in part, by grant DMS-0073770 from the National Science Foundation.

References

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.CrossRefGoogle Scholar
Bonnans, J.F., & Shapiro, A. (2000). Perturbation analysis of optimization problems. New York, NY: Springer-Verlag.CrossRefGoogle Scholar
Browne, M.W. (1974). Generalized least squares estimators in the analysis of covariance structures. South African Statistical Journal, 8, 1–24.Google Scholar
Carroll, J.B. (1993). Human cognitive abilities: A survey of factor analytic research. New York, NY: Cambridge University Press.CrossRefGoogle Scholar
Guttman, L. (1958). To what extent can communalities reduce rank?. Psychometrika, 23, 297–308.CrossRefGoogle Scholar
Harman, H.H. (1967). Modern factor analysis. Chicago, IL: The University of Chicago Press.Google Scholar
Harman, H.H., & Jones, W.H. (1966). Factor analysis by minimizing residuals (minres). Psychometrika, 31, 351–368.CrossRefGoogle ScholarPubMed
Jöreskog, K.G. (1967). Some contributions to maximum likelihood factor analysis. Psychometrika, 31, 443–482.CrossRefGoogle Scholar
Ledermann, W. (1937). On the rank of reduced correlation matrices in multiple factor analysis. Psychometrika, 2, 85–93.CrossRefGoogle Scholar
Rao, C.R. (1973). Linear statistical inference and its applications. New York, NY: Wiley.CrossRefGoogle Scholar
Saigal, R., Vandenberghe, L., & Wolkowicz, H. (2000). Semidefinite programming and applications handbook. Boston, MA: Kluwer Academic Publishers.Google Scholar
Schutz, R.E. (1958). Factorial validity of the Holzinger-Crowder Uni-Factor Tests. Educational and Psychological Measurement, 18, 873–875.CrossRefGoogle Scholar
Shapiro, A. (1982). Rank reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis. Psychometrika, 47, 187–199.CrossRefGoogle Scholar
Shapiro, A. (1985). Identifiability of factor analysis: Some results and open problems. Linear Algebra and its Applications, 70, 1–7.CrossRefGoogle 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.CrossRefGoogle Scholar
ten Berge, J.M.F. (1998). Some recent developments in factor analysis and the search for proper communalities. In Rizzi, A., Vichi, M., & Bock, H.H. (Eds.), Advances in data science and classification (pp. 325–334). Heidelberg, Germany: Springer.CrossRefGoogle Scholar
ten Berge, J.M.F. (2000). Linking reliability and factor analysis:recent developments in some classical psychometric problems. In Hampson, S.E. (Eds.), Advances in personality psychology, Vol. 1 (pp. 138–156). London, England: Routledge.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.CrossRefGoogle 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, 201–213.CrossRefGoogle Scholar