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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

Information

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.

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