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The Rank of Reduced Dispersion Matrices

Published online by Cambridge University Press:  01 January 2025

Paul A. Bekker  and
Jan de Leeuw
Paul A. Bekker
Affiliation:
Department of Econometrics, Tilburg University
Jan de Leeuw*
Affiliation:
Department of Data Theory FSW, University of Leiden
*
Requests for reprints should be sent to Jan de Leeuw, Department of Data Theory FSW/RUL, Middelstegracht 4, 2312 TW Leiden, THE NETHERLANDS.
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Abstract

Psychometricians working in factor analysis and econometricians working in regression with measurement error in all variables are both interested in the rank of dispersion matrices under variation of the diagonal elements. Psychometricians concentrate on cases in which low rank can be attained, preferably rank one, the Spearman case. Econometricians concentrate on cases in which the rank cannot be reduced below the number of variables minus one, the Frisch case. In this paper we give an extensive historial discussion of both fields, we prove the two key results in a more satisfactory and uniform way, we point out various small errors and misunderstandings, and we present a methodological comparison of factor analysis and regression on the basis of our results.

Keywords

factor analysiserrors of measurementstructural regressionfunctional modelscommunalitieserrors in variables
Type
Original Paper
Information
Psychometrika , Volume 52 , Issue 1 , March 1987 , pp. 125 - 135
Copyright
Copyright © 1987 The Psychometric Society

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Footnotes

Financial support by the Netherlands Organization for the Advancement of Pure Research (ZWO) is gratefully acknowledged.

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