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Some Results on Mean Square Error for Factor Score Prediction

Published online by Cambridge University Press:  01 January 2025

Wim P. Krijnen
Wim P. Krijnen*
Affiliation:
University of Amsterdam
*
Requests for reprints should be sent to Wim P. Krijnen, Department of Psychology, Psychological Methods, University of Amsterdam, Roetersstraat 15, 1018 WB Amsterdam, The Netherlands. E-mail: P.W.vanRijn@uva.nl, m-speekenbrink@ucl.ac.uk
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Abstract

For the confirmatory factor model a series of inequalities is given with respect to the mean square error (MSE) of three main factor score predictors. The eigenvalues of these MSE matrices are a monotonic function of the eigenvalues of the matrix Γp = Φ1/2Λ′pΨp−1ΛpΦ1/2. This matrix increases with the number of observable variables p. A necessary and sufficient condition for mean square convergence of predictors is divergence of the smallest eigenvalue of Γp or, equivalently, divergence of signal-to-noise (Schneeweiss & Mathes, 1995). The same condition is necessary and sufficient for convergence to zero of the positive definite MSE differences of factor predictors, convergence to zero of the distance between factor predictors, and convergence to the unit value of the relative efficiencies of predictors. Various illustrations and examples of the convergence are given as well as explicit recommendations on the problem of choosing between the three main factor score predictors.

Keywords

common factor analysisconfirmatory factor analysisrelative efficiencyfactor indeterminacy

Information

Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 2 , June 2006, pp. 395 - 409
Copyright
Copyright © 2006 The Psychometric Society

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Footnotes

The author is obliged to Maarten Speekenbrink and Peter van Rijn for their assistance with plotting the figures. In addition, I am obliged to the referees for their stimulating remarks.

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