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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
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.

References

Anderson, T.W., & Rubin, H. (1956). Statistical inference in factor analysis. Proceedings of the Third Berkeley Symposium, Vol. 5, pp. 111–150.Google Scholar
Apostol, T.M. (1967). Calculus (2nd ed., Vol. 1), New York: Wiley.Google Scholar
Bartlett, M.S. (1937). The statistical conception of mental factors. British Journal of Psychology, 28, 97–104.Google Scholar
Bekker, P.A., Merckens, A., & Wansbeek, T.J. (1994). Identification, equivalent models, and computer algebra. Boston: Academic Press.Google Scholar
Browne, M.W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.CrossRefGoogle ScholarPubMed
Dijkstra, T.K. (1981). Latent variables in linear stochastic models. Amsterdam: Sociometric Research Foundation.Google Scholar
Ferguson, T.S. (1996). A course in large sample theory. London: Chapman & Hall.CrossRefGoogle Scholar
Green, B.F. (1969). Best linear composites with a specified structure. Psychometrika, 34, 301–318.CrossRefGoogle Scholar
Guttman, L. (1955). The determinacy of factor score matrices with implications for five other basic problems of common-factor theory. British Journal of Statistical Psychology, 8, 65–81.CrossRefGoogle Scholar
Hayashi, K., & Bentler, P.M. (2000a). On the relations among regular, equal unique variances and image factor analysis. Psychometrika, 65, 59–72.CrossRefGoogle Scholar
Hayashi, K., & Bentler, P.M. (2000b). The asymptotic covariance matrix of maximum-likelihood estimates in factor analysis: The case of a nearly singular matrix of estimates of unique variances. Linear Algebra and its Applications, 321, 153–173.CrossRefGoogle Scholar
Holzinger, K.J., & Swineford, F. (1939). A study in factor analysis: The stability of a bi-factor solution. Supplementary Educational Monographs, No. 48. Chicago: University of Chicago Press.Google Scholar
Horn, R.A., & Johnson, C.R. (1985). Matrix analysis. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Jöreskog, K.G. (1969). A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 34, 183–202.CrossRefGoogle Scholar
Kano, Y. (1983). Consistency of estimators in factor analysis. Journal of the Japan Statistical Society, 13, 137–144.Google Scholar
Kano, Y. (1984). Construction of additional variables conforming to a common factor model. Statistics & Probability Letters, 2, 241–244.CrossRefGoogle Scholar
Krijnen, W.P. (2002). On the construction of all factors of the model for factor analysis. Psychometrika, 67, 161–172.CrossRefGoogle Scholar
Krijnen, W.P. (2004). Convergence in mean square of factor predictors. British Journal of Mathematical and Statistical Psychology, 57, 311–326.CrossRefGoogle ScholarPubMed
Krijnen, W.P., Wansbeek, T.J., & Ten Berge, J.M.F. (1996). Best linear predictors for factor scores. Communications in Statistics: Theory and Methods, 25, 3013–3025.CrossRefGoogle Scholar
Lawley, D.N., & Maxwell, A.E. (1971). Factor analysis as a statistical method 2nd ed., London: Butterworth.Google Scholar
Lee, S.Y., & Poon, W.Y. (1995). Estimation of factor scores in a two-level confirmatory factor analysis model. Computational Statistics & Data Analysis, 20, 275–284.Google Scholar
Lehmann, E.L. (1999). Elements of large sample theory. New York: Springer-Verlag.CrossRefGoogle Scholar
Lord, M., & Novick, M.R. (1968). Statistical theories of mental test scores. Reading MA: Addison-Wesley.Google Scholar
McDonald, R.P. (1981). Constrained least squares estimators of oblique common factors. Psychometrika, 46, 337–341.CrossRefGoogle Scholar
McDonald, R.P., & Burr, E.J. (1967). A comparison of four methods of constructing factor scores. Psychometrika, 32, 381–401.CrossRefGoogle Scholar
Neudecker, H. (2004). On best affine unbiased covariance-preserving prediction of factor scores. Statistics and Operations Research Transactions, 28, 27–36.Google Scholar
Rao, C.R. (1973). Linear statistical inference and its applications. New York: Wiley.CrossRefGoogle Scholar
Rao, C.R., & Toutenburg, H. (1995). Linear models. New York: Springer-Verlag.CrossRefGoogle Scholar
Rudin, W. (1976). Principles of mathematical analysis 3rd ed., New York: McGraw-Hill.Google Scholar
Schneeweiss, H. (1997). Factors and principal components in the near spherical case. Multivariate Behavioural Research, 32, 375–401.CrossRefGoogle ScholarPubMed
Schneeweiss, H., & Mathes, H. (1995). Factor analysis and principal components. Journal of Multivariate Analysis, 55, 105–124.CrossRefGoogle Scholar
Skrondal, A., & Laake, P. (2001). Regression among factor scores. Psychometrika, 66, 563–576.CrossRefGoogle Scholar
Steiger, J.H. (1979). Factor indeterminacy in the 1930's and the 1970's: Some interesting parallels. Psychometrika, 44, 157–167.CrossRefGoogle Scholar
Ten Berge, J.M.F. (1983). On Green's best linear composites with a specified structure, and oblique estimates of factor scores. Psychometrika, 48, 371–375.CrossRefGoogle Scholar
Ten Berge, J.M.F., Krijnen, W.P., Wansbeek, T.J., & Shapiro, A. (1999). Some new results on correlation preserving factor scores prediction methods. Linear Algebra and its Applications, 289, 311–318.CrossRefGoogle Scholar
Wang, S.G., & Chow, S.C. (1994). Advanced linear models. New York: Marcel Dekker.Google Scholar
Williams, J.S. (1978). A definition for the common-factor analysis model and the elimination of problems of factor score indeterminacy. Psychometrika, 43, 293–306.CrossRefGoogle Scholar