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Necessary Conditions for Mean Square Convergence of the Best Linear Factor Predictor

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 Department of Psychology, Psychological Methods, University of Amsterdam, Roetersstraat 15, 1018 WB Amsterdam, The Netherlands.
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Abstract

Several sufficient conditions are available for mean square convergence of factor predictors. A necessary and sufficient condition is given in the Heywood case with respect to (confirmatory) factor analysis. This condition generalizes that of Krijnen (2006) and performs better than a signal-to-noise type of condition (Schneeweiss & Mathes, 1995).

Keywords

Heywood casesconstrained factor analysisfactor indeterminacycommon factor analysisconfirmatory factor analysistrue factor scores
Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 3 , September 2006 , pp. 593 - 599
Copyright
Copyright © 2006 The Psychometric Society

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Footnotes

The author is obliged to the reviewers for their stimulating remarks.

References

Browne, M.W. (1968). A comparison of factor analytic techniques. Psychometrika, 33, 267–334.CrossRefGoogle ScholarPubMed
Dijkstra, T.K. (1992). On statistical inference with parameter estimates on the boundary of the parameter space. British Journal of Statistical and Mathematical Psychology, 45, 289–309.CrossRefGoogle Scholar
Guttman, L. (1955). The determinacy of factor score matrices with implications for five other basic problems of common-factor theory. The British Journal of Statistical Psychology, 8, 65–81.CrossRefGoogle Scholar
Hayashi, K., Bentler, P.M. (2000). The asymptotic covariance matrix of maximum-likelihood estimates in factor analysis: The case of nearly singular matrix of estimates of unique variances. Linear Algebra and its Applications, 321, 153–173.CrossRefGoogle Scholar
Heywood, H.B. (1931). On finite sequences of real numbers. Proceedings of the Royal Society London, 134, 486–501.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 Japan Statistical Society, 13, 137–144.Google 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. (2006). Some results on mean square error for factor score prediction. Psychometrika, 71, 395–409.CrossRefGoogle ScholarPubMed
Krijnen, W.P., Dijkstra, T.K., Gill, R.D. (1998). Conditions for factor (in)determinacy in factor analysis. Psychometrika, 63, 359–367.CrossRefGoogle Scholar
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
Lee, S.Y. (1980). Estimation of covariance structure models with parameters subject to functional restraints. Psychometrika, 45, 309–324.CrossRefGoogle Scholar
Maraun, M.D. (1996). Metaphor taken as math: Indeterminacy in the factor analysis model. Multivariate Behavioral Research, 31, 517–538.Google Scholar
Neudecker, H. (2004). On best affine unbiased covariance-preserving prediction of factor scores. Statistics and Operations Research Transactions, 28, 27–36.Google Scholar
Pringle, R.M., Rayner, A.A. (1971). Generalized inverse matrices with applications to statistics, Dordrecht: Hafner.Google Scholar
Rudin, W. (1976). Principles of mathematical analysis, (3rd ed.). Dordrecht: McGraw-Hill.Google Scholar
Rudin, W. (1987). Real and complex analysis, (3rd ed.). Dordrecht: 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
Shapiro, A. (1986). Asymptotic theory of overparametrized structural models. Journal of the American Statistician Association, 81, 142–149.CrossRefGoogle Scholar
Skrondal, A., Laake, P. (2001). Regression among factor scores. Psychometrika, 66, 563–576.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
Thomson, G.H. (1951). The factorial analysis of human ability, London: University Press.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