Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-07T18:31:27.376Z Has data issue: false hasContentIssue false
  • English
  • Français

Concise Formulas for the Standard Errors of Component Loading Estimates

Published online by Cambridge University Press:  01 January 2025

Haruhiko Ogasawara
Haruhiko Ogasawara*
Affiliation:
Otaru University of Commerce
*
Request for reprints should be sent to Haxuhiko Ogasawara, Department of Information and Management Science, Otaru University of Commerce, 3-5-21, Midori, Otaxu 047-8501 JAPAN. Email: hogasa@res.otaru-uc.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

Concise formulas for the asymptotic standard errors of component loading estimates were derived. The formulas cover the cases of principal component analysis for unstandardized and standardized variables with orthogonal and oblique rotations. The formulas can be used under any distributions for observed variables as long as the asymptotic covariance matrix for sample covariances/correlations is available. The estimated standard errors in numerical examples were shown to be equivalent to those by the methods using information matrices.

Keywords

asymptotic standard errorsprincipal component analysiscomponent loadingsrotationinformation matrixleast squares
Type
Articles
Information
Psychometrika , Volume 67 , Issue 2 , June 2002 , pp. 289 - 297
Copyright
Copyright © 2002 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author is indebted to anonymous reviewers for the corrections and suggestions on this study, which have led to improvements of earlier versions of this article.

References

Anderson, T.W. (1984). An introduction to multivariate statistical analysis 2nd ed., New York, NY: Wiley.Google Scholar
Archer, C.O., & Jennrich, R.I. (1973). Standard errors for rotated factor loadings. Psychometrika, 38, 581–592.CrossRefGoogle Scholar
Browne, M.W. (1982). Covariance structures. In Hawkins, D.M. (Eds.), Topics in applied multivariate analysis (pp. 72–141). Cambridge, England: Cambridge University Press.CrossRefGoogle 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
Clarkson, D.B. (1979). Estimating the standard errors of rotated factor loadings by jackknifing. Psychometrika, 44, 297–314.CrossRefGoogle Scholar
Dolan, C. (1996). Principal component analysis using LISREL 8. Structural Equation Modeling, 3, 307–322.CrossRefGoogle Scholar
Dolan, C., Bechger, T., & Molenaar, P. (1999). Using structural equation modeling to fit models incorporating principal components. Structural Equation Modeling, 6, 233–261.CrossRefGoogle Scholar
Eckart, C., & Young, G. (1936). The approximation of one matrix by another of lower rank. Psychometrika, 1, 211–218.CrossRefGoogle Scholar
Girshick, M.A. (1939). On the sampling theory of roots of determinantal equations. Annals of Mathematical Statistics, 10, 203–224.CrossRefGoogle Scholar
Harman, H.H. (1976). Modern factor analysis 3rd ed., Chicago, IL: University of Chicago Press.Google Scholar
Hayashi, K., & Yung, Y.F. (1999). Standard errors for the class of orthomax-rotated factor loadings: Some matrix results. Psychometrika, 64, 451–460.CrossRefGoogle Scholar
Ichikawa, M., & Konishi, S. (1995). Application of the bootstrap methods in factor analysis. Psychometrika, 60, 77–93.CrossRefGoogle Scholar
Jennrich, R.I. (1973). Standard errors for obliquely rotated factor loadings. Psychometrika, 38, 593–604.CrossRefGoogle Scholar
Jennrich, R.I. (1974). Simplified formulae for standard errors in maximum likelihood factor analysis. British Journal of Mathematical and Statistical Psychology, 27, 122–131.CrossRefGoogle Scholar
Jennrich, R.I., & Clarkson, D.B. (1980). A feasible method for standard errors of estimate in maximum likelihood factor analysis. Psychometrika, 45, 237–247.CrossRefGoogle Scholar
Kollo, T., & Neudecker, H. (1993). Asymptotics of eigenvalues and unit-length eigenvectors of sample variance and correlation matrices. Journal of Multivariate Analysis, 47, 283–300.CrossRefGoogle Scholar
Kollo, T., & Neudecker, H. (1997). Asymptotics of Pearson-Hotelling principal component vectors of sample variance and correlation matrices. Behaviormetrika, 24, 51–69.CrossRefGoogle Scholar
Meijer, E. (1998). Structural equation models for nonnormal data. Leiden, Netherlands: DSWO Press.Google Scholar
Mooijaart, A. (1985). A note on computational efficiency in asymptotically distribution-free correlational models. British Journal of Mathematical and Statistical Psychology, 38, 112–115.CrossRefGoogle Scholar
Ogasawara, H. (2000). On the standard errors of rotated factor loadings with weights for observed variables. Behaviormetrika, 27, 1–14.CrossRefGoogle Scholar
Ogasawara, H. (2000). Some relationships between factors and components. Psychometrika, 65, 167–185.CrossRefGoogle Scholar
Ogasawara, H. (2000). Asymptotic standard errors of the component loadings for unstandardized and standardized variables. British Journal of Mathematical and Statistical Psychology, 53, 155–174.CrossRefGoogle ScholarPubMed
Steiger, J.H., & Hakstian, A.R. (1982). The asymptotic distribution of elements of a correlation matrix: Theory and application. British Journal of Mathematical and Statistical Psychology, 35, 208–215.CrossRefGoogle Scholar
ten Berge, J.M.F. (1993). Least squares optimization in multivariate analysis. Leiden, Netherlands: DSWO Press.Google Scholar