Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-08T13:24:25.603Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Relations Among Regular, Equal Unique Variances, and Image Factor Analysis Models

Published online by Cambridge University Press:  02 January 2025

Kentaro Hayashi  and
Peter M. Bentler
Kentaro Hayashi
Affiliation:
Bucknell University
Peter M. Bentler*
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to Peter M. Bentler, Department of Psychology, University of California, Los Angeles CA 90095-1563.
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate under what conditions the matrix of factor loadings from the factor analysis model with equal unique variances will give a good approximation to the matrix of factor loadings from the regular factor analysis model. We show that the two models will give similar matrices of factor loadings if Schneeweiss' condition, that the difference between the largest and the smallest value of unique variances is small relative to the sizes of the column sums of squared factor loadings, holds. Furthermore, we generalize our results and discus the conditions under which the matrix of factor loadings from the regular factor analysis model will be well approximated by the matrix of factor loadings from Jöreskog's image factor analysis model. Especially, we discuss Guttman's condition (i.e., the number of variables increases without limit) for the two models to agree, in relation with the condition we have shown, and conclude that Schneeweiss' condition is a generalization of Guttman's condition. Some implications for practice are discussed.

Keywords

factor loadingscommunalitiesscale invarianceprincipal components
Type
Original Paper
Information
Psychometrika , Volume 65 , Issue 1 , March 2000 , pp. 59 - 72
Copyright
Copyright © 2000 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

Kentaro Hayashi is a visiting Assistant Professor, Department of Mathematics, Bucknell University, Lewisburg PA 17837, and Peter M. Bentler is Professor, Departments of Psychology and Statistics, University of California, Los Angeles CA 90095-1563. (Emails: Khayashi@bucknell.edu, bentler@ucla.edu) Parts of this paper were discussed in a session on Factor Analysis (J. ten Berge, Chair) at the IFCS-98 International Conference, Rome, July, 1998. This work was supported by National Institute on Drug Abuse grant DA 01070. The authors thank Professors Hans Schneeweiss and Ke-Hai Yuan, and four anonymous referees, for their invaluable comments which led to an improved version of this paper.

References

Anderson, T. W. (1984). Estimating linear statistical relationships. The Annals of Statistics, 12, 1–45CrossRefGoogle Scholar
Bentler, P. M. (1995). EQS structural equations program manual. Encino, CA: Multivariate SoftwareGoogle Scholar
Bentler, P. M., & Kano, Y. (1990). On the equivalence of factors and components. Multivariate Behavioral Research, 25, 67–74CrossRefGoogle ScholarPubMed
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, 65–81CrossRefGoogle Scholar
Guttman, L. (1956). Best possible systematic estimates of communalities. Psychometrika, 21, 273–285CrossRefGoogle Scholar
Harris, C. W. (1962). Some Rao-Guttman relationships. Psychometrika, 27, 247–263CrossRefGoogle Scholar
Jöreskog, K. G. (1963). Statistical estimation in factor analysis. Uppsala: Almqvist & WiksellGoogle Scholar
Jöreskog, K. G. (1969). Efficient estimation in image factor analysis. Psychometrika, 34, 51–75CrossRefGoogle Scholar
Kaiser, H. F. (1990, October). Outline of EPIC, a new method for factoring a reduced correlation matrix. Paper presented at Society of Multivariate Experimental Psychology, Providence, RI.Google Scholar
Lawley, D. N. (1953). A modified method of estimation in factor analysis and some large sample results. Uppsala symposium on psychological factor analysis, 17–19 March 1953. Uppsala: Almqvist & WiksellGoogle Scholar
Magnus, J. R., & Neudecker, H. (1988). Matrix differential calculus with applications in statistics and econometrics. New York: WileyGoogle Scholar
Muirhead, R. J. (1982). Aspects of multivariate statistical theory. New York: WileyCrossRefGoogle Scholar
Ramsay, J. O., ten Berge, J. M. F., & Styan, G. P. H. (1984). Matrix correlation. Psychometrika, 49, 403–423CrossRefGoogle Scholar
Satorra, A. (1989). Alternative criteria in covariance structure analysis: A unified treatment. Psychometrika, 54, 131–151CrossRefGoogle Scholar
Schneeweiss, H. (1997). Factors and principal components in the near spherical case. Multivariate Behavioral Research, 32, 375–401CrossRefGoogle ScholarPubMed
Schneeweiss, H., & Mathes, H. (1995). Factor analysis and principal components. Journal of Multivariate Analysis, 55, 105–124CrossRefGoogle Scholar
Schott, J. R. (1997). Matrix analysis for statistics. New York: WileyGoogle Scholar
Uhlig, H. (1994). On singular Wishart and singular multivariate beta distributions. The Annals of Statistics, 22, 395–405CrossRefGoogle Scholar
Velicer, W., & Jackson, D. (1990). Component analysis versus common factor analysis: Some issues in selecting an appropriate procedure. Multivariate Behavioral Research, 25, 1–28CrossRefGoogle ScholarPubMed
Whittle, P. (1952). On principal components and least square methods of factor analysis. Skandinavisk Aktuarietidskrift, 35, 223–239Google Scholar
Yuan, K-H., & Bentler, P. M. (1997). Improving parameter tests in covariance structure analysis. Computational Statistics & Data Analysis, 26, 177–198CrossRefGoogle Scholar