Hostname: page-component-89b8bd64d-r6c6k Total loading time: 0 Render date: 2026-05-08T18:58:17.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient Estimation in Image Factor Analysis

Published online by Cambridge University Press:  01 January 2025

K. G. Jöreskog
K. G. Jöreskog*
Affiliation:
Educational Testing Service
Get access Rights & Permissions [Opens in a new window]

Abstract

The image factor analytic model (IFA), as related to Guttman’s image theory, is considered as an alternative to the traditional factor analytic model (TFA). One advantage with IFA, as compared with TFA, is that more factors can be extracted without yielding a perfect fit to the observed data. Several theorems concerning the structural properties of IFA are proved and an iterative procedure for finding the maximum likelihood estimates of the parameters of the IFA-model is given. Substantial experience with this method verifies that Heywood cases never occur. Results of an artificial experiment suggest that IFA may be more factorially invariant than TFA under selection of tests from a large battery.

Information

Type
Original Paper
Information
Psychometrika , Volume 34 , Issue 1 , March 1969 , pp. 51 - 75
Copyright
Copyright © 1969 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.)

Article purchase

Temporarily unavailable

Footnotes

*

The first part of this work was done at the University of Uppsala, Sweden and supported by the Swedish Council for Social Science Research. The second part was done at Educational Testing Service and supported by a grant (NSF-GB-1985) from the National Science Foundation to Educational Testing Service. The author is indebted to Mr. G. Gruvaeus, who wrote many of the computer programs, checked the mathematical derivations and gave other invaluable assistance throughout the work.

References

Aitchison, J., & Silvey, S. D. Maximum likelihood estimation procedures and associated tests of significance. Journal of the Royal Statistical Society, Series B, 1960, 22, 154–171CrossRefGoogle Scholar
Anderson, T. W. An introduction to multivariate statistical analysis, 1958, New York: WileyGoogle Scholar
Anderson, T. W., & Rubin, H. Statistical inference in factor analysis. In Neyman, J. (Eds.), Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. V. Berkeley: University of California Press. 1956, 111–150Google Scholar
Bargmann, R. A study of independence and dependence in multivariate normal analysis. University of North Carolina, Institute of Statistics Mimeo Series No. 186, 1957.Google Scholar
Bellman, R. Introduction to matrix analysis, 1960, New York: McGraw-HillGoogle Scholar
Courant, R. Differential and integral calculus, Vol. II, 1961, London and Glasgow: Blackie & Son LimitedGoogle Scholar
Fletcher, R., & Powell, M. J. D. A rapidly convergent descent method for minimization. Computer Journal, 1963, 2, 163–168CrossRefGoogle Scholar
Green, B. F. The orthogonal approximation of an oblique structure in factor analysis. Psychometrika, 1952, 17, 429–440CrossRefGoogle Scholar
Guttman, L. Image theory for the structure of quantitative variates. Psychometrika, 1953, 18, 277–296CrossRefGoogle Scholar
Guttman, L. Some necessary conditions for common-factor analysis. Psychometrika, 1954, 19, 149–161CrossRefGoogle Scholar
Guttman, L. The determinacy of factor score matrices with implications for five other basic problems of common-factor theory. British Journal of Statistical Psychology, 1955, 8, 65–81CrossRefGoogle Scholar
Guttman, L. Best possible systematic estimates of communalities. Psychometrika, 1956, 21, 273–285CrossRefGoogle Scholar
Guttman, L. An estimate of communalities that is image-wise consistent and structure-free. Berkeley: University of California Research Report No. 20, Contr. No. AF41(657)-76, 1958.Google Scholar
Harman, H. Modern factor analysis, 1960, Chicago: University of Chicago PressGoogle Scholar
Harris, C. W. Some Rao-Guttman relationships. Psychometrika, 1962, 27, 247–263CrossRefGoogle Scholar
Householder, A. S. Principles of numerical analysis, 1953, New York: McGraw-HillGoogle Scholar
Jöreskog, K. G. On the statistical treatment of residuals in factor analysis. Psychometrika, 1962, 27, 335–345CrossRefGoogle Scholar
Jöreskog, K. G. Statistical estimation in factor analysis, 1963, Stockholm: Almqvist & WiksellGoogle Scholar
Jöreskog, K. G. Image factor analysis, 1965, Princeton, N.J.: Educational Testing ServiceGoogle Scholar
Jöreskog, K. G. Testing a simple structure hypothesis in factor analysis. Psychometrika, 1966, 31, 165–178CrossRefGoogle ScholarPubMed
Jöreskog, K. G. UMLFA—A computer program for unrestricted maximum likelihood factor analysis, 1966, Princeton, N.J.: Educational Testing ServiceGoogle Scholar
Jöreskog, K. G. Some contributions to maximum likelihood factor analysis. Psychometrika, 1967, 32, 443–482CrossRefGoogle Scholar
Kaiser, H. F. The best approximation of a common-factor space. Berkeley: University of California Research Report No. 25, Contr. No. AF41(657)-76, 1958.Google Scholar
Kaiser, H. F. Image analysis. In Harris, C. W. (Eds.), Problems in measuring change, 1963, Madison: University of Wisconsin PressGoogle Scholar
Kaiser, H. F., & Caffrey, J. Alpha factor analysis. Psychometrika, 1965, 30, 1–14CrossRefGoogle ScholarPubMed
Lawley, D. N. The estimation of factor loadings by the method of maximum likelihood. Proceedings of the Royal Society of Edinburgh, Section A, 1940, 60, 64–82CrossRefGoogle Scholar
Lawley, D. N., & Maxwell, A. E. Factor analysis as a statistical method, 1963, London: ButterworthsGoogle Scholar
Mattsson, A., Olsson, U., & Rosén, M. The maximum likelihood method in factor analysis with special consideration to the problem of improper solutions, 1966, Sweden: Institute of Statistics, University of UppsalaGoogle Scholar
Rao, C. R. Estimation and tests of significance in factor analysis. Psychometrika, 1955, 20, 93–111CrossRefGoogle Scholar
Thurstone, L. L., & Thurstone, T. G. Factorial studies of intelligence, 1941, Chicago: University of Chicago PressGoogle Scholar
Wilks, S. S. The large-sample distribution of the likelihood ratio for testing composite hypotheses. Annals of Mathematical Statistics, 1938, 9, 60–62CrossRefGoogle Scholar