Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-07T19:03:48.887Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence Theory for Partial Images and Revision of the Definition of Total Images

Published online by Cambridge University Press:  01 January 2025

Peter Findeisen
Peter Findeisen*
Affiliation:
Universität Düsseldorf
*
Requests for reprints should be sent to Peter Findeisen, Psychologisches Institut der Universität Düsseldorf, Lehrstuhl IV, Universittätstrasse 1, 4000 Düsseldorf, WEST GERMANY.
Get access Rights & Permissions [Opens in a new window]

Abstract

Guttman's assumption underlying his definition of “total images” is rejected: Partial images are not generally convergent everywhere. Even divergence everywhere is shown to be possible. The convergence type always found on partial images is convergence in quadratic mean; hence, total images are redefined as quadratic mean-limits. In determining the convergence type in special situations, the asymptotic properties of certain correlations are important, implying, in some cases, convergence almost everywhere, which is also effected by a countable population or multivariate normality or independent variables. The interpretations of a total image as a predictor, and a “common-factor score”, respectively, are made precise.

Keywords

multiple regressionimage analysisconvergence of partial imagescommon-factor scoresregression analysisfactor analysispsychometric sampling
Type
Original Paper
Information
Psychometrika , Volume 44 , Issue 3 , September 1979 , pp. 315 - 328
Copyright
Copyright © 1979 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 referees' comments were greatly appreciated. They led to an improved presentation of results with the addition of the fourth section which was not included in the original manuscript.

References

Bentler, P. M. Some extensions of image analysis. Psychometrika, 1969, 34, 77–83.CrossRefGoogle Scholar
Guttman, L. Image theory for the structure of quantitative variates. Psychometrika, 1953, 18, 277–296.CrossRefGoogle Scholar
Kaiser, H. F. Image analysis. In Harris, C. W. (Eds.), Problems in measuring change, 1963, Madison: The University of Wisconsin Press.Google Scholar
Kaiser, H. F. Image and anti-image covariance matrices from a correlation matrix that may be singular. Psychometrika, 1976, 41, 295–300.CrossRefGoogle Scholar
Loève, M. Probability theory, 1963, Princeton: D. van Nostrand Company, Inc..Google Scholar
McDonald, R. P. Descriptive axioms for common factor theory, image theory and component theory. Psychometrika, 1975, 40, 137–152.CrossRefGoogle Scholar
Tandori, K. Ueber die orthogonalen Funktionen. Acta Scientiarum Mathematicarum, 1957, 18, 57–130.Google Scholar
von Mangoldt, H. & Knopp, K. Einfuehrung in die hoehere Mathematik, 12. Auflage, Stuttgart: Kohlhammer, 1965.Google Scholar
Williams, J. S. A definition for the common-factor analysis model and the elimination of problems of factor score indeterminacy. Psychometrika, 1978, 43, 293–306.CrossRefGoogle Scholar