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Image Theory for the Structure of Quantitative Variates

Published online by Cambridge University Press:  01 January 2025

Louis Guttman
Louis Guttman*
Affiliation:
The Israel Institute of Applied Social Research
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Abstract

A universe of infinitely many quantitative variables is considered, from which a sample of n variables is arbitrarily selected. Only linear least-squares regressions are considered, based on an infinitely large population of individuals or respondents. In the sample of variables, the predicted value of a variable x from the remaining n − 1 variables is called the partial image of x, and the error of prediction is called the partial anti-image of x. The predicted value of x from the entire universe, or the limit of its partial images as n → ∞, is called the total image of x, and the corresponding error is called the total anti-image. Images and anti-images can be used to explain “why” any two variables xj and xk are correlated with each other, or to reveal the structure of the intercorrelations of the sample and of the universe. It is demonstrated that image theory is related to common-factor theory but has greater generality than common-factor theory, being able to deal with structures other than those describable in a Spearman-Thurstone factor space. A universal computing procedure is suggested, based upon the inverse of the correlation matrix.

Type
Original Paper
Information
Psychometrika , Volume 18 , Issue 4 , December 1953 , pp. 277 - 296
Copyright
Copyright © 1953 The Psychometric Society

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Footnotes

*

This paper introduces one of three new structural theories, each of which generalizes common-factor analysis in a different direction. Nodular theory extends common-factor analysis to qualitative data and to data with curvilinear regressions (6). Order-factor theory introduces the notions of order among the observed variables and of separable factors (7). The present image theory is relevant also to the other two.

Attention may be called to empirical results published since this paper was written: Louis Guttman, “Two new approaches to factor analysis,” Annual Technical Report on contract Nonr—731(00). The present research was aided by an uncommitted grant-in-aid from the Ford Foundation.

References

Guttman, Louis. A note on the derivation of formulae for multiple and partial correlation. Ann. math. Statist., 1938, 9, 305–308.CrossRefGoogle Scholar
Guttman, Louis. Multiple rectilinear prediction and the resolution into components. Psychometrika, 1940, 5, 75–99.Google Scholar
Guttman, Louis. Cohen, Joze. Multiple rectilinear prediction and the resolution into components: II. Psychometrika, 1943, 8, 169–183.CrossRefGoogle Scholar
Guttman, Louis. Review of Thurstone's Multiple-factor analysis. J. Amer. statist. Ass., 1947, 42, 651–656.CrossRefGoogle Scholar
Guttman, Louis. The Israel alpha technique for scale analysis: a preliminary statement (stenciled), Jerusalem, Israel: The Israel Institute of Applied Social Research, 1951.Google Scholar
Guttman, Louis. The theory of nodular structures. (In preparation)Google Scholar
Guttman, Louis. A new approach to factor analysis: The Radex. In Lazarsfeld, Paul F. (Eds.), Mathematical thinking in the social sciences, New York: Columbia Univ. Press, 1953.Google Scholar
Guttman, Louis. The matrices of least-squares image analysis. (In preparation)Google Scholar
Guttman, Louis. The existence of total least-squares images and anti-images (In preparation)Google Scholar
Guttman, Louis. A reanalysis of factor analysis. (In preparation)Google Scholar
Jackson, Dunham. The trigonometry of correlation. Amer. math. Monthly, 1924, 31, 275–280.CrossRefGoogle Scholar
Thurstone, L. L. Multiple-factor analysis, Chicago: Univ. Chicago Press, 1947.Google Scholar