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Multiple Group Methods for Common-Factor Analysis: their Basis, Computation, and Interpretation

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

In a previous paper (1) were developed three basic theorems which were shown to provide numerical routines, as well as algebraic proof, for existing common-factor methods. New “multiple” routines were also indicated. The first theorem showed how to extract as many common factors as one wished from the correlation matrix in one operation. The second theorem showed how to do the same from the score matrix. The third proved that factoring the correlation matrix was equivalent to factoring the score matrix. A particular application of these theorems is the multiple group factoring method, which the writer first used in practice on some Army attitude scores during World War II. The present paper explains the basic theorems in more detail with special reference to group factoring. Computations are outlined as consisting of five simple matric operations. The meaning of commonfactor analysis is given in terms of the basic theorems, as well as the relationship to “inverted” factor theory.

Type
Original Paper
Information
Psychometrika , Volume 17 , Issue 2 , June 1952 , pp. 209 - 222
Copyright
Copyright © 1952 The Psychometric Society

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Footnotes

*

Since this article was written, the writer has developed a more general approach to factor analysis than that of common factors, which will tentatively be called the “image” approach (4). Some of the resulting theorems show that the problem of communalities is a most fundamental one which in general cannot be solved by current “approximation” procedures; in many cases it admiks of no parsimonious solution at all. In one of the simplest image structures, the image space is only two-dimensional, yet the minimum possible dimensions of any common-factor space is n – 2. where n is the number of observed variables. By the “image” of an observed variable is meant its projection on all the remaining n – 1 variables, and the image space is the space of all n projections. In general, if there is a parsimonious common-factor solution possible, it will coincide with the image solution as n becomes infinite. But image solutions can be parsimonious even when there is no parsimonious common-factor space at all. The image approach, then, is more fundamental than the common-factor approach, and does not depend on solving for communalitics at all; to the contrary, it clarifies the communality problem and shows how the latter may have no parsimonious solution, even though a parsimonious structure exists.

References

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