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Some Further Inequalities and an Identity in Factor Analysis

Published online by Cambridge University Press:  01 January 2025

John N. Darroch
John N. Darroch*
Affiliation:
The Flinders University of South Australia
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Abstract

Let x denote one of the variables in a factor analysis model. In 1935 Roff proved that the communality of x is greater than or equal to the squared multiple correlation coefficient of x with the other variables in the model. In this paper it is shown that the inequality is the first of an infinite sequence of inequalities, each sharper than the one before, and that the condition for equality is the same for all members of the sequence. Also, the limiting inequality is considered and the condition for equality derived. As this condition is always satisfied in a factor analysis model, the inequality is really an identity.

Type
Original Paper
Information
Psychometrika , Volume 34 , Issue 1 , March 1969 , pp. 45 - 49
Copyright
Copyright © 1969 The Psychometric Society

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Footnotes

*

The research herein reported was supported by HEW OE-6-10-278.

References

Darroch, J. N. A set of inequalities in factor analysis. Psychometrika, 1965, 30, 449–453CrossRefGoogle ScholarPubMed
Madansky, A. On admissible communalities in factor analysis. Psychometrika, 1965, 30, 455–458CrossRefGoogle ScholarPubMed
Roff, M. Some properties of the communality in multiple factor theory. Psychometrika, 1935, 1, 1–6CrossRefGoogle Scholar