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The Mathematical Theory of Factorial Invariance Under Selection

Published online by Cambridge University Press:  01 January 2025

Yrjö Ahmavaara
Yrjö Ahmavaara*
Affiliation:
Helsinki University
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Abstract

It is first demonstrated that Aitken's selection formulas are equivalent to a linear transformation in the factor space. On this basis the Thomson-Ledermann theorem concerning the invariance of the number of common factors under selection, and a theorem concerning the invariance of factor loadings under selection are derived. A mathematical proof of the results of Thurstone, which are concerned with the invariance of simple structure under selection, is given. The paper provides a conclusive answer to the question, considered by Thurstone and Thomson, whether a multivariate selection is always reducible to successive univariate selections.

Type
Original Paper
Information
Psychometrika , Volume 19 , Issue 1 , March 1954 , pp. 27 - 38
Copyright
Copyright © 1954 Psychometric Society

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References

Thurstone, L. L.. Multiple-factor analysis, Chicago: Univ. Chicago Press, 1947Google Scholar
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Aitken, A. C.. Note on selection from a multivariate normal population. Proc. Edin. math. Soc., 1934, 4, 106–110CrossRefGoogle Scholar
Thomson, G. H.. The influence of univariate selection on the factorial analysis of ability. Brit. J. Psychol., 1938, 28, 451–459Google Scholar
Ledermann, W.. Note on Professor Godfrey H. Thomson's article “The influence of univariate selection on factorial analysis of ability. Brit. J. Psychol., 1938, 29, 69–73Google Scholar
Thomson, G. H., Ledermann, W.. The influence of multivariate selection on the factorial analysis of ability. Brit. J. Psychol., 1939, 29, 288–305Google Scholar