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On Component Analyses

Published online by Cambridge University Press:  01 January 2025

William Meredith  and
Roger E. Millsap
William Meredith*
Affiliation:
University of California, Berkeley
Roger E. Millsap
Affiliation:
Baruch College, City University of New York
*
Requests for reprints should be sent to William Meredith, Department of Psychology, Tolman Hall, University of California, Berkeley, CA 94720.
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Abstract

Principal components analysis can be redefined in terms of the regression of observed variables upon component variables. Two criteria for the adequacy of a component representation in this context are developed and are shown to lead to different component solutions. Both criteria are generalized to allow weighting, the choice of weights determining the scale invariance properties of the resulting solution. A theorem is presented giving necessary and sufficient conditions for equivalent component solutions under different choices of weighting. Applications of the theorem are discussed that involve the components analysis of linearly derived variables and of external variables.

Keywords

Principal componentslinear regressionscale invarianceimage analysis
Type
Original Paper
Information
Psychometrika , Volume 50 , Issue 4 , December 1985 , pp. 495 - 507
Copyright
Copyright © 1985 The Psychometric Society

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Footnotes

Preparation of this article was supported in part by National Institute of Aging Grant NIA-AG03164-03 to William Meredith.

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