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Rotation to Simple Loadings Using Component Loss Functions: The Oblique Case

Published online by Cambridge University Press:  01 January 2025

Robert I. Jennrich
Robert I. Jennrich*
Affiliation:
University of California at Los Angeles
*
Requests for reprints should be sent to Robert I. Jennrich, Department of Mathematics, University of California, Los Angeles, CA 90095, USA. E-mail: rij@stat.ucla.edu
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Abstract

Component loss functions (CLFs) similar to those used in orthogonal rotation are introduced to define criteria for oblique rotation in factor analysis. It is shown how the shape of the CLF affects the performance of the criterion it defines. For example, it is shown that monotone concave CLFs give criteria that are minimized by loadings with perfect simple structure when such loadings exist. Moreover, if the CLFs are strictly concave, minimizing must produce perfect simple structure whenever it exists. Examples show that methods defined by concave CLFs perform well much more generally. While it appears important to use a concave CLF, the specific CLF used is less important. For example, the very simple linear CLF gives a rotation method that can easily outperform the most popular oblique rotation methods promax and quartimin and is competitive with the more complex simplimax and geomin methods.

Keywords

component loss criteriafactor analysisgeomingradient projectionloading polishpromaxquartiminsimplimaxsorted absolute loading plots
Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 1 , March 2006 , pp. 173 - 191
Copyright
Copyright © 2006 The Psychometric Society

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Footnotes

The author would like to thank the editor and three reviewers for helpful suggestions and for identifying numerous errors.

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