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A Simple General Procedure for Orthogonal Rotation

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 Statistics, University of California, Los Angeles, CA 90095. E-Mail: [email protected]
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Abstract

A very general algorithm for orthogonal rotation is identified. It is shown that when an algorithm parameter α is sufficiently large the algorithm converges monotonically to a stationary point of the rotation criterion from any starting value. Because a sufficiently largeα is in general hard to find, a modification that does not require it is introduced. Without this requirement the modified algorithm is not only very general, but also very simple. Its implementation involves little more than computing the gradient of the rotation criterion. While the modified algorithm converges monotonically from any starting value, it is not guaranteed to converge to a stationary point. It, however, does so in all of our examples. While motivated by the rotation problem in factor analysis, the algorithms discussed may be used to optimize almost any function of a not necessarily square column-wise orthonormal matrix. A number of these more general applications are considered. Empirical examples show that the modified algorithm can be reasonably fast, but its purpose is to save an investigator's effort rather than that of his or her computer. This makes it more appropriate as a research tool than as an algorithm for established methods.

Keywords

factor analysisquartimaxvarimaxorthomaxsimultaneous rotationsimultaneous diagonalizationsingular value decompositionpairwise methodsglobal convergence
Type
Articles
Information
Psychometrika , Volume 66 , Issue 2 , June 2001 , pp. 289 - 306
Copyright
Copyright © 2001 The Psychometric Society

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Footnotes

The author is indebted to Michael Browne for significant insights and advice on the rotation problem and specific comments on this manuscript, and to the review team for valuable suggestions.

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