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A Simple General Method for Oblique 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: rij@stat.ucla.edu
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Abstract

A simple and very general algorithm for oblique rotation is identified. While motivated by the rotation problem in factor analysis, it may be used to minimize almost any function of a not necessarily square matrix whose columns are restricted to have unit length. The algorithm has two steps. The first is to compute the gradient of the rotation criterion and the second is to project this onto a manifold of matrices with unit length columns. For this reason it is called a gradient projection algorithm. Because the projection step is very simple, implementation of the algorithm involves little more than computing the gradient of the rotation criterion which for many applications is very simple. It is proven that the algorithm is strictly monotone, that is as long as it is not already at a stationary point, each step will decrease the value of the criterion. Examples from a variety of areas are used to demonstrate the algorithm, including oblimin rotation, target rotation, simplimax rotation, and rotation to similarity and simplicity simultaneously. While it may be, the algorithm is not intended for use as a standard algorithm for well established problems, but rather as a tool for investigating new methods where its generality and simplicity may save an investigator substantial effort.

Keywords

factor analysisgradient projectionquartiminoblimintarget rotationsimplimaxsimilarity and simplicitystationary points
Type
Articles
Information
Psychometrika , Volume 67 , Issue 1 , March 2002 , pp. 7 - 19
Copyright
Copyright © 2002 The Psychometric Society

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Footnotes

The author would like to thank the review team for their insights and recommendations.

References

Browne, M.W. (1972). Oblique rotation to a partially specified target. British Journal of Mathematical and Statistical Psychology, 25, 207–212.CrossRefGoogle Scholar
Carroll, D.B. (1957). Biquartimin criterion for rotation to oblique simple structure in factor analysis. Science, 126, 1114–1115.CrossRefGoogle ScholarPubMed
Clarkson, D.B., & Jennrich, R.I. (1988). Quartic rotation criteria and algorithms. Psychometrika, 53, 251–259.CrossRefGoogle Scholar
Cliff, N. (1966). Orthogonal rotation to congruence. Psychometrika, 31, 33–42.CrossRefGoogle Scholar
Crawford, C.B., & Ferguson, G.A. (1970). A general rotation criterion and its use in orthogonal rotation. Psychometrika, 35, 321–332.CrossRefGoogle Scholar
Harman, H.H. (1976). Modern factor analysis 3rd ed., Chicago, IL: University of Chicago Press.Google Scholar
Jennrich, R.I., & Sampson, P.F. (1966). Rotation for simple loadings. Psychometrika, 31, 313–323.CrossRefGoogle ScholarPubMed
Jennrich, R.I. (2001). A simple general procedure for orthogonal rotation. Psychometrika, 66, 289–306.CrossRefGoogle Scholar
Kiers, H.A.L. (1994). SIMPLIMAX: Oblique rotation to an optimal target with simple structure. Psychometrika, 59, 567–579.CrossRefGoogle Scholar
Kiers, H.A.L. (1997). Techniques for rotating two or more loading matrices to optimal agreement and simple structure: Comparison and some technical details. Psychometrika, 62, 545–568.CrossRefGoogle Scholar
Thurstone, L.L. (1947). Multiple factor analysis. Chicago, IL: University of Chicago Press.Google Scholar