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Orthogonal Rotation Algorithms

Published online by Cambridge University Press:  01 January 2025

Robert I. Jennrich
Robert I. Jennrich*
Affiliation:
University of California, Los Angeles
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Abstract

The quartimax and varimax algorithms for orthogonal rotation attempt to maximize particular simplicity criteria by a sequence of two-factor rotations. Derivations of these algorithms have been fairly complex. A simple general theory for obtaining “two factor at a time” algorithms for any polynomial simplicity criteria satisfying a natural symmetry condition is presented. It is shown that the degree of any symmetric criterion must be a multiple of four. A basic fourth degree algorithm, which is applicable to all symmetric fourth degree criteria, is derived and applied using a variety of criteria. When used with the quartimax and varimax criteria the algorithm is mathematically identical to the standard algorithms for these criteria. A basic eighth degree algorithm is also obtained and applied using a variety of eighth degree criteria. In general the problem of writing a basic algorithm for all symmetric criteria of any specified degree reduces to the problem of maximizing a trigonometric polynomial of degree one-fourth that of the criteria.

Keywords

Public PolicyGeneral TheoryStatistical TheorySymmetry ConditionBasic Algorithm
Type
Original Paper
Information
Psychometrika , Volume 35 , Issue 2 , June 1970 , pp. 229 - 235
Copyright
Copyright © 1970 The Psychometric Society

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Footnotes

*

This research was supported by the Bell Telephone Laboratories, Murray Hill, New Jersey and NIH Grant FR-3.

References

Baress, E. H. The numerical solution of polynomial equations and the resultant procedures. In Ralston, A. and Wilf, H. S. (Eds.), Mathematical methods for digital computers. Volume II, Wiley, 1967, 185–214.Google Scholar
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Kaiser, H. F. The varimax criterion for analytic rotation in factor analysis. Psychometrika, 1958, 23, 187–200.CrossRefGoogle Scholar
Newhaus, J. O., & Wrigley, C. The quartimax method: An analytica lapproach to orthogonal simple structure. British Journal of Statistical Psychology, 1954, 7, 81–91.CrossRefGoogle Scholar

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