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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.

References

Buck, R.C. (1956). Advanced calculus. New York, NY: McGraw-Hill.Google Scholar
Carroll, J.B. (1953). An analytical solution for approximating simple structure in factor analysis. Psychometrika, 18, 32–38.CrossRefGoogle Scholar
Cliff, N. (1966). Orthogonal rotation to congruence. Psychometrika, 31, 33–42.CrossRefGoogle Scholar
de Leeuw, J. (1994). Block-relaxation algorithms in statistics. In Bock, H.H., Lenski, W., Richter, M.M. (Eds.), Information systems and data analysis (pp. 308–324). Berlin: Springer.CrossRefGoogle Scholar
de Leeuw, J., Pruzansky, S. (1978). A new computational method to fit the weighted Euclidean distance model. Psychometrika, 43, 479–490.CrossRefGoogle Scholar
Dempster, A.P., Laird, N.M., Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–38.CrossRefGoogle Scholar
Golub, G.H., van Loan, C.F. (1989). Matrix computations. London: The Johns Hopkins University Press.Google Scholar
Harman, H.H. (1960). Modern factor analysis. Chicago, IL: University of Chicago Press.Google Scholar
Heiser, W.J. (1995). Convergent computation by iterative majorization: Theory and applications in multidimensional data analysis. In Krzanowski, W.J. (Eds.), Recent advances in descriptive multivariate analysis (pp. 157–189). Oxford: Oxford University Press.CrossRefGoogle Scholar
Horst, P. (1965). Factor analysis of data matrices. New York, NY: Holt, Rinehart and Winston.Google Scholar
Jamshidian, M., Jennrich, R.I. (1997). Acceleration of the EM algorithm by using quasi-Newton methods. Journal of the Royal Statistical Society, Series B, 59, 569–587.CrossRefGoogle Scholar
Kaiser, H.F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23, 187–200.CrossRefGoogle Scholar
Kaplan, W. (1999). Maxima and minima with applications. New York, NY: Wiley.Google Scholar
Kiers, H.A.L. (1990). Majorization as a tool for optimizing a class of matrix functions. Psychometrika, 55, 417–428.CrossRefGoogle Scholar
Kiers, H.A.L., ten Berge, J.M.F. (1992). Minimization of a class of matrix trace functions by means of refined majorization. Psychometrika, 57, 371–382.CrossRefGoogle Scholar
Kiers, H.A.L., ten Berge, J.M.F., Takane, Y., de Leeuw, J. (1990). A generalization of Takane's algorithm for DEDICOM. Psychometrika, 55, 151–158.CrossRefGoogle Scholar
Loomis, L.H., Sternberg, S. (1968). Advanced calculus. Reading, PA: Addison-Wesley.Google Scholar
Luenberger, D.G. (1984). Linear and nonlinear programming. Reading, PA: Addison-Wesley.Google Scholar
Magnus, J.R., Neudecker, H. (1999). Matrix differential calculus with applications to statistics and econometrics. New York, NY: Wiley.Google Scholar
Mulaik, S.A. (1972). The foundations of factor analysis. New York, NY: McGraw-Hill.Google Scholar
Neuhaus, J.O., Wrigley, C. (1954). The quartimax method: An analytical approach to orthogonal simple structure. British Journal of Statistical Psychology, 7, 81–91.CrossRefGoogle Scholar
Sherin, R.J. (1966). A matrix formulation of Kaiser's varimax criterion. Psychometrika, 34, 535–538.CrossRefGoogle Scholar
Takane, Y. (1985). Diagonal estimation in DEDICOM. Proceedings of the 1985 Annual Meeting of the Behaviormetric Society, 13, 100–101.Google Scholar
ten Berge, J.M.F. (1984). A joint treatment of varimax rotation and the problem of diagonalizing symmetric matrices simultaneously in the least squares sense. Psychometrika, 49, 347–358.CrossRefGoogle Scholar
ten Berge, J.M.F., Knol, D.L., Kiers, H.A.L. (1988). A treatment of the orthomax rotation family in terms of diagonalization, and a re-examination of the singular value approach to varimax rotation. Computational Statistics Quarterly, 4, 207–217.Google Scholar