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Varisim: A New Machine Method for Orthogonal Rotation

Published online by Cambridge University Press:  01 January 2025

Peter H. Schönemann
Peter H. Schönemann*
Affiliation:
Ohio State University
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Abstract

The derivative of Kaiser's Varimax criterion, if set to zero, yields a set of equations which are quite similar to those obtained for a least-squares problem of the “Procrustes” type. This similarity suggested an iterative technique for orthogonal rotation, dubbed “Varisim,” which was programmed for the IBM 7094 in FORTRAN. An empirical comparison between Varimax and Varisim, which was based on a number of data sets taken from the literature yielded three major results so far: (i) Varisim is slower than Varimax, roughly by a factor of 3, (ii) Varisim yields factors which in general contribute more evenly to the common test variance than Varimax factors, and which (iii) line up more closely with oblique configurations obtained with Binormamin than Varimax factors.

Type
Original Paper
Information
Psychometrika , Volume 31 , Issue 2 , June 1966 , pp. 235 - 248
Copyright
Copyright © 1966 Psychometric Society

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Footnotes

*

This paper is based on parts of a thesis submitted to the Graduate College of the University of Illinois in partial fulfillment of the requirements for a degree in Psychology. Some of its contents were read at the joint meeting of the Psychonomic and the Psychometric Society, Niagara Falls, Ontario, 1964. Most of the computer-bound work was carried out while the author was employed by the Statistical Service Unit, University of Illinois; it was written up while the author held a Postdoctoral Fellowship at the Psychometric Laboratory of the University of North Carolina. Although space limitations forbid more complete acknowledgments, I am bound to record my deep gratitude to my advisor, Prof. H. F. Kaiser.

References

Adkins, D. C. and Lyerly, S. B. Factor analysis of reasoning tests, Chapel Hill: Univ. North Carolina Press, 1952.CrossRefGoogle Scholar
Bellman, R. Introduction to matrix analysis, New York: McGraw-Hill, 1960.Google Scholar
Carroll, J. B. Computer applications in the investigation of models in educational research. In Proceedings of a Harvard Symposium on digital computers and their applications, 3–6 April 1961. Harvard Univ. Press, 1962.Google Scholar
Dodd, S. C. The theory of factors. Psychol. Rev., 1928, 35, 211–234.CrossRefGoogle Scholar
Dwyer, P. S. and McPhail, M. S. Symbolic matrix derivatives. Ann. math. Statist., 1948, 19, 517–534.Google Scholar
Green, B. F. The orthogonal approximation of an oblique structure in factor analysis. Psychometrika, 1952, 17, 429–440.CrossRefGoogle Scholar
Harman, H. H. Modern factor analysis, Chicago: Univ. Chicago Press, 1960.Google Scholar
Hendrickson, A. E.and White, P. O. Promax: A quick method of rotation to oblique simple structure. Brit. J. statist. Psychol., 1964, 17, 65–70.CrossRefGoogle Scholar
Hodgson, T. F. The general and primary factors in student evaluation of teaching ability. Technical Report. Principal Investigator: Paul Horst, Univ. Washington, 1958.Google Scholar
Horst, P. H. Matrix algebra for social scientists, New York: Holt, Rinehart and Winston, 1963.Google Scholar
HOW. FORTRAN Subroutine, using non-iterative methods of Householder, Ortega, and Wilkinson, solves for eigenvalues and corresponding eigenvectors of a real symmetric matrix. Program writeup F2 BC HOW. Berkeley Division, Univ. California, 1962.Google Scholar
Humphreys, L. G. Number of cases and number of factors: an example whereN is very large. Educ. psychol. Measmt, 1964, 24, 457–466.CrossRefGoogle Scholar
Hurley, J. R. and Cattell, R. B. Producing direct rotation to test a hypothesized factor structure. Behav. Sci., 1962, 7, 258–262.CrossRefGoogle Scholar
Kaiser, H. F. Relating factors between studies based upon different individuals. Unpublished manuscript, Univ. Illinois, 1960.Google Scholar
Kaiser, H. F. Varimax solution for primary mental abilities. Psychometrika, 1960, 25, 153–157.CrossRefGoogle Scholar
Kaiser, H. F. Computer program for varimax rotation in factor analysis. Educ. psychol. Measmt, 1959, 19, 413–420.CrossRefGoogle Scholar
Kaiser, H. F. The varimax criterion for analytic rotation in factor analysis. Psychometrika, 1958, 23, 187–200.Google Scholar
Kaiser, H. F. and Dickman, K. W. Analytic determination of common factors. Amer. Psychol., 1959, 14, 425–430.Google Scholar
Lawley, D. N. and Maxwell, A. E. Factor analysis as a statistical method, London: Butterworth, 1963.Google Scholar
Mosier, C. I. Determining a simple structure when loadings for certain tests are known. Psychometrika, 1939, 4, 149–162.CrossRefGoogle Scholar
Pinzka, C. and Saunders, D. R. Analytic rotation to simple structure, II: Extension to an oblique solution. Research Bulletin RB-54-31, Princeton, New Jersey: Educational Testing Service, 1954.Google Scholar
Saunders, D. R. The rationale for an “oblimax” method of transformation in factor analysis. Psychometrika, 1961, 26, 317–324.CrossRefGoogle Scholar
Schönemann, P. H. A generalized solution of the orthogonal Procrustes problem. Psychometrika, 1966, 31, 1–10.CrossRefGoogle Scholar
Schönemann, P. H. A solution of the orthogonal Procrustes problem with applications to orthogonal and oblique rotation, Ann Arbor, Michigan: University Microfilms, Inc., 1965.Google Scholar
Schönemann, P. H. A FORTRAN program for Varisim, a machine method for orthogonal rotation to simple structure, and a summary of some empirical results, Chapel Hill: Univ. North Carolina Psychometric Laboratory, 1965.Google Scholar
SSUPAC. Manual of computer programs for statistical analysis, Urbana: Univ. Illinois, 1965.Google Scholar
Taylor, A. Advanced calculus, Boston: Ginn and Company, 1955.Google Scholar
Thurstone, L. L. Multiple-factor analysis, Chicago: Univ. Chicago Press, 1947.Google Scholar
Thurstone, L. L. A factorial study of perception, Chicago: Univ. Chicago Press, 1943.Google Scholar
Thurstone, L. L. Primary mental abilities. Psychometric Monographs, 1938, 1, 1–121.Google Scholar
Tucker, L. R. An objective determination of simple structure in factor analysis. Amer. Psychol., 1953, 8, 448–448.Google Scholar
Tucker, L. R. A method for synthesis of factor analysis studies. A. G. O. Personnel Research Section Rep. No. 984, Dept. Army, 1951.Google Scholar
Tucker, L. R. A semianalytical method of factorial rotation to simple structure. Psychometrika, 1944, 9, 43–68.CrossRefGoogle Scholar
Wrigley, C., Saunders, D. R., and Neuhaus, J. O. Application of the quartimax method of rotation to Thurstone's primary mental abilities study. Psychometrika, 1958, 23, 151–169.CrossRefGoogle Scholar
Wrobleski, W. J. Extension of the Dwyer-MacPhail matrix derivative calculus with applications to estimation problems involving errors-in-variables and errors-in-equations. Technical Report, Univ. Michigan, 1963.Google Scholar