Hostname: page-component-76d6cb85b7-pn7tm Total loading time: 0 Render date: 2026-07-25T18:43:28.740Z Has data issue: false hasContentIssue false
  • English
  • Français

Factor Simplicity Index and Transformations

Published online by Cambridge University Press:  01 January 2025

P. M. Bentler
P. M. Bentler*
Affiliation:
University of California at Los Angeles
*
Requests for reprints should be sent to P. M. Bender, Department of Psychology, University of California, Los Angeles, CA90024.
Get access Rights & Permissions [Opens in a new window]

Abstract

A scale-invariant index of factorial simplicity is proposed as a summary statistic for principal components and factor analysis. The index ranges from zero to one, and attains its maximum when all variables are simple rather than factorially complex. A factor scale-free oblique transformation method is developed to maximize the index. In addition, a new orthogonal rotation procedure is developed. These factor transformation methods are implemented using rapidly convergent computer programs. Observed results indicate that the procedures produce meaningfully simple factor pattern solutions.

Keywords

multivariate analysisorthogonal rotationoblique transformation

Information

Type
Original Paper
Information
Psychometrika , Volume 42 , Issue 2 , June 1977 , pp. 277 - 295
Copyright
Copyright © 1977 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This investigation was supported in part by a Research Scientist Development Award (K02-DA00017) and research grants (MH24149 and DA01070) from the U. S. Public Health Service. The assistance of Andrew L. Comrey, Henry F. Kaiser, Bonnie Barron, Marion Hee, and several anonymous reviewers is gratefully acknowledged.

References

References Notes

Carroll, J. B. IBM 704 program for generalized analytic rotation solution in factor analysis. Unpublished manuscript, Harvard University, 1960.Google Scholar
Kaiser, H. F. & Dickman, K. W. Analytic determination of common factors. Unpublished manuscript, University of Illinois, 1959.Google Scholar
Saunders, D. R. An analytic method for rotation to orthogonal simple structure, 1953, Princeton, New Jersey: Educational Testing Service.CrossRefGoogle Scholar
Saunders, D. R. Transvarimax: Some properties of the ratiomax and equamax criteria for blind orthogonal rotation. Paper delivered at the American Psychological Association convention, 1962.Google Scholar

References

Balloun, J. L. & Kearns, J. An approach to an orthogonal simple structure solution by maximizing test-factor interaction effects among squared factor loadings. British Journal of Mathematical and Statistical Psychology, 1975, 28, 63–70.CrossRefGoogle Scholar
Bentler, P. M. Clustran, a program for oblique transformation. Behavioral Science, 1971, 16, 183–185.Google Scholar
Bentler, P. M. Multistructure statistical model applied to factor analysis. Multivariate Behavioral Research, 1976, 11, 3–25.CrossRefGoogle ScholarPubMed
Bentler, P. M. & Lee, S. Y. Some extensions of matrix calculus. General Systems, 1975, 20, 145–150.Google Scholar
Bentler, P. M. & Wingard, J. Function-invariant and parameter scale-free transformation methods. Psychometrika, 1977, 42, 221–240.CrossRefGoogle Scholar
Carroll, J. B. An analytical solution for approximating simple structure in factor analysis. Psychometrika, 1953, 18, 23–38.CrossRefGoogle Scholar
Cattell, R. B. “Parallel proportional profiles” and other principles for determining the choice of factors by rotation. Psychometrika, 1944, 9, 267–283.CrossRefGoogle Scholar
Comrey, A. L. Tandem criteria for analytic rotation in factor analysis. Psychometrika, 1967, 32, 143–154.CrossRefGoogle ScholarPubMed
Crawford, C. A comparison of the direct oblimin and primary parsimony methods of oblique rotation. British Journal of Mathematical and Statistical Psychology, 1975, 28, 201–213.CrossRefGoogle Scholar
Crawford, C. B. & Ferguson, G. A. A general rotation criterion and its use in orthogonal rotation. Psychometrika, 1970, 35, 321–332.CrossRefGoogle Scholar
Cureton, E. E. & Mulaik, S. A. The weighted varimax rotation and the promax rotation. Psychometrika, 1975, 40, 183–195.CrossRefGoogle Scholar
Eber, H. W. Toward oblique simple structure: Maxplane. Multivariate Behavioral Research, 1966, 1, 112–125.CrossRefGoogle Scholar
Ferguson, G. A. The concept of parsimony in factor analysis. Psychometrika, 1954, 19, 281–290.CrossRefGoogle Scholar
Hakstian, A. R. The development of a class of oblique factor solutions. British Journal of Mathematical and Statistical Psychology, 1974, 27, 100–114.CrossRefGoogle Scholar
Hakstian, A. R. & Abell, R. A. A further comparison of oblique factor transformation methods. Psychometrika, 1974, 39, 429–444.CrossRefGoogle Scholar
Harman, H. H. Modern factor analysis, 2nd ed., Chicago: University of Chicago Press, 1967.Google Scholar
Harris, C. W. & Kaiser, H. F. Oblique factor analytic solutions by orthogonal transformations. Psychometrika, 1964, 29, 347–362.CrossRefGoogle Scholar
Horn, J. L. On extension analysis and its relation to correlations between variables and factor scores. Multivariate Behavioral Research, 1973, 8, 477–489.CrossRefGoogle ScholarPubMed
Jackson, D. N. & Skinner, H. A. Univocal varimax: An orthogonal factor rotation program for optimal simple structure. Educational and Psychological Measurement, 1975, 35, 663–665.CrossRefGoogle Scholar
Jennrich, R. I. Orthogonal rotation algorithms. Psychometrika, 1970, 35, 229–235.CrossRefGoogle Scholar
Jennrich, R. I. & Sampson, P. J. Rotation for simple loadings. Psychometrika, 1966, 31, 313–323.CrossRefGoogle ScholarPubMed
Kaiser, H. F. The varimax criterion for analytic rotation in factor analysis. Psychometrika, 1958, 23, 187–200.CrossRefGoogle Scholar
Kaiser, H. F. An index of factorial simplicity. Psychometrika, 1974, 39, 31–36.CrossRefGoogle Scholar
Katz, J. O. & Rohlf, F. J. Functionplane—a new approach to simple structure rotation. Psychometrika, 1974, 39, 37–51.CrossRefGoogle Scholar
Katz, J. O. & Rohlf, F. J. Primary product functionplane: An oblique rotation to simple structure. Multivariate Behavioral Research, 1975, 10, 219–231.CrossRefGoogle Scholar
McDonald, R. P. & Swaminathan, H. A simple matrix calculus with applications to multivariate analysis. General Systems, 1973, 18, 37–54.Google Scholar
Mulaik, S. A. The foundations of factor analysis, 1972, New York: McGraw-Hill.Google Scholar
Neuhaus, J. O. & Wrigley, C. The quartimax method: An analytical approach to orthogonal simple structure. British Journal of Statistical Psychology, 1954, 7, 81–91.CrossRefGoogle 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. Varisim: A new machine method for orthogonal rotation. Psychometrika, 1966, 31, 235–248.CrossRefGoogle ScholarPubMed
Thurstone, L. L. Multiple-factor analysis, 1947, Chicago: University of Chicago Press.Google Scholar