Advertisement

Psychometrika

, Volume 42, Issue 2, pp 277–295 | Cite as

Factor simplicity index and transformations

  • P. M. Bentler
Article

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.

Key words

multivariate analysis orthogonal rotation oblique transformation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References Notes

  1. Carroll, J. B.IBM 704 program for generalized analytic rotation solution in factor analysis. Unpublished manuscript, Harvard University, 1960.Google Scholar
  2. Kaiser, H. F. & Dickman, K. W.Analytic determination of common factors. Unpublished manuscript, University of Illinois, 1959.Google Scholar
  3. Saunders, D. R.An analytic method for rotation to orthogonal simple structure (Research Bulletin RB 53-10). Princeton, New Jersey: Educational Testing Service, 1953.Google Scholar
  4. 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

  1. 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.Google Scholar
  2. Bentler, P. M. Clustran, a program for oblique transformation.Behavioral Science, 1971,16, 183–185.Google Scholar
  3. Bentler, P. M. Multistructure statistical model applied to factor analysis.Multivariate Behavioral Research, 1976,11, 3–25.Google Scholar
  4. Bentler, P. M. & Lee, S. Y. Some extensions of matrix calculus.General Systems, 1975,20, 145–150.Google Scholar
  5. Bentler, P. M. & Wingard, J. Function-invariant and parameter scale-free transformation methods.Psychometrika, 1977,42, 221–240.Google Scholar
  6. Carroll, J. B. An analytical solution for approximating simple structure in factor analysis.Psychometrika, 1953,18, 23–38.Google Scholar
  7. Cattell, R. B. “Parallel proportional profiles” and other principles for determining the choice of factors by rotation.Psychometrika, 1944,9, 267–283.Google Scholar
  8. Comrey, A. L. Tandem criteria for analytic rotation in factor analysis.Psychometrika, 1967,32, 143–154.Google Scholar
  9. 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.Google Scholar
  10. Crawford, C. B. & Ferguson, G. A. A general rotation criterion and its use in orthogonal rotation.Psychometrika, 1970,35, 321–332.Google Scholar
  11. Cureton, E. E. & Mulaik, S. A. The weighted varimax rotation and the promax rotation.Psychometrika, 1975,40, 183–195.Google Scholar
  12. Eber, H. W. Toward oblique simple structure: Maxplane.Multivariate Behavioral Research, 1966,1, 112–125.Google Scholar
  13. Ferguson, G. A. The concept of parsimony in factor analysis.Psychometrika, 1954,19, 281–290.Google Scholar
  14. Hakstian, A. R. The development of a class of oblique factor solutions.British Journal of Mathematical and Statistical Psychology, 1974,27, 100–114.Google Scholar
  15. Hakstian, A. R. & Abell, R. A. A further comparison of oblique factor transformation methods.Psychometrika, 1974,39, 429–444.Google Scholar
  16. Harman, H. H.Modern factor analysis (2nd ed.) Chicago: University of Chicago Press, 1967.Google Scholar
  17. Harris, C. W. & Kaiser, H. F. Oblique factor analytic solutions by orthogonal transformations.Psychometrika, 1964,29, 347–362.Google Scholar
  18. Horn, J. L. On extension analysis and its relation to correlations between variables and factor scores.Multivariate Behavioral Research, 1973,8, 477–489.Google Scholar
  19. 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.Google Scholar
  20. Jennrich, R. I. Orthogonal rotation algorithms.Psychometrika, 1970,35, 229–235.Google Scholar
  21. Jennrich, R. I. & Sampson, P. J. Rotation for simple loadings.Psychometrika, 1966,31, 313–323.Google Scholar
  22. Kaiser, H. F. The varimax criterion for analytic rotation in factor analysis.Psychometrika, 1958,23, 187–200.Google Scholar
  23. Kaiser, H. F. An index of factorial simplicity.Psychometrika, 1974,39, 31–36.Google Scholar
  24. Katz, J. O. & Rohlf, F. J. Functionplane—a new approach to simple structure rotation.Psychometrika, 1974,39, 37–51.Google Scholar
  25. Katz, J. O. & Rohlf, F. J. Primary product functionplane: An oblique rotation to simple structure.Multivariate Behavioral Research, 1975,10, 219–231.Google Scholar
  26. McDonald, R. P. & Swaminathan, H. A simple matrix calculus with applications to multivariate analysis.General Systems, 1973,18, 37–54.Google Scholar
  27. Mulaik, S. A.The foundations of factor analysis. New York: McGraw-Hill, 1972.Google Scholar
  28. Neuhaus, J. O. & Wrigley, C. The quartimax method: An analytical approach to orthogonal simple structure.British Journal of Statistical Psychology, 1954,7, 81–91.Google Scholar
  29. Saunders, D. R. The rationale for an “oblimax” method of transformation in factor analysis.Psychometrika, 1961,26, 317–324.Google Scholar
  30. Schönemann, P. H. Varisim: A new machine method for orthogonal rotation.Psychometrika, 1966,31, 235–248.Google Scholar
  31. Thurstone, L. L.Multiple-factor analysis. Chicago: University of Chicago Press, 1947.Google Scholar

Copyright information

© Psychometric Society 1977

Authors and Affiliations

  • P. M. Bentler
    • 1
  1. 1.Department of PsychologyUniversity of CaliforniaLos Angeles

Personalised recommendations