Advertisement

Psychometrika

, Volume 42, Issue 1, pp 127–140 | Cite as

On rotating to smooth functions

  • James Arbuckle
  • Michael L. Friendly
Article

Abstract

Tucker has outlined an application of principal components analysis to a set of learning curves, for the purpose of identifying meaningful dimensions of individual differences in learning tasks. Since the principal components are defined in terms of a statistical criterion (maximum variance accounted for) rather than a substantive one, it is typically desirable to rotate the components to a more interpretable orientation. “Simple structure” is not a particularly appealing consideration for such a rotation; it is more reasonable to believe that any meaningful factor should form a (locally) smooth curve when the component loadings are plotted against trial number. Accordingly, this paper develops a procedure for transforming an arbitrary set of component reference curves to a new set which are mutually orthogonal and, subject to orthogonality, are as smooth as possible in a well defined (least squares) sense. Potential applications to learning data, electrophysiological responses, and growth data are indicated.

Key words

factor analysis principal components rotation factor transformation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference note

  1. Carroll, J. D. & Chang, J. J. A general index of nonlinear correlation and its application to the problem of relating physical and psychological dimensions. Paper presented at the meeting of the American Psychological Association, Los Angeles, California, September, 1964.Google Scholar

References

  1. Arbuckle, J. & Friendly, M. L. A program for rotation to smooth functions.Behavior Research Methods and Instrumentation, 1975,7, 474.Google Scholar
  2. Bock, R. D.Multivariate statistical methods in behavioral research. New York: McGraw-Hill, 1975.Google Scholar
  3. Cleary, P. J. Description of individual differences in autonomic reactions.Psychological Bulletin, 1974,81, 934–944.Google Scholar
  4. Cliff, N. Adverbs multiply adjectives.Psychological Review, 1959,66, 27–44.Google Scholar
  5. Sheth, J. N. Using factor analysis to estimate parameters.Journal of the American Statistical Association, 1969,64, 808–822.Google Scholar
  6. Tucker, L. R. Determination of parameters of a functional relationship by factor analysis.Psychometrika, 1958,23, 19–23.Google Scholar
  7. Tucker, L. R. Learning theory and multivariate experiment: Illustration by determination of generalized learning curves. In R. B. Cattell (Ed.),Handbook of multivariate experimental psychology. Chicago: Rand McNally, 1966.Google Scholar
  8. Weitzman, R. A. A factor analytic method for investigating differences between groups of individual learning curves.Psychometrika, 1963,28, 69–80.Google Scholar

Copyright information

© Psychometric Society 1977

Authors and Affiliations

  • James Arbuckle
    • 2
  • Michael L. Friendly
    • 1
  1. 1.York UniversityCanada
  2. 2.Department of PsychologyTemple UniversityPhiladelphia

Personalised recommendations