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On Rotating to Smooth Functions

Published online by Cambridge University Press:  01 January 2025

James Arbuckle  and
Michael L. Friendly
James Arbuckle*
Affiliation:
Temple University
Michael L. Friendly
Affiliation:
York University
*
Requests for reprints should be sent to James Arbuckle, Department of Psychology, Temple University, Philadelphia, Pennsylvania 19122.
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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.

Keywords

factor analysisprincipal componentsrotationfactor transformation
Type
Original Paper
Information
Psychometrika , Volume 42 , Issue 1 , March 1977 , pp. 127 - 140
Copyright
Copyright © 1977 The Psychometric Society

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Footnotes

Portions of this research were supported by the National Research Council of Canada, Grant A8615 to the second author. We thank Jagdeth Sheth for supplying his raw data.

References

Reference Note

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