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.
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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.
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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.
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Arbuckle, J., Friendly, M.L. On rotating to smooth functions. Psychometrika 42, 127–140 (1977). https://doi.org/10.1007/BF02293749
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DOI: https://doi.org/10.1007/BF02293749