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Solutions to Some Nonlinear Equations from Nonmetric Data

Published online by Cambridge University Press:  01 January 2025

Stanley J. Rule
Stanley J. Rule*
Affiliation:
University of Alberta
*
Request for reprints should be sent to Stanely J. Rule, Department of Psychology, University of Alberta, Edmonton, Alberta, CANADA, T6G 2E9.
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Abstract

A method is presented to provide estimates of parameters of specified nonlinear equations from ordinal data generated from a crossed design. The analytic method, NOPE, is an iterative method in which monotone regression and the Gauss-Newton method of least squares are applied alternatively until a measure of stress is minimized. Examples of solutions from artificial data are presented together with examples of applications of the method to experimental results.

Keywords

monotone regressionGauss-Newton methodordinal dataparameter estimation

Information

Type
Original Paper
Information
Psychometrika , Volume 44 , Issue 2 , June 1979, pp. 143 - 155
Copyright
Copyright © 1979 The Psychometric Society

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Footnotes

This work was begun while the author was on sabbatical leave during 1970-71 at the Department of Mathematical Psychology, University of Nijmegen, the Netherlands, where discussions with E. E. Roskam on the problem were very helpful. Support was provided by Grant A0151 from the Natural Sciences and Engineering Council, Canada.

References

Anderson, N. H. Algebraic models in perception. In Carterette, E. C. & Friedman, M. P.(Eds.), Handbook of perception (Vol. 2), 1974, New York: Academic Press.Google Scholar
Curtis, D. W., & Mullin, L. C. Judgments of average magnitude: Analyses in terms of the functional measurement and two-stage models. Perception & Psychophysics, 1975, 18, 299–308.CrossRefGoogle Scholar
Curtis, D. W., & Rule, S. J. The binocular processing of brightness information: A vector-sum model. Journal of Experimental Psychology: Human Perception and Performance, 1978, 4, 132–143.Google ScholarPubMed
Hartley, H. O. The modified Gauss-Newton method for the fitting of non-linear regression functions by least squares. Technometrics, 1961, 3, 269–280.CrossRefGoogle Scholar
Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. Foundations of measurement. (Vol. 1), 1971, New York: Academic Press.Google Scholar
Kruskal, J. B. Nonmetric multidimensional scaling: A numerical method. Psychometrika, 1964, 29, 115–129.CrossRefGoogle Scholar
Kruskal, J. B. Analysis of factorial experiments by estimating monotone transformations of the data. The Journal of the Royal Statistical Society, Series B, 1965, 27, 251–263.CrossRefGoogle Scholar
Lingoes, J. C. The Guttman-Lingoes nonmetric program series, 1973, Ann Arbor: Mathesis Press.Google Scholar
Roskam, E. E. Metric analysis of ordinal data in psychology, 1968, Voorschoten, The Netherlands: Vam.Google Scholar
Roskam, E. E. A general system for nonmetric data analysis. In Lingoes, J. C.(Eds.), Geometric representations of relational data, 1977, Ann Arbor: Mathesis Press.Google Scholar
Roskam, E. E. A survey of the Michigan-Israel-Netherlands-integrated series. In Lingoes, J. C.(Eds.), Geometric representations of relational data, 1977, Ann Arbor: Mathesis Press.Google Scholar
Rule, S. J., & Curtis, D. W. Conjoint scaling of subjective number and weight. Journal of Experimental Psychology, 1973, 97, 305–309.CrossRefGoogle ScholarPubMed
Rule, S. J., Curtis, D. W., & Markley, R. P. Input and output transformations from magnitude estimation. Journal of Experimental Psychology, 1970, 86, 343–349.CrossRefGoogle ScholarPubMed
Stevens, S. S. The psychophysics of sensory function. In Rosenblith, W. A.(Eds.), Sensory communication, 1961, New York: Wiley.Google Scholar
Stevens, S. S. Issues in psychophysical measurement. Psychological Review, 1971, 78, 426–450.CrossRefGoogle Scholar
Takane, Y., Young, F. W., & De Leeuw, J. Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features. Psychometrika, 1977, 42, 7–67.CrossRefGoogle Scholar
Young, F. W. A model for polynomial conjoint analysis algorithms. In Shepard, R. N., Romney, A. K., & Nerlove, S. B.(Eds.), Multidimensional scaling: Theory and applications in the behavioral science. (Vol. 1), 1972, New York: Seminar Press.Google Scholar