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Probabilistic, Multidimensional Unfolding Analysis

Published online by Cambridge University Press:  01 January 2025

Joseph L. Zinnes  and
Richard A. Griggs
Joseph L. Zinnes
Affiliation:
Indiana University
Richard A. Griggs
Affiliation:
Indiana University
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Abstract

A probabilistic, multidimensional version of Coombs' unfolding model is obtained by assuming that the projections of each stimulus and each individual on each axis are normally distributed. Exact equations are developed for the single dimensional case and an approximate one for the multidimensional case. Both types of equations are expressed solely in terms of univariate normal distribution functions and are therefore easy to evaluate. A Monte Carlo experiment, involving 9 stimuli and 3 subjects in a 2 dimensional space, was run to determine the degree of accuracy of the multidimensional equation and the feasibility of using iterative methods to obtain maximum likelihood estimates of the stimulus and subject coordinates. The results reported here are gratifying in both respects.

Type
Original Paper
Information
Psychometrika , Volume 39 , Issue 3 , September 1974 , pp. 327 - 350
Copyright
Copyright © 1974 The Psychometric Society

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References

Abdel-Aty, S. H.. Approximate formulae for the percentage points and the probability integral of the non-central x 2 distribution. Biometrika, 1954, 41, 538–540Google Scholar
Bechtel, G. G.. Folded and unfolded scaling from preferential paried comparisons. Journal of Mathematical Psychology, 1968, 5, 333–357CrossRefGoogle Scholar
Bulgren, W. G.. On representations of the doubly non-central F distribution. Journal of the American Statistical Association, 1971, 66, 184–186Google Scholar
Carroll, J. D.. Individual differences and multidimensional scaling. In Shepard, R. N., Romney, A. K., Nerlove, S. (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences, Vol. I. New York: Seminar Press. 1972, 105–155Google Scholar
Chandler, J. P.. STEPIT—Finds local minima of a smooth function of several parameters. Behavioral Science, 1969, 14, 81–82Google Scholar
Coombs, C. H.. Psychological scaling without a unit of measurement. Psychological Review, 1950, 57, 145–158CrossRefGoogle ScholarPubMed
Coombs, C. H.. A theory of data, 1964, New York: WileyGoogle Scholar
Coombs, C. H., Greenberg, M., Zinnes, J. L.. A double law of comparative judgment for the analysis of preferential choice and similarities data. Psychometrika, 1961, 26, 165–171CrossRefGoogle Scholar
Greene, P. E., Rao, V. R.. Applied multidimensional scaling, 1972, New York: Holt, Rinehart & WinstonGoogle Scholar
Hefner, R. A. Extensions of the law of comparative judgment to discriminable and multidimensional stimuli. Unpublished doctoral dissertation, University of Michigan, 1958.Google Scholar
Luce, R. D.. Detection and recognition. In Luce, R. D., Bush, R. R., Galanter, E. (Eds.), Handbook of mathematical psychology, Vol. 1. New York: Wiley. 1963, 103–189Google Scholar
Patnaik, P. B.. The non-central x 2- and F- distributions and their applications. Biometrika, 1949, 36, 202–232Google Scholar
Price, R.. Some non-central F-distributions expressed in closed form. Biometrika, 1964, 51, 107–122CrossRefGoogle Scholar
Ramsay, J. O.. Some statistical considerations in multidimensional scaling. Psychometrika, 1969, 34, 167–182CrossRefGoogle Scholar
Schönemann, P. H.. On metric multidimensional unfolding. Psychometrika, 1970, 35, 349–366CrossRefGoogle Scholar
Schönemann, P. H., Wang, M. M.. An individual difference model for the multidimensional analysis of preference data. Psychometrika, 1972, 37, 275–309CrossRefGoogle Scholar
Shepard, R. N.. Stimulus and response generalization: A stochastic model relating generalization to distance in a psychological space. Psychometrika, 1957, 22, 325–345CrossRefGoogle Scholar
Sixtl, F.. Probabilistic unfolding. Psychometrika, 1973, 38, 235–248CrossRefGoogle Scholar
Suppes, P., Zinnes, J. L.. Basic measurement theory. In Luce, R. D., Bush, R. R., Galanter, E. (Eds.), Handbook of mathematical psychology, Vol. I. New York: Wiley. 1963, 1–76Google Scholar
Tang, P. C.. Power of the F-test in terms of the non-central F distribution. Statistics Research Memoirs, 1938, 2, 126–149Google Scholar
Thurstone, L. L.. Psychophysical analysis. American Journal of Psychology, 1927, 38, 368–389CrossRefGoogle Scholar
Tiku, M. L.. Series expansions for the doubly non-central F-distribution. Australian Journal of Statistics, 1965, 7, 78–89CrossRefGoogle Scholar
Torgerson, W. S.. Multidimensional scaling: I. Theory and method. Psychometrika, 1952, 17, 401–419CrossRefGoogle Scholar
Torgerson, W. S.. Theory and methods of scaling, 1958, New York: WileyGoogle Scholar
Zinnes, J. L. A probabilistic theory of preferential choice. Unpublished doctoral dissertation, University of Michigan, 1959.Google Scholar