Abstract
The Thurstone and the Bradley-Terry models, both initially advanced on intuitive grounds, have proved useful in the analysis of paired comparisons. The psychological meaning of these models and their relation to one another is unclear, but they fit data. Stevens has observed that there may be two basic mechanisms of discrimination 1) additive and 2) substitutive. We advance two corresponding mathematical models: that experienced sensation is 1) the sum of a large number of independent signals and 2) the maximum of a large number of independent signals. These assumptions yield 1) Thurstone's model and 2) the model of Bradley-Terry. Psychological interpretations of the various parameters, in terms of sensation, present themselves in a natural manner. Thus this paper presents a theory which unifies and interprets two paired comparison models that have proved useful in fitting experimental data.
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References
Audley, R. J. A stochastic model for individual choice behavior.Psychological Review, 1960,67, 1–15.
Block, H. D., and Marshak, J. Random orderings and stochastic theories of responses.Contributions to probability and statistics, easays in honor of Harold Hotelling. Edited by Olkin and others. Stanford University Press, 1960.
Bradley, R. A., and Terry, M. E. The rank analysis of incomplete block designs I. The method of paired comparisons.Biometrika, 1952,39, 324–345.
Bradley, R. A. Some statistical methods in taste testing and quality evaluation.Biometrics, 1953,9, 22–38.
Bradley, R. A. Another interpretation of a model for paired comparisons.Psychometrika, 1966,30, 315–318.
Brunk, H. D. Mathematical models for ranking from paired comparisons.Journal of the American Statistical Association, 1960,55, 503–520.
Cramér, H.Mathematical methods of statistics. Princeton University Press, 1946.
David, H. A.The method of paired comparisons. Hafner, New York, 1963.
Drever, J.A dictionary of psychology. Penguin books, Baltimore, Md., 1963.
Fechner, G. T.Elemente der Psychophysik. Breitkopf and Härtel, Leipzig, 1960.
Fisher, R. A., and Tippett, L. H. C. Limiting forms of the frequency distribution of the largest or smallest member of a sample.Proceedings Cambridge Philosophical Society, 1928,24, 180–190.
Gnedenko, B. V. Sur la distribution limite du terme maximum d'une série aléatoire.Ann. Math. 1943,44, 423–453.
Gumbel, E. G.Statistics of extremes. Columbia University Press, New York, 1958.
Luce, R. D.Individual choice behavior. Wiley, New York, 1959.
Mises, R. von. La distribution de la plus grande de n valeurs.Revue mathématique de l'union Interbalkanique t. I, f. 1, 1939, 141–160.
Pfanzagl, J. Über die stochastische fundierung des psychophysischen gesetzes.Biometrische Zeitschrift, 1962,4, 1–14.
Stevens, S. S. On the psychophysical law.Psychological Review, 1957,64, 153–181.
Stevens, S. S. A metric for the social consensus.Science, 1966,151, 530–541.
Thurstone, L. L. Psychophysical analysis.American Journal of Psychology, 1927,38, 368–389.
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This research was supported in part by a National Science Foundation Grant GP 3807 and by the Army, Navy, and Air Force under ONR Contract # NONR-988(08), Task Order NR 042-004 with the Florida State University.
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Thompson, W.A., Singh, J. The use of limit theorems in paired comparison model building. Psychometrika 32, 255–264 (1967). https://doi.org/10.1007/BF02289589
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DOI: https://doi.org/10.1007/BF02289589