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Circularity in Conjoint Paired Comparisons

  • Thorn Bezembinder
Part of the Recent Research in Psychology book series (PSYCHOLOGY)

Abstract

Let a dyad denote an unordered pair of objects interpreted as, say, a similarity. A pair of dyads is said to be conjoint if its two dyads have precisely one object in common. For a set Z of n ≥ 3 objects a system of conjoint paired comparisons (CPC-system) is the set of all paired comparisons of nonidentical conjoint dyads that may be formed from Z. It is claimed that a CPC-system is usually highly circular. This is shown analytically and by computer simulation for CPC-systems in which all pairwise choices are equally likely. The claim is shown by computer simulation for CPC-systems that arise from a stochastic pairwise choice model enjoying simple scalability as well as for conditional proximity matrices (a special form of CPC-systems). The bearing of this circularity on the use of multidimensional scaling is indicated.

Keywords

Choice Model Multidimensional Scaling Paired Comparison Strong Component Unordered Pair 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Ashby, F.G. & Perrin, N.A. (1988). Toward a unified theory of similarity and recognition. Psychological Review, 95, 124–150.CrossRefGoogle Scholar
  2. Bezembinder, Th. (1981). Circularity and consistency in paired comparisons. British Journal of Mathematical & Statistical Psychology, 34, 16–37.CrossRefGoogle Scholar
  3. Bossuyt, P. (1990). A comparison of probabilistic unfolding theories for paired comparisons data. Berlin: Springer.Google Scholar
  4. Coombs, C.H. (1964). A theory of data. New York: Wiley.Google Scholar
  5. Coxon, A.P.M. & Davies, P.M. (1982). The User’s Guide to Multidimensional Scaling: with special reference to the MDS(X) Library of Computer Programs. London: Heinemann.Google Scholar
  6. David, H.A. (1988). The method of paired comparisons. London: Griffin.Google Scholar
  7. Falmagne, J.-C. & Iverson, G. (1979). Conjoint Weber laws and additivity. Journal of Mathematical Psychology, 20, 164–183.CrossRefGoogle Scholar
  8. Fishbum P.C. (1982). Nontransitive measurable utility. Journal of Mathematical Psychology, 26, 31–67.CrossRefGoogle Scholar
  9. Harary, F., Norman, R.Z. & Cartwright, D. (1965). Structural models: an introduction to the theory of directed graphs. New York: Wiley.Google Scholar
  10. Harary, F. (1972). Graph theory. Reading (Mass.): Addison-Wesley.Google Scholar
  11. Iverson, G. & Falmagne, J.-C. (1985). Statistical issues in measurement. Mathematical Social Sciences, 10, 131–153.CrossRefGoogle Scholar
  12. Kendall, M.G. (1955). Rank correlation methods. London: Griffin (4th ed.).Google Scholar
  13. Lingoes, J.C. (1972). A general survey of the Guttman-Lingoes nonmetric program series. In: R.N. Shepard, A.K. Romney & S.B. Nerlove (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences.Volume I. New York: Seminar Press (p.49–68).Google Scholar
  14. Luce, R.D. (1959). Individual choice behavior. New York: Wiley.Google Scholar
  15. Luce, R.D. (1961). A choice theory analysis of similarity judgments. Psychometrika, 26, 151–163.CrossRefGoogle Scholar
  16. Luce, R.D. & Galanter, E. (1963). Discrimination. In: R.D. Luce, R.R. Bush & E. Galanter (Eds.). Handbook of mathematical psychology. Volume I. New York: Wiley (p.191–243).Google Scholar
  17. Luce, R.D. & Suppes, P. (1965). Preference, utility, and subjective probability. In: R.D. Luce, R.R. Bush & E. Galanter (Eds.), Handbook of mathematical psychology. Volume III. New York: Wiley (p. 249–410).Google Scholar
  18. May, K.O. (1954). Intransitivity, utility and the aggregation of preference patterns. Econometrica, 22, 1–13.CrossRefGoogle Scholar
  19. Moon, J.W. (1968). Topics on tournaments. New York: Holt, Rinehart & Winston.Google Scholar
  20. Morrison, H.W. (1963). Testable conditions for triads of paired comparison choices. Psychometrika, 28, 369–390.CrossRefGoogle Scholar
  21. Reid, K.B. & Beineke, L.W. (1978). Tournaments. In: L.W. Beineke & R.J. Wilson (Eds.), Selected Topics in Graph Theory. New York: Academic Press (p. 169–204).Google Scholar
  22. Roberts, F.S. (1979). Measurement theory: with applications to decisionmaking, utility, and the social sciences. Reading (Mass.): Addison-Wesley.Google Scholar
  23. Roskam, E.E. (1982). MINICPA version 3.20. In MDS(X) Library of Computer Programs. Edinburgh: University of Edinburgh.Google Scholar
  24. Russo, J.E. & Dosher, B.A. (1983). Strategies for multiattribute binary choice. Journal of experimental Psychology: Learning, Memory and Cognition, 9, 676–696.CrossRefGoogle Scholar
  25. Shafir, E.B., Osherson, D.N. & Smith, E.E. (1989). An advantage model of choice. Journal of Behavioral Decision Making, 2, 1–23.CrossRefGoogle Scholar
  26. Shepard, R.N. (1986). Discrimination and generalization in identification and classification: comment on Nosofsky. Journal of Experimental Psychology: General 115, 58–61.CrossRefGoogle Scholar
  27. Slater, P. (1961). Inconsistencies in a schedule of paired comparisons. Biometrika, 48, 303–312.Google Scholar
  28. Thurstone, L.L. (1927). A law of comparative judgement. Psychological Review, 34, 273–286.CrossRefGoogle Scholar
  29. Torgerson, W.S. (1962). Theory and methods of scaling. New York: Wiley (2nd. ed.).Google Scholar
  30. Tversky, A. (1969). Intransitivity of preferences. Psychological Review, 76, 31–48.CrossRefGoogle Scholar
  31. Tversky, A. & Russo, J.E. (1969). Substitutability and similarity in binary choices. Journal of Mathematical Psychology, 6, 1–12.CrossRefGoogle Scholar
  32. Heerman, D.W. (1986). Computer simulation methods in theoretical physics. Berlin: Springer.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1991

Authors and Affiliations

  • Thorn Bezembinder
    • 1
  1. 1.Nijmegen Institute of Cognition research and Information technology (NICI), Department of Mathematical PsychologyUniversity of NijmegenNijmegenThe Netherlands

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