On the Babington Smith Class of Models for Rankings

  • Harry Joe
  • Joseph S. Verducci
Part of the Lecture Notes in Statistics book series (LNS, volume 80)

Abstract

In 1950, Babington Smith proposed a general family of probability models for rankings based on a paired comparisons idea. Mallows [9] studied several simple subclasses of the Babington Smith models, but the full class was considered computationaly intractible for practical application at that time. With modern computers, the models are simple to use. With this incentive, we investigate various properties of the Babington Smith models, including their characterization as maximum entropy models, the relationships among different parametrizations of the models, and the conditions under which various forms of stochastic transitivity, unimodality and consensus are obtained. The maximum entropy characterization suggests models that are nested within the Babington Smith models and models that are more general. Computational details for the models are briefly discussed. The models are illustrated with examples where words are ranked in accordance to their perceived degree of association with a target word.

Keywords

Entropy 

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References

  1. [1]
    L. Brown. Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. IMS Lecture Notes — Monograph Series, 9, 1987.Google Scholar
  2. [2]
    D. E. Critchlow and Fligner, M. A. and J. S. Verducci. Probability models on rankings. Journal of Mathematical Psychology, 35:294–318, 1991.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    H.A. David. The Method of Paired Comparisons. Griffin, London, second edition, 1988.MATHGoogle Scholar
  4. [4]
    M. A. Fligner and J. S. Verducci. Distance Based ranking Models. J. Amer. Statist. Ass. B, 83:359–369, 1986.Google Scholar
  5. [5]
    M. A. Fligner and J.S. Verducci. Aspects of Two Group Concordance. Communications in Statistics: Theory and Methods, 16:1479–1503, 1987.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    M. A. Fligner and J. S. Verducci. Multi-stage ranking models. J. Amer. Statist. Ass., 83:892–901, 1988.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    M. G. Kendall. Rank Correlation Methods. Griffin, London, 1970.MATHGoogle Scholar
  8. [8]
    R. D. Luce. Individual Choice Behavior. Wiley, New York, 1959.MATHGoogle Scholar
  9. [9]
    C. L. Mallows. Non-null ranking models. Biometrika, 44:114–130, 1957.MathSciNetMATHGoogle Scholar
  10. [10]
    B. Babington Smith. Discussion of Professor Ross’s paper. J. Roy. Statist. Soc. B, 12:53–56, 1950.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • Harry Joe
    • 1
  • Joseph S. Verducci
    • 2
  1. 1.Department of StatisticsUniversity of British ColumbiaCanada
  2. 2.Department of StatisticsOhio State UniversityColumbusUSA

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