Psychometrika

, Volume 65, Issue 3, pp 281–299 | Cite as

Bayesian analysis of order-statistics models for ranking data

  • Philip L. H. Yu
Article

Abstract

In this paper, a class of probability models for ranking data, the order-statistics models, is investigated. We extend the usual normal order-statistics model into one where the underlying random variables follow a multivariate normal distribution. Bayesian approach and the Gibbs sampling technique are used for parameter estimation. In addition, methods to assess the adequacy of model fit are introduced. Robustness of the model is studied by considering a multivariate-t distribution. The proposed method is applied to analyze the presidential election data of the American Psychological Association (APA).

Key words

data augmentation Gibbs sampling order-statistics model ranking data 

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References

  1. Albert, J.H., & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data.Journal of the American Statistical Association, 88, 669–679.Google Scholar
  2. Albert, J.H. & Chib, S. (1995). Bayesian residual analysis for binary response regression models.Biometrika, 82, 747–759.Google Scholar
  3. Anderson, T.W. (1984).An introduction to multivariate statistical analysis (2nd ed.). New York: John Wiley.Google Scholar
  4. Andrews, D.F., & Mallows, C.L. (1974). Scale mixtures of normality.Journal of Royal Statistical Society, Series B, 36, 99–102.Google Scholar
  5. Arbuckle, J., & Nugent, J.H. (1973). A general procedure for parameter estimation for the law of comparative judgement.British Journal of Mathematical and Statistical Psychology, 26, 240–260.Google Scholar
  6. Beggs, S., Cardell, S., & Hausman, J. (1981). Assessing the potential demand for electric cars.Journal of Econometrics, 17, 1–19.CrossRefGoogle Scholar
  7. Besag, J. Green, P., Higdon, D., & Mengersen, K. (1995). Bayesian computation and stochastic systems.Statistical Science, 10, 3–66.Google Scholar
  8. Böckenholt, U. (1993). Applications of Thurstonian models to ranking data. In M.A. Fligner & J.S. Verducci (Eds.),Probability models and statistical analyses for ranking data. New York: Springer-Verlag.Google Scholar
  9. Box, G.E.P., & Tiao, G.C. (1973).Bayesian inference in statistical analysis. New York: John Wiley.Google Scholar
  10. Bradley, R.A., & Terry, M.E. (1952). Rank analysis of incomplete block designs, I: The method of paired comparisons.Biometrika, 39, 324–345.Google Scholar
  11. Brook, D., & Upton, G.J.G. (1974). Biases in local government elections due to position on the ballot paper.Applied Statistics, 23, 414–419.Google Scholar
  12. Bunch, D.S. (1991). Estimability in the multinomial probit model.Transportation Research, Series B, 25(1), 1–12.Google Scholar
  13. Carlin, B.P., & Polson, N.G. (1991). Inference for nonconjugate Bayesian models using the Gibbs sampler.Canadian Journal of Statistics, 19, 399–405.Google Scholar
  14. Chapman, R.G., & Staelin, R. (1982). Exploiting rank ordered choice set data within the stochastic utility model.Journal of Marketing Research, 19, 288–301.Google Scholar
  15. Chintagunta, P.K. (1992). Estimating a multinomial probit model of brand choice using the method of simulated moments.Marketing Science, 11(4), 386–407.Google Scholar
  16. Cohen, A., & Mallows, C.L. (1983). Assessing goodness of fit of ranking models to data.The Statistician, 32, 361–373.Google Scholar
  17. Critchlow, D.E., & Fligner, M.A. (1993). Ranking models with item covariates. In M.A. Fligner & J.S. Verducci (Eds.),Probability models and statistical analyses for ranking data (pp. 1–19). New York: Springer-Verlag.Google Scholar
  18. Critchlow, D.E., Fligner, M.A., & Verducci, J.S. (1991). Probability models on rankings.Journal of Mathematical Psychology, 35, 294–318.CrossRefGoogle Scholar
  19. Daniels, H.E. (1950). Rank correlation and population models.Biometrika, 33, 129–135.Google Scholar
  20. Dansie, B.R. (1985). Parameter estimability in the multinomial probit model.Transportation Research, Series B, 19(6), 526–528.Google Scholar
  21. Dansie, B.R. (1986). Normal order statistics as permutation probability models.Applied Statistics, 35, 269–275.Google Scholar
  22. Devroye, L. (1986).Non-uniform random variate generation. New York: Springer-Verlag.Google Scholar
  23. Diaconis, P. (1988).Group Representations in Probability and Statistics (IMS Lecture Notes, Volume 11). Hayward, CA: Institute of Mathematical Statistics.Google Scholar
  24. Diaconis, P. (1989). A generalization of spectral analysis with application to ranked data.Annals of Statistics, 17, 949–979.Google Scholar
  25. Elrod, T., & Keane, M.P. (1995). A factor-analytic probit model for representing the market structure in panel data.Journal of Marketing Research, XXXII, 1–16.Google Scholar
  26. Gelman, A., Meng, X.L., & Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies.Statistica Sinica, 6(4), 733–760.Google Scholar
  27. Geweke, J. (1992). Efficient simulation from the multivariate normal and student-t distributions subject to linear constraints.Computer Science and Statistics: Proceedings of the Twenty-Third Symposium on the Interface (pp. 571–578). Alexandria, VA: American Statistical Association.Google Scholar
  28. Hajivassiliou, V. (1993). Simulation estimation methods for limited dependent variable models. In G.S. Maddala, C.R. Rao, & H.D. Vinod (Eds.),Handbook of Statistics (Econometrics) (Volume 11, pp. 519–543). Amersterdam: North-Holland.Google Scholar
  29. Hajivassiliou, V., McFadden, D., & Ruud, P. (1996). Simulation of multivariate normal rectangle probabilities and their derivatives: Theoretical and computational results.Journal of Econometrics, 72, 85–134.CrossRefGoogle Scholar
  30. Hausman, J.A., & Wise, D.A. (1978). A conditional probit model for qualitative choice: Discrete decisions recognizing interdependence and heterogeneous preferences.Econometrica, 46(2), 403–426.Google Scholar
  31. Henery, R.J. (1981). Permutation probabilities as models for horse races.Journal of Royal Statistical Society, Series B, 43, 86–91.Google Scholar
  32. Henery, R.J. (1983). Permutation probabilities for gamma random variables.Journal of Applied Probability, 20, 822–834.Google Scholar
  33. Johnson, M.E. (1987).Multivariate statistical simulation. New York: John Wiley.Google Scholar
  34. Keane, M.P. (1994). A computationally practical simulation estimator for panel data.Econometrica, 62, 95–116.Google Scholar
  35. Koop, G., & Poirier, D.J. (1994). Rank-ordered logit models: An empirical analysis of Ontario voter preferences.Journal of Applied Econometrics, 9, 369–388.Google Scholar
  36. Lo, V.S.Y., Bacon-Shone, J., & Busche, K. (1995). The application of ranking probability models to racetrack betting.Management Science, 41, 1048–1059.Google Scholar
  37. Luce, R.D. (1959).Individual choice behavior. New York: Wiley.Google Scholar
  38. Marden, J.L. (1995).Analyzing and modeling rank data. New York: Chapman and Hall.Google Scholar
  39. McCullagh, P. (1993). Permutations and regression models. In M.A. Fligner & J.S. Verducci (Eds.),Probability models and statistical analyses for ranking data (pp. 196–215). New York: Springer-Verlag.Google Scholar
  40. McCulloch, R.E., & Rossi, P.E. (1994). An exact likelihood analysis of the multinomial probit model.Journal of Econometrics, 64, 207–240.CrossRefGoogle Scholar
  41. Mosteller, F. (1951). Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations.Psychometrika, 16, 3–9.Google Scholar
  42. Stern, H. (1990a). A continum of paired comparisons models.Biometrika, 77, 265–273.Google Scholar
  43. Stern, H. (1990b). Models for distributions on permutations.Journal of the American Statistical Association, 85, 558–564.Google Scholar
  44. Stern, H. (1993). Probability models on rankings and the electoral process. In M.A Fligner & J.S. Verducci (Eds.),Probability models and statistical analyses for ranking data (pp. 173–195). New York: Springer-Verlag.Google Scholar
  45. Tallis, G.M., & Dansie, B.R. (1983). An alternative approach to the analysis of permutations.Applied Statistics, 32, 110–114.Google Scholar
  46. Tanner, T., & Wong, W. (1987). The calculation of posterior distributions by data augmentation.Journal of the American Statistical Association, 82, 528–549.Google Scholar
  47. Thurstone, L.L. (1927). A law of comparative judgement.Psychological Review, 34, 273–287.Google Scholar
  48. Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion and rejoinder).Annals of Statistics, 22, 1701–1762.Google Scholar
  49. Tversky, A. (1972). Elimination by aspects: a theory of choice.Psychological Review, 79, 281–299.Google Scholar
  50. Venables, W.N., & Ripley, B.D. (1994).Modern applied statistics with S-Plus. New York: Springer-Verlag.Google Scholar
  51. Yai, T., Iwakura, S., & Morichi, S. (1997). Multinomial probit with structured covariance for route choice behavior.Transportation Research, Series B, 31(3), 195–207.Google Scholar
  52. Yao, K.G., & Böckenholt, U. (1999). Bayesian estimation of Thurstonian ranking models based on the Gibbs sampler.British Journal of Mathematical and Statistical Psychology, 52(1), 79–92.CrossRefGoogle Scholar
  53. Yellot, J. (1977). The relationship between Luce's choice axiom, Thurstone's theory of comparative judgment, and the double exponential distribution.Journal of Mathematical Psychology, 15, 109–144.Google Scholar

Copyright information

© The Psychometric Society 2000

Authors and Affiliations

  • Philip L. H. Yu
    • 1
  1. 1.Department of Statistics and Actuarial ScienceThe University of Hong KongHong Kong

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