BAYESIAN ANALYSIS OF THE VAN BAAREN MODEL FOR PAIRED COMPARISON

Abstract

The technique of paired comparison is being commonly studied thesedays because of its attractive applications for the comparison of several objects, simultaneously. This technique permits the ranking of theobjects by means of a score, which reflects the merit of the items ona linear scale. The present study is concerned with the Bayesian analysis of a paired comparison model, namely the van Baaren model VIusing the informative and the conjugate priors. For this purpose, aninclusive elicitation technique to evaluate the hyperparameters of theprior distributions has also been elaborated. The joint posterior distribution for the parameters of the model, their marginal distributionsand their inferences are obtained via programming in the SAS package.The model is also tested for its appropriateness.

Keywords

• Bayesian hypothesis testing
• Conjugate prior
• Informative prior
• Posterior distribution
• Predictive probability

References

1. Aslam, M. Bayesian analysis for paired comparisons data. Ph.D. Dissertation, Dept. of Mathematics. University of Wales. Aberystwyth, 1996.
2. Aslam, M. Bayesian analysis for paired comparisons models allowing ties and not allowing ties. Pakistan Journal of Statistics 18, 53-69, 2002.
3. Aslam, M. An application of prior predictive distribution to elicit the prior density. Journal of Statistical Theory and Applications 2, 70-83, 2003.
4. Aslam, M. Bayesian comparison of the paired comparison models allowing ties. Journal of Statistical Theory and Applications 4, 161-171, 2005.
5. Berger, J. O. Statistical decision theory and Bayesian analysis, 2nd edn. (New York: Springer- Verlag, 1985).
6. Bradley, R. A., and Terry. M.E. Rank analysis of incomplete block design: I. The method of paired comparisons. Biometrika 39, 324-345, 1952.
7. Clarke, B. Implications of reference priors for prior information and for sample size. Journal of the American Statistical Association 91, 173-184, 1996.
8. Davidson, R.R. and Beaver, R.J. On extending the Bradley-Terry model to incorporate within pair order effects. Biometrics 33, 693-702, 1977.

9. Davidson, R.R. and Solomon, D.L. A Bayesian approach to paired comparison experimentation. Biometrika 60, 477-87, 1973.
10. Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. Bayesian data analysis, 2 nd edn. (London: Chapman and Hall, 2003).
11. Glickman, M. E. Bayesian Locally Optimal Design of Knockout Tournaments. Journal of Statistical Planning and Inference 138, 2117 – 2127, 2008.
12. Kadane, J. B. and Wolfson, L. J. Experiences in elicitation. The Statistician 47, 3-19, 1998. Kim, D. H. and Kim, H. J. A Bayesian approach to paired comparison of several products of Poisson rates. Proceedings of the Autumn Conference, Korean Statistical Society, 2004. Kim, H. J. A Bayesian approach to paired comparison. Rankings based on a graphical model. Computational Statistics & Data Analysis 48, 269-290, 2005.
13. Li, L. and Kim, K. Estimating driver’s crash risks based on extended Bradley-Terry model: An induced exposure method. Journal of Royal Statistical Society A 163, 227-240, 2000.
14. Merrick, J. R. W., van Dorp, J. R. and Singh, A. Analysis of correlated expert judgments from extended pairwise comparisons. Decision Analysis 2, 17-29, 2005.
15. Neuman, K. and Watson, B. G. Application of paired comparison methodology in measuring Canadian’s forest values. The Public Opinion Quarterly 57, 437-464, 1993.
16. Szwed , P., van Dorp, J. R., Merrick , J.R.W., Mazzuchi, T.A. and Singh, A. A Bayesian paired comparison approach for relative accident probability assessment with covariate information. European Journal of Operations Research 169, 157-177, 2006. van Baaren. On a class of extensions to the Bradley-Terry model in paired comparisons. Statistica Neerlandica 32, 57-66, 1978.