Paired Comparison Modeling with Bayesian Analysis

Abstract

The method of paired comparisons is a techniques in which the treatments or stimuli are presented in pairs to some respondents, who depending upon sensory evaluation, pick the better one. The experiment is repeatedly executed for all the treatments with a number of respondents to reach certain ranking of the treatments. In this study, an attempt is made to develop paired comparison models, namely the Pareto model and the Cauchy model. The Cauchy model is also extended to accommodate ties. Bayesian analysis is a modern inferential technique which endeavors to estimate parameters of an underlying posterior distribution based on a prior and an observed distribution. The developed models, along with the Stern (1990a) Chi-square model, are analyzed in Bayesian framework using noninformative and informative priors. The priors are compared on behalf of the Lindley-Shannon information. As an illustration, we use a real data set for the years 2000-2006 to rank five top-ranked one-day-international cricket teams, namely those of Australia, India, New Zealand Pakistan and South Africa. The analysis comprises finding the posterior means, joint posterior modes, marginal posterior distributions, preference probabilities, predictive probabilities and posterior probabilities of hypotheses. The plausibility of the models is also tested. The models are also analyzed through the simulated data sets of different sizes, and the results are compared. The renowned Bradley-Terry model is also studied for comparison purpose. Extensive computer programs are developed in C, SAS, Microsoft Excel, Mathematica and Maple softwares to draw results. At last, the conclusions are drawn about the entire study.