Bayesian Analysis Of The Extended Paired Comparison Models

Paired nature of psychological competitions of many kinds in which the items are compared by the assessors by making very fine sensory discriminations has made paired comparison experiments a widely opted method in simultaneously comparing of several objects. Bradley and Terry (1952) proposed a very simple and the most commonly used paired comparison model. However, the Bradley-Terry model lacks the ability to address the tied responses and with-in pair order of presentation effects. These discrepancies are further addressed by the researchers to accommodate the additional information in the basic paired comparison model of Bradley and Terry. Among others, Baaren (1978) extended the basic Bradley-Terry model and proposed six new extensions through different substitutions using the worth parameters, the tie parameter and with-in pair order effect parameter.This study focuses on some of the extensions namely model-Ⅲ and model-Ⅳ of the Baaren paired comparison models and analyzes those models using Bayesian inference. First, we use model-Ⅳ to examine a data set of the three packaged food mixes. The posterior distribution is obtained for the case of uniform prior and the Jeffreys prior. The complex nature of the posterior distribution motivates us to use the numerical integration. Specifically, the Gibbs sampling method is used to approximate the findings. The graphs of the posterior marginal densities are obtained and visually follow the symmetrical pattern around the posterior estimates. The posterior mean, standard deviations and coefficient of variation are obtained and compared for the two models. Both of the models agree on ranking the second food mix on the top followed by first and third brand of food mixes. However on the basis of the results of the coefficient of variation, model-Ⅲ comes out to be slightly better. The results produced for both models under both of the prior coincide and give confidence to use either of the models and either of the prior for further analysis.Secondly, three extensions of the Baaren paired comparison models namely model-Ⅰ, model-Ⅳ and model-Ⅳ have further been modified. Ignoring the with-in pair order effect the original models include tie parameter along with the worth parameters. In paired comparison experiments such a situation may arise when the items under study are infact undistinguishable i.e. genuinely similar and sometimes the assessor cannot distinguish among the items under consideration. This fact is taken into consideration and the tie parameter in the original models is parted to incorporate the genuine and vague tie effects. The three amended paired comparison models are studied through the Bayesian analysis using the uniform prior. The posterior estimates obtained for the data comparing fragrances of four different brand of perfumes help to decide that the fragrance of perfume four has outclass the other fragrances and the fragrance of perfume one has received the least preference. However for the middle ranking amended model-I and amended model-IV agrees to put the fragrance of third perfume on second spot but reverse is noticed for the amended model-VI. The mixed pattern of coefficient of variation has been observed so no model has been preferred to the other model. The95%confidence intervals have been obtained and tie parameter of the amended model-VI has comparatively the wider credible interval. The graphs of the marginal posterior densities have been obtained for the three amended models that again appeared to be symmetrical. The preference probabilities, predictive probabilities and the posterior probabilities have further validated the findings of the ranking of the items. The appropriateness of the amended models have been performed and resulted in the positive findings. Last but not the least the amended models are compared for their combined posterior variances and the amended model-I have appeared to carry the least combined variance. It has however been observed that the estimates of both of the tie parameters are more or less similar; it gives a positive indication of splitting the tie into its components.