Developing and Comparing of Paired Comparisons Models Under Bayesian Principle.

Abstract

The paired comparisons (PCs) is applicable in the situation, when items/ treatments are ranked to obtain the response, treatments presented in pairs. This study is carried out to develop some new paired comparison models named as the Rayleigh PC model, the Maxwell PC model, and the Nakagami PC model. Moreover, an Amendment is suggested in the existing van-Baaren VI PC model to introduce two-tie parameters. The Rayleigh PC model is also modified by introducing the parameter of no preference. The models are analyzed under Bayesian method. The posterior distributions for parameters are derived under non-informative and informative priors. The hyperparameters are elicited through the technique of prior predictive distributions. The ranking for treatments are obtained by the posterior estimates. To obtain posterior summaries, the following loss functions are considered in this study: quadratic loss function, weighted loss function, squared error loss function. The graphical depiction of marginal posterior distributions is given. Furthermore, the preference and predictive probabilities are evaluated for a current and future single PC, respectively. The posterior probabilities of the hypotheses are computed for comparing two parameters. Model appropriateness is checked through the χ2 test. The Lindley-Shannon information is enumerated for the amount of information in the priors. The Akaike information criterion and the Bayesian information criterion are measured for the model selection. The analysis is performed on the two real life data sets of five cigarette brands: Goldleaf, Marlboro, Dunhill, Benson & Hedges and Davidoff and four drinking water brands: Aquafina, Nestle, Kinley and Springley. The posterior summaries are obtained by the Gibbs sampling. Mostly SAS package is used for finding results.