Statistical Analysis of Paired Comparison Models Through Bayesian Approach

Abstract

Bayesian statistics provides a theory of inference which enables us to narrate the results of observation with hypothetical predictions and it provides the only generic tool for incorporating new experimental evidence and updating the existing information. In most of the pragmatic situations in Statistics, we have to deals with comparisons. One such comparing technique is the paired comparisons. The method of paired comparison has been widely employed to remove some of the difficulties involved in the simultaneous comparison of several objects. This method is being used in experimentation and research methodologies in which subjective judgment is involved. So it has become demanding to tract the attention of many of the Bayesian analytics. In recent years, many models for paired comparisons have been devised. The present study contributes to the theory of Bayesian Statistics by presenting Bayesian analysis for four different paired comparison models: the Davidson model with order effect, the Rao-Kupper model with order effect, the van Barren model VI and the amended Davidson model. For the analysis, both the noninformative and informative priors are used. The joint posterior distributions and the marginal posterior distributions of the parameters of the models are derived, the posterior estimates (means and modes) of the parameters, the predictive probabilities for future single paired comparison and the posterior probabilities for comparing the two parameters are calculated. The use of the Gibbs sampling procedure is also given in this study. The analysis has been performed for three and four treatments. An interesting amendment has been made in the Davidson model to accommodate the no preference category for those respondents who genuinely have no preference as well as those who have not been able to distinguish between the two treatments/objects. We give the Bayesian analysis of the amended model using both the noninformative and the informative priors. For using the informative prior, the hyperparameters are elicited through the prior predictive distribution. Those values of the hyperparameters are elicited at which the difference between the confidence levels characterized by the hyperparameters in prior predictive distribution and the elicited confidence levels of expert is the minimum. For the analysis, the entire calculation of the posterior estimates, the predictive and the preference probabilities and the marginal distributions along with their graphical presentations as well as the posterior probabilities for testing of hypotheses of comparing parameters is carried out mainly in SAS package. For the novelty of our work, an assessment that has been done by comparing the posterior estimates, the predictive probabilities for future single paired comparison and the posterior probabilities of hypotheses for comparing parameters of the said three models has also been included. The small data set is also considered for the analysis of the models. Finally, some ideas for future research has also been proposed and appendices carrying some important programs designed in SAS and Mathematica packages have been added.