Bayesian Multidimensional Scaling Model for Ordinal Preference Data

The model within the present study incorporated Bayesian Multidimensional Scaling and Markov Chain Monte Carlo methods to represent individual preferences and threshold parameters as they relate to the influence of survey items popularity and their interrelationships. The model was used to interpret two independent data samples of ordinal consumer preference data related to purchasing behavior. The objective of the procedure was to provide an understanding and visual depiction of consumers' likelihood of having a strong affinity toward one of the survey choices, and how other survey choices relate to it. The study also aimed to derive the joint spatial representation of the subjects and products represented by the dissimilarity preference data matrix within a reduced dimensionality. This depiction would aim to enable interpretation of the preference structure underlying the data and potential demand for each product. Model simulations were created both from sampling the normal distribution, as well as incorporating Lambda values from the two data sets and were analyzed separately. Posterior checks were used to determine dimensionality, which were also confirmed within the simulation procedures. The statistical properties generated from the simulated data confirmed that the true parameter values (loadings, utilities, and latititudes) were recovered. The model effectiveness was contrasted and evaluated both within real data samples and a simulated data set. The two data sets analyzed were confirmed to have differences in their underlying preference structures that resulted in differences in the optimal dimensionality in which the data should be represented. The Biases and MSEs of the lambdas and alphas provide further understanding of the data composition and Analysis of variance (ANOVA) confirmed the differences in MSEs related to changes in dimensions were statistically significant.