A Bayesian Multidimensional Scaling Model for Partial Rank Preference Data

There has been great advancement on research for preferential choice in field of marketing. When we look at preferential choice data, there are two components to consider: the individuals and the items. Coombs (1950; 1964) introduced the unfolding technique on preferential choice data. In 1960, Bennett and Hays went on to create a multidimensional unfolding model. Hojo (1997;1998) showed rank data could be used in multidimensional scaling, however he did not implement a Bayesian technique. In 2010, Fong, DeSarbo, Park, and Scott proposed a new Bayesian vector Multidimensional Scaling (MDS) model which was applied to data from a five-point Likert scale survey. This paper focused on Bayesian approach choice behavior multidimensional space model for the analysis of partially ranked data (rank top 3 from J data) to provide a joint space of individuals and products, using MCMC procedure. The procedure is similar to what Fong, DeSarbo, Park, and Scott (2010) did but this study used partial rank data instead of Likert scale data. The goal of this study was to create a probability-based model that calculates the average product utility which indicates how popular the product is. Lambdas or the item loadings are the direction of the products and thetas are the direction for the individuals. In addition, this study dealt with rotational invariance by calculating the optimal lambda values for each iteration and each dimension by flipping the sign so it approaches the average value. To determine the number of dimensions of the datasets, the sum of squared loadings were calculated. We applied the MCMC procedure to simulated data in which we sampled the loadings from the normal distribution as well as loadings from the real datasets. In addition, we applied the MCMC procedure to the real dataset and created a multidimensional space for the products.