A Stochastic Multidimensional Scaling Vector Threshold Model for the Spatial Representation of “Pick Any/N” Data

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Abstract
This paper presents a new stochastic multidimensional scaling vector threshold model designed to analyze “pick any/n” choice data (e.g., consumers rendering buy/no buy decisions concerning a number of actual products). A maximum likelihood procedure is formulated to estimate a joint space of both individuals (represented as vectors) and stimuli (represented as points). The relevant psychometric literature concerning the spatial treatment of such binary choice data is reviewed. The nonlinear probit type model is described, as well as the conjugate gradient procedure used to estimate parameters. Results of Monte Carlo analyses investigating the performance of this methodology with synthetic choice data sets are presented. An application concerning consumer choices for eleven competitive brands of soft drinks is discussed. Finally, directions for future research are presented in terms of further applications and generalizing the model to accommodate three-way choice data.