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multidimensional scaling, nonmetric scaling, maximum likelihood estimation, complete and incomplete data, noncentral chi-square approximations,

A probabilistic multidimensional scaling model that estimates both location and variance parameters for proximity and preference data is described and compared to a deterministic scaling model. Simulated and empirical choice data are used to compare models. Variance estimates from the probabilistic model are used to test a hypothesis about the homogeneity of stimulus perception under alternative modes of stimulus presentation.probabilistic model, preference data, proximity data, multidimensional scaling

Computing unsegmented product maps from preference data by means of single ideal point models is commonly thought to be impossible because of indeterminacy problems. The authors show that this mathematical indeterminacy can be overcome by incorporating dependent sampling assumptions into a probabilistic multidimensional scaling (MDS) model. As a result, product space maps can be estimated for single markets from preference data alone. If desired, dissimilarity data can be combined with preference data to produce jointly estimated product space maps. The authors illustrate the advantages of the proposed approach with real and simulated data. They also make comparisons to both internal and external deterministic models. The results are favorable.

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