Schonemann P. H. and Wang, M.M.
An individual difference model for the multidimensional analysis of preference data
Psychometrika, 1972, 37, 275-309
A model for the analysis of paired comparison data is presented which combines features of the BTL-model with features of the Unfolding model.
The model is metric, mathematically tractable, and has an exact algebraic solution. Since it is multidimensional and allows for individual differences, it is thought to be more realistic for some choice situations than either the Thurstone model or the BTL-model.
No claim is made that the present model will be appropriate for all conceivable situations. Rather, it is argued that the fact that it is falsifiable is a point in its favor.
This paper is an extension of my algebraic solution of the metric unfolding problem.
After solving this problem, I was bothered by the question: "where do the euclidean distances come from?" The original non-metric version of Coombs' one-dimensional unfolding paradigm appealed me to intuitively because the data requirements were weak (all that was required were permutations of the choice objects). This intuitive appeal dissipated once the problem was generalized to m > 1 dimension and euclidean distances had to be introduced to render it algebraically soluble.
The model presented in this paper was meant to remedy this plausibility problem by combining two different scaling models, the 1-dimensional BTL model and the Coombs' unfolding model. The BTL model supplies a number of subject-specific 1-dimensional preference scales with a minimum of ad hoc metric assumption: All that was needed were pairwise choice probabilities. The Coombsian unfolding model then accounts for individual differences in a possibly multidimensional space. The key idea was that the invariance property of the algebraic solution of the the metric unfolding problem permitted to eliminate the arbitrary origins of the logged BTL scales and thus to tie them together in a multidimensional euclidean space.
A fairly detailed application of this model to voting data can be found in Wang, Schonemann, and Rusk (1975).
Finally, it might be noted that the underlying model,
pik = K exp(-dik2)
where dik2 is the squared euclidean distance between the k'th ideal point and the i'th choice object, is symmetric in i and k. This means that this model can also be given an interpretation in Signal Detection Theory: Instead of centering the multinormal preference contours at the ideal points, they could be centered at the choice objects. From this point of view, one obtains a multidimensional signal detection model that postulates that a subject's choice is a monotone function of the likelihood ratio given by the two stimulus ordinates under the multinormal preference contours. These ordinates covary with the distances between ideal and choice points.