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psychometrikavol. 70, no. 1, 7198
march 2005DOI: 10.1007/s11336-001-0908-1AVOIDING DEGENERACY IN MULTIDIMENSIONAL UNFOLDING BY PENALIZING
ON THE COEFFICIENT OF VARIATIONFrank M.T.A. Busingleiden universityPatrick J.K. Groenenerasmus university rotterdamWillem J. Heiserleiden universityMultidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by
approximately equal distances between points from different sets. A definition of an absolutely degenerate
solution is proposed, which makes clear that these solutions only occur when an intercept is present in
the transformation function. Many solutions for the degeneracy problem have been proposed and tested,
but with little success so far. In this paper, we offer a substantial modification of an approach initiated by
Kruskal and Carroll (1969) that introduced a normalization factor based on the variance in the usual least
squares loss function. Heiser (unpublished thesis, 1981) and De Leeuw (1983) showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies. The factor
proposed in the present paper, based on the coefficient of variation, discourages or penalizes (nonmetric)
transformations of the proximities with small variation, so that the procedure steers away from solutions
with small variation in the interpoint distances. An algorithm is described for minimizing the re-adjusted
loss function, based on iterative majorization. The results of a simulation study are discussed, in which
the optimal range of the penalty parameters is determined. Two empirical data sets are analyzed by our
method, clearly showing the benefits of the proposed loss function.Key words: unfolding, degeneracy, penalty, Stress, iterative majorization, PREFSCAL.IntroductionNonmetric multidimensional unfolding has been an idea that defeated most if not all
attempts so far to develop it into a regular analysis method for preference rankings or two-mode
proximity data. In Coombs original formulation of unidimensional unfolding (Coombs, 1950,
1964), we have rankings of n individuals over m stimulus objects, and the objective is to find a
common dimension called the quantitative J scale (joint scale), which contains ideal points representing the individuals and stimulus points representing the stimulus objects. On the J scale, each
individuals preference ordering corresponds to the rank order of the distances of the stimulus
points from the ideal point, the nearest being the most preferred.Analytical procedures for fitting a quantitative...