Multidimensional 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 Heiser (unpublished thesis, 1981) and De Leeuw (1983) showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies, The factor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.

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Erasmus Research Institute of Management