A Monte Carlo Evaluation of Three Nonmetric Multidimensional Scaling Algorithms

As a consequence of the complexity of the iterative optimum seeking procedures used by practical nonmetric multidimensional scaling algorithms, many of their computational properties have not been well understood. In particular, the following questions are of interest: (a) from the user’s point of view, are there significant differences between alternative available programs, (b) are suboptimal solutions frequently encountered, and how does this depend on the characteristics of the algorithm? Using Monte Carlo techniques to generate dissimilarity matrices with known underlying configurations, Kruskal’s M-D-SCAL, Guttman-Lingoes’ SSA-I, and Young-Torgerson’s TORSCA-9 programs were compared. It was found: (a) that differences between the solutions obtained by the algorithms were typically so small as to be of little practical importance, (b) deviant solutions were occasionally produced by each of the algorithms, but most often by M-D-SCAL, and furthermore, most frequently in one dimension. The likelihood of being trapped in a nonoptimal position is reduced by a good choice of initial configuration, and also possibly by constructing the iterative process to favor the possibility of stepping over small local depressions when far from the location of the optimum. The conclusions of this study are based on a total of 2160 scaling solutions.