GOODNESS OF FIT FOR RANDOM RANKINGS IN KRUSKAL'S NONMETRIC SCALING PROCEDURE *

in order to establish null hypotheses for the goodness-of-fit measure. Various numbers of hypothetical stimuli were analyzed using the Euclidean distance metric in various dimensionalities, and the effects of tied ranks were studied. Kruskal (1964a, 1964b) presented a method of multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. His method is an elaboration and refinement of the analysis of proximities proposed by Shepard (1962). In Kruskal's procedure, N points are positioned in an jw-dimensional space in such a way as to achieve the best possible approximation to a monotonic relationship be/N\ tween the ( . 1 interpoint distances and an experimentally obtained ranking of the dissimilarities among all pairs of N objects. For a given configuration of points, interpoint distance (d) is expressed as a function of ranked dissimilarity (8). The degree to which d is not a monotonic function of 8 for a given configuration is expressed by a measure called stress (S). Stress can be thought of as an