A Quasi-Nonmetric Method for Multidimensional Scaling Via an Extended Euclidean Model

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Abstract
An Extended Two-Way Euclidean Multidimensional Scaling (MDS) model which assumes both common and specific dimensions is described and contrasted with the “standard” (Two-Way) MDS model. In this Extended Two-Way Euclidean model the n stimuli (or other objects) are assumed to be characterized by coordinates on R common dimensions. In addition each stimulus is assumed to have a dimension (or dimensions) specific to it alone. The overall distance between object i and object j then is defined as the square root of the ordinary squared Euclidean distance plus terms denoting the specificity of each object. The specificity, sj, can be thought of as the sum of squares of coordinates on those dimensions specific to object i, all of which have nonzero coordinates only for object i. (In practice, we may think of there being just one such specific dimension for each object, as this situation is mathematically indistinguishable from the case in which there are more than one.)We further assume that δij =F(dij) +eij where δij is the proximity value (e.g., similarity or dissimilarity) of objects i and j, dij is the extended Euclidean distance defined above, while eij is an error term assumed i.i.d. N(0, σ2). F is assumed either a linear function (in the metric case) or a monotone spline of specified form (in the quasi-nonmetric case). A numerical procedure alternating a modified Newton-Raphson algorithm with an algorithm for fitting an optimal monotone spline (or linear function) is used to secure maximum likelihood estimates of the paramstatistics) can be used to test hypotheses about the number of common dimensions, and/or the existence of specific (in addition to R common) dimensions.This approach is illustrated with applications to both artificial data and real data on judged similarity of nations.