Nonmetric maximum likelihood multidimensional scaling from directional rankings of similarities

Abstract

A maximum likelihood estimation procedure is developed for multidimensional scaling when (dis)similarity measures are taken by ranking procedures such as the method of conditional rank orders or the method of triadic combinations. The central feature of these procedures may be termed directionality of ranking processes. That is, rank orderings are performed in a prescribed order by successive first choices. Those data have conventionally been analyzed by Shepard-Kruskal type of nonmetric multidimensional scaling procedures. We propose, as a more appropriate alternative, a maximum likelihood method specifically designed for this type of data. A broader perspective on the present approach is given, which encompasses a wide variety of experimental methods for collecting dissimilarity data including pair comparison methods (such as the method of tetrads) and the pick-M method of similarities. An example is given to illustrate various advantages of nonmetric maximum likelihood multidimensional scaling as a statistical method. At the moment the approach is limited to the case of one-mode two-way proximity data, but could be extended in a relatively straightforward way to two-mode two-way, two-mode three-way or even three-mode three-way data, under the assumption of such models as INDSCAL or the two or three-way unfolding models.