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Multidimensional scaling for \(n\)-tuples. (English) Zbl 0726.92031

Summary: Wide use has been made of multidimensional scaling (MDS) techniques since the pioneering papers of R. N. Shepard [Psychometrika 27, 125–140 and 219–246 (1962; Zbl 0129.121)] and J. B. Kruskal [ibid. 29, 1–29 and 115–129 (1964; Zbl 0123.368)]. In the main, dissimilarities used in the various MDS techniques are derived for pairs of objects or stimuli. This is termed 2-way, 1-mode data, meaning pairs of objects within a single set are considered. Some MDS techniques are designed for 3-way or even higher, and for 2-mode, 3-mode or more. One such example is the Candecomp model which can deal with \(n\)-way, \(m\)-mode data where \(3\leq n\leq 7\) and \(2\leq m\leq 7\). This model considers \(n\)-tuples of objects at a time, selecting these from \(m\) different sets.
To date there are no models which consider \(n\)-way, 1-mode data, where \(n\geq 3\). An attempt is made in this paper to cater for this situation by an extension of the Shepard-Kruskal approach to nonmetric multidimensional scaling which deals with dissimilarities defined for three or more objects. A computer program has been written using the new model to produce a configuration in Euclidean space to represent the objects. Some historical voting data and some artificial data are then analyzed.

MSC:

91C15 One- and multidimensional scaling in the social and behavioral sciences
62P15 Applications of statistics to psychology
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