Linear Multidimensional Scaling of Choice1

Abstract

A multidimensional scaling analysis is presented for replicated layouts of pairwise choice responses. In most applications the replicates will represent individuals who respond to all pairs in some set of objects. The replicates and the objects are scaled in a joint space by means of a linear model which assigns weights to each of the dimensions of the space. Least squares estimates of the replicates' and objects' coordinates, and of unscalability parameters, are obtained through a manipulation of the error sum of squares for fitting the linear model. The solution involves the reduction of a three‐way least squares problem to two subproblems, one trivial and the other solvable by classical least squares matrix factorization. The analytic technique is illustrated with political preference data and is contrasted with multidimensional unfolding in the domain of preferential choice.

Measuring individuals' priorities for national goals: A methodology and empirical example

“…Other possible extensions of the method might include: (i) use of the composition paradigm of Bechtel [2] to develop scale values with elevation left in; (ii) use of the quantity [~(~9) -~2))211/2 involved in the F value equation (5) as a rational and theoretically meaningful distance function between individuals for metric multidimensional scaling of individuals [6]; and (iii) using a scaling procedure developed by Bechtel, Tucker, and Chang [4] to multidimensionally scale stimuli and individuals. This technique would yield dimensions underlying the scale value ratings which then could be interpreted.…”