The optimal number of tetrads for measurement of dissimilarities in nonmetric multidimensional scaling

The direct determination of dissimilarities is the most popular and most frequently used method for raising input data in nonmetric multidimensional scaling, i.e. when variables are measured on an ordinal scale (e.g. in preference studies). Most methods for the direct measurement of similarities, including ranking, sorting, pairwise comparison, conditional ranking of similarities are, however, very laborious, especially when a large number of objects is tested. Thus, the research described in this article is based on the tetrad method, which is uncomplicated and less burdensome for the respondents. In the proposed method, respondents are asked to evaluate four-element subsets (tetrads) from a set of n objects. The respondent is asked to indicate the pair with the most and the least similar elements in each tetrad. As the number of tetrads rapidly increases along with the number of objects, it becomes necessary to use the incomplete variant of the method, in which only some four-element subsets are presented to the respondents for evaluation. The aim of the research presented in the article is to determine the size of the tetrad set that is sufficient to create a dissimilarity matrix used to perform nonmetric multidimensional scaling. The study was based on four distance matrices for 7, 9, 11 and 13 objects that were randomly selected voivodship capitals in Poland. The distances between the cities were expressed in kilometres. The Procrustes analysis and Spearman’s rank correlation were used in the study. The findings show that the use of the tetrad method for the measurement of dissimilarities produces beneficial results already at the point when each pair of objects appears in the set of tetrads only once, which allows the number of opinions provided by the respondents to be significantly reduced.