Unidimensional data: A graphical analysis

The purpose of this article is three-fold: to demonstrate the interface between the theory of scaling and graph theory, to determine the class of unidimensional graphs, and to provide a method for analyzing data which consists cf both dominance and indifference relations. The most definitive work on the subject of data analysis is Coombs’ (1976) A Theory of Data in which eight types of data are described. The eight data types arise from three defining dichotomies, one of which separates the data in terms of whether the obtained relations demonstrate dominance or indifference (termed proximity by Coombs). Let us assume, without loss of generality, that we have decided upon the method of paired comparisons for data collection. Under usual circumstances, when presented with two stimuli (or sets of stimuli, hereafter referred to simply as stimuli) the subject is asked to answer one of two questions. Either the subject must indicate which of the two stimuli dominates (is preferred to, is greater than, contains more of some aspect, etc.) the other, or the subject is asked whether or not the stimuli are proximal (similar, indistinguishable, within some indifference threshold of one another, etc.). There ..:ire numerous ways of handling data thus obtained (cf. Coombs, 1976 for a complcrte treatment). Scaling analysis has primarily relied upon data cclnsisting of either dominance or indifference relations, but not both. It is sometimes the case that both types are collected, but only one analyzed. In one such study by Coombs and Coombs (1977) both types of data were collected and separately anialyzed using different methods. It was found that similar scales were obtained under both circumstances. Two implications of this experiment and its results should be mentioned. First, both types of data are meaningful in the sense that they are compatibly generated from the same underlying scale. And second, the me(.tsures of quality of the scales