Reconstructing One-Mode Three-way Asymmetric Data for Multidimensional Scaling

Abstract

Some models have been proposed to analyze one-mode three-way data [e.g. De Rooij and Gower (J Classification 20:181–220, 2003), De Rooij and Heiser (Br J Math Stat Psychol 53:99–119, 2000)]. These models usually assume triadic symmetric relationships. Therefore, it is general to transform asymmetric data into symmetric proximity data when one-mode three-way asymmetric proximity data are analyzed using multidimensional scaling. However, valuable information among objects is lost by symmetrizing asymmetric proximity data. It is necessary to devise this transformation so that valuable information among objects is not lost. In one-mode two-way asymmetric data, a method that the overall sum of the rows and columns are equal was proposed by Harshman et al. (Market Sci 1:205–242, 1982). Their method is effective to analyze the data that have differences among the overall sum of the rows and columns caused by external factors. Therefore, the present study proposes a method that extends (Harshman et al., Market Sci 1:205–242, 1982) method to one-mode three-way asymmetric proximity data. The proposed method reconstructs one-mode three-way asymmetric data so that the overall sum of the rows, columns and depths is made equal.