Approaches to asymmetric multidimensional scaling with external information

Asymmetric relationships contained in square data matrices like proximities (e.g. similarity ratings), preferences (e.g. socio-matrices), flow data (e.g.import-export, brand switching), can be represented in low-dimensional spaces byscalar product or Euclidean distance models (MDS models), opportunely modifiedby increasing the number of parameters (see e.g. Zielman and Heiser, 1996). In manyapplications additional information (external information) on the objects is availablethat could be conveniently incorporated in the data analysis. For instance, thisallows to analyze the contribution of variables suggested from theoretical knowledgeto the explanation of the relationships in the data. To this aim many methods wereproposed in the context of symmetric MDS (see e.g. Borg and Groenen 1997, chapter10), while a lack of proposals seems to characterize asymmetric MDS. In thiscommunication some possible approaches to asymmetric multidimensional scalingwith external information to analyze graphically asymmetric proximity matricesare discussed. In particular, a method based on the unique decomposition of thedata matrix in its symmetric and skew-symmetric components, recently proposedby Bove and Rocci (2004), and a proposal to incorporate external information in biplotmethod are considered. The presented methods allow joint or separate analyzesof symmetry and skew-symmetry.