The Distance Perspective of Generalized Biadditive Models: Scalings and Transformations

Abstract

Recently two articles studied scalings in biplot models, and concluded that these have little impact on the interpretation. In this article again scalings are studied for generalized biadditive models and correspondence analysis, that is, special cases of the general biplot family, but from a different perspective. The generalized biadditive models, but also correspondence analysis, are often used for Gaussian ordination. In Gaussian ordination one takes a distance perspective for the interpretation of the relationship between a row and a column category. It is shown that scalings—but also nonsingular transformations—have a major impact on this interpretation. So, depending on the perspective one takes, the inner product or distance perspective, scalings and transformations do have (distance) or do not have (inner-product) impact on the interpretation. If one is willing to go along with the assumption of the author that diagrams are in practice often interpreted by a distance rule, the findings in this article influence all biplot models.

Key Words:

• Correspondence analysis
• Euclidean distance
• Gaussian ordination
• Multidimensional scaling
• Multidimensional unfolding