Correspondence analysis from the viewpoint of compositional tables

Abstract

Correspondence analysis (CA), a well-known method for analyzing the relationships between rows and columns of a table, has been reformulated to link to the logratio methodology of compositional data by using the limiting case of the power transformation. The resulting methodology investigates relative rather than absolute information, and it is invariant with respect to rescaling rows or columns. The latter properties also hold for the analysis of compositional tables, where the table is first decomposed into an independent and an interaction part. It is shown that the analysis of the interaction part is equivalent to CA, but in addition, the variance contributions can be determined. Both concepts also allow for an inclusion of weights to suppress undesirable variance, and it is shown that the equivalence between weighted CA and the analysis of weighted compositional tables again holds. This equivalence allows us to make use of the mathematical framework of weighted compositional tables, the so-called Bayes spaces, to get a deeper understanding of CA and to construct extensions to multi-factorial tables (cubes, etc.).