Dependence measures based on partial and total orderings

The aim of this paper is to propose a new operational measure for evaluating the degree of dependence existing between two nominal categorical variables. Given an r×c table T, representing bivariate statistical data, our approach to measure the strength of this relation is based on the consideration of the class F of all contingency tables with the same given margins as T. Once a partial or total ordering of dependence in F (as defined in Greselin and Zenga [2004b]) has been given, the relative position assumed by T in F can be a meaningful measure of dependence. Some desirable properties of these indexes are presented: by construction they are normalized, coherent with each level of ordering and attain extreme values in extreme dependence situations. They are invariant to permutation of rows and columns in the table and to transposition (as qualitative variables classification requires), and, finally they show a sort of stability behaviour with respect to similar populations. Furthermore, their straightforward interpretability is compared with the classical interpretation of some well-known normalized indexes. Interesting remarks arise when the comparison is carried out on the discussion of their values, particularly on the extreme dependence situations