Partial and total orderings of dependence on tables with given margins

The purpose of this paper is the study of dependence for r×s tables in the class of all bivariate tables with given margins. First of all, a dependence partial ordering will be defined, based on the signs and the absolute values of r×s contingencies. Then, to increase the number of couples of comparable tables, a new partial ordering, called 'intensity of dependence ordering' and based only on the absolute value of contingencies, is introduced. To allow the comparison of all pairs of tables, a dependence total ordering is needed, therefore a broad class of functions - able to induce total synthetic dependence orderings is introduced, their arguments being the absolute values of relative contingencies. Among them, Mortara's and Pearson's indexes can be identified. Finally, the dependence degree of a given table can be expressed by its relative position in a chosen dependence ordering of all the tables in the same class. This methodology is applied, as an example, to four classes of tables.