Cross Association Measures and Optimal Clustering

Abstract

The purpose of this short paper is to show how the introduction of the concept of ‘Paired Comparisons’ allows us to linearize most of the cross association criteria on Contingency Tables in Statistics. These statistical cross association measures derived from contingency tables amount to ‘at least’ quadratic mathematical, formulations in the classical statistical approach of cross association for categorical variables (qualitative nominal variables). This possibility of linearizing some of these criteria allows us to transform ‘classification problems’ into Linear Programming ones without any type of hypothesis of cluster size fixation. The ‘paired comparisons principle’ plays a prominent part in this paper. Let us mention here that the first mathematician who spoke about ‘paired comparisons’ was A. Condorcet in 1785, in order to solve the difficult problem of ‘Collective Choice’ in Preferences Aggregation. We shall not speak, once more, about the axiomatic of the Condorcet’s criterion, widely discussed in [15], nor about the solving of the associated aggregation problems, for which we shall find details in [14]. We shall only focus on the ‘central’ part of the paired comparisons principle’ in a ‘broad sense’, thus enlarging the possibility of this concept in a context not reserved to Similarity aggregation but to a very large theory, called “Maximal Association”.1 In the paired comparisons approach, we compare the partitions induced by two modality variables noted: Y and C, pairs of objects with pairs of objects two by two. y ij and cij, that needs to consider, starting from n objects, data tableaux of size n 2. On the contrary, in the statistical approach, for two categorical variables, one builds a contingency’ table, crossing both the variables Y and C. The distribution of values in this table n„v represents the number of objects having simultaneously the modality u of C and the modality v of Y. Then, these tables are of size p × q, where p is the number of modalities of C and q the one of Y.