Maximal association structure for the sum of squares of a contingency table entries: A programmed algorithmic solution

This problem is considered as very difficult-and even quasi-impossible-to resolve. The authors begin by analyzing the approach that they call ‘analytic’ where a non exact bound is given by a mathematic formula, this with respect to the two margins. The authors show that this approach is necessarily based on an application of Schwarz inequality, conceived in a logical context. The best bound which can be obtained by this method is in fact less accurate than the one determined from the ‘best’ margin. The original solution that they propose to construct the optimal configuration and the associated exact bound, is based on the notion of recursive algorithm. The starting point of application of the algorithm is the couple of margins of the contingency table. Most important part of this paper is devoted to the study of this solution. Specific notions are introduced. On the other hand, mathematical justification of the algorithm is provided as deeply as possible. The present solution enables the exact and logical normalization of a large family of associated coefficients between two nominal qualitative variables. On the other hand, this solution builds an optimal statistical transition between two partitions.