Maximal structure for the sum of squares of a contingency with fixed margins; a programmed algorithmic solution. (French) Zbl 0639.90101

Lerman, I. C.; Peter, Ph.

RAIRO, Rech. Opér. 22, No. 2, 83-136 (1988).

This problem is considered as very difficult - and even quasi-impossible - to resolve. We begin by analyzing the approach that we call “analytic” where a non-exact bound is given by a mathematical formula with respect to the two margins. We show that this approach is necessarily based on an application of the 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 we 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. The most important part of this paper is devoted to the study of this solution. Specific notions are introduced. On the other hand, a mathematical justification of the algorithm is provided as deeply as possible.
Our solution enables the exact and logical normalization of a large family of association coefficients between two nominal qualitative variables. On the other hand, this solution builds an optimal statistical transition between two partitions.

MSC:

90B99	Operations research and management science
62H17	Contingency tables
90C90	Applications of mathematical programming
90C27	Combinatorial optimization

Keywords:

data structuring; data analysis; normalization of association coefficients; optimal configuration; exact bound; recursive algorithm; contingency table; nominal qualitative variables