The association in a two-way contingency table through log odds ratio analysis: the case of Sarno river pollution

In this paper we are proposing a general framework for the analysis of the complete set of log Odds Ratios (ORs) generated by a two-way contingency table. Starting from the RC (M) association model and hypothesizing a Poisson distribution for the counts of the two-way contingency table we are obtaining the weighted Log Ratio Analysis that we are extending to the study of log ORs. Particularly we are obtaining an indirect representation of the log ORs and some synthesis measures. Then for studying the matrix of log ORs we are performing a generalized Singular Value Decomposition that allows us to obtain a direct representation of log ORs. We also expect to get summary measures of association too. We have considered the matrix of complete set of ORs, because, it is linked to the two-way contingency table in terms of variance and it allows us to represent all the ORs on a factorial plan. Finally, a two-way contingency table, which crosses pollution of the Sarno river and sampling points, is to be analyzed to illustrate the proposed framework.