Using Collapsing and Multiple Comparisons to Detect Association in Two Way Contingency Tables

In order to test the presence of association between two categorical variables we can use the $\chi^2$ test, the $G^2$-test or generalizations of the Fisher's exact test.  The $\chi^2$ and $G^2$-test can be applied to two way tables of any size and their sampling distribution is approximated, under the null hypothesis of independence, by $\chi^2$ distributions. On the other hand, Fisher's exact test is defined for 2 $\times$ 2 tables and enables to compute exact $p$-values. In this paper we analyze the properties of a procedure which collapses the original contingency table in 2 $\times$ 2 tables and to each of them applies the Fisher's exact test.  The result of the procedure enables us to highlight the contribution to the association of each single entry of the original two way table. We compare this approach with the standard one which is based on the inertia decomposition.