A Measures of Association for Two-way Contingency Tables by Maximum Association

Pearson $X^2$ statistics can be used as the measure of association for two-way contingency tables. However, it's magnitude depends on the sample size N, the normalized index ${\phi}^2 = X^2\;/\;N$ has. been widely used in many researches instead of that. In this study, we firstly point out that the actual upper bound of ${\phi}^2$ can be affected by the given marginals of the tables, so that ${\phi}^2$ should be adjusted the actual maximum to get the proper range [ 0, 1]. Also we point out that the maximum could be found easily by using several algorithms like Hu and Mukerjee(2002). With the actual maximum, we propose the relative measure of association which is normalized by it.