Information Identities and Testing Hypotheses: Power Analysis for Contingency Tables

An information theoretic approach to the evaluation of 2 x 2 contingency tables is proposed. By investigating the relationship between the Kullback-Leibler divergence and the maximum likelihood estimator, information identities are established for testing hypotheses, in particular, testing independence. These identities not only validate the calibration of p values, but also yield unified power analysis for the likelihood ratio test, Fisher’s exact test and Pearson-Yates’ chi-square test. It is shown that a widely discussed exact unconditional test for the equality of binomial parameters is ill-posed for testing independence, and that using this test to criticize Fisher’s exact test as being conservative is logically flawed.