We study exact tests for (2 x 2) and (2 x 3) contingency tables, in particular exact chi-squared tests and exact tests of Fisher type. In practice, these tests are typically carried out without randomization, leading to reproducible results but not exhausting the significance level. We discuss that this can lead to methodological and practical issues in a multiple testing framework when many tables are simultaneously under consideration as in genetic association studies.Realized randomized p-values are proposed as a solution which is especially useful for data-adaptive (plug-in) procedures. These p-values allow to estimate the proportion of true null hypotheses much more accurately than their non-randomized counterparts. Moreover, we address the problem of positively correlated p-values for association by considering techniques to reduce multiplicity by estimating the "effective number of tests" from the correlation structure.An algorithm is provided that bundles all these aspects, efficient computer implementations are made available, a small-scale simulation study is presented and two real data examples are shown.
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