Tests of Independence for a $2 \times 2$ Contingency Table with Random Margins

Fisher's exact test is commonly used for testing the hypothesis of independence between the row and column variables in a $r \times c$ contingency table. It is a ``small-sample'' test since it is used when the sample size is not large enough for the Pearsonian chi-square test to be valid. Fisher's exact test conditions on both margins of a $2 \times 2$ table leading to a hypergeometric distribution of the cell counts under independence. Moreover, it is conservative in the sense that its actual significance level falls short of the nominal level. In this paper, we modify Fisher's exact test by lifting the restriction of fixed margins and allow the margins to be random. In doing so, we propose two new tests - a likelihood ratio test in a frequentist framework and a Bayes factor test in a Bayesian framework, both of which are based on a new multinomial distributional framework. We apply the three tests on data from the Worcester Heart Attack study and compare their power functions in assessing gender difference in the therapeutic management of patients with acute myocardial infarction (AMI).