Testing association with Fisher's Exact test

WHAT IS FISHER’S EXACT TEST? Undoubtedly the most widely known test of association between two binary variables is the 2×2 Chi-square (χ) test. However, many readers will also have learned about Fisher’s Exact test at some point – most likely in a basic statistics course – that Fisher’s Exact test is the advised, or in fact the obligatory, alternative to the 2×2 χ test in the situation that ‘the sample size is small’. It might seem surprising then that Fisher Exact tests have been used for all analyses of association in the article by Akintomide et al., even though the n available for analysis is >100 in all analyses reported, and despite the fact that the crosstabulations are not 2×2, but 3×3 or, in one case, 3×4. The fact is, Fisher’s Exact test of association between two categorical (classification) variables is much more widely applicable than basic statistics courses have led learners to believe. There is an historical reason why it has been so ‘overlooked’, and that is because of the torturous arithmetic calculations that are required to achieve the Fisher Exact test for a cross-tabulation with large overall n, even more so to complete tests analogous to 2×2 Fisher Exact test, for tables of larger dimension (R×C rather than 2×2). The calculations necessary would be pretty much impossible using a calculator, and have not even been much available in statistical software for personal computers. It is only with recent improvements in desktop computing power that the necessary procedures have come to be added into statistical software packages. Fisher’s Exact test (or an analogous test for tables larger than 2×2) enables, for any cross-classified R×C table, calculation of the exact probability of obtaining a set of cell frequencies at least as extreme as the observed data. Reflection on the size of this calculated probability then allows evaluation of the null hypothesis of no association (or equivalently, of independence) between the two classification variables.