More enlightenment on the essence of applying Fisher's Exact test when testing for statistical significance using small sample data presented in a 2 x 2 table

Abstract

The paper illustrates the inconsistent behaviour of the Chi-Square test when applied onto 2 x 2 tables involving independent samples and a small overall sample size. The appropriateness of applying this test to a given 2 x 2 table was judged by the degree of agreement between the probability associated with a value of Chi-Square for that table and that obtained by direct application of Fisher's Exact Test. Data from two different studies were used for the illustration. For one of the tables, the two tests led to divergent conclusions about the Null Hypothesis (H degrees) being tested; and the two probabilities were found to differ quite significantly. When the same procedures were repeated for the data in another 2 x 2 table (also involving a small sample size) the two tests led to the same conclusion about Ho; and in this latter case, the two probabilities were approximately equal: thus implying good approximate accuracy of the Chi-Square test relative to the Exact Test. To avoid these unantisipated variations of results from the Chi-Square test when it is applied onto 2 x 2 tables involving small sample sizes, Fisher's Exact Test is further recommended as a definitive test, always to be resorted to for fourfold tables involving small sample sizes and independent samples.