Chi-square and Fisher’s exact tests

The final step is to sum the probability of the observed table and the more extreme tables (ie, those with probabilities < the observed table) (.2803 + .2337 + .0543 + .0045 = .5728). Thus, the resulting rounded P value is .57, which indicates a high level of compatibility between the data and the null hypothesis of no association. The decision is to fail to reject the null hypothesis and the conclusion is that the evidence does not support an association among lung morphology and Parkinson disease. In other words, there is insufficient evidence to claim that the proportion of Parkinson disease differs between the focal and nonfocal ARDS patients (0% vs 5%, P = .57). This matches the P value reported by Mrozek for this association.

The first objective of this article was to identify scenarios in which a chi-square or Fisher’s exact test should be considered. The general setting discussed was an investigation of the association between two categorical variables. Use of each test specifically depends on whether the assumptions have been met. Both of the examples used in our discussion happened to be binary, but that is not a restriction. Categorical variables can have more than 2 levels. All of the methods demonstrated for 2 × 2 tables can be generalized to r × c tables. 

The second objective of this article was to recognize when test assumptions have been violated. For simplicity, most researchers adhere to the following: if ≤ 20% of expected cell counts are less than 5, then use the chi-square test; if > 20% of expected cell counts are less than 5, then use Fisher’s exact test. Both methods assume that the observations are independent. Could one use the exact test when the chi-square assumptions are met? Yes, but it is more computationally expensive as it uses all possible fixed margin tables and their probabilities. If the chi-square assumptions are met, then the sample size is typically larger and these calculations become numerous. Also, it does not have to be that large of a sample for the chi-square to be a good approximation and do it very quickly.

The final objective of this article was to test claims made regarding the association of 2 independent categorical variables. We included examples from the medical literature showing step-by-step calculations of both the large sample approximation (chi-square) and exact (Fisher’s) methodologies providing insight into how these tests are conducted as well as when they are appropriate.