Chi-square test
On the use of Chi-square test and approximate the distribution: Small expected frequencies leads to poor approximation of the distribution of the chi-square statistic with the chi-square distribution. That is because when small frequencies are present, the possible values of the chi-square statistic are “much” discrete. Therefore, the continuous chi-square distribution becomes not a good fit.
A general rule is to at least have 5 as the smallest frequency in your table, however this rule seems to be chosen at random (Cochran,1952). Yates proposed a correction (implemented in R).
Nowadays, given computing capabilities the correction is less needed. So what should we do?
Fisher’s exact test is a solid alternative but if your design allows the assumption of “fixed margins”. More over Fisher’s exact test can be used in larger tables than 2 x 2. If the expected frequencies are small (<5) then one can use the Monte Carlo simulated p-value.
Citation
@online{cohen,
author = {Cohen, Achraf},
title = {On the {Chi-square} Test with Small Expected Frequencies},
url = {https://acohenstat.github.io/main/posts/chisquare_test},
langid = {en}
}