A review of classic non-asymptotic methods for comparing two proportions by means of independent sampLES

The 2 × 2 tables used to present the data in an experiment for comparing two proportions by means of two observations of two independent binomial distributions may appear simple but are not. The debate about the best method to use is unending, and has divided statisticians into practically irreconcilable groups. In this article, all the available non-asymptotic tests are reviewed (except the Bayesian methodology). The author states which is the optimal (for each group), referring to the tables and programs that exist for them, and contrast the arguments used by supporters of each of the options. They also sort the tangle of solutions into "families", based on the methodology used and/or prior assumptions, and point out the most frequent methodological mistakes committed when comparing the different families.