Optimal correction for continuity and conditions for validity in the unconditional chi-squared test

The most usual asymptotic unconditional method for comparing two independent binomial proportions is the chi-squared test. There are various versions of it (with or without continuity corrections), but there has been no exhaustive study about how they behave, about their optimality, nor about the conditions under which the approximation is considered reliable. In this paper these queries are examined by analysing a total of 20 different methods (some of them new), proving that two of the new methods are the optimals, that the same method is not the best in every situation, and that the condition for validity (which depends on the minimum expected quantity, E, and on the imbalance between the samples) is not unique, although a conservative rule for the optimal method (related to that of Haber (1982)) is that the method is valid if the sample sizes are approximately equal or if E > 4.5.