On the validity condition of the chi-squared test in 2 x 2 tables

A 2 x 2 contingency table is usually analyzed by using the chi-squaxed asymptotic test, with Yates' continuity correction (c = n/2, where n is the total size of sample). This correction is the correct one when the chi-squared test is an approximation to Fisher's exact (conditional) test. When the chi-squared test is used as an approximation to Barnard's exact (unconditional) test for comparing two independent proportions (two samples of size n(1). and n(2)), or for contrasting independence (one sample with a size of n), the correction c is different (c = 1 if n(1) not equal n(2) or c = 2 if n(1) = n(2) in the first case; c = 0.5 in the second). Whatever the case, it is traditional to affirm that the asymptotic test is valid when E > 5, where E is the minimum expected quantity. Today it is recognized that this condition is too general and may not be appropriate. In the case of Yates' correction, Martin Andres and Herranz Tejedor (2000) proved that the validity condition must be of the type E > E* -where E* is a known function depending on the marginals of the table- and that checking the validity of the asymptotic test is equivalent to checking the asymmetry of the base statistic (a hypergeometric random variable). In the present article the authors prove that this argument is valid for the other two continuity corrections, and moreover, that the value E* is obtained for all three cases. Given that the function E* reaches an absolute maximum, it can be affirmed that the three chi-squared tests referred to are valid when E > 19.2, 14.9 or 6.2 (or E > 8.1, 7.7 or 3.9 if n <= 500) respectively for the three previous models (although for the first model and the two-tailed test E > 0 is sufficient).