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

A 2×2 contingency table is usually analyzed by using the chi-squared asymptotic test, with Yates' continuity correction (c=n/2, wheren 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 sizen ₁ andn ₂) or for contrasting independence (one sample with a size ofn), the correctionc is different (c=1 ifn ₁≠n ₂ orc=2 ifn 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 whenE>5, whereE 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. Martín Andrés and Herranz Tejedor (2000) proved that the validity condition must be of the typeE>E ^* —whereE ^* 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 valueE ^* is obtained for all three cases. Given that the functionE ^* reaches an absolute maximum, it can be affirmed that the three chisquared tests referred to are valid whenE<19.2, 14.9 or 6.2 (orE>8.1, 7.7 or 3.9 ifn≤500) respectively for the three previous models (although for the first model and the two-tailed testE>0 is sufficient). This research was supported by the Dirección General de Investigación (Spain). Grant BFM2000-1472.