Testing Independence in 2 × 2 Contingency Tables

Three of the potential models for the 2 × 2 contingency table are discussed: the hyper geometric Independence Trial, the double-binomial Comparative Trial, and the multinomial Double Dichotomy Trial. Then the critical regions of seven statistical tests are evaluated within each of these models. The uncorrected Pearson’s Chi-Square test and Upton’s correction are found to be overly liberal in all three models, particularly for the hypergeometric. A continuity correction, when properly applied, is shown to be an extremely good approximation to Fisher’s Exact test, which employs the hypergeometric distribution to evaluate outcome probabilities—but is an extremely conservative approximation to the double binomial and multinomial models. Haber’s test is recommended for these two cases, but the amount of work required may be prohibitive for many investigators.