Accurate and efficient power calculations for 2 x m tables in unmatched case-control designs

The aim of this article was to derive an approximation of the distribution of Pearson's statistic for 2 x m contingency tables to calculate power for case-control studies accurately and efficiently in terms of computer time. We first prove that, rather than a non-central chi-square distribution, Pearson's statistic is asymptotically equivalent with the sum of squares of m independent normal random variables whose variances are typically different under the alternative hypothesis. Based on this asymptotically equivalent (AE) we derive a cost-effective approximation (CE). Numerical results show that CE was almost as precise as AE, but computationally more efficient. Although somewhat less costly in terms of CPU time, we show that a commonly used approximation using the non-central chi-square distribution can be very inaccurate and overestimate power in some scenarios and underestimate it in others. Simulations and exact distributions, on the other hand, are accurate but computationally very intensive compared to our CE. The CE reached its lowest accuracy under the null hypothesis where it has a chi-square distribution with m - 1 degree of freedom. This suggests that CE can be used to calculate power for tables where researchers would feel comfortable using Pearson's chi-square test.