A revisit to sample size and power calculations for testing odds ratio in two independent binomials

We reexamine the subject of sample size determination (SSD) when testing logarithm of odds ratio (OR) against zero in two independent binomials. Four common approaches are considered: a closed-form SS formula based on the Wald test (nW), closed-form formulas that meet SS requirement by score and exact tests respectively (nS and nE), and a numerical approach to calculating SS based on likelihood ratio (LR) tests (nL). Several practically useful findings are presented. First, nW is a strictly convex function of OR for OR >1 and OR <1, respectively, implying that SS calculated by nW does not necessarily decrease as OR gets further away from 1. However, minimum SS often occurs at OR values that are deemed relatively extreme and rare in real life. nS, nE, and nL decrease monotonically as OR diverges from 1. Secondly, the optimal sampling ratio (OSR) between two independent binomials that yields maximum power for a given total SS is not always 1:1 but depends on the odds of outcome in each arm. nW benefits the most from the application of OSR in that total SS can be significantly reduced as compared to the commonly used 1:1 sampling ratio. Savings in SS by OSR in nS, nL and nE are relatively immaterial from a practical perspective. Finally, we use simulation studies to examine the power loyalty of each SS approach and explore penalized likelihood as a remedy for undermined power loyalty.