A Revisit to Sample Size and Power Calculations for Testing Odds Ratio in Two Independent Binomials

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Abstract

Summary

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 (math formula), closed-form formulas that meet SS requirement by score and exact tests respectively (math formula and math formula), and a numerical approach to calculating SS based on likelihood ratio (LR) tests (math formula). Several practically useful findings are presented. First, math formula is a strictly convex function of OR for OR math formula and OR math formula, respectively, implying that SS calculated by math formula 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. math formula, and math formula 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. math formula 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 math formula, math formula and math formula 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.

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