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A Revisit to Sample Size and Power Calculations for Testing Odds Ratio in Two Independent Binomials
Author(s) -
Liu Fang
Publication year - 2013
Publication title -
biometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.298
H-Index - 130
eISSN - 1541-0420
pISSN - 0006-341X
DOI - 10.1111/biom.12026
Subject(s) - sample size determination , statistics , logarithm , odds , odds ratio , mathematics , power function , regular polygon , monotonic function , power (physics) , score test , sampling (signal processing) , function (biology) , sample (material) , statistical hypothesis testing , computer science , biology , mathematical analysis , physics , logistic regression , geometry , filter (signal processing) , quantum mechanics , evolutionary biology , computer vision , thermodynamics
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 ( n W ), closed‐form formulas that meet SS requirement by score and exact tests respectively ( n S and n E ), and a numerical approach to calculating SS based on likelihood ratio (LR) tests ( n L ). Several practically useful findings are presented. First, n W is a strictly convex function of OR for OR > 1 and OR < 1 , respectively, implying that SS calculated by n W 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. n S , n E , and n L 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. n W 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 n S , n L and n E 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.