Effect Sizes for 2×2 Contingency Tables

Sample size calculations are an important part of research to balance the use of resources and to avoid undue harm to participants. Effect sizes are an integral part of these calculations and meaningful values are often unknown to the researcher. General recommendations for effect sizes have been proposed for several commonly used statistical procedures. For the analysis of 2×2 tables, recommendations have been given for the correlation coefficient φ for binary data; however, it is well known that φ suffers from poor statistical properties. The odds ratio is not problematic, although recommendations based on objective reasoning do not exist. This paper proposes odds ratio recommendations that are anchored to φ for fixed marginal probabilities. It will further be demonstrated that the marginal assumptions can be relaxed resulting in more general results.