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The odds ratio (OR) is a measure of effect size commonly used inobservational research. OR reflects statistical association between a binaryoutcome, such as the presence of a health condition, and a binary predictor,such as an exposure to a pollutant. Statistical inference and intervalestimation for OR are often performed on the logarithmic scale, due toasymptotic convergence of log(OR) to a normal distribution. Here, we propose anew normalized measure of effect size, $\gamma'$, and derive its asymptoticdistribution. We show that the new statistic, based on the $\gamma'$distribution, is more powerful than the traditional one for testing thehypothesis $H_0$: log(OR)=0. The new normalized effect size is termed `gammaprime' in the spirit of $D'$, a normalized measure of genetic linkagedisequilibrium, which ranges from -1 to 1 for a pair of genetic loci. Thenormalization constant for $\gamma'$ is based on the maximum range of thestandardized effect size, for which we establish a peculiar connection to theLaplace Limit Constant. Furthermore, while standardized effects are of littlevalue on their own, we propose a powerful application, in which standardizedeffects are employed as an intermediate step in an approximate, yet accurateposterior inference for raw effect size measures, such as log(OR) and$\gamma'$.

Put the odds on your side: a new measure for epidemiological associations