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

TL;DR

This paper introduces a new normalized measure of epidemiological association, gamma prime, which offers increased power for hypothesis testing and aids in more accurate posterior inference of effect sizes like the odds ratio.

Contribution

The paper proposes gamma prime, a novel normalized effect size measure with an established asymptotic distribution, improving hypothesis testing and inference in epidemiological studies.

Findings

Gamma prime is more powerful than traditional OR for testing effects.

The asymptotic distribution of gamma prime is derived and connected to the Laplace Limit Constant.

Standardized effects can improve posterior inference of raw effect sizes.

Abstract

The odds ratio (OR) is a measure of effect size commonly used in observational research. OR reflects statistical association between a binary outcome, such as the presence of a health condition, and a binary predictor, such as an exposure to a pollutant. Statistical inference and interval estimation for OR are often performed on the logarithmic scale, due to asymptotic convergence of log(OR) to a normal distribution. Here, we propose a new normalized measure of effect size, $\gamma'$, and derive its asymptotic distribution. We show that the new statistic, based on the $\gamma'$ distribution, is more powerful than the traditional one for testing the hypothesis $H_0$: log(OR)=0. The new normalized effect size is termed `gamma prime' in the spirit of $D'$, a normalized measure of genetic linkage disequilibrium, which ranges from -1 to 1 for a pair of genetic loci. The normalization constant for $\gamma'$ is based on the maximum range of the standardized effect size, for which we establish a peculiar connection to the Laplace Limit Constant. Furthermore, while standardized effects are of little value on their own, we propose a powerful application, in which standardized effects are employed as an intermediate step in an approximate, yet accurate posterior inference for raw effect size measures, such as log(OR) and $\gamma'$.