An Analytical Expression for the Distribution of the Odds Ratio with applications to Coronary Heart Disease risk

Abstract The concept of the odds ratio introduces a powerful measurement that extends the usefulness of odds by incorporating the correlation between two risk factors. Odds ratio, defined as the relative strength of the association between two risks, has the ability to reveal the most influential risk factors associated with a given outcome. This knowledge has the potential to shape preventive strategies and improve public health interventions. Across a diverse spectrum of risk factors and outcomes, odds ratio is an adaptable tool that enriches the arsenal of researchers and public health practitioners. The focal point of this paper is to introduce an analytical expression governing the distribution of the odds ratio in the cases of two independent risks, each following a classical Beta distribution. We provide explicit formulations for the moments of the proposed distribution. This provides a comprehensive understanding of the statistical properties of the distribution. Through simulations under various parameter settings, we elucidate the characteristics of the proposed distribution. Our aim is to showcase its versatility in capturing diverse scenarios. To illustrate its application, we use data from an ongoing cardiovascular study based in Framingham, Massachusetts. The dataset contains 4,238 records and covers 15 demographic, behavioural, and medical risk factors, this study serves as a practical example. Our analysis demonstrates that the proposed distribution appropriately models the relationship between the risk of coronary heart disease and the range of factors involved, including age, sex, education level, and smoking. Therefore, this paper introduces the distribution of the odds ratio as a powerful metric while offering an analytical foundation for its distribution. By demonstrating its efficacy in a substantial cardiovascular study, we underline its practical value in understanding and quantifying the interaction between risk factors and health outcomes.