The relations among three popular indices of risks

The relative risk, risk difference, and odds ratio are three major indices of differences in risks of diseases between different groups. Although widely used in research and practice in biomedical and epidemiologic research, misconceptions are not uncommon about their relationships. Many publications offer contradicting advices in how to use them in studies. Some biomedical researchers believe that these indices are related in a monotone fashion, and, thus, changes in one direction in one of the indices can be interpreted as same directional changes in the other two. Misconceptions about these three indices such as the monotone relationship are so prevalent in the biomedical and epidemiologic research that clarifications of such popular beliefs are warranted. In this paper, we take a systematic approach to characterize the relationships among the indices. We develop key results to elucidate the intricate relationships between the indices. Our findings speak to the need for investigators to carefully consider the different indices before using them in their studies, since they are not interchangeable and results based on one index are generally not translatable into any of the others.