Inconsistency of three indices in measuring the association between the risk factor and the risk of a disease

The relative risk (r), risk difference (d), and odds ratio $\left(\theta \right)$ are three commonly used indices in epidemiology to quantify the association between the risk of a disease and the exposure to a risk factor. However, it has been reported in [C. Feng, B. Wang, and H. Wang, The relations among three popular indices of risks, Stat. Med. 38(2019), pp. 4772-4787.] that there is no monotonic relationship between any two of these three indices. In fact, our research shows that even if two of these indices change in the same direction, the third one may change in the opposite direction. This indicates that there is an internal inconsistency among these three indices in measuring the association between the risk of the disease and the risk factor. Therefore, the sizes of these indices cannot be interpreted as the strength of the association. We have derived some limiting behaviors of these indices and discussed the approximation of risk ratio by odds ratio. Our results clarify some misconceptions about these three widely used indices. In summary, our research highlights the limitations of using only one of these indices to measure the association between the risk of a disease and the exposure to a risk factor. To fully understand the nature of the association, it is important to consider all three indices and their relationships.