Distributional interaction: Interpretational problems when using incidence odds ratios to assess interaction

It is well known that the incidence odds ratio approximates the risk ratio when the disease of interest is rare, but increasingly overestimates the risk ratio as the disease becomes more common. However when assessing interaction, incidence odds ratios may not approximate risk ratios even when the disease is rare. We use the term "distributional interaction" to refer to interaction that appears when using incidence odds ratios that does not appear, or appears to a lesser degree, when using risk ratios. The interpretational problems that arise from this discrepancy can have important implications in epidemiologic research. Therefore, quantification of the relationship between the interaction odds ratio and the interaction risk ratio is warranted. In this paper, we provide a formula to quantify the differences between incidence odds ratios and risk ratios when they are used to estimate effect modification on a multiplicative scale. Using this formula, we examine the conditions under which these two estimates diverge. Furthermore, we expand this discussion to the implications of using incidence odds ratios to assess effect modification on an additive scale. Finally, we illustrate how distributional interaction arises and the problems that it causes using an example from the literature. Whenever the risk of the outcome variable is non-negligible, distributional interaction is possible. This is true even when the disease is rare (e.g., disease risk is less than 5%). Therefore, when assessing interaction on either an additive or multiplicative scale, caution should be taken in interpreting interaction estimates based on incidence odds ratios.