Moving from two- to multi-way interactions among binary risk factors on the additive scale

Many studies have focused on investigating deviations from additive interaction of two dichotomous risk factors on a binary outcome. There is, however, a gap in the literature with respect to interactions on the additive scale of >2 risk factors. In this paper, we present an approach for examining deviations from additive interaction among three or more binary exposures. The relative excess risk due to interaction (RERI) is used as measure of additive interaction. First, we concentrate on three risk factors - we propose to decompose the total RERI to: the RERI owned to the joint presence of all three risk factors and the RERI of any two risk factors, given that the third is absent. We then extend this approach, to >3 binary risk factors. For illustration, we use a sample from data from the Greek EPIC cohort and we investigate the association with overall mortality of Mediterranean diet, body mass index , and smoking. Our formulae enable better interpretability of any evidence for deviations from additivity owned to more than two risk factors and provide simple ways of communicating such results from a public health perspective by attributing any excess relative risk to specific combinations of these factors. Abbreviations: BMI: Body Mass Index; ERR: excess relative risk; EPIC: European Prospective Investigation into Cancer and nutrition; MD: Mediterranean diet; RERI: relative excess risk due to interaction; RR: relative risk; TotRERI: total relative excess risk due to interaction.