Measuring additive interaction using odds ratios

Interaction measured on the additive scale has been argued to be better correlated with biologic interaction than when measured on the multiplicative scale. Measures of interaction on the additive scale have been developed using risk ratios. However, in studies that use odds ratios as the sole measure of effect, the calculation of these measures of additive interaction is usually performed by directly substituting odds ratios for risk ratios. Yet assessing additive interaction based on replacing risk ratios by odds ratios in formulas that were derived using the former may be erroneous. In this paper, we evaluate the extent to which three measures of additive interaction - the interaction contrast ratio (ICR), the attributable proportion due to interaction (AP), and the synergy index (S), estimated using odds ratios versus using risk ratios differ as the incidence of the outcome of interest increases in the source population and/or as the magnitude of interaction increases. Our analysis shows that the difference between the two depends on the measure of interaction used, the type of interaction present, and the baseline incidence of the outcome. Substituting odds ratios for risk ratios, when calculating measures of additive interaction, may result in misleading conclusions. Of the three measures, AP appears to be the most robust to this direct substitution. Formulas that use stratum specific odds and odds ratios to accurately calculate measures of additive interaction are presented.