Sensitivity analysis for interactions under unmeasured confounding

We develop a sensitivity analysis technique to assess the sensitivity of interaction analyses to unmeasured confounding. We give bias formulas for sensitivity analysis for interaction under unmeasured confounding on both additive and multiplicative scales. We provide simplified formulas in the case in which either one of the two factors does not interact with the unmeasured confounder in its effects on the outcome. An interesting consequence of the results is that if the two exposures of interest are independent (e.g., gene-environment independence), even under unmeasured confounding, if the estimate of the interaction is nonzero, then either there is a true interaction between the two factors or there is an interaction between one of the factors and the unmeasured confounder; an interaction must be present in either scenario. We apply the results to two examples drawn from the literature.