Generalized Cornfield conditions for the risk difference

A central question in causal inference with observational studies is the sensitivity of conclusions to unmeasured confounding. The classical Cornfield condition allows us to assess whether an unmeasured binary confounder can explain away the observed relative risk of the exposure on the outcome. It states that for an unmeasured confounder to explain away an observed relative risk, the association between the unmeasured confounder and the exposure and the association between the unmeasured confounder and the outcome must both be larger than the observed relative risk. In this paper, we extend the classical Cornfield condition in three directions. First, we consider analogous conditions for the risk difference and allow for a categorical, not just a binary, unmeasured confounder. Second, we provide more stringent thresholds that the maximum of the above-mentioned associations must satisfy, rather than weaker conditions that both must satisfy. Third, we show that all the earlier results on Cornfield conditions hold under weaker assumptions than previously used. We illustrate the potential applications by real examples, where our new conditions give more information than the classical ones.