  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
also that 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
which the maximum of the above-mentioned associations must satisfy, rather than
simply weaker conditions that both must satisfy. Third, we show that all
previous results on Cornfield conditions hold under weaker assumptions than
previously used. We illustrate their potential applications by real examples,
where our new conditions give more information than the classical ones.
