  Unmeasured confounding may undermine the validity of causal inference with
observational studies. Sensitivity analysis provides an attractive way to
partially circumvent this issue by assessing the potential influence of
unmeasured confounding on the causal conclusions. However, previous sensitivity
analysis approaches often make strong and untestable assumptions such as having
a confounder that is binary, or having no interaction between the effects of
the exposure and the confounder on the outcome, or having only one confounder.
Without imposing any assumptions on the confounder or confounders, we derive a
bounding factor and a sharp inequality such that the sensitivity analysis
parameters must satisfy the inequality if an unmeasured confounder is to
explain away the observed effect estimate or reduce it to a particular level.
Our approach is easy to implement and involves only two sensitivity parameters.
Surprisingly, our bounding factor, which makes no simplifying assumptions, is
no more conservative than a number of previous sensitivity analysis techniques
that do make assumptions. Our new bounding factor implies not only the
traditional Cornfield conditions that both the relative risk of the exposure on
the confounder and that of the confounder on the outcome must satisfy, but also
a high threshold that the maximum of these relative risks must satisfy.
Furthermore, this new bounding factor can be viewed as a measure of the
strength of confounding between the exposure and the outcome induced by a
confounder.
