  In observational studies, identification of ATEs is generally achieved by
assuming that the correct set of confounders has been measured and properly
included in the relevant models. Because this assumption is both strong and
untestable, a sensitivity analysis should be performed. Common approaches
include modeling the bias directly or varying the propensity scores to probe
the effects of a potential unmeasured confounder. In this paper, we take a
novel approach whereby the sensitivity parameter is the "proportion of
unmeasured confounding:" the proportion of units for whom the treatment is not
as good as randomized even after conditioning on the observed covariates. We
consider different assumptions on the probability of a unit being unconfounded.
In each case, we derive sharp bounds on the average treatment effect as a
function of the sensitivity parameter and propose nonparametric estimators that
allow flexible covariate adjustment. We also introduce a one-number summary of
a study's robustness to the number of confounded units. Finally, we explore
finite-sample properties via simulation, and apply the methods to an
observational database used to assess the effects of right heart
catheterization.
