  In causal inference, sensitivity models assess how unmeasured confounders
could alter causal analyses, but the sensitivity parameter -- which quantifies
the degree of unmeasured confounding -- is often difficult to interpret. For
this reason, researchers sometimes compare the sensitivity parameter to an
estimate for measured confounding. This is known as calibration. Although
calibration can aid interpretation, it is typically conducted post hoc, and
uncertainty in the point estimate for measured confounding is rarely accounted
for. To address these limitations, we propose novel calibrated sensitivity
models, which directly bound the degree of unmeasured confounding by a multiple
of measured confounding. The calibrated sensitivity parameter is interpretable
as an intuitive unit-less ratio of unmeasured to measured confounding, and
uncertainty due to estimating measured confounding can be incorporated.
Incorporating this uncertainty shows causal analyses can be less or more robust
to unmeasured confounding than would have been suggested by standard
approaches. We develop efficient estimators and inferential methods for bounds
on the average treatment effect with three calibrated sensitivity models,
establishing parametric efficiency and asymptotic normality under doubly robust
style nonparametric conditions. We illustrate our methods with a data analysis
of the effect of mothers' smoking on infant birthweight.
