In causal inference, sensitivity models assess how unmeasured confounders could alter causal analyses. However, the sensitivity parameter in these models -- which quantifies the degree of unmeasured confounding -- is often difficult to interpret. For this reason, researchers will sometimes compare the magnitude of the sensitivity parameter to an estimate for measured confounding. This is known as calibration. We propose novel calibrated sensitivity models, which directly incorporate measured confounding, and bound the degree of unmeasured confounding by a multiple of measured confounding. We illustrate how to construct calibrated sensitivity models via several examples. We also demonstrate their advantages over standard sensitivity analyses and calibration; in particular, the calibrated sensitivity parameter is an intuitive unit-less ratio of unmeasured divided by measured confounding, unlike standard sensitivity parameters, and one can correctly incorporate uncertainty due to estimating measured confounding, which standard calibration methods fail to do. By incorporating uncertainty due to measured confounding, we observe that causal analyses can be less robust or more robust to unmeasured confounding than would have been shown with standard approaches. We develop efficient estimators and methods for inference for bounds on the average treatment effect with three calibrated sensitivity models, and establish that our estimators are doubly robust and attain parametric efficiency and asymptotic normality under nonparametric conditions on their nuisance function estimators. We illustrate our methods with data analyses on the effect of exposure to violence on attitudes towards peace in Darfur and the effect of mothers' smoking on infant birthweight.

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