Calibrated sensitivity models

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.