Sensitivity of weighted least squares estimators to omitted variables

Abstract

We introduce tools for assessing the sensitivity, to unobserved confounding, of a common estimator of causal effects that employs weights: the weighted linear regression of the outcome on the treatment and observed covariates. This estimator's bias is a function of two intuitive weighted partial $R^2$ values: (i) the proportion of weighted variance in the treatment that unobserved confounding explains given the covariates and (ii) the proportion of weighted variance in the outcome that unobserved confounding explains given the covariates and the treatment. Following previous work, we define sensitivity statistics for routine reporting, derive formal bounds on the strength of unobserved confounding with (multiples of) the strength of certain covariates, and propose adjusted inference procedures. A key choice we make is to examine only how the outcome model is influenced by unobserved confounding, instead of how the weights have been affected. One benefit of this choice is that our tools apply with any (non-negative) weights (e.g., inverse propensity score, matching, or covariate balancing). Another benefit is that we can rely on omitted variable bias approaches that impose no distributional assumptions on the data or unobserved confounding, and can address misspecification bias. The tools are available in the sensewls package for R.