Bayesian inference for treatment effects under nested subsets of controls
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When constructing a model to estimate the causal effect of a treatment, it is necessary to control for other factors which may have confounding effects. Because the ignorability assumption is not testable, however, it is usually unclear which set of controls is appropriate, and effect estimation is generally sensitive to this choice. A common approach in this case is to fit several models, each with a different set of controls, but it is difficult to reconcile inference under the multiple resulting posterior distributions for the treatment effect. Therefore we propose a two-stage approach to measure the sensitivity of effect estimation with respect to control specification. In the first stage, a model is fit with all available controls using a prior carefully selected to adjust for confounding. In the second stage, posterior distributions are calculated for the treatment effect under nested sets of controls by propagating posterior uncertainty in the original model. We demonstrate how our approach can be used to detect the most significant confounders in a dataset, and apply it in a sensitivity analysis of an observational study measuring the effect of legalized abortion on crime rates.
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