Confounder selection via penalized credible regions

When estimating the effect of an exposure or treatment on an outcome it is important to select the proper subset of confounding variables to include in the model. Including too many covariates increases mean square error on the effect of interest while not including confounding variables biases the exposure effect estimate. We propose a decision-theoretic approach to confounder selection and effect estimation. We first estimate the full standard Bayesian regression model and then post-process the posterior distribution with a loss function that penalizes models omitting important confounders. Our method can be fit easily with existing software and in many situations without the use of Markov chain Monte Carlo methods, resulting in computation on the order of the least squares solution. We prove that the proposed estimator has attractive asymptotic properties. In a simulation study we show that our method outperforms existing methods. We demonstrate our method by estimating the effect of fine particulate matter (PM2.5) exposure on birth weight in Mecklenburg County, North Carolina.