The effect of the prior distribution in the Bayesian Adjustment for Confounding algorithm

The effect of the prior distribution of the outcome and exposure models’ covariate inclusion indicators in the Bayesian Adjustment for Confounding (BAC) algorithm is studied. The investigative approach is to analytically describe the posterior probabilities of the outcome models in terms of the integrated likelihoods for the outcome and exposure models. It is shown that BAC gives a posteriori   progressively more weight to outcome models nesting all exposure models receiving non negligible support from the data as ωω increases. Then, relying on the causal graphical framework and additional assumptions on the set of potential confounding covariates, further theoretical justifications for BAC are given. In particular, it is explained why outcome models nesting all likely exposure models are unbiased for the causal effect of exposure, and why favoring these outcome models in BAC increases the variance of the exposure effect estimator. Using the R package BACprior, the performance of two cross-validation procedures for selecting an ωω value that minimizes the mean square error of the BAC exposure effect estimator is examined. A bootstrap procedure is subsequently studied. It is found that the performance of the resampling procedures examined is sensitive to the underlying data.