A Bayesian Approach to the G-Formula via Iterative Conditional Regression

In longitudinal observational studies with time-varying confounders, the generalized computation algorithm formula (g-formula) is a principled tool to estimate the average causal effect of a treatment regimen. However, the standard non-iterative g-formula implementation requires specifying both the conditional distribution of the outcomes and the joint distribution of all time-varying covariates. This process can be cumbersome to implement and is prone to model misspecification bias. As an alternative, the iterative conditional expectation (ICE) g-formula estimator solely depends on a series of nested outcome regressions and avoids the need for specifying the full distribution of all time-varying covariates. This simplicity lends itself to the natural integration of flexible machine learning techniques to develop more robust average causal effect estimators with time-varying treatments. In this work, we introduce a Bayesian approach that includes parametric regressions and Bayesian Additive Regression Trees to flexibly model a series of outcome surfaces. We fit the ICE g-formula and develop a sampling algorithm to obtain samples from the posterior distribution of the final causal effect estimator. We illustrate the performance characteristics of the Bayesian ICE estimator and the associated variations via simulation studies and applications to two real world data examples.