Uncertainty in the design stage of two-stage Bayesian propensity score analysis

The two-stage process of propensity score analysis (PSA) includes a design stage where propensity scores (PSs) are estimated and implemented to approximate a randomized experiment and an analysis stage where treatment effects are estimated conditional on the design. This article considers how uncertainty associated with the design stage impacts estimation of causal effects in the analysis stage. Such design uncertainty can derive from the fact that the PS itself is an estimated quantity, but also from other features of the design stage tied to choice of PS implementation. This article offers a procedure for obtaining the posterior distribution of causal effects after marginalizing over a distribution of design-stage outputs, lending a degree of formality to Bayesian methods for PSA that have gained attention in recent literature. Formulation of a probability distribution for the design-stage output depends on how the PS is implemented in the design stage, and propagation of uncertainty into causal estimates depends on how the treatment effect is estimated in the analysis stage. We explore these differences within a sample of commonly used PS implementations (quantile stratification, nearest-neighbor matching, caliper matching, inverse probability of treatment weighting, and doubly robust estimation) and investigate in a simulation study the impact of statistician choice in PS model and implementation on the degree of between- and within-design variability in the estimated treatment effect. The methods are then deployed in an investigation of the association between levels of fine particulate air pollution and elevated exposure to emissions from coal-fired power plants.