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The two-stage process of propensity score analysis (PSA) includes a designstage where propensity scores are estimated and implemented to approximate arandomized experiment and an analysis stage where treatment effects areestimated conditional upon the design. This paper considers how uncertaintyassociated with the design stage impacts estimation of causal effects in theanalysis stage. Such design uncertainty can derive from the fact that thepropensity score itself is an estimated quantity, but also from other featuresof the design stage tied to choice of propensity score implementation. Thispaper offers a procedure for obtaining the posterior distribution of causaleffects after marginalizing over a distribution of design-stage outputs,lending a degree of formality to Bayesian methods for PSA (BPSA) that havegained attention in recent literature. Formulation of a probabilitydistribution for the design-stage output depends on how the propensity score isimplemented in the design stage, and propagation of uncertainty into causalestimates depends on how the treatment effect is estimated in the analysisstage. We explore these differences within a sample of commonly-used propensityscore implementations (quantile stratification, nearest-neighbor matching,caliper matching, inverse probability of treatment weighting, and doubly robustestimation) and investigate in a simulation study the impact of statisticianchoice in PS model and implementation on the degree of between- andwithin-design variability in the estimated treatment effect. The methods arethen deployed in an investigation of the association between levels of fineparticulate air pollution and elevated exposure to emissions from coal-firedpower plants.

Uncertainty in the Design Stage of Two-Stage Bayesian Propensity Score Analysis