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.

中文翻译：

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两阶段贝叶斯倾向得分分析设计阶段的不确定性。

倾向得分分析 (PSA) 的两阶段过程包括一个设计阶段，其中倾向得分 (PS) 被估计和实施以近似随机实验，以及一个分析阶段，其中治疗效果根据设计进行估计。本文考虑了与设计阶段相关的不确定性如何影响分析阶段因果效应的估计。这种设计不确定性可能源于 PS 本身是一个估计量这一事实，也可能源于与 PS 实施选择相关的设计阶段的其他特征。本文提供了一种在设计阶段输出的分布边缘化后获得因果效应后验分布的程序，为最近文献中引起关注的 PSA 的贝叶斯方法提供了一定程度的形式化。设计阶段输出的概率分布的制定取决于 PS 在设计阶段如何实施，不确定性传播到因果估计中取决于在分析阶段如何估计处理效果。我们在常用 PS 实现的样本（分位数分层、最近邻匹配、卡尺匹配、治疗加权的逆概率和双重稳健估计）中探索这些差异，并在模拟研究中调查统计学家选择在 PS 模型和对估计治疗效果的设计间和设计内可变性程度的实施。然后将这些方法用于调查细颗粒物空气污染水平与燃煤电厂排放量增加之间的关系。