Propensity Score Propagation: A General Framework for Design-Based Inference with Unknown Propensity Scores
Siyu Heng, Yanxin Shen, Zijian Guo
TL;DR
This paper addresses design-based inference when propensity scores are unknown, a scenario common in observational studies, real-world surveys, and missing-data analyses. It introduces propensity score propagation, a regeneration-and-union framework that explicitly propagates PS-estimation uncertainty into downstream design-based inference, accommodating both parametric and nonparametric PS models. The authors establish theoretical guarantees ensuring nominal coverage as the number of regeneration runs $M$ grows, and demonstrate favorable finite-sample performance in simulations, with CI lengths only modestly larger than the oracle. The framework is shown to be broadly applicable across causal inference, survey sampling, missing data, Fisher randomization tests, and DID analyses, and it provides natural avenues for sensitivity analysis to hidden bias. Overall, propensity score propagation makes design-based inference feasible and reliable in settings with unknown propensity scores, while preserving the core design-based principles and offering plug-and-play compatibility with existing procedures.
Abstract
Design-based inference, also known as randomization-based or finite-population inference, provides a principled framework for causal and descriptive analyses that attribute randomness solely to the design mechanism (e.g., treatment assignment, sampling, or missingness) without imposing distributional or modeling assumptions on the outcome data of study units. Despite its conceptual appeal and long history, this framework becomes challenging to apply when the underlying design probabilities (i.e., propensity scores) are unknown, as is common in observational studies, real-world surveys, and missing-data settings. Existing plug-in or matching-based approaches either ignore the uncertainty stemming from estimated propensity scores or rely on the post-matching uniform-propensity condition (an assumption typically violated when there are multiple or continuous covariates), leading to systematic under-coverage. Finite-population M-estimation partially mitigates these issues but remains limited to parametric propensity score models. In this work, we introduce propensity score propagation, a general framework for valid design-based inference with unknown propensity scores. The framework introduces a regeneration-and-union procedure that automatically propagates uncertainty in propensity score estimation into downstream design-based inference. It accommodates both parametric and nonparametric propensity score models, integrates seamlessly with standard tools in design-based inference with known propensity scores, and is universally applicable to various important design-based inference problems, such as observational studies, real-world surveys, and missing-data analyses, among many others. Simulation studies demonstrate that the proposed framework restores nominal coverage levels in settings where conventional methods suffer from severe under-coverage.