[Paper Review] Propensity score regression for causal inference with treatment heterogeneity

Peng Wu, Shasha Han+2 more|arXiv (Cornell University)|Sep 16, 2021

Advanced Causal Inference Techniques4 citations

TL;DR

This paper proposes a nonparametric propensity score regression (PSR) method for estimating heterogeneous treatment effects that is robust to extreme propensity scores and high-dimensional confounders. By combining two nonparametric regressions—first on propensity scores and covariates of interest, then regressing the result on the covariates alone—the PSR achieves consistent, asymptotically normal estimates with an explicit variance estimator, outperforming existing methods in simulations and real-world analysis of flu vaccination and sick leave effects.

ABSTRACT

Understanding how treatment effects vary on individual characteristics is critical in the contexts of personalized medicine, personalized advertising and policy design. When the characteristics are of practical interest are only a subset of full covariate, non-parametric estimation is often desirable; but few methods are available due to the computational difficult. Existing non-parametric methods such as the inverse probability weighting methods have limitations that hinder their use in many practical settings where the values of propensity scores are close to 0 or 1. We propose the propensity score regression (PSR) that allows the non-parametric estimation of the heterogeneous treatment effects in a wide context. PSR includes two non-parametric regressions in turn, where it first regresses on the propensity scores together with the characteristics of interest, to obtain an intermediate estimate; and then, regress the intermediate estimates on the characteristics of interest only. By including propensity scores as regressors in the non-parametric manner, PSR is capable of substantially easing the computational difficulty while remain (locally) insensitive to any value of propensity scores. We present several appealing properties of PSR, including the consistency and asymptotical normality, and in particular the existence of an explicit variance estimator, from which the analytical behaviour of PSR and its precision can be assessed. Simulation studies indicate that PSR outperform existing methods in varying settings with extreme values of propensity scores. We apply our method to the national 2009 flu survey (NHFS) data to investigate the effects of seasonal influenza vaccination and having paid sick leave across different age groups.

Motivation & Objective

• To address the challenge of estimating conditional average treatment effects that vary across key characteristics like age in personalized medicine and policy design.
• To overcome limitations of inverse probability weighting and AIPW methods when propensity scores are near zero or one, which cause unstable estimates.
• To develop a nonparametric method that flexibly models treatment effect heterogeneity without requiring parametric assumptions on the outcome or propensity score.
• To reduce computational burden and improve robustness in high-dimensional settings by leveraging the balancing property of propensity scores.
• To provide a method with theoretical guarantees, including consistency, asymptotic normality, and an explicit variance estimator for inference.

Proposed method

• The PSR method uses two-stage nonparametric regression: first, it regresses the outcome on the propensity score and the covariates of interest to obtain an intermediate estimate.
• Second, it regresses this intermediate estimate on the covariates of interest alone, effectively integrating out the high-dimensional confounders.
• The method uses kernel-based nonparametric regression with bandwidth selection to estimate conditional expectations, ensuring flexibility in modeling heterogeneous effects.
• It leverages the balancing property of propensity scores, which ensures that covariate distributions are equal across treatment groups at each score level, thus controlling for confounding from high-dimensional $X^{-l}$.
• The approach is robust to extreme propensity scores because it uses a continuous, bounded function of the score rather than inverse weighting.
• An explicit variance estimator is derived, enabling analytical precision assessment and valid inference for the estimated treatment effects.

Experimental results

Research questions

• RQ1 How can we nonparametrically estimate treatment effects that vary across key characteristics like age, while remaining robust to extreme propensity scores?
• RQ2 What are the finite-sample and asymptotic properties of a two-stage nonparametric regression method that uses propensity scores as regressors?
• RQ3 How does the proposed PSR method compare to inverse probability weighting and AIPW in terms of bias, variance, and robustness when propensity scores are near zero or one?
• RQ4 Can the PSR method effectively handle high-dimensional confounders without suffering from the curse of dimensionality?
• RQ5 What is the impact of using a continuous function of the propensity score instead of inverse weighting on estimation stability?

Key findings

• The PSR method achieves consistency and asymptotic normality under regularity conditions, with an explicit variance estimator enabling valid inference.
• In simulation studies, PSR outperforms existing methods—especially inverse probability weighting and AIPW—when propensity scores are near zero or one, showing lower bias and mean squared error.
• The method maintains good performance even with high-dimensional confounders, avoiding the curse of dimensionality by leveraging the balancing property of propensity scores.
• In the NHFS application, PSR reveals that seasonal flu vaccination increases doctor visits for individuals over 65, while having paid sick leave reduces doctor visits among those under 33 or aged 41–64, with no significant effect for ages 33–40.
• The PSR method is robust to extreme propensity scores, as demonstrated by its stable performance across simulations with scores ranging from 0.001 to 0.984.
• Theoretical extensions show the method remains valid for discrete $X^l$ and under semiparametric models such as single-index models, broadening its applicability.