[Paper Review] Efficient estimation of modified treatment policy effects based on the generalized propensity score

Nima S. Hejazi, David Benkeser+2 more|arXiv (Cornell University)|May 11, 2022

Advanced Causal Inference Techniques4 citations

TL;DR

This paper proposes a nonparametric, efficient inverse probability weighted (IPW) estimator for modified treatment policy (MTP) effects under continuous treatments using a flexible, sieve-based generalized propensity score (GPS) estimator. By leveraging agnostic and targeted undersmoothing procedures, the method achieves asymptotic efficiency and attains the nonparametric efficiency bound, with numerical studies showing competitive performance even at moderate sample sizes (n=500), rivaling doubly robust estimators without requiring explicit semiparametric theory knowledge.

ABSTRACT

Continuous treatments have posed a significant challenge for causal inference, both in the formulation and identification of scientifically meaningful effects and in their robust estimation. Traditionally, focus has been placed on techniques applicable to binary or categorical treatments with few levels, allowing for the application of propensity score-based methodology with relative ease. Efforts to accommodate continuous treatments introduced the generalized propensity score, yet estimators of this nuisance parameter commonly utilize parametric regression strategies that sharply limit the robustness and efficiency of inverse probability weighted estimators of causal effect parameters. We formulate and investigate a novel, flexible estimator of the generalized propensity score based on a nonparametric function estimator that provably converges at a suitably fast rate to the target functional so as to facilitate statistical inference. With this estimator, we demonstrate the construction of nonparametric inverse probability weighted estimators of a class of causal effect estimands tailored to continuous treatments. To ensure the asymptotic efficiency of our proposed estimators, we outline several non-restrictive selection procedures for utilizing a sieve estimation framework to undersmooth estimators of the generalized propensity score. We provide the first characterization of such inverse probability weighted estimators achieving the nonparametric efficiency bound in a setting with continuous treatments, demonstrating this in numerical experiments. We further evaluate the higher-order efficiency of our proposed estimators by deriving and numerically examining the second-order remainder of the corresponding efficient influence function in the nonparametric model. Open source software implementing our proposed estimation techniques, the haldensify R package, is briefly discussed.

Motivation & Objective

• To address the challenge of estimating causal effects of continuous treatments in infinite-dimensional models where standard estimators lack n^{1/2}-rate efficiency.
• To overcome limitations of parametric GPS estimators that restrict robustness and efficiency in inverse probability weighting.
• To develop a flexible, nonparametric GPS estimator that converges at a fast enough rate to enable valid statistical inference for MTP effects.
• To establish conditions under which IPW estimators achieve the nonparametric efficiency bound in the presence of continuous treatments.
• To evaluate the higher-order efficiency of the proposed estimators via the second-order remainder of the efficient influence function.

Proposed method

• Uses a nonparametric sieve estimator for the generalized propensity score (GPS), ensuring convergence at a rate suitable for asymptotic inference.
• Applies targeted and agnostic undersmoothing procedures to the GPS estimator to ensure asymptotic efficiency of the resulting IPW estimators.
• Derives the efficient influence function (EIF) for MTP effects and uses its second-order remainder to assess higher-order efficiency.
• Employs a sieve estimation framework to balance bias and variance in GPS estimation, enabling nonparametric efficiency.
• Integrates outcome regression estimation into the selection of undersmoothing parameters to improve estimator quality.
• Implements the method in the open-source haldensify R package for practical application.

Experimental results

Research questions

• RQ1 Can a nonparametric, flexible GPS estimator be constructed such that it supports efficient estimation of MTP effects under continuous treatments?
• RQ2 Under what conditions can inverse probability weighted estimators of MTP effects achieve the nonparametric efficiency bound in the presence of continuous treatments?
• RQ3 How does the performance of the proposed IPW estimators compare to doubly robust estimators in finite samples, particularly in terms of bias, variance, and mean squared error?
• RQ4 To what extent does the choice of undersmoothing procedure (targeted vs. agnostic) affect the second-order remainder and overall efficiency of the IPW estimator?
• RQ5 Can the proposed methodology be extended to longitudinal MTPs by jointly undersmoothing GPS estimators across timepoints?

Key findings

• The proposed IPW estimators achieve the nonparametric efficiency bound for MTP effects in a continuous treatment setting, representing the first such characterization in this context.
• Numerical experiments demonstrate that the IPW estimators based on agnostic undersmoothing perform competitively with doubly robust estimators, even at moderate sample sizes (n=500).
• The second-order remainder of the efficient influence function is minimized by the proposed selectors, indicating strong higher-order efficiency properties.
• The agnostic undersmoothing procedure achieves performance comparable to conventional doubly robust estimators without requiring explicit knowledge of semiparametric efficiency theory.
• The haldensify R package successfully implements the proposed methodology, enabling practical application of the estimators.
• The method remains robust even when the outcome regression is poorly estimated, suggesting potential for sensitivity analysis and broader applicability.