Principal Stratification with Continuous Post-Treatment Variables: Nonparametric Identification and Semiparametric Estimation

TL;DR

This paper develops a nonparametric identification and semiparametric estimation framework for principal stratification involving continuous post-treatment variables, addressing a key gap in causal inference methodology.

Contribution

It introduces a novel theoretical approach for continuous post-treatment variables, including doubly robust estimators and an R package for implementation.

Findings

Established nonparametric identification for continuous post-treatment variables

Derived efficient influence functions for causal effect estimation

Provided practical estimators with doubly robust properties

Abstract

Post-treatment variables often complicate causal inference. They appear in many scientific problems, including noncompliance, truncation by death, mediation, and surrogate endpoint evaluation. Principal stratification is a strategy to address these challenges by adjusting for the potential values of the post-treatment variables, defined as the principal strata. It allows for characterizing treatment effect heterogeneity across principal strata and unveiling the mechanism of the treatment's impact on the outcome related to post-treatment variables. However, the existing literature has primarily focused on binary post-treatment variables, leaving the case with continuous post-treatment variables largely unexplored. This gap persists due to the complexity of infinitely many principal strata, which present challenges to both the identification and estimation of causal effects. We fill this gap by providing nonparametric identification and semiparametric estimation theory for principal stratification with continuous post-treatment variables. We propose to use working models to approximate the underlying causal effect surfaces and derive the efficient influence functions of the corresponding model parameters. Based on the theory, we construct doubly robust estimators and implement them in an R package.