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

arXiv: https://arxiv.org/abs/2309.12425 · HTML: https://mlsanity.com/paper/2309.12425/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Sep 2023.

## Built on by (most cited first, 3 of 3)

- [Treatment response as a latent variable](https://mlsanity.com/paper/2502.08776/) (2025)
- [What Does a Skill Actually Do? Estimands and Evaluation Validity for Tool and Skill Use in LLM Agents: A Critical Review](https://mlsanity.com/paper/2609.33153/) (2026)
- [Counterfactually Safe Reinforcement Learning](https://mlsanity.com/paper/2605.25114/) (2026)

## Abstract (arXiv)

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.