# Estimating Treatment Effects with Observed Confounders and Mediators

arXiv: https://arxiv.org/abs/2003.11991 · HTML: https://mlsanity.com/paper/2003.11991/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Mar 2020.

## Built on by (most cited first, 8 of 8)

- [Estimating long-term causal effects from short-term experiments and long-term observational data with unobserved confounding](https://mlsanity.com/paper/2302.10625/) (2023)
- [Efficient Online Estimation of Causal Effects by Deciding What to Observe](https://mlsanity.com/paper/2108.09265/) (2021)
- [Partial Identification with Noisy Covariates: A Robust Optimization Approach](https://mlsanity.com/paper/2202.10665/) (2022)
- [Linear-Time Algorithms for Front-Door Adjustment in Causal Graphs](https://mlsanity.com/paper/2211.16468/) (2022)
- [Front-door Adjustment Beyond Markov Equivalence with Limited Graph Knowledge](https://mlsanity.com/paper/2306.11008/) (2023)
- [PCM Selector: Penalized Covariate-Mediator Selection Operator for Evaluating Linear Causal Effects](https://mlsanity.com/paper/2412.18180/) (2024)
- [Conditional Front-door Adjustment for Heterogeneous Treatment Assignment Effect Estimation Under Non-adherence](https://mlsanity.com/paper/2505.05677/) (2025)
- [What You Observe Determines How You Identify Causal Effects: Evaluating Causal Models across Observational Views](https://mlsanity.com/paper/2609.36881/) (2026)

## Abstract (arXiv)

Given a causal graph, the do-calculus can express treatment effects as functionals of the observational joint distribution that can be estimated empirically. Sometimes the do-calculus identifies multiple valid formulae, prompting us to compare the statistical properties of the corresponding estimators. For example, the backdoor formula applies when all confounders are observed and the frontdoor formula applies when an observed mediator transmits the causal effect. In this paper, we investigate the over-identified scenario where both confounders and mediators are observed, rendering both estimators valid. Addressing the linear Gaussian causal model, we demonstrate that either estimator can dominate the other by an unbounded constant factor. Next, we derive an optimal estimator, which leverages all observed variables, and bound its finite-sample variance. We show that it strictly outperforms the backdoor and frontdoor estimators and that this improvement can be unbounded. We also present a procedure for combining two datasets, one with observed confounders and another with observed mediators. Finally, we evaluate our methods on both simulated data and the IHDP and JTPA datasets.