# Regression-Based Estimation of Causal Effects in the Presence of Selection Bias and Confounding

arXiv: https://arxiv.org/abs/2503.20546 · HTML: https://mlsanity.com/paper/2503.20546/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 51 references that are in the map.
- First posted on arXiv 26 Mar 2025.

## Builds on (most cited first, 12 of 12)

- [Causal Effect Inference with Deep Latent-Variable Models](https://mlsanity.com/paper/1705.08821/) (2017)
- [Causality matters in medical imaging](https://mlsanity.com/paper/1912.08142/) (2019)
- [Identifying Causal Effects With Proxy Variables of an Unmeasured Confounder](https://mlsanity.com/paper/1609.08816/) (2016)
- [Graphical Models for Processing Missing Data](https://mlsanity.com/paper/1801.03583/) (2018)
- [Causal Calculus in the Presence of Cycles, Latent Confounders and Selection Bias](https://mlsanity.com/paper/1901.00433/) (2019)
- [The Importance of Modeling Data Missingness in Algorithmic Fairness: A Causal Perspective](https://mlsanity.com/paper/2012.11448/) (2020)
- [Kernel methods for long term dose response curves](https://mlsanity.com/paper/2201.05139/) (2022)
- [Doubly Robust Inference for Targeted Minimum Loss Based Estimation in Randomized Trials with Missing Outcome Data](https://mlsanity.com/paper/1704.01538/) (2017)
- [Correcting for Selection Bias and Missing Response in Regression using Privileged Information](https://mlsanity.com/paper/2303.16800/) (2023)
- [Regression-Based Proximal Causal Inference](https://mlsanity.com/paper/2402.00335/) (2024)
- [BETS: The dangers of selection bias in early analyses of the coronavirus disease (COVID-19) pandemic](https://mlsanity.com/paper/2004.07743/) (2020)
- [Recovery and inference of causal effects with sequential adjustment for confounding and attrition](https://mlsanity.com/paper/2401.16990/) (2024)

## Abstract (arXiv)

We consider the problem of estimating the expected causal effect E[Y|do(X)] for a target variable Y when treatment X is set by intervention, focusing on continuous random variables. In settings without selection bias or confounding, E[Y|do(X)] = E[Y|X], which can be estimated using standard regression methods. However, regression fails when systematic missingness induced by selection bias, or confounding distorts the data. Proxy variables unaffected by the selection process can, under certain constraints, be used to correct for selection bias to recover E[Y|X], and hence E[Y|do(X)], reliably. When data is additionally affected by confounding, recovering the causal effect from selection-biased data is more challenging and requires access to proxies to both correct for confounding and for the selection mechanism. Assuming access to such proxies from external unbiased observational data, we derive theoretical conditions ensuring identifiability and recoverability of causal effects. We further introduce a linear two-step regression estimator (TSR), which can be extended through adding non-linear basis functions, capable of exploiting proxy variables to adjust for selection bias while accounting for confounding. We show that TSR is consistent with previous estimators when confounding is absent, but achieves a lower variance. Extensive simulation studies validate TSR's correctness for scenarios that include both selection bias and confounding with proxy variables.