# On the definition of a confounder

arXiv: https://arxiv.org/abs/1304.0564 · HTML: https://mlsanity.com/paper/1304.0564/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Apr 2013.

## Built on by (most cited first, 19 of 19)

- [Local Discovery by Partitioning: Polynomial-Time Causal Discovery Around Exposure-Outcome Pairs](https://mlsanity.com/paper/2310.17816/) (2023)
- [Annotation Curricula to Implicitly Train Non-Expert Annotators](https://mlsanity.com/paper/2106.02382/) (2021)
- [Quant 4.0: Engineering Quantitative Investment with Automated, Explainable and Knowledge-driven Artificial Intelligence](https://mlsanity.com/paper/2301.04020/) (2022)
- [Local Causal Discovery for Structural Evidence of Direct Discrimination](https://mlsanity.com/paper/2405.14848/) (2024)
- [Causal Inference in medicine and in health policy, a summary](https://mlsanity.com/paper/2105.04655/) (2021)
- [Heterogeneous Treatment Effect Estimation using machine learning for Healthcare application: tutorial and benchmark](https://mlsanity.com/paper/2109.12769/) (2021)
- [Detecting and Measuring Confounding Using Causal Mechanism Shifts](https://mlsanity.com/paper/2409.17840/) (2024)
- [IV-ICL: Bounding Causal Effects with Instrumental Variables via In-Context Learning](https://mlsanity.com/paper/2605.12924/) (2026)
- [A survey on bias in machine learning research](https://mlsanity.com/paper/2308.11254/) (2023)
- [Reexamining Racial Disparities in Automatic Speech Recognition Performance: The Role of Confounding by Provenance](https://mlsanity.com/paper/2407.13982/) (2024)
- [An Empirical Examination of Balancing Strategy for Counterfactual Estimation on Time Series](https://mlsanity.com/paper/2408.08815/) (2024)
- [Comparing Two Proxy Methods for Causal Identification](https://mlsanity.com/paper/2512.00175/) (2025)
- [Investigating the Robustness of Natural Language Generation from Logical Forms via Counterfactual Samples](https://mlsanity.com/paper/2210.08548/) (2022)
- [Confound-leakage: Confound Removal in Machine Learning Leads to Leakage](https://mlsanity.com/paper/2210.09232/) (2022)
- [Causal Discovery and Inference towards Urban Elements and Associated Factors](https://mlsanity.com/paper/2503.06395/) (2025)
- [Causal Inference, Biomarker Discovery, Graph Neural Network, Feature Selection](https://mlsanity.com/paper/2511.13295/) (2025)
- [Age Predictors Through the Lens of Generalization, Bias Mitigation, and Interpretability: Reflections on Causal Implications](https://mlsanity.com/paper/2603.16377/) (2026)
- [Causal Representation Learning on High-Dimensional Data: Benchmarks, Reproducibility, and Evaluation Metrics](https://mlsanity.com/paper/2603.17405/) (2026)
- [ConfoundingSHAP: Quantifying confounding strength in causal inference](https://mlsanity.com/paper/2605.10533/) (2026)

## Abstract (arXiv)

The causal inference literature has provided a clear formal definition of confounding expressed in terms of counterfactual independence. The literature has not, however, come to any consensus on a formal definition of a confounder, as it has given priority to the concept of confounding over that of a confounder. We consider a number of candidate definitions arising from various more informal statements made in the literature. We consider the properties satisfied by each candidate definition, principally focusing on (i) whether under the candidate definition control for all "confounders" suffices to control for "confounding" and (ii) whether each confounder in some context helps eliminate or reduce confounding bias. Several of the candidate definitions do not have these two properties. Only one candidate definition of those considered satisfies both properties. We propose that a "confounder" be defined as a pre-exposure covariate C for which there exists a set of other covariates X such that effect of the exposure on the outcome is unconfounded conditional on (X,C) but such that for no proper subset of (X,C) is the effect of the exposure on the outcome unconfounded given the subset. We also provide a conditional analogue of the above definition; and we propose a variable that helps reduce bias but not eliminate bias be referred to as a "surrogate confounder." These definitions are closely related to those given by Robins and Morgenstern [Comput. Math. Appl. 14 (1987) 869-916]. The implications that hold among the various candidate definitions are discussed.