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On the definition of a confounder
Open accessAbstract
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.
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“A great deal of variability in cellular responses to treatment may be attributable to underlying confounding variation10.”
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BackgroundMethodSeasonality of depressive symptoms during pregnancy
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“This aspect has been well-studied in other applications of WOLS, and is known as omitted variable bias , confounding effects , and spurious correlation [VanderWeele and Shpitser 2013].”
Karlsson, P. E., Pleijel, H., Fowler, P., Farahat, E. A., Linderholm, H. W., Engardt, M., & Andersson, C. (2023) · Forest Ecology and Management · doi:10.1016/j.foreco.2023.121455 · Cited by 3
Porter, A. K., Kleinschmidt, S. E., Andres, K. L., Reusch, C. N., Krisko, R. M., Taiwo, O. A., Olsen, G. W., & Longnecker, M. P. (2024) · Global Epidemiology · doi:10.1016/j.gloepi.2024.100137 · Cited by 3
“Such interaction would not be a pre-exposure covariate, and thus not a confounder [65].”
BackgroundMartin, G. L., Petri, C. F., Rozenberg, J., Simon‐Tillaux, N., Hajage, D., Kirchgesner, J., Tubach, F., Létinier, L., & Dechartres, A. (2024) · Journal of Clinical Epidemiology · doi:10.1016/j.jclinepi.2024.111305 · Cited by 3
Nadendla, H., Etha, P. P., & Chowriappa, P. (2024) · IEEE Access · doi:10.1109/access.2024.3372369 · Cited by 3
“Confounding bias in CD [14] is when one feature which can act as a confounder influence both independent and other dependent features causing a spurious association.”
BackgroundProtocols for Observational Studies: Methods and Open Problems
Small, D. S. (2024) · Statistical Science · doi:10.1214/24-sts945 · Cited by 4
Event extraction for visual: De-biasing with causality-guided attention mechanism
Wang, C., Yang, J., & He, Q. (2025) · Neurocomputing · doi:10.1016/j.neucom.2025.130783 · Cited by 3
Double moderated mediation models: problems and (part) remedies
Chryssochoidis, G. (2017) · Journal of Modelling in Management · doi:10.1108/jm2-06-2016-0053 · Cited by 3
““Confounding” has been defined primarily as non-modelling model-relevant variables (“confounders”; VanderWeele and Shpitser, 2013) resulting in inaccurate estimates (Antonakis et al., 2010).”
BackgroundA More Efficient Causal Mediator Model Without the No-Unmeasured-Confounder Assumption
Brandt, H. (2019) · Multivariate Behavioral Research · doi:10.1080/00273171.2019.1656051 · Cited by 5
Improving Fair Predictions Using Variational Inference In Causal ModelsPreprint
Helwegen, R., Louizos, C., & Forré, P. D. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2008.10880 · Cited by 3
Copula-based Sensitivity Analysis for Multi-Treatment Causal Inference with Unobserved ConfoundingPreprint
Zheng, J., D’Amour, A., & Franks, A. M. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2102.09412 · Cited by 3
“These two densities are the same if and only if there are no confounders (VanderWeele and Shpitser, 2013).”
BackgroundMorcos, M., Celis‐Aguilar, E., Zhou, A. S., Ahanotu, A., Teng, S. E., Randolph, G. W., Schmalbach, C. E., & Shin, J. J. (2024) · Otolaryngology · doi:10.1002/ohn.1078 · Cited by 2
Understanding and Managing Confounders, Mediators and Colliders in Research
Younas, A., & Inayat, S. (2025) · Journal of Evaluation in Clinical Practice · doi:10.1111/jep.70298 · Cited by 2
Stokes, K., Piaggio, D., Micco, F. D., Zarro, M., Benedictis, A. D., Tambone, V., Moon, M., Maccaro, A., & Pecchia, L. (2024) · Infectious Disease Reports · doi:10.3390/idr16030039 · Cited by 2
“It is good practice to address confounders of studies, as they can result in a biased estimate of the exposure effect by causing the emergence of spurious associations [33,34].”
BackgroundRichey, M. M., Maciejewski, M. L., Zepel, L., Arterburn, D. E., Kawatkar, A. A., Sloan, C. E., & Smith, V. A. (2024) · BMC Medical Research Methodology · doi:10.1186/s12874-024-02391-3 · Cited by 2
Coatanéa, É., Nagarajan, H. P. N., Panicker, S., & Mokhtarian, H. (2022) · Journal of Integrated Design and Process Science · doi:10.3233/jid-221009 · Cited by 2
Rosenbaum, C. L., Yu, Q., Buzhardt, S., Sutton, E. F., & Chapple, A. G. (2023) · Pharmaceutical Statistics · doi:10.1002/pst.2323 · Cited by 2
Liu, C., Gan, L., Kuang, K., & Wu, F. Q. (2022) · Conference on Empirical Methods in Natural Language Processing (EMNLP) · doi:10.18653/v1/2022.emnlp-main.370 · Cited by 3
Decoupling global biases and local interactions between cell biological variablesPreprint
Zaritsky, A., Obolski, U., Gan, Z., Reis, C. R., Kadlecová, Z., Du, Y., Schmid, S. L., & Danuser, G. M. (2016) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/038059 · Cited by 2
Double Robust Representation Learning for Counterfactual PredictionPreprint
Zeng, S., Assaad, S., Tao, C., Datta, S., Carin, L., & Li, F. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2010.07866 · Cited by 2
“The main hurdle to causal inference is confounding, i.e., factors that affect both the outcome and the treatment assignment (VanderWeele & Shpitser, 2013).”
BackgroundThiese, M. S., Hegmann, K. T., Page, G. B., & Weames, G. G. (2021) · Journal of Occupational and Environmental Medicine · doi:10.1097/jom.0000000000002315 · Cited by 2
Helicobacter Pylori e Pressão Alta
Durães, A. R. (2021) · Arquivos Brasileiros de Cardiologia · doi:10.36660/abc.20210629 · Cited by 2
A Multiverse Analysis on the relationship between interdisciplinary research and scientific impactPreprint
Cantone, G. G., & Nighintigale, P. (2023) · doi:10.31222/osf.io/jy8mz · Cited by 2
Huang, L., Ma, J., Qiu, X., & Suo, T. (2021) · Scientific Programming · doi:10.1155/2021/9618161 · Cited by 1
Causal Reasoning and Inference in Epidemiology
Didelez, V. (2024) · Springer eBooks · doi:10.1007/978-1-4614-6625-3_74-1 · Cited by 1
Confounder-Aware Medical Data Selection for Fine-Tuning Pretrained Vision Models
Ji, A., Kang, Q., Xu, W., Wang, C., Li, K., & Lao, Q. (2025) · IEEE International Symposium on Biomedical Imaging (ISBI) · doi:10.1109/isbi60581.2025.10980785 · Cited by 2
Xue, C., Gao, J., & Gu, Z. (2025) · IEEE Transactions on Artificial Intelligence · doi:10.1109/tai.2025.3574299 · Cited by 2
Hoffmann, S., Michalski, N., Dragano, N., & Spallek, J. (2025) · European Journal of Public Health · doi:10.1093/eurpub/ckaf082 · Cited by 1
“However, in line with the research question and to reduce bias, we first consider individual-level family SEP together with area-level deprivation in the models and then conduct an analysis by stratifying according to individual-level family SEP [38].”
Hansson, S. O. (2025) · Studies in History and Philosophy of Science Part A · doi:10.1016/j.shpsa.2025.07.002 · Cited by 1
From Archives to Artifacts: A Forensic Analysis of Noisy Data
Woods, D. (2025) · Chinese Political Science Review · doi:10.1007/s41111-025-00309-y · Cited by 1
Metadata
- Authors
- VanderWeele, T. J., & Shpitser, I.
- Affiliations
- Harvard University (US)
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 2013-02-01
- Volume
- 41
- Issue
- 1
- Pages
- 196-220
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- PMC
- PMC4276366
- OpenAlex
- W2067177016
- Cited by
- 268
- References
- 46
- Citations by year
- 2026: 18 · 2025: 23 · 2024: 28 · 2023: 25 · 2022: 23 · 2021: 46 · 2020: 33 · 2019: 20 · 2018: 17 · 2017: 15 · 2016: 11 · 2015: 5 · 2014: 3 · 2013: 1
- FWCI
- 5.87
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Statistics (publishedVersion)
- PubMed (publishedVersion)
- Europe PMC (PubMed Central) (acceptedVersion)
- PubMed Central (acceptedVersion)
- arXiv (Cornell University) (submittedVersion)
- Collection of Biostatistics Research Archive (submittedVersion)
- Digital Access to Scholarship at Harvard (DASH) (Harvard University) (submittedVersion, cc-by)
- Project Euclid (Cornell University) (submittedVersion)
- ePrints Soton (University of Southampton) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Institutes of Health (ES017876)
- National Institute of Environmental Health Sciences (R01ES017876, R56ES017876)
- Eunice Kennedy Shriver National Institute of Child Health and Human Development (R03HD060696)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Epidemiology
- Keywords
- confounding, causal inference, unconfoundedness, covariate control
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
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