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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.”
From Causal identification of single-cell experimental perturbation effects with CINEMA-OT (2023) · cites this paper as [10]
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“Selection of confounders was based on recommendations in the published epidemiology literature.(26,27) We adjusted for variables that were pre-existing and were associated with aspirin use and the outcome or solely the outcome including: age, gender, BMI, race, hypertension, diabetes mellitus, coronary artery disease,…”
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Statistical Control Requires Causal Justification
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Commentary: Causal Inference for Social Exposures
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ResultStatistical Control Requires Causal JustificationPreprint
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Zhang, M., Michos, E. D., Wang, G., Wang, X., & Mueller, N. T. (2020) · JAMA Network Open · doi:10.1001/jamanetworkopen.2020.19046 · Cited by 23
Barberio, J., Ahern, T. P., MacLehose, R. F., Collin, L. J., Cronin-Fenton, D. P., Damkier, P., Sørensen, H. T., & Lash, T. L. (2021) · Clinical Epidemiology · doi:10.2147/clep.s313613 · Cited by 23
“in a minimally sufficient set of variables that will leave no open backdoor path between the exposure and the outcome.(2) Valid estimation of a causal effect of an”
BackgroundA calibration approach to transportability and data‐fusion with observational data
Josey, K., Yang, F., Ghosh, D., & Raghavan, S. (2022) · Statistics in Medicine · doi:10.1002/sim.9523 · Cited by 37
“If both Assumptions 8 and 9 are violated, then Si behaves like a surrogate confounder (VanderWeele and Shpitser, 2013) and omitting it from either the potential outcome models or the propensity score model may lead to biased effect estimates.”
BackgroundNuzzo, J. L., Finn, H. T., & Herbert, R. (2019) · Sports Medicine · doi:10.1007/s40279-019-01131-8 · Cited by 23
Jordan, S. E., Bromley, R., Damase‐Michel, C., Given, J. E., Komninou, S., Loane, M. A., Marfell, N., & Dolk, H. M. (2022) · International Breastfeeding Journal · doi:10.1186/s13006-022-00494-5 · Cited by 22
“The definition of confounderglossary most widely adopted is: “a factor associated with both the exposure and the outcome, and not part of the causal pathway from exposure to outcome” [8], blurring effects [7, 69, 70] (Table 1).”
BackgroundConfound-leakage: confound removal in machine learning leads to leakage
Hamdan, S., Love, B. C., von Polier, G. G., Weis, S., Schwender, H. R., Eickhoff, S. B., & Patil, K. R. (2022) · GigaScience · doi:10.1093/gigascience/giad071 · Cited by 23
“It is important to note that we use a practitioner-oriented operational definition of confounds as a set of variables suspected to share an unwanted effect with both the features and target, which does not imply a causal link as in more formal definitions [18].”
Xia, X., Jiang, B., Li, J., & Zhang, W. (2015) · Lifetime Data Analysis · doi:10.1007/s10985-015-9350-z · Cited by 21
“2009) that may be regarded as confounders (VanderWeele and Shpitser 2013).”
BackgroundA counterfactual approach to bias and effect modification in terms of response types
Suzuki, E., Mitsuhashi, T., Tsuda, T., & Yamamoto, E. (2013) · BMC Medical Research Methodology · doi:10.1186/1471-2288-13-101 · Cited by 21
“Recently, VanderWeele and Shpitser [29,30] further discussed the definition of a confounder.”
BackgroundInfluentialHe, J., Zhong, Y., Song, Y., Luo, J., Lin, C., Wu, Y., Pan, L., Cen, Y., Zhao, J., Gou, S., et al. (2025) · BMC Public Health · doi:10.1186/s12889-025-23383-y · Cited by 22
“Confounders (or covariates) are variables that impact both the outcome and grouping variables [32].”
Testing group differences for confounder selection in nonrandomized studies: flawed practice
Sourial, N., Vedel, I., Berre, M. L., & Schuster, T. (2019) · Canadian Medical Association Journal · doi:10.1503/cmaj.190085 · Cited by 20
CoCoA-diff: counterfactual inference for single-cell gene expression analysis
Park, Y., & Kellis, M. (2021) · Genome biology · doi:10.1186/s13059-021-02438-4 · Cited by 18
“We define that a variable can confound a disease label (W) and gene expressions (λ and Y) if (1) it is associated with the disease and gene expression variables and (2) it is still associated with the expression even after conditioning on the disease label [26].”
BackgroundEffects of statins and aspirin on HCC risk in alcohol-related cirrhosis: nationwide emulated trials
Kraglund, F., Christensen, D. H., Eiset, A. H., Villadsen, G. E., West, J., & Jepsen, P. (2023) · Hepatology Communications · doi:10.1097/hc9.0000000000000013 · Cited by 21
“Furthermore, most studies have not utilized a new user design, and prior studies have not specified the minimally sufficient adjustment set of confounders.[19] Two prior studies have investigated the effects of both statins and aspirin, and in both cases, the effect of statins was more protective.”
BackgroundSet-Based Tests for the Gene–Environment Interaction in Longitudinal Studies
He, Z., Zhang, M., Lee, S., Smith, J. A., Kardia, S. L. R., Roux, A. V. D., & Mukherjee, B. (2016) · Journal of the American Statistical Association · doi:10.1080/01621459.2016.1252266 · Cited by 20
Decoupling global biases and local interactions between cell biological variables
Zaritsky, A., Obolski, U., Gan, Z., Reis, C. R., Kadlecová, Z., Du, Y., Schmid, S. L., & Danuser, G. M. (2017) · eLife · doi:10.7554/elife.22323 · Cited by 20
“6 (VanderWeele and Shpitser, 2013) that must be eliminated for more accurate assessment of the true local interaction, but ignore the possibility that the global bias contains by-itself valuable mechanistic information to cell behavior.”
BackgroundTichelman, E., Warmink-Perdijk, W. D. B., Henrichs, J., Peters, L., Schellevis, F., Berger, M. Y., & Burger, H. (2021) · Midwifery · doi:10.1016/j.midw.2021.103045 · Cited by 21
Zawadzki, R. S., Grill, J. D., Gillen, D. L., & Initiative, A. F. T. A. D. N. (2023) · BMC Medical Research Methodology · doi:10.1186/s12874-023-01936-2 · Cited by 19
“[17][18] Alternatively, we may identify the treatment effect through an IV denoted as Z.”
Causal Etiology of the Research of James M. Robins
Richardson, T. S., & Rotnitzky, A. G. (2014) · Statistical Science · doi:10.1214/14-sts505 · Cited by 19
“As one example, he offered a precise mathematical definition for the informal epidemiologic concept of a “confounding variable” that has apparently stood the test of time (see VanderWeele and Shpitser, 2013).”
BackgroundWilson, A., Zigler, C. M., Patel, C. J., & Dominici, F. (2018) · Biometrics · doi:10.1111/biom.12860 · Cited by 18
“We consider a minimal set of covariates sufficient to satisfy (2) as the confounders (VanderWeele and Shpitser, 2013), denoted by C∗ ⊆ C.”
BackgroundMalec, S. A., Taneja, S. B., Albert, S. M., Shaaban, C. E., Karim, H. T., Levine, A. S., Munro, P., Callahan, T. J., & Boyce, R. D. (2023) · Journal of Biomedical Informatics · doi:10.1016/j.jbi.2023.104368 · Cited by 19
Conditional or unconditional logistic regression for frequency matched case‐control design?
Wan, F. (2022) · Statistics in Medicine · doi:10.1002/sim.9313 · Cited by 17
Methods Commentary: Uncovering Unobserved Data Patterns With Latent Variable Mixture Modeling
Berlin, K. S., Ankney, R. L., & Rybak, T. M. (2018) · Journal of Pediatric Psychology · doi:10.1093/jpepsy/jsy042 · Cited by 16
Yu, Y., Li, H., Sun, X., Liu, X., Yang, F., Hou, L., Liu, L., Yan, R., Yu, Y., Jing, M., et al. (2020) · American Journal of Epidemiology · doi:10.1093/aje/kwaa172 · Cited by 15
Chan, G., Sun, T., Stjepanović, D., Vu, G. T., Hall, W., Connor, J. P., & Leung, J. (2024) · Addiction · doi:10.1111/add.16442 · Cited by 14
Educational attainment by children with parental alcohol problems in Denmark and Finland
Raitasalo, K., Østergaard, J., & Andrade, S. B. (2020) · Nordic Studies on Alcohol and Drugs · doi:10.1177/1455072520968343 · Cited by 15
Leviton, A., Dammann, O., Allred, E. N., Joseph, R. M., Fichorova, R. N., O’Shea, T. M., & Kuban, K. (2018) · International Journal of Developmental Neuroscience · doi:10.1016/j.ijdevneu.2018.01.001 · Cited by 16
“Inflammation is a broad and complex phenomenon (VanderWeele and Shpitser, 2013; Zak and Aderem, 2015) and we have assessed only a very small part of it.”
BackgroundLong-term causal effects of far-right terrorism in New Zealand
Bulbulia, J. A., Afzali, M. U., Yogeeswaran, K., & Sibley, C. G. (2023) · PNAS Nexus · doi:10.1093/pnasnexus/pgad242 · Cited by 15
Using negative controls to adjust for unmeasured confounding bias in time series studies
Hu, J. K., Tchetgen, E. J. T., & Dominici, F. (2023) · Nature Reviews Methods Primers · doi:10.1038/s43586-023-00249-4 · Cited by 16
Patterns of resilient functioning in early life: Identifying distinct groups and associated factors
Cahill, S., Hager, R., & Shryane, N. M. (2023) · Development and Psychopathology · doi:10.1017/s0954579423001165 · Cited by 16
“All ACE predictors were entered into each regression model to capture the cumulative impact of all ACEs on resilience (VanderWeele & Shpitser, 2013).”
MethodAlMaghlouth, I. A., Pullenayegum, E., Gladman, D. D., Urowitz, M. B., & Johnson, S. R. (2020) · The Journal of Rheumatology · doi:10.3899/jrheum.200254 · Cited by 14
Marcello, M., Denham, J. W., Kennedy, A., Haworth, A., Steigler, A., Greer, P. B., Holloway, L. C., Dowling, J., Jameson, M., Roach, D., et al. (2020) · Frontiers in Oncology · doi:10.3389/fonc.2020.01174 · Cited by 14
“Age, prescribed dose, disease risk, cancer stage, baseline PSA concentration and number of treatment beams were patient baseline characteristics investigated as potential control variables in each model, attempting to eliminate their confounding influence at each voxel (30, 31).”
BackgroundMalec, S. A., Wei, P., Bernstam, E. V., Boyce, R. D., & Cohen, T. (2021) · Journal of Biomedical Informatics · doi:10.1016/j.jbi.2021.103719 · Cited by 14
Aarabi, G., Walther, C., Kaymaz, K., Spinler, K., Buczak‐Stec, E., König, H., & Hajek, A. (2021) · Aging Clinical and Experimental Research · doi:10.1007/s40520-021-02051-2 · Cited by 16
“In directed acyclic graphs (DAG) notation, our aim was to block the backdoor path (i.e., a non-causal path from X to Y ; usually common causes of X and Y [20]) between personality factors and regular dental visit attendance via sociodemographic and health-related factors.”
tipr: An R package for sensitivity analyses for unmeasured confounders
McGowan, L. D. (2022) · The Journal of Open Source Software · doi:10.21105/joss.04495 · Cited by 14
Marcello, M., Ebert, M. A., Haworth, A., Steigler, A., Kennedy, A., Joseph, D. J., & Denham, J. W. (2017) · Radiotherapy and Oncology · doi:10.1016/j.radonc.2017.10.021 · Cited by 14
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
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