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Confounding and Collapsibility in Causal Inference
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Consideration of confounding is fundamental to the design and analysis of studies of causal effects. Yet, apart from confounding in experimental designs, the topic is given little or no discussion in most statistics texts. We here provide an overview of confounding and related concepts based on a counterfactual model for causation. Special attention is given to definitions of confounding, problems in control of confounding, the relation of confounding to exchangeability and collapsibility, and the importance of distinguishing confounding from noncollapsibility.
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“Both models account for potential biases and confounding factors [20, 21].”
From Health and Socio-Economic Impacts of Climate-Related Displacement in Bangladesh’s Chars: Causal Evidence From a Household Survey (2026) · cites this paper as [20]
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“The causal interpretation of the estimated treatment effects relies on the standard identification assumptions used in the program-evaluation literature [16, 20, 21].”
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Oxblom, A., Hedlund, H., Nemes, S., Brismar, B., Felländer‐Tsai, L., & Rolfson, O. (2019) · Acta Orthopaedica · doi:10.1080/17453674.2019.1604343 · Cited by 22
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“1 is correctly specified, then parameter β1 is equal to the average treatment effect E [Y(1)-Y(0)] [21].”
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Dashti, S. G., Lee, K. J., Simpson, J. A., White, I. R., Carlin, J. B., & Moreno‐Betancur, M. (2024) · American Journal of Epidemiology · doi:10.1093/aje/kwae012 · Cited by 26
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“Quantitatively, the marginal and conditional ORs from logistic regressions do not always converge owing to the non-collapsibility property of ORs [34,35].”
BackgroundPropensity Score Methods and Unobserved Covariate Imbalance: Comments on “Squeezing the Balloon”
Ali, M. S., Groenwold, R. H. H., & Klungel, O. H. (2014) · Health Services Research · doi:10.1111/1475-6773.12152 · Cited by 22
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Life course models: improving interpretation by consideration of total effects
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Poor Oral Health Predicts Higher 10-Year Cardiovascular Risk
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“In an I × J × K contingency table representing the joint distribution of three discrete variables (call these treatment X , outcome Y and confounder Z), a measure of association between X and Y is considered collapsible across strata (subtables) of Z if it is both constant across strata, and if this constant value is…”
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“Such a variable, known as a confounding factor, is both associated with the intervention allocation and the outcome and may lead to a mis-estimation of the intervention effect [43].”
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“To compare whether spending reductions come disproportionately from low-value spending, we compare the magnitudes of these low-value and total spending coefficients in each category with a Z-statistic developed to test for differences in regression coefficients computed from different models using the same sample that…”
MethodThe Importance of Spatial Effects for Environmental Health Policy and Research
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“Finally, in this study, we did not control confounding variables, which might affect the true relationship between the independent and dependent variables [53].”
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Loh, W. W., Moerkerke, B., Loeys, T., & Vansteelandt, S. (2020) · Epidemiologic Methods · doi:10.1515/em-2020-0023 · Cited by 17
“But specifying an outcomemodel that is coherentwith the posited effectmodel can be difficult or impossible for nonlinear effect models due to noncollapsibility (Greenland, Robins, and Pearl 1999).”
BackgroundConditional or unconditional logistic regression for frequency matched case‐control design?
Wan, F. (2022) · Statistics in Medicine · doi:10.1002/sim.9313 · Cited by 17
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“Consequently, RD and RR, but not OR, are said to be collapsible [37].”
BackgroundAdjusting for confounding by neighborhood using complex survey data
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“Unlike the risk difference and relative risk, the odds ratio is not collapsible even assuming no confounding [14], and our outcome is relatively prevalent (the estimated percent of adults with health insurance coverage is 82.”
BackgroundPhosphoglucose isomerase gene expression as a prognostic biomarker of gastric cancer
Huang, H.-C., Wen, X., Xue, H., Chen, R., Ji, J., & Xu, L. (2019) · Chinese Journal of Cancer Research · doi:10.21147/j.issn.1000-9604.2019.05.07 · Cited by 17
“Reprogrammed metabolism is no longer considered a mere consequence of oncogenic transformation, but a critical hallmark of cancer (6,7).”
BackgroundMethodEstimating population effects of vaccination using large, routinely collected data
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“There are several confounding factors that must be taken into consideration, such as variable exchangeability and causality [47].”
BackgroundCovariate adjustment in randomized clinical trials with missing covariate and outcome data
Chang, C., Song, Y., Li, F., & Wang, R. (2023) · Statistics in Medicine · doi:10.1002/sim.9840 · Cited by 20
“Non-collapsibility of odds ratio is a well-known issue for binary outcomes (Greenland et al., 1999).”
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Al‐Hassany, L., MaassenVanDenBrink, A., & Kurth, T. (2024) · JAMA Network Open · doi:10.1001/jamanetworkopen.2024.40577 · Cited by 16
Quantitative Falsification of Instrumental Variables Assumption Using Balance Measures
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“In other words, the population of alters connected to obese egos may not be exchangeable with the population of alters connected to nonobese egos.(41) thus, while this agentbased model provides information on the properties of the simulation under different computational interventions, one”
BackgroundMeta‐analysis of rare events under the assumption of a homogeneous treatment effect
Piaget‐Rossel, R., & Taffé, P. (2019) · Biometrical Journal · doi:10.1002/bimj.201800381 · Cited by 17
“This is because, under this particular framework, the three ES are collapsible (Greenland et al., 1999; Guo and Geng, 2005; Hernan et al., 2011).”
BackgroundUanhoro, J., Wang, Y., & O'Connell, A. A. (2019) · The Journal of Experimental Education · doi:10.1080/00220973.2019.1693328 · Cited by 16
Multiple Treatment Effect Estimation using Deep Generative Model with Task Embedding
Saini, S. K., Dhamnani, S., Ibrahim, A. A., & Chavan, P. (2019) · International World Wide Web Conference (WWW) · doi:10.1145/3308558.3313744 · Cited by 16
“This can lead to misleading estimates of treatment effect [12].”
BackgroundStatistical Approaches for Modeling Radiologists' Interpretive Performance
Miglioretti, D. L., Haneuse, S., & Anderson, M. L. (2009) · Academic Radiology · doi:10.1016/j.acra.2008.07.022 · Cited by 16
“The distinction in the interpretations of marginal and conditional parameters is closely related to the notion of non-collapsibility of non-linear models (46).”
BackgroundOn some “disadvantages” of the population approach
Nedelman, J. R. (2005) · The AAPS Journal · doi:10.1208/aapsj070238 · Cited by 16
Keller, B., & Branson, Z. (2024) · Asia Pacific Education Review · doi:10.1007/s12564-024-09957-2 · Cited by 20
A typology of four notions of confounding in epidemiology
Suzuki, E., Mitsuhashi, T., Tsuda, T., & Yamamoto, E. (2016) · Journal of Epidemiology · doi:10.1016/j.je.2016.09.003 · Cited by 16
“ðq1 þ q2Þ ðq1 þ q3Þ 1⁄4 ðp1 þ p2Þ ðq1 þ q3Þ ⇔ðp1 þ p2Þ 1⁄4 ðq1 þ q2Þ; (2)”
Ritz, C. (2020) · European Journal of Clinical Nutrition · doi:10.1038/s41430-020-00750-z · Cited by 15
“results are not directly comparable [50], e.”
Stuurman, A. L., Levi, M., Beutels, P., Bricout, H., Descamps, A., Santos, G. D., McGovern, I., Mira‐Iglesias, A., Nauta, J. J. P., Torcel‐Pagnon, L., et al. (2022) · Influenza and Other Respiratory Viruses · doi:10.1111/irv.13087 · Cited by 15
Webb-Vargas, Y., Chen, S., Fisher, A., Mejia, A. F., Xu, Y., Crainiceanu, C. M., Caffo, B. S., & Lindquist, M. A. (2017) · Statistics in Biosciences · doi:10.1007/s12561-017-9195-y · Cited by 15
“For example, one assumption needed to make causal conclusions is that of ‘no unmeasured confounding of the relation between the stimulus and the outcome’ [34,35].”
von Cube, M., Schumacher, M., & Wolkewitz, M. (2019) · Journal of the Royal Statistical Society Series A (Statistics in Society) · doi:10.1111/rssa.12486 · Cited by 15
Analytic epidemiology and periodontal diseases
Heaton, B., & Dietrich, T. (2011) · Periodontology 2000 · doi:10.1111/j.1600-0757.2011.00419.x · Cited by 15
Summary of relationships between exchangeability, biasing paths and bias
Flanders, W. D., & Eldridge, R. C. (2014) · European Journal of Epidemiology · doi:10.1007/s10654-014-9915-2 · Cited by 15
On stratification, minimization and protection against types 1 and 2 error
Bracken, M. B. (2001) · Journal of Clinical Epidemiology · doi:10.1016/s0895-4356(00)00286-9 · Cited by 15
How Choice of Effect Measure Influences Minimally Sufficient Adjustment Sets for External Validity
Webster‐Clark, M., & Keil, A. P. (2023) · American Journal of Epidemiology · doi:10.1093/aje/kwad041 · Cited by 14
A comparison of marginal odds ratio estimators
Loux, T. M., Drake, C., & Smith‐Gagen, J. (2014) · Statistical Methods in Medical Research · doi:10.1177/0962280214541995 · Cited by 14
“Greenland et al.(6) discuss confounding and collapsibility in depth, extending the concept of collapsibility to a non-constant conditional effect by necessitating that the collapsed effect measure equal the appropriately weighted average of the conditional measures.”
BackgroundHutton, H. E., Lesko, C. R., Chander, G., Lau, B. M., Wand, G. S., & McCaul, M. E. (2017) · Drug and Alcohol Dependence · doi:10.1016/j.drugalcdep.2017.05.021 · Cited by 14
“We used inverse probability weights to control for possible confounders, which allows us to report marginal estimates of association (risk ratios and rate ratios are not dependent on the other variables in the model) (Greenland et al., 1999).”
Greenland, S., Fay, M. P., Brittain, E., Shih, J. H., Follmann, D. A., Gabriel, E. E., & Robins, J. M. (2019) · The American Statistician · doi:10.1080/00031305.2019.1575771 · Cited by 14
Self-citationConformal Sensitivity Analysis for Individual Treatment Effects
Yin, M., Shi, C., Wang, Y., & Blei, D. M. (2022) · Journal of the American Statistical Association · doi:10.1080/01621459.2022.2102503 · Cited by 19
“In practice, this assumption can be difficult to accept for many observational studies (Greenland, Pearl, and Robins 1999), and violations of unconfoundedness will introduce hidden biases into the estimation of the ITE.”
BackgroundUncovering selection bias in case–control studies using Bayesian post‐stratification
Geneletti, S., Best, N., Toledano, M. B., Elliott, P., & Richardson, S. T. (2013) · Statistics in Medicine · doi:10.1002/sim.5722 · Cited by 14
“Other explanations are that the logistic regression estimates an odds ratio conditional on the SIV having fixed values, rather than an MOR; there could be an interaction between the exposure and the SIV, or it could be a product of the non-collapsibility of the odds ratio [38].”
BackgroundGoodman, S. N., & Samet, J. M. (2006) · Oxford University Press eBooks · doi:10.1093/acprof:oso/9780195149616.003.0001 · Cited by 14
Antenatal Programming of Hypertension: Paradigms, Paradoxes, and How We Move Forward
South, A. M., & Allen, N. B. (2022) · Current Hypertension Reports · doi:10.1007/s11906-022-01227-z · Cited by 13
Cox, L. A., Thompson, W. J., & Mundt, K. A. (2024) · Critical Reviews in Toxicology · doi:10.1080/10408444.2024.2337435 · Cited by 15
“As noted by Greenland et al. (1999), “Causal effects are defined only for comparisons of treatment levels.””
BackgroundAn Organizational Schema for Epidemiologic Causal Effects
Gatto, N. M., Campbell, U. B., & Schwartz, S. (2013) · Epidemiology · doi:10.1097/ede.0000000000000005 · Cited by 13
Asiimwe, I. G., Pushpakom, S. P., Turner, R. M., Kolamunnage‐Dona, R., Jorgensen, A., & Pirmohamed, M. (2022) · British Journal of Clinical Pharmacology · doi:10.1111/bcp.15331 · Cited by 13
“However, and even though adjustment may account for some confounders present in observational studies, it does not account for nonmeasured confounders and pooling of adjusted estimates may be problematic.(8,9) For more reliable evidence, therefore, a decision was made a priori to focus on RCTs, the current gold standa…”
BackgroundAustin, P. C., Schuster, T., & Platt, R. W. (2015) · BMC Medical Research Methodology · doi:10.1186/s12874-015-0081-3 · Cited by 13
“Marginal effects can be contrasted with conditional effects, which are the average effect of treatment at the individual level [8, 9].”
BackgroundGreenland, S. (2010) · The American Statistician · doi:10.1198/tast.2010.10006 · Cited by 13
“Noncollapsibility is often called Simpson’s paradox, although the phenomenon was illustrated by many authors before Simpson (see Greenland, Robins, and Pearl 1999 for a history) and the term “Simpson’s paradox” is sometimes restricted to association reversal upon stratification.”
Self-citationCovariate-Adjusted Putative Placebo Analysis in Active-Controlled Clinical Trials
Zhang, Z. (2009) · Statistics in Biopharmaceutical Research · doi:10.1198/sbr.2009.0034 · Cited by 13
“…effect and the model specification, one may not be able to derive the marginal effect directly from the conditional effect due to the issue of noncollapsibility (Gail, Wieand, and Piantadosi 1984; Greenland, Robins, and Pearl 1999), even if the conditional effect is assumed constant across trials.”
BackgroundELSHORBAGY, A. A., Wagener, T., Razavi, S., & Sauchyn, D. (2016) · Water Resources Research · doi:10.1002/2016wr018985 · Cited by 13
Metadata
- Authors
- Greenland, S., Pearl, J., & Robins, J. M.
- Venue
- Statistical Science
- Publisher
- Institute of Mathematical Statistics
- Published
- 1999-02-01
- Volume
- 14
- Issue
- 1
- Pages
- 29-46
- ISSN
- 0883-4237 (ISSN-L)
- 2168-8745
- OpenAlex
- W1996016456
- Cited by
- 933
- References
- 108
- Citations by year
- 2026: 43 · 2025: 47 · 2024: 61 · 2023: 53 · 2022: 65 · 2021: 75 · 2020: 48 · 2019: 40 · 2018: 40 · 2017: 36 · 2016: 35 · 2015: 44 · 2014: 52 · 2013: 47 · 2012: 47
- FWCI
- 13.67
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Statistical Science (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Epidemiology
- Social Sciences › Arts and Humanities › History and Philosophy of Science › Philosophy of Biology and Scientific Explanation
- Keywords
- confounding, collapsibility, counterfactual modeling, causal inference, exchangeability, causal effects, observational studies
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