Confounding and Collapsibility in Causal Inference

Greenland, Sander; Pearl, Judea; Robins, James M.

doi:10.1214/ss/1009211805

Cited by 933Advanced Causal Inference TechniquesStatistical Methods in EpidemiologyPhilosophy and History of Science

Confounding and Collapsibility in Causal Inference

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Greenland, Sander, Pearl, Judea, Robins, James M.·1999·Statistical Science

DOI 10.1214/ss/1009211805

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Abstract

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.
“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]
Section 2.5 (Regression Formulations; page unavailable) explains that including confounders in a regression model can reduce bias in estimating a treatment effect. The Conclusion (section 7; page unavailable) cautions that measurement, selection, missing-data, and model-specification biases remain separate problems. The paper therefore supports accounting for confounding-related bias, but not the broader assertion that both unspecified models account for potential biases generally. (LLM, conf=0.85)
Experimental — accuracy not yet measured
The causal interpretation of the estimated treatment effects relies on standard identification assumptions used in the program-evaluation literature.
“The causal interpretation of the estimated treatment effects relies on the standard identification assumptions used in the program-evaluation literature [16, 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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Combining information on multiple instrumental variables in Mendelian randomization: comparison of allele score and summarized data methods
Burgess, S., Dudbridge, F., & Thompson, S. G. (2016) · Statistics in Medicine · doi:10.1002/sim.6835 · Cited by 1,153 · see citing works
“With a positive causal effect, power to reject the null was reduced compared with a continuous outcome, and the median estimates were attenuated towards the null; this is a known phenomenon and relates to the non-collapsibility of odds ratios [22,46].”
BackgroundMethod
The Table 2 Fallacy: Presenting and Interpreting Confounder and Modifier Coefficients
Westreich, D., & Greenland, S. (2013) · American Journal of Epidemiology · doi:10.1093/aje/kws412 · Cited by 1,276 · see citing works
“represent changes in the log odds of the outcome) and we assume the outcome is uncommon at all covariate levels to avoid complications arising from noncollapsibility of odds ratios (9, 10).”
BackgroundSelf-citation
Endogenous Selection Bias: The Problem of Conditioning on a Collider Variable
Elwert, F., & Winship, C. (2014) · Annual Review of Sociology · doi:10.1146/annurev-soc-071913-043455 · Cited by 1,168 · see citing works
Mendelian randomisation for mediation analysis: current methods and challenges for implementation
Carter, A. R., Sanderson, E., Hammerton, G., Richmond, R. C., Davey Smith, G., Heron, J., Taylor, A. E., Davies, N. M., & Howe, L. D. (2021) · European Journal of Epidemiology · doi:10.1007/s10654-021-00757-1 · Cited by 1,215 · see citing works
“This means the association between an exposure and outcome would not be constant on the odds-ratio scale by strata of categorical covariate [34, 35].”
Background
Doubly robust estimation of causal effects
Funk, M. J., Westreich, D., Wiesen, C., Stürmer, T., Brookhart, M. A., & Davidian, M. (2011) · American Journal of Epidemiology · doi:10.1093/aje/kwq439 · Cited by 1,158 · see citing works
“If the distributions of confounders are the same within each exposure group, then there is no longer an association between the confounders and exposure, making the exposed and unexposed exchangeable (15).”
Background
An introduction to instrumental variables for epidemiologists
Greenland, S. (2000) · International Journal of Epidemiology · doi:10.1093/ije/29.4.722 · Cited by 1,156 · see citing works
Self-citation
Mendelian randomization as an instrumental variable approach to causal inference
Didelez, V., & Sheehan, N. (2007) · Statistical Methods in Medical Research · doi:10.1177/0962280206077743 · Cited by 923 · see citing works
“Greenland et al. [ 30 ] discuss this problem referring to it as non{collapsibility of the logistic regression model.”
Sparse data bias: a problem hiding in plain sight
Greenland, S., Mansournia, M. A., & Altman, D. G. (2016) · BMJ · doi:10.1136/bmj.i1981 · Cited by 898 · see citing works
Self-citation
Mendelian Randomization: Concepts and Scope
Richmond, R. C., & Davey Smith, G. (2022) · Cold Spring Harbor Perspectives in Medicine · doi:10.1101/cshperspect.a040501 · Cited by 841 · see citing works
Multivariable Mendelian Randomization and MediationPreprint
Sanderson, E. C. M. (2020) · Cold Spring Harbor Perspectives in Medicine · doi:10.1101/cshperspect.a038984 · Cited by 829 · see citing works
“(Greenland et al. 1999; VanderWeele and Vansteelandt 2010; Burgess 2017) In practice this means that the total and direct effects estimated may differ because the direct effect is estimated conditional on the potential mediator, even if the potential mediator is not actually a mediator of the…”
Background
Odds Ratios for Mediation Analysis for a Dichotomous Outcome
VanderWeele, T. J., & Vansteelandt, S. (2010) · American Journal of Epidemiology · doi:10.1093/aje/kwq332 · Cited by 809 · see citing works
“Second, the methods described above require a rare outcome; this was necessary in the derivations and also circumvents collapsibility issues with odds ratios (39); some existing work considers or could be adapted for non-rare outcomes (16, 40); future work will consider settings in which the outcome is not rare and co…”
Estimating predicted probabilities from logistic regression: different methods correspond to different target populations
Muller, C. J., & MacLehose, R. F. (2014) · International Journal of Epidemiology · doi:10.1093/ije/dyu029 · Cited by 775 · see citing works
A survey of controlled experiments in software engineering
Sjoeberg, D. I. K., Hannay, J. E., Hansen, O., Kampenes, V. B., Karahasanović, A., Liborg, N.-K., & Rekdal, A. C. (2005) · IEEE Transactions on Software Engineering · doi:10.1109/tse.2005.97 · Cited by 684 · see citing works
“For further discussions (including formal definitions) of concepts of confounding, see [21].”
Background
Sample size and power calculations in Mendelian randomization with a single instrumental variable and a binary outcome
Burgess, S. (2014) · International Journal of Epidemiology · doi:10.1093/ije/dyu005 · Cited by 640 · see citing works
“This is due to the non-collapsibility of the odds ratio, meaning that the parameter estimate depends on the choice of covariate adjustment.(29) This is a general property of odds ratios, and not a specific feature of the ratio method.”
Background
AI recognition of patient race in medical imaging: a modelling study
Gichoya, J. W., Banerjee, I., Bhimireddy, A. R., Burns, J. L., Celi, L. A., Chen, L.-C., Correa, R., Dullerud, N., Ghassemi, M., Huang, S.-C., et al. (2022) · The Lancet Digital Health · doi:10.1016/s2589-7500(22)00063-2 · Cited by 611 · see citing works
“16,17 Many published AI models have conceptually amounted to simple bivariate analyses (ie, image features and their ability to predict clinical outcomes).”
Quantifying Biases in Causal Models: Classical Confounding vs Collider-Stratification Bias
Greenland, S. (2003) · Epidemiology · doi:10.1097/01.ede.0000042804.12056.6c · Cited by 810 · see citing works
Self-citation
The C-Word: Scientific Euphemisms Do Not Improve Causal Inference From Observational Data
Hernán, M. Á. (2018) · American Journal of Public Health · doi:10.2105/ajph.2018.304337 · Cited by 570 · see citing works
A comparison of propensity score methods: a case‐study estimating the effectiveness of post‐AMI statin use
Austin, P. C., & Mamdani, M. (2005) · Statistics in Medicine · doi:10.1002/sim.2328 · Cited by 560 · see citing works
“This is frequently, though mistakenly, thought to be indicative of the presence of unmeasured confounding, but this need not be the case [28, 29].”
Background
Converting an odds ratio to a range of plausible relative risks for better communication of research findings
Grant, R. L. (2014) · BMJ · doi:10.1136/bmj.f7450 · Cited by 515 · see citing works
Best (but oft-forgotten) practices: mediation analysis ,
Fairchild, A. J., & McDaniel, H. L. (2017) · American Journal of Clinical Nutrition · doi:10.3945/ajcn.117.152546 · Cited by 512 · see citing works
“(29) and VanderWeele and Shpitser (30).”
Handling drop‐out in longitudinal studies
Hogan, J. W., Roy, J., & Korkontzelou, C. (2004) · Statistics in Medicine · doi:10.1002/sim.1728 · Cited by 404 · see citing works
“Violations of the exogeneity condition can arise from multiple sources, including measurement error and confounding; see References [38–41] for a detailed discussion.”
Background
Interpreting and Understanding Logits, Probits, and Other Nonlinear Probability Models
Breen, R., Karlson, K. B., & Holm, A. (2018) · Annual Review of Sociology · doi:10.1146/annurev-soc-073117-041429 · Cited by 468 · see citing works
“In contrast to odds ratios, risk ratios are not affected by controlling for orthogonal confounders (Greenland et al. 1999).”
Background
Estimating causal effects
Maldonado, G., & Greenland, S. (2002) · International Journal of Epidemiology · doi:10.1093/intjepid/31.2.422 · Cited by 396 · see citing works
Self-citation
Confounding in Health Research
Greenland, S., & Morgenstern, H. (2001) · Annual Review of Public Health · doi:10.1146/annurev.publhealth.22.1.189 · Cited by 390 · see citing works
“As discussed below, confounding is also possible in randomized experiments because of systematic elements in treatment allocation, administration, and compliance and because of random differences between comparison groups (22, 30, 36, 38, 87).”
BackgroundMethodInfluentialSelf-citation
Methods for dealing with time‐dependent confounding
Daniel, R., Cousens, S. N., Stavola, B. L. D., Kenward, M. G., & Sterne, J. A. C. (2012) · Statistics in Medicine · doi:10.1002/sim.5686 · Cited by 378 · see citing works
“Because of the non-collapsibility of the odds ratio [24,25], it is not possible for P.”
Background
Testing for baseline differences in randomized controlled trials: An unhealthy research behavior that is hard to eradicate
de Boer, M. R., Waterlander, W. E., Kuijper, L. D., Steenhuis, I. H., & Twisk, J. W. (2015) · International Journal of Behavioral Nutrition and Physical Activity · doi:10.1186/s12966-015-0162-z · Cited by 370 · see citing works
“This means that crude and adjusted effect estimates have different interpretations when analyzing OR’s or HR’s [6,17].”
Background
ipw: AnRPackage for Inverse Probability Weighting
van der Wal, W. M., & Geskus, R. B. (2011) · Journal of Statistical Software · doi:10.18637/jss.v043.i13 · Cited by 364 · see citing works
“When using IPW to fit an MSM, there is minimal risk of (1) adjusting away part of the effect (Robins 1997; Robins, Greenland, and Hu 1999), (2) non-collapsibility, (Greenland, Robins, and Pearl 1999), or (3) Berksons bias (Hernán, Hernández-Dı́az, and Robins 2004).”
Method
Causal Inference Without Counterfactuals
Dawid, A. P. (2000) · Journal of the American Statistical Association · doi:10.1080/01621459.2000.10474210 · Cited by 399 · see citing works
Handling time varying confounding in observational research
Mansournia, M. A., Etminan, M., Danaei, G., Kaufman, J. S., & Collins, P. G. S. (2017) · BMJ · doi:10.1136/bmj.j4587 · Cited by 343 · see citing works
Causal Inference without Counterfactuals
Dawid, A. P. (2001) · Applied logic series · doi:10.1007/978-94-017-1586-7_3 · Cited by 343 · see citing works
Causality and causal inference in epidemiology: the need for a pluralistic approach
Vandenbroucke, J. P., Broadbent, A., & Pearce, N. E. (2016) · International Journal of Epidemiology · doi:10.1093/ije/dyv341 · Cited by 328 · see citing works
“That is perhaps a reason why the language of ‘potential outcomes’ is often preferred.(23)One simply imagines that one has knowledge about the outcome of two different treatments (levels of exposure) in the same individual(17)—and avoids the philosophical complexities of ‘possible worlds’.”
Background
Methods for Estimating Adjusted Risk Ratios
Cummings, P. (2009) · The Stata Journal Promoting communications on statistics and Stata · doi:10.1177/1536867x0900900201 · Cited by 326 · see citing works
A Review of Challenges and Opportunities in Machine Learning for Health
Ghassemi, M., Naumann, T., Schulam, P., Beam, A. L., Chen, I. Y., & Ranganath, R. (2018) · PubMed Central · doi:10.48550/arxiv.1806.00388 · Cited by 323 · see citing works
Estimating high-dimensional intervention effects from observational data
Maathuis, M. H., Kalisch, M., & Bühlmann, P. (2009) · The Annals of Statistics · doi:10.1214/09-aos685 · Cited by 318 · see citing works
“This task coincides with inferring causal eﬀects, a well-established area in Statistics (e.g., [5, 8, 10, 11, 13, 18, 24, 25, 26] and [31]).”
Background
Commentary: Considerations for Use of Racial/Ethnic Classification in Etiologic Research
Kaufman, J. S. (2001) · American Journal of Epidemiology · doi:10.1093/aje/154.4.291 · Cited by 405 · see citing works
Advancing functional connectivity research from association to causation
Reid, A., Headley, D. B., Mill, R. D., Sánchez-Romero, R., Uddin, L. Q., Marinazzo, D., Lurie, D. J., Valdes-Sosa, P. A., Hanson, S. J., Biswal, B. B., et al. (2019) · Nature Neuroscience · doi:10.1038/s41593-019-0510-4 · Cited by 303 · see citing works
An overview of relations among causal modelling methods
Greenland, S., & Brumback, B. (2002) · International Journal of Epidemiology · doi:10.1093/ije/31.5.1030 · Cited by 406 · see citing works
Self-citation
Ecologic versus individual-level sources of bias in ecologic estimates of contextual health effects.
Greenland, S. (2001) · International Journal of Epidemiology · doi:10.1093/ije/30.6.1343 · Cited by 317 · see citing works
Self-citation
Marginalized multilevel models and likelihood inference (with comments and a rejoinder by the authors)
Heagerty, P. J., & Zeger, S. L. (2000) · Statistical Science · doi:10.1214/ss/1009212671 · Cited by 298 · see citing works
Invited Commentary: Variable Selection versus Shrinkage in the Control of Multiple Confounders
Greenland, S. (2007) · American Journal of Epidemiology · doi:10.1093/aje/kwm355 · Cited by 281 · see citing works
Self-citation
On the definition of a confounder
VanderWeele, T. J., & Shpitser, I. (2013) · The Annals of Statistics · doi:10.1214/12-aos1058 · Cited by 268 · see citing works
“Hernán (2008) accordingly speaks of a confounder as a variable that "can be used to block a backdoor path between exposure and outcome" (p.”
Simpson's Paradox, Lord's Paradox, and Suppression Effects are the same phenomenon – the reversal paradox
Tu, Y., Gunnell, D., & Gilthorpe, M. S. (2008) · Emerging Themes in Epidemiology · doi:10.1186/1742-7622-5-2 · Cited by 261 · see citing works
“Within epidemiology, Simpson's paradox is closely linked to the concepts of confounding [9] and incollapsibility [10].”
Background
Propensity scores: From naïve enthusiasm to intuitive understanding
Williamson, E., Morley, R., Lucas, A., & Carpenter, J. (2012) · Statistical Methods in Medical Research · doi:10.1177/0962280210394483 · Cited by 248 · see citing works
“In the odds ratio setting, however, the definition of the causal estimand becomes much more complex due to a property called the ‘non-collapsibility’ of the odds ratio.(40) For example, if p1 denotes the proportion of the whole population who would have the outcome of interest if the whole population were exposed and…”
Background
A further critique of the analytic strategy of adjusting for covariates to identify biologic mediation
Kaufman, J. S., MacLehose, R. F., & Kaufman, S. (2004) · Epidemiologic Perspectives & Innovations · doi:10.1186/1742-5573-1-4 · Cited by 253 · see citing works
“It is now possible to specify exchangeability conditions that are sufficient to guarantee that there is no confounding, meaning that associational measures and causal effects are equivalent[28].”
Automated versus Do-It-Yourself Methods for Causal Inference: Lessons Learned from a Data Analysis Competition
Dorie, V., Hill, J., Shalit, U., Scott, M. A., & Cervone, D. (2019) · Statistical Science · doi:10.1214/18-sts667 · Cited by 253 · see citing works
Causal inference based on counterfactuals
Höfler, M. (2005) · BMC Medical Research Methodology · doi:10.1186/1471-2288-5-28 · Cited by 246 · see citing works
Analysis of Survey Data
Chambers, R. L., & Skinner, C. J. (2020) · Wiley eBooks · doi:10.1002/0470867205 · Cited by 236 · see citing works
Causal Inference About the Effects of Interventions From Observational Studies in Medical Journals
Dahabreh, I. J., & Bibbins-Domingo, K. (2024) · JAMA · doi:10.1001/jama.2024.7741 · Cited by 242 · see citing works
Conditioning on the propensity score can result in biased estimation of common measures of treatment effect: a Monte Carlo study
Austin, P. C., Grootendorst, P. V., Normand, S. T., & Anderson, G. M. (2006) · Statistics in Medicine · doi:10.1002/sim.2618 · Cited by 230 · see citing works
“This phenomenon is also known as non-collapsibility of the odds ratio [1], but is apparent in all non-linear regression models and generalized linear models with a link function other than the identity link (linear models) or log-link function [2].”
Background
Social network analysis and agent-based modeling in social epidemiology
El‐Sayed, A. M., Scarborough, P., Seemann, L., & Galea, S. (2012) · Epidemiologic Perspectives & Innovations · doi:10.1186/1742-5573-9-1 · Cited by 224 · see citing works
“As social factors may interact in complex ways to determine health and disease risk, the current “risk factor” approach to epidemiology, which emphasizes decontextualized, independent effect measures for exposures may not be appropriate [11,12].”
BackgroundInfluential
The Simpson's paradox unraveled
Hernán, M. Á., Clayton, D. G., & Keiding, N. (2011) · International Journal of Epidemiology · doi:10.1093/ije/dyr041 · Cited by 222 · see citing works
Graphical Causal Models
Elwert, F. (2013) · Handbooks of Sociology and Social Research · doi:10.1007/978-94-007-6094-3_13 · Cited by 325 · see citing works
The Relationship between Prenatal PCB Exposure and Intelligence (IQ) in 9-Year-Old Children
Stewart, P. W., Lonky, E., Reihman, J., Pagano, J., Gump, B. B., & Darvill, T. (2008) · Environmental Health Perspectives · doi:10.1289/ehp.11058 · Cited by 214 · see citing works
“To address this, we employed a strategy partly informed by Greenland and Robins (1999) with an emphasis on efficient modeling and a minimal covariate set.”
Adjusting for Covariates in Studies of Diagnostic, Screening, or Prognostic Markers: An Old Concept in a New Setting
Janes, H., & Pepe, M. S. (2008) · American Journal of Epidemiology · doi:10.1093/aje/kwn099 · Cited by 212 · see citing works
Dissecting Causal Pathways Using Mendelian Randomization with Summarized Genetic Data: Application to Age at Menarche and Risk of Breast Cancer.
Burgess, S., Thompson, D. J., Rees, J. M. B., Day, F. R., Perry, J. R., & Ong, K. K. (2017) · Genetics · doi:10.1534/genetics.117.300191 · Cited by 197 · see citing works
“Odds ratios are non-collapsible, meaning that they do not average intuitively, and they depend on the choice of covariate adjustment even in the absence of confounding (so conditional odds ratios differ in magnitude to marginal odds ratios) [Greenland et al., 1999].”
Background
Propensity Score Methods in Health Technology Assessment: Principles, Extended Applications, and Recent Advances
Ali, M. S., Prieto-Alhambra, D., Lopes, L. C., Ramos, D., Bispo, N., Ichihara, M. Y., Pescarini, J. M., Williamson, E., Fiaccone, R. L., Barreto, M. L., et al. (2019) · Frontiers in Pharmacology · doi:10.3389/fphar.2019.00973 · Cited by 200 · see citing works
“…refers to a phenomenon in which, in the presence of a non-null treatment effect, the marginal (overall) treatment effect estimate is different from the conditional (stratum-specific) treatment effect estimate, even in the absence of confounding (Greenland et al., 1999; Austin, 2008b).”
Result
Contextualizing selection bias in Mendelian randomization: how bad is it likely to be?
Gkatzionis, A., & Burgess, S. (2019) · International Journal of Epidemiology · doi:10.1093/ije/dyy202 · Cited by 182 · see citing works
“33,34 Bias in mean association estimates due to selection was towards the null and increased with c U .”
Disentangled Representation Learning
Wang, X., Chen, H., Tang, S., Wu, Z., & Zhu, W.-W. (2022) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2024.3420937 · Cited by 266 · see citing works
“Let G i denote the i -th generative factor and C be the confounders in the causal learning literature [87].”
Background
Structural Nested Models and G-estimation: The Partially Realized Promise
Vansteelandt, S., & Joffe, M. (2014) · Statistical Science · doi:10.1214/14-sts493 · Cited by 173 · see citing works
On regression adjustment for the propensity score
Vansteelandt, S., & Daniel, R. (2014) · Statistics in Medicine · doi:10.1002/sim.6207 · Cited by 180 · see citing works
“However, because of noncollapsibility of the odds ratio [13,14], this generally differs from the conditional exposure effect P ( Y1 = 1|c)P (Y0 = 0|c) P ( Y1 = 0|c)P (Y0 = 1|c) ,”
Background
Propensity Score Methods: Theory and Practice for Anesthesia Research
Schulte, P. J., & Mascha, E. J. (2018) · Anesthesia & Analgesia · doi:10.1213/ane.0000000000002920 · Cited by 180 · see citing works
Ignoring the matching variables in cohort studies – when is it valid and why?
Sjölander, A., & Greenland, S. (2013) · Statistics in Medicine · doi:10.1002/sim.5879 · Cited by 179 · see citing works
“The difference is that the odds ratio and hazard ratio are noncollapsible, meaning that they may change upon stratification by a perfectly balanced (and thus nonconfounding) factor, yet the unstratified (marginal) ratio is still unbiased for the population effect standardized to the exposed [7, 12].”
BackgroundSelf-citation
Case-control matching: effects, misconceptions, and recommendations
Mansournia, M. A., Jewell, N. P., & Greenland, S. (2018) · European Journal of Epidemiology · doi:10.1007/s10654-017-0325-0 · Cited by 175 · see citing works
Self-citation
Imputation Strategies for the Estimation of Natural Direct and Indirect Effects
Vansteelandt, S., Bekaert, M., & Lange, T. (2012) · Epidemiologic Methods · doi:10.1515/2161-962x.1014 · Cited by 177 · see citing works
“A major drawback of this approach is that, as a result of noncollapsibility of the odds ratio (Greenland, Robins, and Pearl, 1999), β ′a will typically differ from βa, even when the exposure does not affect the mediator and thus even when there is no indirect effect.”
Background
On model selection and model misspecification in causal inference.
Vansteelandt, S., Bekaert, M., & Claeskens, G. (2012) · Statistical Methods in Medical Research · doi:10.1177/0962280210387717 · Cited by 172 · see citing works
Matching Weights to Simultaneously Compare Three Treatment Groups
Yoshida, K., Hernández–Dı́az, S., Solomon, D. H., Jackson, J. W., Gagne, J. J., Glynn, R. J., & Franklin, J. M. (2017) · Epidemiology · doi:10.1097/ede.0000000000000627 · Cited by 169 · see citing works
Making apples from oranges: Comparing noncollapsible effect estimators and their standard errors after adjustment for different covariate sets
Daniel, R., Zhang, J., & Farewell, D. (2021) · Biometrical Journal · doi:10.1002/bimj.201900297 · Cited by 167 · see citing works
“2 NOTATION AND FRAMEWORK Let X denote a binary exposure or treatment variable (X = 1 for the exposed/treated, and X = 0 for the unexposed/untreated), let Y denote the binary or right-censored time-to-event outcome, according to the context, and C a set of covariates. When Y is a right-censored time-to-event outcome, l…”
Background
MaAsLin 3: refining and extending generalized multivariable linear models for meta-omic association discovery
Nickols, W. A., Kuntz, T. M., Shen, J., Maharjan, S., Mallick, H., Franzosa, E. A., Thompson, K. N., Nearing, J. T., & Huttenhower, C. (2026) · Nature Methods · doi:10.1038/s41592-025-02923-9 · Cited by 165 · see citing works
A substantial and confusing variation exists in handling of baseline covariates in randomized controlled trials: a review of trials published in leading medical journals
Austin, P. C., Manca, A., Zwarenstein, M., Juurlink, D. N., & Stanbrook, M. B. (2009) · Journal of Clinical Epidemiology · doi:10.1016/j.jclinepi.2009.06.002 · Cited by 158 · see citing works
“[152,153].”
Matched designs and causal diagrams
Mansournia, M. A., Hernán, M. Á., & Greenland, S. (2013) · International Journal of Epidemiology · doi:10.1093/ije/dyt083 · Cited by 161 · see citing works
Self-citation
Instrumental variables and inverse probability weighting for causal inference from longitudinal observational studies
Hogan, J. W., & Lancaster, T. (2004) · Statistical Methods in Medical Research · doi:10.1191/0962280204sm351ra · Cited by 154 · see citing works
Identifiability, exchangeability and confounding revisited
Greenland, S., & Robins, J. M. (2009) · Epidemiologic Perspectives & Innovations · doi:10.1186/1742-5573-6-4 · Cited by 155 · see citing works
“To describe the situation more generally (as in [2]) for a binary exposure X, let U1A and U0A be some summary of the outcomes in group A when everyone in A is exposed (X = 1) and when everyone in A is unexposed (X = 0); similarly for group B, let U1B and U0B be the summaries when everyone in group B is exposed and whe…”
BackgroundInfluentialSelf-citation
Applied Bayesian Modeling And Causal Inference From Incomplete-Data Perspectives
Gelman, A., & Xiaoli, M. (2014) · Cited by 152 · see citing works
Biostatistical Methods in Epidemiology
Newman, S. C. (2002) · Technometrics · doi:10.1198/004017002320256693 · Cited by 146 · see citing works
Statistics and causal inference: A review
Pearl, J. (2003) · Test · doi:10.1007/bf02595718 · Cited by 144 · see citing works
“…been avoided in much of that literature (e.g., Bishop, 1971; Whittemore, 1978; Grayson, 1987; Hauck et al., 1991; Becher, 1992)6, major efforts would be required to assess the relevance of this impressive literature to the modern conception of confounding as effect bias Greenland et al. (1999b)7.”
BackgroundMethodInfluentialSelf-citation
Ecological Inference for 2 × 2 Tables
Wakefield, J. (2004) · Journal of the Royal Statistical Society Series A (Statistics in Society) · doi:10.1111/j.1467-985x.2004.02046_1.x · Cited by 141 · see citing works
The Impact of Prior Distributions for Uncontrolled Confounding and Response Bias: A Case Study of the Relation of Wire Codes and Magnetic Fields to Childhood Leukemia
Greenland, S. (2003) · Journal of the American Statistical Association · doi:10.1198/01621450338861905 · Cited by 141 · see citing works
“Because the disease is rare the latter collapsibility condition implies no confounding by U (Greenland, Robins, and Pearl 1999).”
BackgroundSelf-citation
Adversity exposure during sensitive periods predicts accelerated epigenetic aging in children
Marini, S., Davis, K. A., Soare, T. W., Zhu, Y., Suderman, M. J., Simpkin, A. J., Smith, A. D. A. C., Wolf, E. J., Relton, C. L., & Dunn, E. C. (2019) · Psychoneuroendocrinology · doi:10.1016/j.psyneuen.2019.104484 · Cited by 140 · see citing works
Applying Propensity Score Methods in Medical Research: Pitfalls and Prospects
Luo, Z.-H., Gardiner, J., & Bradley, C. (2010) · Medical Care Research and Review · doi:10.1177/1077558710361486 · Cited by 134 · see citing works
Full loss of residual renal function causes higher mortality in dialysis patients; findings from a marginal structural model
van der Wal, W. M., Noordzij, M., Dekker, F. W., Boeschoten, E. W., Krediet, R. T., Korevaar, J. C., Geskus, R. B., & for The Netherlands Cooperative Study on the Adequacy of Dialysis Study Group (NECOSAD) (2011) · Nephrology Dialysis Transplantation · doi:10.1093/ndt/gfq856 · Cited by 135 · see citing works
“Noncollapsibility means that the effect estimate for GFR, as estimated from a Cox proportional hazards model such as in our study, changes by adding covariates that are completely independent from GFR [14].”
Method
Causal Inference
Kuang, K., Li, L., Geng, Z., Xu, L., Zhang, K., Liao, B., Huang, H., Ding, P., Miao, W., & Jiang, Z. (2020) · Engineering · doi:10.1016/j.eng.2019.08.016 · Cited by 133 · see citing works
“[87,88] review the concepts of confounding, and Refs.”
Background
Philosophy of Statistics
Bandyopadhyay, P. S., & Forster, M. R. (2011) · Elsevier eBooks · doi:10.1016/b978-0-444-51862-0.50001-0 · Cited by 132 · see citing works
“The problem can be dealt with by allowing the potential outcomes to refer to distributions (or parameters) instead of realizations [Greenland et al., 1999].”
Background
Discussion of research using propensity‐score matching: Comments on ‘A critical appraisal of propensity‐score matching in the medical literature between 1996 and 2003’ by Peter Austin, Statistics in Medicine
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“We can define a confounder as a variable that predicts both the treatment and the outcome (see [3] for a precise, thorough, and understandable discussion of confounding including a technical exception to the definition just provided).”
Background
Individual participant data meta-analyses should not ignore clustering
Abo-Zaid, G., Guo, B., Deeks, J. J., Debray, T. P. A., Steyerberg, E. W., Moons, K. G. M., & Riley, R. D. (2013) · Journal of Clinical Epidemiology · doi:10.1016/j.jclinepi.2012.12.017 · Cited by 130 · see citing works
“For logistic regression, this phenomenon is also known as noncollapsibility of the odds ratio [40] as conditional odds ratios are typically larger than marginal odds ratios after conditioning on important covariates, with the increase becoming higher as the true odds ratio increases and the number of included importan…”
Background
Causes and consequences of child growth faltering in low-resource settings
Mertens, A., Benjamin-Chung, J., Colford, J. M., Coyle, J., van der Laan, M. J., Hubbard, A. E., Rosete, S., Malenica, I., Hejazi, N., Sofrygin, O., et al. (2023) · Nature · doi:10.1038/s41586-023-06501-x · Cited by 130 · see citing works
The role of causal criteria in causal inferences: Bradford Hill's "aspects of association"
Ward, A. C. (2009) · Epidemiologic Perspectives & Innovations · doi:10.1186/1742-5573-6-2 · Cited by 120 · see citing works
“Further, let us suppose that "μ will equal μA1 if x1 is applied to population A, and will equal μA0 if x0 is applied to that population" [30].”
Systematic differences in treatment effect estimates between propensity score methods and logistic regression
Martens, E. P., Pestman, W. R., de Boer, A., Belitser, S. V., & Klungel, O. H. (2008) · International Journal of Epidemiology · doi:10.1093/ije/dyn079 · Cited by 126 · see citing works
G-computation, propensity score-based methods, and targeted maximum likelihood estimator for causal inference with different covariates sets: a comparative simulation study
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The Role of Prediction Modeling in Propensity Score Estimation: An Evaluation of Logistic Regression, bCART, and the Covariate-Balancing Propensity Score
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“Although binary outcomes are more common in pharmacoepidemiology, we chose to simulate a continuous outcome to simplify the analysis and to avoid issues with the noncollapsibility of the odds ratio (22).”
Background
Studying noncollapsibility of the odds ratio with marginal structural and logistic regression models
Pang, M., Kaufman, J. S., & Platt, R. W. (2013) · Statistical Methods in Medical Research · doi:10.1177/0962280213505804 · Cited by 123 · see citing works
“’’(34) We emphasize that the counterfactual definition of confounding depends on the target population and focuses on causal inference, while collapsibility depends on the selected parameter and has no definitive implication for causality or confounding.(4,7)”
Background
Clarifying Associations between Childhood Adversity, Social Support, Behavioral Factors, and Mental Health, Health, and Well-Being in Adulthood: A Population-Based Study
Sheikh, M. A., Abelsen, B., & Olsen, J. A. (2016) · Frontiers in Psychology · doi:10.3389/fpsyg.2016.00727 · Cited by 122 · see citing works
“…assessing mediation (Pearl, 2012; Sheikh et al., 2014), as the direct effect is overestimated, and the indirect effect is underestimated (Jiang and VanderWeele, 2015) due to the property of non-collapsibility (Miettinen and Cook, 1981; Greenland, 1987; Greenland et al., 1999; Pang et al., 2013).”
Background
Conducting a Reproducible Mendelian Randomization Analysis Using the R Analytic Statistical Environment
Rasooly, D., & Patel, C. J. (2019) · Current Protocols in Human Genetics · doi:10.1002/cphg.82 · Cited by 122 · see citing works
“…associations are inﬂuenced by biases such as measured and unmeasured confounds—which can occur when an outside variable is associated with the exposure and the disease—and reverse causality and therefore can lack ability to establish a directional effect (Greenland, Robins, & Pearl, 1999).”
Background
On collapsibility and confounding bias in Cox and Aalen regression models
Martinussen, T., & Vansteelandt, S. (2013) · Lifetime Data Analysis · doi:10.1007/s10985-013-9242-z · Cited by 117 · see citing works
“Indeed, nl(t) = 0 reflects a difference between subgroup effects and population-averaged effects which is typical of nonlinear association measures and is commonly referred to as noncollapsibility (Greenland et al. 1999) or non-linearity (Janes et al. 2010).”
BackgroundInfluential
Insights into different results from different causal contrasts in the presence of effect-measure modification
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“We assume a prevalence of atrial fibrillation of 10% in this elderly population.(16) Without antithrombotic therapy, the risk of stroke is five times higher in individuals with atrial fibrillation (15%) than in individuals without atrial fibrillation (3%).”
BackgroundInfluential

Authors: Greenland, S., Pearl, J., & Robins, J. M.
ORCID: Greenland, Sander
Robins, James 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
DOI: 10.1214/ss/1009211805
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 and History of Science
Keywords: confounding, collapsibility, counterfactual modeling, causal inference, exchangeability, causal effects, observational studies

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Confounding and Collapsibility in Causal Inference

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