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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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Citations
Cites
The central role of the propensity score in observational studies for causal effects
ROSENBAUM, P. R., & RUBIN, D. B. (1983) · Biometrika · doi:10.1093/biomet/70.1.41 · Cited by 31,241
Specification Tests in Econometrics
Hausman, J. A. (1978) · Econometrica · doi:10.2307/1913827 · Cited by 18,820
Hume, D. A. (1739) · Cambridge University Press eBooks · doi:10.1017/cbo9780511620409.040 · Cited by 9,910
The Statistical Analysis of Failure Time Data.
Aitkin, M. A., Kalbfleisch, J. D., & Prentice, R. L. (1981) · Biometrics · doi:10.2307/2530581 · Cited by 9,875
Estimating causal effects of treatments in randomized and nonrandomized studies
Rubin, D. B. (1974) · Journal of Educational Psychology · doi:10.1037/h0037350 · Cited by 9,436
Statistics and Causal Inference
Holland, P. W. (1986) · Journal of the American Statistical Association · doi:10.1080/01621459.1986.10478354 · Cited by 5,293
Discrete Multivariate Analysis: Theory and Practice.
Beniger, J. R., Bishop, Y. M. M., Feinberg, S. E., Holland, P. W., Light, R. J., & Mosteller, F. C. (1975) · Contemporary Sociology A Journal of Reviews · doi:10.2307/2063625 · Cited by 5,119
Fisher, R. A. (1936) · BMJ · doi:10.1136/bmj.1.3923.554-a · Cited by 4,980
Causal Diagrams for Epidemiologic Research
Greenland, S., Pearl, J., & Robins, J. M. (1999) · Epidemiology · doi:10.1097/00001648-199901000-00008 · Cited by 3,728
Self-citationAir Pollution and Hospital Admissions for Heart Disease in Eight U.S. Counties
Schwartz, J. D. (1999) · Epidemiology · doi:10.1097/00001648-199901000-00005 · Cited by 3,045
Research works printed but not linked: 9
- Clayton, D. and Hills, M. (1993). Statistical Models in Epidemiology. Oxford Univ. Press.
- Feynman, R. P. (1963). Lectures on Physics. Addison-Wesley, Reading, MA.
- Gail, M. H. (1986). Adjusting for covariates that have the same distribution in exposed and unexposed cohorts. In Modern Statistical Methods in Chronic Disease Epidemiology (S. H. Moolgavkar, and R. L. Prentice, eds.) 3-18. Wiley, New York.
- MacMahon, B. and Pugh, T. F. (1967). Causes and entities of disease. In Preventive Medicine (D. W. Clark and B. MacMahon, eds.) 11-18. Little Brown, Boston.
- McKechnie, J. L. (ed.) (1979). Webster's New Twentieth Century Dictionary. Simon and Schuster, New York.
- Neyman, J. (1935). Statistical problems in agricultural experimentation (with discussion). J. Roy. Statist. Soc. Suppl. 2 107-180. · DOI
- Pearl, J. and Robins, J. M. (1995). Probabilitic evaluation of sequential plans from causal model with hidden variables. In Uncertainty in Artificial Intelligence (P. Besnard and S. Hanks, eds.) 11 444-453. Morgan-Kaufman, San Francisco.
- Robins, J. M. (1998). Correction for non-compliance in equivalence trials. Statistics in Medicine 17 269-302. · DOI
- Rothman, K. J. and Greenland, S. (1998). Modern Epidemiology, 2nd ed. Lippincott-Raven, Philadelphia.
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VanderWeele, T. J. (2016) · Annual Review of Public Health · doi:10.1146/annurev-publhealth-032315-021402 · Cited by 1,985
“…the correct causal structure ov i it y A cce B = γλ γ + λ − . can alter the direct and indirect effects) we can take our direct and indirect effects estimates and their confidence intervals from the observed data and divide the direct effect estimate and both limits of its confidence interval by…”
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Burgess, S., Small, D. S., & Thompson, S. G. (2015) · Statistical Methods in Medical Research · doi:10.1177/0962280215597579 · Cited by 1,769
“Noncollapsibility is the violation of this property.(62) The relative risk and absolute risk difference are collapsible measures of association.”
BackgroundMoons, K. G. M., Wolff, R. F., Riley, R. D., Whiting, P. F., Westwood, M., Collins, G. S., Reitsma, J. B., Kleijnen, J., & Mallett, S. (2019) · Annals of Internal Medicine · doi:10.7326/m18-1377 · Cited by 1,669
Hernan, M. A. (2002) · American Journal of Epidemiology · doi:10.1093/aje/155.2.176 · Cited by 1,445
Principles of confounder selection
VanderWeele, T. J. (2019) · European Journal of Epidemiology · doi:10.1007/s10654-019-00494-6 · Cited by 1,535
Burgess, S., Dudbridge, F., & Thompson, S. G. (2016) · Statistics in Medicine · doi:10.1002/sim.6835 · Cited by 1,153
“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].”
BackgroundMethodThe 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
“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-citationEndogenous 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
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
“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].”
BackgroundDoubly 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
“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).”
BackgroundAn introduction to instrumental variables for epidemiologists
Greenland, S. (2000) · International Journal of Epidemiology · doi:10.1093/ije/29.4.722 · Cited by 1,156
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Didelez, V., & Sheehan, N. (2007) · Statistical Methods in Medical Research · doi:10.1177/0962280206077743 · Cited by 923
“Greenland et al. [ 30 ] discuss this problem referring to it as non{collapsibility of the logistic regression model.”
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Greenland, S., Mansournia, M. A., & Altman, D. G. (2016) · BMJ · doi:10.1136/bmj.i1981 · Cited by 898
Self-citationMendelian 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
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“(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…”
BackgroundOdds 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
“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…”
Muller, C. J., & MacLehose, R. F. (2014) · International Journal of Epidemiology · doi:10.1093/ije/dyu029 · Cited by 775
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
“For further discussions (including formal definitions) of concepts of confounding, see [21].”
BackgroundBurgess, S. (2014) · International Journal of Epidemiology · doi:10.1093/ije/dyu005 · Cited by 637
“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.”
BackgroundAI 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
“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
Self-citationThe 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
Austin, P. C., & Mamdani, M. (2005) · Statistics in Medicine · doi:10.1002/sim.2328 · Cited by 560
“This is frequently, though mistakenly, thought to be indicative of the presence of unmeasured confounding, but this need not be the case [28, 29].”
BackgroundGrant, R. L. (2014) · BMJ · doi:10.1136/bmj.f7450 · Cited by 515
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 548
“(29) and VanderWeele and Shpitser (30).”
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
“In contrast to odds ratios, risk ratios are not affected by controlling for orthogonal confounders (Greenland et al. 1999).”
BackgroundHandling drop‐out in longitudinal studies
Hogan, J. W., Roy, J., & Korkontzelou, C. (2004) · Statistics in Medicine · doi:10.1002/sim.1728 · Cited by 404
“Violations of the exogeneity condition can arise from multiple sources, including measurement error and confounding; see References [38–41] for a detailed discussion.”
BackgroundMaldonado, G., & Greenland, S. (2002) · International Journal of Epidemiology · doi:10.1093/intjepid/31.2.422 · Cited by 396
Self-citationConfounding in Health Research
Greenland, S., & Morgenstern, H. (2001) · Annual Review of Public Health · doi:10.1146/annurev.publhealth.22.1.189 · Cited by 390
“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-citationMethods 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
“Because of the non-collapsibility of the odds ratio [24,25], it is not possible for P.”
Backgroundde 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
“This means that crude and adjusted effect estimates have different interpretations when analyzing OR’s or HR’s [6,17].”
Backgroundipw: 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
“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).”
MethodCausal Inference Without Counterfactuals
Dawid, A. P. (2000) · Journal of the American Statistical Association · doi:10.1080/01621459.2000.10474210 · Cited by 414
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
Causal Inference without Counterfactuals
Dawid, A. P. (2001) · Applied logic series · doi:10.1007/978-94-017-1586-7_3 · Cited by 343
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 355
“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’.”
BackgroundMethods for Estimating Adjusted Risk Ratios
Cummings, P. (2009) · The Stata Journal Promoting communications on statistics and Stata · doi:10.1177/1536867x0900900201 · Cited by 326
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Ghassemi, M., Naumann, T., Schulam, P., Beam, A. L., Chen, I. Y., & Ranganath, R. (2018) · PubMed · doi:10.48550/arxiv.1806.00388 · Cited by 368
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Maathuis, M. H., Kalisch, M., & Bühlmann, P. (2009) · The Annals of Statistics · doi:10.1214/09-aos685 · Cited by 309
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BackgroundCommentary: 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
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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
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Greenland, S., & Brumback, B. (2002) · International Journal of Epidemiology · doi:10.1093/ije/31.5.1030 · Cited by 406
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Self-citationHeagerty, P. J., & Zeger, S. L. (2000) · Statistical Science · doi:10.1214/ss/1009212671 · Cited by 298
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Greenland, S. (2007) · American Journal of Epidemiology · doi:10.1093/aje/kwm355 · Cited by 281
Self-citationOn the definition of a confounder
VanderWeele, T. J., & Shpitser, I. (2013) · The Annals of Statistics · doi:10.1214/12-aos1058 · Cited by 268
“Hernán (2008) accordingly speaks of a confounder as a variable that "can be used to block a backdoor path between exposure and outcome" (p.”
Tu, Y., Gunnell, D., & Gilthorpe, M. S. (2008) · Emerging Themes in Epidemiology · doi:10.1186/1742-7622-5-2 · Cited by 261
“Within epidemiology, Simpson's paradox is closely linked to the concepts of confounding [9] and incollapsibility [10].”
BackgroundPropensity 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
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BackgroundKaufman, J. S., MacLehose, R. F., & Kaufman, S. (2004) · Epidemiologic Perspectives & Innovations · doi:10.1186/1742-5573-1-4 · Cited by 207
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Dorie, V., Hill, J., Shalit, U., Scott, M. A., & Cervone, D. (2019) · Statistical Science · doi:10.1214/18-sts667 · Cited by 311
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Höfler, M. (2005) · BMC Medical Research Methodology · doi:10.1186/1471-2288-5-28 · Cited by 249
Chambers, R. L., & Skinner, C. J. (2020) · Wiley eBooks · doi:10.1002/0470867205 · Cited by 236
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Dahabreh, I. J., & Bibbins-Domingo, K. (2024) · JAMA · doi:10.1001/jama.2024.7741 · Cited by 242
Austin, P. C., Grootendorst, P. V., Normand, S. T., & Anderson, G. M. (2006) · Statistics in Medicine · doi:10.1002/sim.2618 · Cited by 230
“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].”
BackgroundSocial 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 243
“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].”
BackgroundInfluentialThe 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
Elwert, F. (2013) · Handbooks of Sociology and Social Research · doi:10.1007/978-94-007-6094-3_13 · Cited by 325
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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
“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.”
Janes, H., & Pepe, M. S. (2008) · American Journal of Epidemiology · doi:10.1093/aje/kwn099 · Cited by 212
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
“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].”
BackgroundAli, 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
“…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).”
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Gkatzionis, A., & Burgess, S. (2019) · International Journal of Epidemiology · doi:10.1093/ije/dyy202 · Cited by 182
“33,34 Bias in mean association estimates due to selection was towards the null and increased with c U .”
Disentangled Representation Learning
Wang, X. Y., Chen, H., Tang, S., Wu, Z., & Zhu, W. (2024) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2024.3420937 · Cited by 265
“Let G i denote the i -th generative factor and C be the confounders in the causal learning literature [87].”
BackgroundStructural Nested Models and G-estimation: The Partially Realized Promise
Vansteelandt, S., & Joffe, M. (2014) · Statistical Science · doi:10.1214/14-sts493 · Cited by 173
On regression adjustment for the propensity score
Vansteelandt, S., & Daniel, R. (2014) · Statistics in Medicine · doi:10.1002/sim.6207 · Cited by 180
“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) ,”
BackgroundPropensity Score Methods: Theory and Practice for Anesthesia Research
Schulte, P. J., & Mascha, E. J. (2018) · Anesthesia & Analgesia · doi:10.1213/ane.0000000000002920 · Cited by 177
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Sjölander, A., & Greenland, S. (2013) · Statistics in Medicine · doi:10.1002/sim.5879 · Cited by 179
“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-citationCase-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
Self-citationImputation 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
“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.”
BackgroundOn 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
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 175
Daniel, R., Zhang, J., & Farewell, D. (2021) · Biometrical Journal · doi:10.1002/bimj.201900297 · Cited by 167
“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…”
BackgroundNickols, 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 218
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 163
“[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
Self-citationHogan, J. W., & Lancaster, T. (2004) · Statistical Methods in Medical Research · doi:10.1191/0962280204sm351ra · Cited by 154
Identifiability, exchangeability and confounding revisited
Greenland, S., & Robins, J. M. (2009) · Epidemiologic Perspectives & Innovations · doi:10.1186/1742-5573-6-4 · Cited by 161
“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-citationApplied Bayesian Modeling And Causal Inference From Incomplete-Data Perspectives
Gelman, A., & Xiaoli, M. (2014) · Cited by 152
Biostatistical Methods in Epidemiology
Newman, S. C. (2002) · Technometrics · doi:10.1198/004017002320256693 · Cited by 146
Statistics and causal inference: A review
Pearl, J. (2003) · Test · doi:10.1007/bf02595718 · Cited by 144
“…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-citationEcological 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 140
Greenland, S. (2003) · Journal of the American Statistical Association · doi:10.1198/01621450338861905 · Cited by 141
“Because the disease is rare the latter collapsibility condition implies no confounding by U (Greenland, Robins, and Pearl 1999).”
BackgroundSelf-citationAdversity 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 145
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
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
“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].”
MethodKuang, 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 147
“[87,88] review the concepts of confounding, and Refs.”
BackgroundBandyopadhyay, P. S., & Forster, M. R. (2011) · Elsevier eBooks · doi:10.1016/b978-0-444-51862-0.50001-0 · Cited by 132
“The problem can be dealt with by allowing the potential outcomes to refer to distributions (or parameters) instead of realizations [Greenland et al., 1999].”
BackgroundHill, J. (2008) · Statistics in Medicine · doi:10.1002/sim.3245 · Cited by 131
“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).”
BackgroundIndividual 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 135
“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…”
BackgroundCauses 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
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
“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].”
Martens, E., Pestman, W., de Boer, A., Belitser, S., & Klungel, O. (2008) · International Journal of Epidemiology · doi:10.1093/ije/dyn079 · Cited by 133
Chatton, A., Leborgne, F., Leyrat, C., Gillaizeau, F., Rousseau, C., Barbin, L., Laplaud, D. A., Léger, M., Giraudeau, B., & Foucher, Y. (2020) · Scientific Reports · doi:10.1038/s41598-020-65917-x · Cited by 137
Wyss, R., Ellis, A. R., Brookhart, M. A., Girman, C. J., Funk, M. J., LoCasale, R. J., & Stürmer, T. (2014) · American Journal of Epidemiology · doi:10.1093/aje/kwu181 · Cited by 119
“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).”
BackgroundStudying 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
“’’(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)”
BackgroundSheikh, M. A., Abelsen, B., & Olsen, J. A. (2016) · Frontiers in Psychology · doi:10.3389/fpsyg.2016.00727 · Cited by 129
“…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).”
BackgroundRasooly, D., & Patel, C. J. (2019) · Current Protocols in Human Genetics · doi:10.1002/cphg.82 · Cited by 121
“…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).”
BackgroundOn 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 124
“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).”
BackgroundInfluentialStürmer, T., Rothman, K. J., & Glynn, R. J. (2006) · Pharmacoepidemiology and Drug Safety · doi:10.1002/pds.1231 · Cited by 120
“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
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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