The DOI matches its Crossref record; 45 of the 45 references it cites were found in a registry or an index.
Bias Formulas for Sensitivity Analysis for Direct and Indirect Effects
Open accessAbstract
A key question in many studies is how to divide the total effect of an exposure into a component that acts directly on the outcome and a component that acts indirectly, ie, through some intermediate. For example, one might be interested in the extent to which the effect of diet on blood pressure is mediated through sodium intake and the extent to which it operates through other pathways. In the context of such mediation analysis, even if the effect of the exposure on the outcome is unconfounded, estimates of direct and indirect effects will be biased if control is not made for confounders of the mediator-outcome relationship. Often data are not collected on such mediator-outcome confounding variables; the results in this paper allow researchers to assess the sensitivity of their estimates of direct and indirect effects to the biases from such confounding. Specifically, the paper provides formulas for the bias in estimates of direct and indirect effects due to confounding of the exposure-mediator relationship and of the mediator-outcome relationship. Under some simplifying assumptions, the formulas are particularly easy to use in sensitivity analysis. The bias formulas are illustrated by examples in the literature concerning direct and indirect effects in which mediator-outcome confounding may be present.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
Baron, R. M., & Kenny, D. A. (1986) · Journal of Personality and Social Psychology · doi:10.1037/0022-3514.51.6.1173 · Cited by 91,532
Introduction to Statistical Mediation Analysis
MacKinnon, D. P. (2012) · doi:10.4324/9780203809556 · Cited by 5,618
A general approach to causal mediation analysis.
Imai, K., Keele, L., & Tingley, D. (2010) · Psychological Methods · doi:10.1037/a0020761 · Cited by 4,143
A structural approach to selection bias
Hernán, M. A., Hernández-Díaz, S., & Robins, J. M. (2004) · Epidemiology · doi:10.1097/01.ede.0000135174.63482.43 · Cited by 2,844
Robins, J. (1986) · Mathematical Modelling · doi:10.1016/0270-0255(86)90088-6 · Cited by 2,451
Process Analysis: Estimating Mediation in Treatment Evaluations
Judd, C. M., & Kenny, D. A. (1981) · Evaluation Review · doi:10.1177/0193841x8100500502 · Cited by 2,442
Causal diagrams for empirical research
PEARL, J. (1995) · Biometrika · doi:10.1093/biomet/82.4.669 · Cited by 2,355
Identifiability and exchangeability for direct and indirect effects
Robins, J. M., & Greenland, S. (1992) · Epidemiology · doi:10.1097/00001648-199203000-00013 · Cited by 1,793
On the importance—and the unimportance— of birthweight
Wilcox, A. J. (2001) · International Journal of Epidemiology · doi:10.1093/ije/30.6.1233 · Cited by 889
Conceptual issues concerning mediation, interventions and composition
VanderWeele, T. J., & Vansteelandt, S. (2009) · Statistics and Its Interface · doi:10.4310/sii.2009.v2.n4.a7 · Cited by 758
Self-citation
Flagged first, then by citations
Research works printed but not linked: 3
- Pearl, 2001, Proceedings of the Seventeenth Conference on Uncertainty and Artificial Intelligence., 411
- Hafeman, 2008, Opening the Black Box: A Reassessment of Mediation From a Counterfactual Perspective [dissertation].
- Avin, 2005, Proceedings of the International Joint Conferences on Artificial Intelligence., 357
Cited by
Sorted by citations
Items 401–434 of 434
Analyzing Causal Mechanisms: A Review of Theory and Practices in Political SciencePreprint
Mohamed, A. E. (2022) · SSRN Electronic Journal · doi:10.2139/ssrn.4125543
Lamp, S. J., Alvarez-Bartolo, D., Luecken, L. J., & MacKinnon, D. P. (2023) · Child maltreatment solutions network · doi:10.1007/978-3-031-33739-0_5
Bias and sensitivity analyses for linear front-door models
Thoemmes, F. J., & Kim, Y. (2023) · Methodology · doi:10.5964/meth.9205
“VanderWeele (2010) and VanderWeele and Arah (2011) argue that every time untestable causal assumptions are invoked, a sensitivity analysis is advisable.”
BackgroundThe importance of translating genetic partitioning into causal language
Diemer, E. W. (2024) · International Journal of Epidemiology · doi:10.1093/ije/dyae036
Hu, A. (2024) · Springer eBooks · doi:10.1007/978-981-97-6398-6_5
Grubic, N., Maximova, K., Chioléro, A., Siddiqi, A., Carsley, S., Batomen, B., Harris, K. M., & Carmeli, C. (2025) · CJC Open · doi:10.1016/j.cjco.2025.08.002 · Cited by 1
“129,130 Information biases, such as recall bias and social desirability bias, may influence the self-reported measures of movement behaviours.”
Miočević, M., & Talbot, D. (2025) · Structural Equation Modeling A Multidisciplinary Journal · doi:10.1080/10705511.2025.2559270
“…confounders, researchers are advised to perform sensitivity analyses to evaluate the magnitude of the relationship of the unmeasured confounder with the mediator and outcome required to completely account for the observed mediated effect (Cox et al., 2013; Imai et al., 2010; VanderWeele, 2010).”
Definition, identification, and estimation of the direct and indirect number needed to treat
Vancak, V., & Sjölander, A. (2025) · Epidemiologic Methods · doi:10.1515/em-2025-0018
“Indeed, for both models, these issues were no longer observed at sample sizes of n = 800 and above.”
Kim, J., Kim, M., & Lee, W. (2025) · Prevention Science · doi:10.1007/s11121-025-01858-z
“…can draw on existing methodological work to construct sensitivity analyses by positing hypothetical confounders along the exposure–mediator, mediator–outcome, or exposure–outcome pathways and specifying the plausible strength of their associations (VanderWeele, 2010; VanderWeele & Chiba, 2014).”
Chen, S., Wang, X., Lee, B. K., & Gardner, R. M. (2023) · Figshare · doi:10.6084/m9.figshare.24522559
Lindmark, A., Norrving, B., & Eriksson, M. (2020) · Figshare · doi:10.6084/m9.figshare.12197436
Valdes, A. M., Louca, P., Visconti, A., Asnicar, F., Bermingham, K. M., Nogal, A., Wong, K., Michelotti, G. A., Wolf, J., Segata, N., et al. (2024) · Figshare · doi:10.6084/m9.figshare.26758532
Additional file 1 of CoCoA-diff: counterfactual inference for single-cell gene expression analysis
Park, Y., & Kellis, M. (2021) · Figshare · doi:10.6084/m9.figshare.15184832
Lindmark, A., Norrving, B., & Eriksson, M. (2020) · Figshare · doi:10.6084/m9.figshare.12197445
Additional file 2 of CoCoA-diff: counterfactual inference for single-cell gene expression analysis
Park, Y., & Kellis, M. (2021) · Figshare · doi:10.6084/m9.figshare.15184835
Lindmark, A., Norrving, B., & Eriksson, M. (2020) · Figshare · doi:10.6084/m9.figshare.12197451
Designing Multisite Randomized Trials to Detect (Conditional) Indirect Effects
Bai, F., Kelcey, B., Ataneka, A., Xie, Y., Cox, K., & Dong, N. (2025) · American Journal of Evaluation · doi:10.1177/10982140251394367
“Prior studies have emphasized the importance of contextual effects — that is, site-level processes that operate in complement to individual-level changes (e.g., Pituch & Stapleton, 2012; Skinns et al., 2020).”
InfluentialAssessing surrogate heterogeneity in real world data using meta-learners
Knowlton, R. K., & Parast, L. M. (2026) · Journal of Causal Inference · doi:10.1515/jci-2025-0033
Overview of mediation methods: from estimands to estimations
Lepage, B., Colineaux, H., Garès, V., Bodinier, B., Delpierre, C., & Chadeau‐Hyam, M. (2025) · Exposome · doi:10.1093/exposome/osag010
Xiong, Y., Ong, C. H., Wang, Z., & Liu, Z. (2026) · Frontiers in Sustainable Food Systems · doi:10.3389/fsufs.2026.1757799
Elmer, J., DiFiore, S., Guyette, F. X., Patterson, C. G., Ventz, S., Callaway, C. W., Anderson, M. W., Callaway, C. W., Condle, J. P., Coppler, P. J., et al. (2026) · Resuscitation Plus · doi:10.1016/j.resplu.2026.101307
Saito, R., & Watanabe, K. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6699078
Xu, H., Zhu, Y., & Coffman, D. L. (2026) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.1093/jrsssc/qlag029
Pacheco, J. N., Poček, J., Turró, A., & Urbano, D. (2026) · Business Strategy and the Environment · doi:10.1002/bse.71071
“Unlike the traditional Baron and Kenny (1986) approach, this framework provides formal definitions of natural direct and indirect effects that allow for proper total effect decomposition even when the exposure and the mediator interact in their effect on the outcome (VanderWeele 2010).”
Ololade, S. O., Zúñiga, M. L., Morojele, N. K., Garfein, R. S., Mattson, S. N., Kiene, S. M., & Chambers, C. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6948787
Bayesian mediation analysis for zero-inflated mediator
Cui, J., Zhou, X., Smith, M. J., Sims, A., Leach, J. M., Long, D. L., & Yi, N. (2026) · Statistical Methods in Medical Research · doi:10.1177/09622802261468022 · Cited by 1
“5,37 Under the Bayesian framework, an unmeasured confounder can be incorporated by defining a prior distribution that affects X − M , M − Y and/or X − Y pathways.”
Zhang, H., & Jetley, G. (2026) · Production and Operations Management · doi:10.1177/10591478261474395
Nowicka, N., Remsö, A., & Renström, E. A. (2026) · Scandinavian Journal of Psychology · doi:10.1111/sjop.70138
Liu, X., & McCann, C. (2026) · British Journal of Mathematical and Statistical Psychology · doi:10.1111/bmsp.70072
Jia, Z., Yang, Y., Wang, Z., Ming, X., Ge, S., & Xu, H. (2026) · Atmosphere · doi:10.3390/atmos17100960
“…[44] is particularly demanding in urban settings, where unmeasured factors such as soil quality and irrigation infrastructure may confound the mediator– outcome relationship; the reported mediation proportions should therefore be treated as suggestive pending sensitivity analyses on real data [45].”
InfluentialPotential Outcomes Effects in Mediation Designs with Three Experimental Conditions
Alvarez-Bartolo, D., Smyth, H. L., Brincks, A. M., & MacKinnon, D. P. (2026) · Prevention Science · doi:10.1007/s11121-026-01983-3
Digital Trade Rules and Embodied Carbon Emissions: A Scale–Composition–Technique Analysis
Zhou, Y., & Dong, M. (2026) · Sustainability · doi:10.3390/su182010322
Rojas, J. D., Huarachi-Napa, C. R., Salazar-Buitron, S. Z., Quillo-Diaz, A. P., & Calderon, R. I. (2026) · Vaccine · doi:10.1016/j.vaccine.2026.129286
Metadata
- Authors
- VanderWeele, T. J.
- Affiliations
- Harvard University (US)
- Corresponding author
- VanderWeele, Tyler J.
- Venue
- Epidemiology
- Published
- 2010-07
- Volume
- 21
- Issue
- 4
- Pages
- 540-551
- ISSN
- 1044-3983 (ISSN-L)
- PMC
- PMC4231822
- OpenAlex
- W2075031358
- Cited by
- 435
- References
- 42
- Citations by year
- 2026: 20 · 2025: 16 · 2024: 15 · 2023: 25 · 2022: 32 · 2021: 28 · 2020: 29 · 2019: 38 · 2018: 21 · 2017: 23 · 2016: 34 · 2015: 32 · 2014: 33 · 2013: 27 · 2012: 34
- FWCI
- 23.96
- Type
- journal article
- Language
- en
- Open access
- green
- www.ncbi.nlm.nih.gov
- Locations
- Epidemiology (publishedVersion)
- PubMed (publishedVersion)
- PubMed Central (acceptedVersion)
- Funders
- National Institute of Environmental Health Sciences (R01ES017876)
- 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 in Epidemiology
- Social Sciences › Decision Sciences › Statistics, Probability and Uncertainty › Meta-analysis and systematic reviews
- Keywords
- direct and indirect effects, mediation analysis, sensitivity analysis, confounding, bias correction
- MeSH terms
- Brain, Cognition Disorders, Humans, Lead Poisoning, Organ Size, Regression Analysis, Risk Factors, Sensitivity and Specificity, Statistics as Topic, Models, Statistical, Bias
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
