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.
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Detecting moderator effects using subgroup analyses
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“…ways to deal with the mathematics of determining this ‘‘cut-point,’’ including average causal mediation (e.g., Imai, Keele, & Tingley, 2010), the confounder bias method (VanderWeele, 2010); left out variables error method (e.g., Cox, Kisbu-Sakarya, Miočević, & MacKinnon, 2013; Mauro, 1990).”
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“Further, to be able to claim that the effects obtained are causal, the following assumptions need to be fulfilled: (i) no unmeasured confounding between the exposure– mediator relationship; (ii) no unmeasured confounding between the mediator–outcome relationship; (iii) no unmeasured confounding between the exposure–ou…”
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Self-citationCollider Bias Is Only a Partial Explanation for the Obesity Paradox
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“Here the terms direct and indirect are used in their infectious disease sense, which differs from that typical in the causal inference literature.(52).”
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VanderWeele, T. J. (2013) · Biometrics · doi:10.1111/biom.12071 · Cited by 125 · see citing works
Self-citationAnanth, C. V., & VanderWeele, T. J. (2011) · American Journal of Epidemiology · doi:10.1093/aje/kwr045 · Cited by 123 · see citing works
“In order to test the impact of potential unmeasured confounders on the direct and indirect effects of the abruption-mortality relation, we performed a series of sensitivity analyses (4).”
BackgroundMethodInfluentialVanderWeele, T. J., Valeri, L., & Ogburn, E. L. (2012) · Epidemiology · doi:10.1097/ede.0b013e318258f5e4 · Cited by 123 · see citing works
Self-citationDunn, G., Emsley, R., Liu, H., Landau, S., Green, J., White, I. R., & Pickles, A. R. (2015) · Health Technology Assessment · doi:10.3310/hta19930 · Cited by 122 · see citing works
Does selection bias explain the obesity paradox among individuals with cardiovascular disease?
Banack, H. R., & Kaufman, J. S. (2015) · Annals of Epidemiology · doi:10.1016/j.annepidem.2015.02.008 · Cited by 119 · see citing works
“To explore the possibility that the protective effect of obesity on mortality among individuals with CVD is due to unmeasured confounding of the CVD-mortality relationship by CRF (Figure 1b), we conducted sensitivity analyses for the controlled direct effect(18, 47).”
BackgroundMethodInfluentialStanton, A. L., Luecken, L. J., MacKinnon, D. P., & Thompson, E. H. (2013) · Journal of Consulting and Clinical Psychology · doi:10.1037/a0028833 · Cited by 114 · see citing works
Breastfeeding and Oral Health: Evidence and Methodological Challenges
Peres, K. G., Chaffee, B. W., Feldens, C. A., Flores‐Mir, C., Moynihan, P. J., & Rugg-Gunn, A. J. (2017) · Journal of Dental Research · doi:10.1177/0022034517738925 · Cited by 117 · see citing works
Parent–child interaction, maternal depressive symptoms and preterm infant cognitive function
McManus, B. M., & Poehlmann, J. (2012) · Infant Behavior and Development · doi:10.1016/j.infbeh.2012.04.005 · Cited by 114 · see citing works
“…interval between the exposure (maternal clinical depressive symptoms), mediators (infant emotional/ behavioral regulation and parenting quality), and outcome (child’s cognitive function) separately for each social support category using methods described by Hayes (2010) and Vanderweele (2010).”
MethodBécares, L., Nazroo, J. Y., & Kelly, Y. J. (2015) · Social Science & Medicine · doi:10.1016/j.socscimed.2015.08.025 · Cited by 112 · see citing works
“Assumptions (a), (b), and (c) also require an assumption of temporal ordering (D. Cole and Maxwell, 2003; VanderWeele, 2010), whereby (a) the assumption of no unmeasured confounding of the exposure-outcome association implicitly assumes that the treatment temporally precedes the outcome; (b) the…”
BackgroundInfluentialConfounding in statistical mediation analysis: What it is and how to address it.
Valente, M. J., Pelham, W. E., Smyth, H. L., & MacKinnon, D. P. (2017) · Journal of Counseling Psychology · doi:10.1037/cou0000242 · Cited by 113 · see citing works
“Cox et al. (2014) described three methods for sensitivity analyses to test the robustness of the mediated effect to the presence of unmeasured confounders: the L.O.V.E. method (Mauro, 1990), the correlated residuals method (Imai, Keele, & Tingley, 2010), and VanderWeele’s method (VanderWeele, 2010).”
MethodRueda, D., & Stegmueller, D. (2019) · Cambridge University Press eBooks · doi:10.1017/9781108681339 · Cited by 113 · see citing works
Principal Stratification -- Uses and Limitations
VanderWeele, T. J. (2011) · The International Journal of Biostatistics · doi:10.2202/1557-4679.1329 · Cited by 109 · see citing works
“…cation of these principal strata effects from observed data but sensitivity analysis techniques can be used to address principal strata direct effects (Sjölander et al., 2009; VanderWeele, 2010a; Chiba, 2010) and Bayesian methods have also been employed (Gallop et al., 2009; Elliott et al., 2010).”
BackgroundMethodSelf-citationRecent Developments in Causal Inference and Machine Learning
Brand, J. E., Zhou, X., & Xie, Y. F. (2023) · Annual Review of Sociology · doi:10.1146/annurev-soc-030420-015345 · Cited by 124 · see citing works
“In practice, to assess the robustness of mediation analysis results to different forms of unobserved confounding, one can employ a set of general-purpose bias formulas developed by VanderWeele (2010) and VanderWeele & Arah (2011) (for a recent sociological application, see Brand et al. 2019b).”
MethodThe Combined Effects of Measurement Error and Omitting Confounders in the Single-Mediator Model
Fritz, M. S., Kenny, D. A., & MacKinnon, D. P. (2016) · Multivariate Behavioral Research · doi:10.1080/00273171.2016.1224154 · Cited by 104 · see citing works
“In addition, multiple authors have explored methods to investigate the sensitivity of results to confounding (Cox, Kisbu-Sakarya, Miočević, &MacKinnon, 2014; Imai et al., 2010; Imai & Yamamoto, 2013; Liu et al., 2013;MacKinnon&Pirlott, 2015; VanderWeele, 2008, 2010, 2013).”
BackgroundInvestigating and Remediating Selection Bias in Geriatrics Research: The Selection Bias Toolkit
Banack, H. R., Kaufman, J. S., Wactawski‐Wende, J., Troen, B. R., & Stovitz, S. D. (2019) · Journal of the American Geriatrics Society · doi:10.1111/jgs.16022 · Cited by 105 · see citing works
“Sensitivity analysis or bias analysis are analytic tools that can be used to understand whether study results are affected by selection bias.(26-30) These are supplementary analyses that involve empirically examining the influence of selection factors on study results.”
MethodMammographic density and breast cancer risk: a mediation analysis
Rice, M. S., Bertrand, K. A., VanderWeele, T. J., ROSNER, B. A., Liao, X., Adami, H., & Tamimi, R. M. (2016) · Breast Cancer Research · doi:10.1186/s13058-016-0750-0 · Cited by 103 · see citing works
“We further used sensitivity analyses to assess how robust evidence for mediation was to potential unmeasured common causes of MD and breast cancer risk [9, 22].”
BackgroundMethodSelf-citationG-computation demonstration in causal mediation analysis
Wang, A., & Arah, O. A. (2015) · European Journal of Epidemiology · doi:10.1007/s10654-015-0100-z · Cited by 100 · see citing works
“In the presence of possible violation of this assumption, sensitivity analysis are needed [3, 20].”
BackgroundLapointe‐Shaw, L., Bouck, Z., Howell, N. A., Lange, T., Orchanian‐Cheff, A., Austin, P. C., Ivers, N. M., Redelmeier, D. A., & Bell, C. M. (2018) · BMC Medical Research Methodology · doi:10.1186/s12874-018-0578-7 · Cited by 97 · see citing works
Practical challenges in mediation analysis: a guide for applied researchers
Schuler, M. S., Coffman, D. L., Stuart, E. A., Nguyen, T. Q., Vegetabile, B., & McCaffrey, D. F. (2024) · Health Services and Outcomes Research Methodology · doi:10.1007/s10742-024-00327-4 · Cited by 95 · see citing works
“VanderWeele proposed a sensitivity analysis for the no-unmeasured-confounding assumptions used in identifying controlled direct effects (VanderWeele 2010) as well as sensitivity analyses for natural direct and indirect effects regarding mediator-outcome confounders affected by the exposure (i.e.,…”
Ozer, E. J., Fernald, L. C. H., Weber, A. M., Flynn, E. P., & VanderWeele, T. J. (2011) · International Journal of Epidemiology · doi:10.1093/ije/dyr103 · Cited by 94 · see citing works
Self-citationTrani, J., Moodley, J., Anand, P., Graham, L., & Maw, M. T. T. (2020) · Social Science & Medicine · doi:10.1016/j.socscimed.2020.113449 · Cited by 93 · see citing works
“We controlled for the effect of these covariates on the exposure-mediator confounding, mediator-outcome confounding and exposure-outcome confounding using sensitivity analysis techniques to explore and rule out alternative assumptions and biases (Vanderweele 2010; Vanderweele and Chiba 2014).”
MethodControlled Direct and Mediated Effects: Definition, Identification and Bounds
VanderWeele, T. J. (2010) · Scandinavian Journal of Statistics · doi:10.1111/j.1467-9469.2010.00722.x · Cited by 92 · see citing works
“A complementary approach to direct and indirect effects using sensitivity analysis rather than bounds to assess unmeasured confounding is presented elsewhere (VanderWeele, 2010).”
BackgroundSelf-citationOn Causal Mediation Analysis with a Survival Outcome
Tchetgen, E. J. T. (2011) · The International Journal of Biostatistics · doi:10.2202/1557-4679.1351 · Cited by 87 · see citing works
“…of techniques have previously been proposed for conducting sensitivity analyses for unmeasured confounding in the context of mediation, for example, VanderWeele (2010), Imai et al (2010a), Tchetgen Tchetgen and Shpitser (2011a,b), none of the existing techniques apply to mediation in the survival…”
VanderWeele, T. J., Hong, G., Jones, S. M., & Brown, J. L. (2013) · Journal of the American Statistical Association · doi:10.1080/01621459.2013.779832 · Cited by 85 · see citing works
“Specifically, our work extends beyond the results of VanderWeele (2010a) by (i) giving counterfactual definitions for within-classroom and spillover mediated effects and showing that not only does the total effect of the intervention decompose into a direct effect and an indirect effect but also…”
BackgroundMethodResultInfluentialSelf-citationLife Course Socioeconomic Position and Mid-Late Life Cognitive Function in Eastern Europe
Horvat, P., Richards, M., Malyutina, S. K., Pajak, A., Kubínová, R., Tamošiūnas, A., Pikhart, H., Peasey, A., Marmot, M. G., & Bobák, M. (2014) · The Journals of Gerontology Series B · doi:10.1093/geronb/gbu014 · Cited by 82 · see citing works
“Finally, estimation of direct and indirect effects in mediation analysis assumes there is no unmeasured confounding of the exposure–outcome, mediator–outcome, and exposure–mediator relationships (vanderWeele, 2010).”
BackgroundMediation analysis for a survival outcome with time‐varying exposures, mediators, and confounders
Lin, S., Young, J. G., Logan, R. W., & VanderWeele, T. J. (2017) · Statistics in Medicine · doi:10.1002/sim.7426 · Cited by 78 · see citing works
Self-citationle Cessie, S., Debeij, J., Rosendaal, F. R., Cannegieter, S. C., & Vandenbroucke, J. P. (2012) · Epidemiology · doi:10.1097/ede.0b013e318254f5de · Cited by 81 · see citing works
“Bias formulas for sensitivity analysis for controlled and natural direct-effect and indirect-effect odds ratios have been derived by VanderWeele.(13)”
MethodGene-environment dependencies lead to collider bias in models with polygenic scores
Akimova, E. T., Breen, R., Brazel, D. M., & Mills, M. C. (2021) · Scientific Reports · doi:10.1038/s41598-021-89020-x · Cited by 80 · see citing works
Misclassification of the mediator matters when estimating indirect effects
Blakely, T. A., McKenzie, S., & Carter, K. N. (2013) · Journal of Epidemiology & Community Health · doi:10.1136/jech-2012-201813 · Cited by 76 · see citing works
Sharp sensitivity bounds for mediation under unmeasured mediator-outcome confounding
Ding, P., & VanderWeele, T. J. (2016) · Biometrika · doi:10.1093/biomet/asw012 · Cited by 74 · see citing works
Self-citationLin, S., Young, J. G., Logan, R. W., Tchetgen, E. J. T., & VanderWeele, T. J. (2016) · Epidemiology · doi:10.1097/ede.0000000000000609 · Cited by 72 · see citing works
Self-citationWhen Is the Difference Method Conservative for Assessing Mediation?
Jiang, Z., & VanderWeele, T. J. (2015) · American Journal of Epidemiology · doi:10.1093/aje/kwv059 · Cited by 72 · see citing works
Self-citationRacial Residential Segregation and Preterm Birth
Anthopolos, R., Kaufman, J. S., Messer, L. C., & Miranda, M. L. (2014) · Epidemiology · doi:10.1097/ede.0000000000000079 · Cited by 71 · see citing works
Robust mediation analysis based on median regression.
Yuan, Y., & MacKinnon, D. P. (2013) · Psychological Methods · doi:10.1037/a0033820 · Cited by 71 · see citing works
“More information on recent developments in causal inference for mediation models can be obtained elsewhere (Imai, Keele, & Tingley, 2010; Jo et al., 2011; Muthen, 2011; Pearl, 2010; VanderWeele, 2010; West, 2011).”
A Tutorial in Bayesian Potential Outcomes Mediation Analysis
Miočević, M., González, O., Valente, M. J., & MacKinnon, D. P. (2017) · Structural Equation Modeling A Multidisciplinary Journal · doi:10.1080/10705511.2017.1342541 · Cited by 70 · see citing works
“…some solutions to violations of Assumption 2 (Coffman & Zhong, 2012; Cox, Kisbu-Sakarya, Mio č evi ć , & MacKinnon, 2014; MacKinnon & Pirlott, 2015; VanderWeele, 2010) and Assumption 4 (Daniel, De Stavola, Cousens, & Vansteelandt, 2015; Imai & Yamamoto, 2013; Tchetgen & VanderWeele, 2014; Robins &…”
Sensitivity Plots for Confounder Bias in the Single Mediator Model
Cox, M. G., Kisbu‐Sakarya, Y., Miočević, M., & MacKinnon, D. P. (2013) · Evaluation Review · doi:10.1177/0193841x14524576 · Cited by 70 · see citing works
“The first approach described by VanderWeele (2010) is based on the relation of the confounder to Y and the difference in proportion of persons with the confounder prevalence between treatment groups at the same level of the mediator.”
BackgroundMethodInfluentialA principled approach to mediation analysis in perinatal epidemiology
Ananth, C. V., & Brandt, J. S. (2022) · American Journal of Obstetrics and Gynecology · doi:10.1016/j.ajog.2021.10.028 · Cited by 69 · see citing works
VanderWeele, T. J., & Chiba, Y. (2022) · Epidemiology Biostatistics and Public Health · doi:10.2427/9027 · Cited by 69 · see citing works
“If we had unmeasured mediator-outcome confounding variables that were not affected by exposure we could use sensitivity analysis techniques in VanderWeele [10].”
MethodResultInfluentialSelf-citationCoen, V. D., Vansteelandt, S., Maes, L., Huybrechts, I., Bourdeaudhuij, I. D., & Vereecken, C. A. (2012) · Appetite · doi:10.1016/j.appet.2012.03.024 · Cited by 69 · see citing works
Bias Analysis for Uncontrolled Confounding in the Health Sciences
Arah, O. A. (2015) · Annual Review of Public Health · doi:10.1146/annurev-publhealth-032315-021644 · Cited by 69 · see citing works
Montoya, A. K. (2022) · Multivariate Behavioral Research · doi:10.1080/00273171.2022.2077287 · Cited by 68 · see citing works
“Though this discussion is relatively brief, this is a very active area of research and other resources cover these topics more in-depth (e.g., Bind et al., 2016; Imai, Keele, & Tingley, 2010; Josephy et al., 2015; VanderWeele, 2010, 2015).”
BackgroundStraatmann, V. S., Lai, E. T. C., Lange, T., Campbell, M., Wickham, S. L., Andersen, A. N., Strandberg‐Larsen, K., & Taylor-Robinson, D. C. (2019) · Journal of Epidemiology & Community Health · doi:10.1136/jech-2019-212367 · Cited by 67 · see citing works
A brief primer on conducting regression-based causal mediation analysis.
Li, Y., Yoshida, K., Kaufman, J. S., & Mathur, M. B. (2023) · Psychological Trauma Theory Research Practice and Policy · doi:10.1037/tra0001421 · Cited by 67 · see citing works
Inoue, K., Yan, Q. J., Arah, O. A., Paul, K. C., Walker, D. I., Jones, D. P., & Ritz, B. R. (2020) · Current Environmental Health Reports · doi:10.1007/s40572-020-00284-3 · Cited by 66 · see citing works
Romero, R. J., & Korzeniewski, S. J. (2012) · American Journal of Obstetrics and Gynecology · doi:10.1016/j.ajog.2012.12.026 · Cited by 66 · see citing works
Alegria, M. M., Shrout, P. E., CANINO, G. J., Álvarez, K., Wang, Y., Bird, H., Markle, S. L., Ramos-Olazagasti, M. A., Vila, D., Cook, B. L., et al. (2019) · World Psychiatry · doi:10.1002/wps.20671 · Cited by 66 · see citing works
VanderWeele, T. J. (2013) · European Journal of Epidemiology · doi:10.1007/s10654-013-9770-6 · Cited by 65 · see citing works
“[1] give an example of such sensitivity analysis in their paper using ideas presented in VanderWeele [8].”
InfluentialSelf-citationRichardson, A., Meyer, K. A., Howard, A., Boone-Heinonen, J., Popkin, B., Evenson, K., Shikany, J., Lewis, C., & Gordon-Larsen, P. (2015) · Health and Place · doi:10.1016/j.healthplace.2015.09.003 · Cited by 57 · see citing works
“We assumed our estimates were not confounded by unmeasured factors, but sensitivity methods to address unmeasured bias (VanderWeele, 2010) do not exist for longitudinal SEMs.”
BackgroundGenome-wide analyses of sparse mediation effects under composite null hypotheses
Huang, Y. (2019) · The Annals of Applied Statistics · doi:10.1214/18-aoas1181 · Cited by 63 · see citing works
“Sensitivity analyses may be conducted to evaluate the influence by the potential confounding (Imai et al., 2010; VanderWeele, 2010).”
BackgroundMendola, P., Mumford, S. L., Männistö, T. I., Holston, A. M., Reddy, U. M., & Laughon, S. K. (2014) · Epidemiology · doi:10.1097/ede.0000000000000213 · Cited by 61 · see citing works
Mediation Analysis with Survival Outcomes: Accelerated Failure Time vs. Proportional Hazards Models
Gelfand, L. A., MacKinnon, D. P., DeRubeis, R. J., & Baraldi, A. N. (2016) · Frontiers in Psychology · doi:10.3389/fpsyg.2016.00423 · Cited by 59 · see citing works
“…article, methods for performing mediation analysis with other types of data, such as dichotomous (MacKinnon and Dwyer, 1993; MacKinnon et al., 2007; VanderWeele and Vansteelandt, 2010) or longitudinal dependent variables (see MacKinnon, 2008 for review) are now receiving increasing attention.”
MethodNonparametric Bounds and Sensitivity Analysis of Treatment Effects
Richardson, A., Hudgens, M. G., Gilbert, P. B., & Fine, J. P. (2014) · Statistical Science · doi:10.1214/14-sts499 · Cited by 55 · see citing works
“For further details regarding bounds and sensitivity analysis in mediation analysis, see Imai, Keele and Yamamoto (2010), VanderWeele (2010) and Hafeman (2011).”
BackgroundMethod
- 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
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