Home Medicine Heterogeneous indirect effects for multiple mediators using interventional effect models
Article
Licensed
Unlicensed Requires Authentication

Heterogeneous indirect effects for multiple mediators using interventional effect models

  • EMAIL logo , , and
Published/Copyright: January 2, 2021
Epidemiologic Methods
From the journal Epidemiologic Methods Volume 9 Issue 1

Abstract

Decomposing an exposure effect on an outcome into separate natural indirect effects through multiple mediators requires strict assumptions, such as correctly postulating the causal structure of the mediators, and no unmeasured confounding among the mediators. In contrast, interventional indirect effects for multiple mediators can be identified even when – as often – the mediators either have an unknown causal structure, or share unmeasured common causes, or both. Existing estimation methods for interventional indirect effects require calculating each distinct indirect effect in turn. This can quickly become unwieldy or unfeasible, especially when investigating indirect effect measures that may be modified by observed baseline characteristics. In this article, we introduce simplified estimation procedures for such heterogeneous interventional indirect effects using interventional effect models. Interventional effect models are a class of marginal structural models that encode the interventional indirect effects as causal model parameters, thus readily permitting effect modification by baseline covariates using (statistical) interaction terms. The mediators and outcome can be continuous or noncontinuous. We propose two estimation procedures: one using inverse weighting by the counterfactual mediator density or mass functions, and another using Monte Carlo integration. The former has the advantage of not requiring an outcome model, but is susceptible to finite sample biases due to highly variable weights. The latter has the advantage of consistent estimation under a correctly specified (parametric) outcome model, but is susceptible to biases due to extrapolation. The estimators are illustrated using publicly available data assessing whether the indirect effects of self-efficacy on fatigue via self-reported post-traumatic stress disorder symptoms vary across different levels of negative coping among health care workers during the COVID-19 outbreak.

Keywords: conditional effects; direct and indirect effects; effect decomposition; effect modification; natural effect models
Corresponding author: Wen Wei Loh, Department of Data Analysis, Ghent University, Gent, Belgium, E-mail:

Funding source: Research Foundation - Flanders (FWO)

Award Identifier / Grant number: G019317N

Acknowledgments

The authors would like to thank the Editor, and three reviewers, for their comments on prior versions of this manuscript. Computational resources and services were provided by the VSC (Flemish Supercomputer Center), funded by the FWO and the Flemish Government - department EWI.

  1. Research funding: This research was supported by the Research Foundation - Flanders (FWO) under Grant G019317N.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission. The content is solely the responsibility of the authors and does not represent the official views of the authors’ institutions or FWO.

  3. Competing interests: Authors state no conflict of interest.

  4. Informed consent: Not applicable.

References

Albert, J. M., J. I. Cho, Y. Liu, and S. Nelson. 2019. “Generalized Causal Mediation and Path Analysis: Extensions and Practical Considerations.” Statistical Methods in Medical Research 28 (6): 1793–807, https://doi.org/10.1177/0962280218776483.Search in Google Scholar PubMed PubMed Central

Andrews, R. M., and V. Didelez. 2020. “Insights into the “Cross-World” Independence Assumption of Causal Mediation Analysis.” arXiv Preprint, arXiv:2003.10341.10.1097/EDE.0000000000001313Search in Google Scholar PubMed

Avin, C., I. Shpitser, and J. Pearl. 2005. “Identifiability of Path-Specific Effects.” In Proceedings of the 19th International Joint Conference on Artificial Intelligence, 357–63. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.Search in Google Scholar

Dai, J. Y., J. L. Stanford, and M. LeBlanc. 2020. “A Multiple-Testing Procedure for High-Dimensional Mediation Hypotheses.” Journal of the American Statistical Association: 1–16, https://doi.org/10.1080/01621459.2020.1765785.Search in Google Scholar PubMed PubMed Central

Daniel, R. M., B. L. De Stavola, S. N. Cousens, and S. Vansteelandt. 2015. “Causal Mediation Analysis with Multiple Mediators.” Biometrics 71 (1): 1–14, https://doi.org/10.1111/biom.12248.Search in Google Scholar PubMed PubMed Central

Derkach, A., S. C. Moore, S. M. Boca, and J. N. Sampson. 2020. “Group Testing in Mediation Analysis.” Statistics in Medicine 39 (18): 2423–36, https://doi.org/10.1002/sim.8546.Search in Google Scholar PubMed PubMed Central

Didelez, V., A. P. Dawid, and S. Geneletti. 2006. “Direct and Indirect Effects of Sequential Treatments.” In Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence, 138–46. Arlington, VA, USA: AUAI Press.Search in Google Scholar

Efron, B., and R. J. Tibshirani. 1994. An Introduction to the Bootstrap. New York, NY: Chapman and Hall/CRC.10.1201/9780429246593Search in Google Scholar

Geneletti, S. 2007. “Identifying Direct and Indirect Effects in a Non‐counterfactual Framework.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 69: 199–215, https://doi.org/10.1111/j.1467-9868.2007.00584.x.Search in Google Scholar

Greenland, S., J. M. Robins, and J. Pearl. 1999. “Confounding and Collapsibility in Causal Inference.” Statistical Science 14 (1): 29–46.10.1214/ss/1009211805Search in Google Scholar

Hong, G. 2010. “Ratio of Mediator Probability Weighting for Estimating Natural Direct and Indirect Effects.” In Proceedings of the American Statistical Association, Biometrics Section, edited by American Statistical Association, 2401–15. Alexandria, VA, USA.Search in Google Scholar

Hou, T., W. Dong, R. Zhang, X. Song, F. Zhang, W. Cai, Y. Liu, and G. Deng. 2020. “Self-Efficacy and Fatigue Among Health Care Workers During COVID-19 Outbreak: A Moderated Mediation Model of Posttraumatic Stress Disorder Symptoms and Negative Coping.” Preprint (Version 1). Available at Research Square, https://doi.org/10.21203/rs.3.rs-23066/v1.Search in Google Scholar

Huang, Y.-T., and W.-C. Pan. 2016. “Hypothesis Test of Mediation Effect in Causal Mediation Model with High-Dimensional Continuous Mediators.” Biometrics 72 (2): 402–13, https://doi.org/10.1111/biom.12421.Search in Google Scholar PubMed

Imai, K., and M. Ratkovic. 2013. “Estimating Treatment Effect Heterogeneity in Randomized Program Evaluation.” Annals of Applied Statistics 7 (1): 443–70, https://doi.org/10.1214/12-aoas593.Search in Google Scholar

Imai, K., and M. Ratkovic. 2015. “Robust Estimation of Inverse Probability Weights for Marginal Structural Models.” Journal of the American Statistical Association 110 (511): 1013–23, https://doi.org/10.1080/01621459.2014.956872.Search in Google Scholar

Jackson, J. W., and T. J. VanderWeele. 2018. “Decomposition Analysis to Identify Intervention Targets for Reducing Disparities.” Epidemiology 29 (6): 825–35, https://doi.org/10.1097/EDE.0000000000000901.Search in Google Scholar PubMed PubMed Central

Kennedy, E. H. 2020. “Optimal Doubly Robust Estimation of Heterogeneous Causal Effects.” arXiv preprint arXiv:2004.14497.Search in Google Scholar

Lange, T., M. Rasmussen, and L. C. Thygesen. 2013. “Assessing Natural Direct and Indirect Effects Through Multiple Pathways.” American Journal of Epidemiology 179 (4): 513–8, https://doi.org/10.1093/aje/kwt270.Search in Google Scholar PubMed

Lange, T., S. Vansteelandt, and M. Bekaert. 2012. “A Simple Unified Approach for Estimating Natural Direct and Indirect Effects.” American Journal of Epidemiology 176 (3): 190–5, https://doi.org/10.1093/aje/kwr525.Search in Google Scholar PubMed

Lin, S.-H., and T. VanderWeele. 2017. “Interventional Approach for Path-Specific Effects.” Journal of Causal Inference 5 (1), https://doi.org/10.1515/jci-2015-0027.Search in Google Scholar

Lok, J. J. Mar 2019. “Causal Organic Direct and Indirect Effects: Closer to Baron and Kenny.” arXiv Preprint, art. arXiv:1903.04697.Search in Google Scholar

Meng, X.-L. 1994. “Multiple-Imputation Inferences with Uncongenial Sources of Input.” Statistical Science 9 (4): 538–58, https://doi.org/10.1214/ss/1177010269.Search in Google Scholar

Micali, N., R. M. Daniel, G. B. Ploubidis, and B. L. De Stavola. 2018. “Maternal Prepregnancy Weight Status and Adolescent Eating Disorder Behaviors: A Longitudinal Study of Risk Pathways.” Epidemiology 29 (4): 579–89, https://doi.org/10.1097/ede.0000000000000850.Search in Google Scholar

Moreno-Betancur, M., and J. B. Carlin. 2018. “Understanding Interventional Effects: A More Natural Approach to Mediation Analysis?” Epidemiology 29 (5): 614–7, https://doi.org/10.1097/EDE.0000000000000866.Search in Google Scholar PubMed

Moreno-Betancur, M., P. Moran, D. Becker, G. Patton, and J. B. Carlin. July 2020. “Mediation Effects that Emulate a Target Randomised Trial: Simulation-Based Evaluation of Ill-Defined Interventions on Multiple Mediators.” arXiv e-prints, art. arXiv:1907.06734.10.1177/0962280221998409Search in Google Scholar PubMed PubMed Central

Naimi, A. I., J. S. Kaufman, and R. F. MacLehose. 2014. “Mediation Misgivings: Ambiguous Clinical and Public Health Interpretations of Natural Direct and Indirect Effects.” International Journal of Epidemiology 43 (5): 1656–61, https://doi.org/10.1093/ije/dyu107.Search in Google Scholar PubMed

Nguyen, T. Q., I. Schmid, and E. A. Stuart. Apr 2019. “Clarifying Causal Mediation Analysis for the Applied Researcher: Defining Effects Based on What We Want to Learn.” arXiv Preprint, art. arXiv:1904.08515.10.1037/met0000299Search in Google Scholar PubMed PubMed Central

Nie, X., and S. Wager. 2020. “Quasi-Oracle Estimation of Heterogeneous Treatment Effects.” arXiv preprint arXiv:1712.04912.10.1093/biomet/asaa076Search in Google Scholar

Noguchi, Y. 2020. Pandemic Affects Mental Health of Frontline Health Workers. Also available at https://www.npr.org/2020/04/22/841925658/pandemic-affects-mental-health-of-frontline-health-workers.Search in Google Scholar

Pearl, J. 2001. “Direct and Indirect Effects.” In Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence, 411–20. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.10.1145/3501714.3501736Search in Google Scholar

Pearl, J. 2009. Causality: Models, Reasoning and Inference, 2nd ed. New York, NY, USA: Cambridge University Press.10.1017/CBO9780511803161Search in Google Scholar

Petersen, M. L., S. E. Sinisi, and M. J. van der Laan. 2006. “Estimation of Direct Causal Effects.” Epidemiology 17 (3): 276–84, https://doi.org/10.1097/01.ede.0000208475.99429.2d.Search in Google Scholar PubMed

Robins, J. M., and S. Greenland. 1992. “Identifiability and Exchangeability for Direct and Indirect Effects.” Epidemiology 3 (2): 143–55, https://doi.org/10.1097/00001648-199203000-00013.Search in Google Scholar PubMed

Robins, J. M., and T. S. Richardson. 2010. Alternative Graphical Causal Models and the Identification of Direct Effects, 103–58. New York, NY, USA: Oxford University Press. ISBN 9780199754649.10.1093/oso/9780199754649.003.0011Search in Google Scholar

Robins, J. M. 2000. Marginal Structural Models versus Structural Nested Models as Tools for Causal Inference, 95–133. New York, NY, USA: Springer-Verlag.10.1007/978-1-4612-1284-3_2Search in Google Scholar

Santacatterina, M., C. García-Pareja, R. Bellocco, A. Sönnerborg, A. M. Ekström, and M. Bottai. 2019. “Optimal Probability Weights for Estimating Causal Effects of Time-Varying Treatments with Marginal Structural Cox Models.” Statistics in Medicine 38 (10): 1891–902, https://doi.org/10.1002/sim.8080.Search in Google Scholar PubMed

Snowden, J. M., S. Rose, and K. M. Mortimer. 2011. “Implementation of G-Computation on a Simulated Data Set: Demonstration of a Causal Inference Technique.” American Journal of Epidemiology 173 (7): 731–8, https://doi.org/10.1093/aje/kwq472.Search in Google Scholar PubMed PubMed Central

Steen, J., T. Loeys, B. Moerkerke, and S. Vansteelandt. 2017. “Flexible Mediation Analysis with Multiple Mediators.” American Journal of Epidemiology 186 (2): 184–93, https://doi.org/10.1093/aje/kwx051.Search in Google Scholar PubMed

Taguri, M., J. Featherstone, and J. Cheng. 2018. “Causal Mediation Analysis with Multiple Causally Non-Ordered Mediators.” Statistical Methods in Medical Research 27 (1): 3–19, https://doi.org/10.1177/0962280215615899.Search in Google Scholar PubMed PubMed Central

Tchetgen Tchetgen, E. J. 2014. “A Note on Formulae for Causal Mediation Analysis in an Odds Ratio Context.” Epidemiologic Methods 2 (1): 21–31, https://doi.org/10.1515/em-2012-0005.Search in Google Scholar PubMed PubMed Central

VanderWeele, T. J., and E. J. Tchetgen Tchetgen. 2017. “Mediation Analysis with Time Varying Exposures and Mediators.” Journal of the Royal Statistical Society: Series B 79 (3): 917–38, https://doi.org/10.1111/rssb.12194.Search in Google Scholar PubMed PubMed Central

VanderWeele, T. J., and S. Vansteelandt. 2010. “Odds Ratios for Mediation Analysis for a Dichotomous Outcome.” American Journal of Epidemiology 172 (12): 1339–48, https://doi.org/10.1093/aje/kwq332.Search in Google Scholar PubMed PubMed Central

VanderWeele, T. J., S. Vansteelandt, and J. M. Robins. 2014. “Effect Decomposition in the Presence of an Exposure-Induced Mediator-Outcome Confounder.” Epidemiology 25 (2): 300, https://doi.org/10.1097/EDE.0000000000000034.Search in Google Scholar PubMed PubMed Central

Vansteelandt, S., M. Bekaert, and T. Lange. 2012. “Imputation Strategies for the Estimation of Natural Direct and Indirect Effects.” Epidemiologic Methods 1 (1): 131–58, https://doi.org/10.1515/2161-962x.1014.Search in Google Scholar

Vansteelandt, S., and R. M. Daniel. 2017. “Interventional Effects for Mediation Analysis with Multiple Mediators.” Epidemiology 28 (2): 258–65, https://doi.org/10.1097/EDE.0000000000000596.Search in Google Scholar PubMed PubMed Central

Vansteelandt, S., and O. Dukes. 2020. “Assumption-Lean Inference for Generalised Linear Model Parameters.” arXiv preprint arXiv:2006.08402.Search in Google Scholar

Vansteelandt, S., and N. Keiding. 2011. “Invited Commentary: G-Computation–Lost in Translation?” American Journal of Epidemiology 173 (7): 739–42, https://doi.org/10.1093/aje/kwq474.Search in Google Scholar PubMed

Vansteelandt, S., and T. J. VanderWeele. 2012. “Natural Direct and Indirect Effects on the Exposed: Effect Decomposition under Weaker Assumptions.” Biometrics 68 (4): 1019–27, https://doi.org/10.1111/j.1541-0420.2012.01777.x.Search in Google Scholar PubMed PubMed Central

Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/em-2020-0023).

Received: 2020-07-15
Accepted: 2020-12-08
Published Online: 2021-01-02

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Editorial
  2. The mean prevalence
  3. Research Articles
  4. Heterogeneous indirect effects for multiple mediators using interventional effect models
  5. Sleep habits and their association with daytime sleepiness among medical students of Tanta University, Egypt
  6. Population attributable fractions for continuously distributed exposures
  7. A real-time search strategy for finding urban disease vector infestations
  8. Disease mapping models for data with weak spatial dependence or spatial discontinuities
  9. A comparison of cause-specific and competing risk models to assess risk factors for dementia
  10. A simple index of prediction accuracy in multiple regression analysis
  11. A comparison of approaches for estimating combined population attributable risks (PARs) for multiple risk factors
  12. Posterior predictive treatment assignment methods for causal inference in the context of time-varying treatments
  13. Random effects tumour growth models for identifying image markers of mammography screening sensitivity
  14. Extrapolating sparse gold standard cause of death designations to characterize broader catchment areas
  15. Extending balance assessment for the generalized propensity score under multiple imputation
  16. Regression analysis of unmeasured confounding
  17. The Use of Logic Regression in Epidemiologic Studies to Investigate Multiple Binary Exposures: An Example of Occupation History and Amyotrophic Lateral Sclerosis
  18. Meeting the Assumptions of Inverse-Intensity Weighting for Longitudinal Data Subject to Irregular Follow-Up: Suggestions for the Design and Analysis of Clinic-Based Cohort Studies
https://doi.org/10.1515/em-2020-0023
Keywords for this article
conditional effects; direct and indirect effects; effect decomposition; effect modification; natural effect models

Articles in the same Issue

  1. Editorial
  2. The mean prevalence
  3. Research Articles
  4. Heterogeneous indirect effects for multiple mediators using interventional effect models
  5. Sleep habits and their association with daytime sleepiness among medical students of Tanta University, Egypt
  6. Population attributable fractions for continuously distributed exposures
  7. A real-time search strategy for finding urban disease vector infestations
  8. Disease mapping models for data with weak spatial dependence or spatial discontinuities
  9. A comparison of cause-specific and competing risk models to assess risk factors for dementia
  10. A simple index of prediction accuracy in multiple regression analysis
  11. A comparison of approaches for estimating combined population attributable risks (PARs) for multiple risk factors
  12. Posterior predictive treatment assignment methods for causal inference in the context of time-varying treatments
  13. Random effects tumour growth models for identifying image markers of mammography screening sensitivity
  14. Extrapolating sparse gold standard cause of death designations to characterize broader catchment areas
  15. Extending balance assessment for the generalized propensity score under multiple imputation
  16. Regression analysis of unmeasured confounding
  17. The Use of Logic Regression in Epidemiologic Studies to Investigate Multiple Binary Exposures: An Example of Occupation History and Amyotrophic Lateral Sclerosis
  18. Meeting the Assumptions of Inverse-Intensity Weighting for Longitudinal Data Subject to Irregular Follow-Up: Suggestions for the Design and Analysis of Clinic-Based Cohort Studies
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Winner of the OpenAthens UX Award 2026
Winner of the OpenAthens UX Award 2026
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 23.3.2026 from https://www.degruyterbrill.com/document/doi/10.1515/em-2020-0023/html
Scroll to top button