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Targeted Maximum Likelihood Estimation of Natural Direct Effects

Author

Listed:
  • Zheng Wenjing

    (University of California, Berkeley)

  • van der Laan Mark J.

    (University of California, Berkeley)

Abstract

In many causal inference problems, one is interested in the direct causal effect of an exposure on an outcome of interest that is not mediated by certain intermediate variables. Robins and Greenland (1992) and Pearl (2001) formalized the definition of two types of direct effects (natural and controlled) under the counterfactual framework. The efficient scores (under a nonparametric model) for the various natural effect parameters and their general robustness conditions, as well as an estimating equation based estimator using the efficient score, are provided in Tchetgen Tchetgen and Shpitser (2011b). In this article, we apply the targeted maximum likelihood framework of van der Laan and Rubin (2006) and van der Laan and Rose (2011) to construct a semiparametric efficient, multiply robust, substitution estimator for the natural direct effect which satisfies the efficient score equation derived in Tchetgen Tchetgen and Shpitser (2011b). We note that the robustness conditions in Tchetgen Tchetgen and Shpitser (2011b) may be weakened, thereby placing less reliance on the estimation of the mediator density. More precisely, the proposed estimator is asymptotically unbiased if either one of the following holds: i) the conditional mean outcome given exposure, mediator, and confounders, and the mediated mean outcome difference are consistently estimated; (ii) the exposure mechanism given confounders, and the conditional mean outcome are consistently estimated; or (iii) the exposure mechanism and the mediator density, or the exposure mechanism and the conditional distribution of the exposure given confounders and mediator, are consistently estimated. If all three conditions hold, then the effect estimate is asymptotically efficient. Extensions to the natural indirect effect are also discussed.

Suggested Citation

  • Zheng Wenjing & van der Laan Mark J., 2012. "Targeted Maximum Likelihood Estimation of Natural Direct Effects," The International Journal of Biostatistics, De Gruyter, vol. 8(1), pages 1-40, January.
    • Handle: RePEc:bpj:ijbist:v:8:y:2012:i:1:n:3
    DOI: 10.2202/1557-4679.1361
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    References listed on IDEAS

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    1. van der Laan Mark J. & Gruber Susan, 2010. "Collaborative Double Robust Targeted Maximum Likelihood Estimation," The International Journal of Biostatistics, De Gruyter, vol. 6(1), pages 1-71, May.
    2. van der Laan Mark J. & Rubin Daniel, 2006. "Targeted Maximum Likelihood Learning," The International Journal of Biostatistics, De Gruyter, vol. 2(1), pages 1-40, December.
    3. Gruber Susan & van der Laan Mark J., 2010. "A Targeted Maximum Likelihood Estimator of a Causal Effect on a Bounded Continuous Outcome," The International Journal of Biostatistics, De Gruyter, vol. 6(1), pages 1-18, August.
    4. Mark van der Laan & Maya Petersen, 2004. "Estimation of Direct and Indirect Causal Effects in Longitudinal Studies," U.C. Berkeley Division of Biostatistics Working Paper Series 1155, Berkeley Electronic Press.
    5. van der Laan Mark J. & Petersen Maya L, 2008. "Direct Effect Models," The International Journal of Biostatistics, De Gruyter, vol. 4(1), pages 1-29, October.
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    Cited by:

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    2. Lucia Babino & Andrea Rotnitzky & James Robins, 2019. "Multiple robust estimation of marginal structural mean models for unconstrained outcomes," Biometrics, The International Biometric Society, vol. 75(1), pages 90-99, March.
    3. Martin Huber & Michael Lechner & Giovanni Mellace, 2016. "The Finite Sample Performance of Estimators for Mediation Analysis Under Sequential Conditional Independence," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(1), pages 139-160, January.
    4. R. M. Daniel & B. L. De Stavola & S. N. Cousens & S. Vansteelandt, 2015. "Causal mediation analysis with multiple mediators," Biometrics, The International Biometric Society, vol. 71(1), pages 1-14, March.
    5. Kara E. Rudolph & Iván Díaz, 2022. "When the ends do not justify the means: Learning who is predicted to have harmful indirect effects," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 185(S2), pages 573-589, December.
    6. Martin Huber & Anna Solovyeva, 2020. "Direct and Indirect Effects under Sample Selection and Outcome Attrition," Econometrics, MDPI, vol. 8(4), pages 1-25, December.
    7. Helmut Farbmacher & Martin Huber & Lukáš Lafférs & Henrika Langen & Martin Spindler, 2022. "Causal mediation analysis with double machine learning [Mediation analysis via potential outcomes models]," The Econometrics Journal, Royal Economic Society, vol. 25(2), pages 277-300.
    8. Brooks Jordan C. & van der Laan Mark J. & Singer Daniel E. & Go Alan S., 2013. "Targeted Minimum Loss-Based Estimation of Causal Effects in Right-Censored Survival Data with Time-Dependent Covariates: Warfarin, Stroke, and Death in Atrial Fibrillation," Journal of Causal Inference, De Gruyter, vol. 1(2), pages 235-254, November.
    9. Samuel D. Lendle & Meenakshi S. Subbaraman & Mark J. van der Laan, 2013. "Identification and Efficient Estimation of the Natural Direct Effect among the Untreated," Biometrics, The International Biometric Society, vol. 69(2), pages 310-317, June.
    10. Eva Deuchert & Martin Huber & Mark Schelker, 2019. "Direct and Indirect Effects Based on Difference-in-Differences With an Application to Political Preferences Following the Vietnam Draft Lottery," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(4), pages 710-720, October.
    11. Martin Huber, 2015. "Causal Pitfalls in the Decomposition of Wage Gaps," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(2), pages 179-191, April.
    12. Hsu Yu-Chin & Huber Martin & Lai Tsung-Chih, 2019. "Nonparametric estimation of natural direct and indirect effects based on inverse probability weighting," Journal of Econometric Methods, De Gruyter, vol. 8(1), pages 1-20, January.
    13. Isaac Meza & Rahul Singh, 2021. "Nested Nonparametric Instrumental Variable Regression: Long Term, Mediated, and Time Varying Treatment Effects," Papers 2112.14249, arXiv.org, revised Mar 2024.
    14. Xiang Zhou, 2022. "Semiparametric estimation for causal mediation analysis with multiple causally ordered mediators," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(3), pages 794-821, July.

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