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Cited by 1,628Advanced Causal Inference TechniquesStatistical Methods and Bayesian InferenceBayesian Modeling and Causal Inference

The DOI matches its Crossref record; 44 of the 45 references it cites were found in a registry or an index; none of the 26 sources its claims cite is checked for support yet.

Identification, Inference and Sensitivity Analysis for Causal Mediation Effects

Open access
Imai, Kosuke, Keele, Luke, Yamamoto, Teppei·2010·Statistical Science
DOI 10.1214/10-sts321
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Abstract

Causal mediation analysis is routinely conducted by applied researchers in a variety of disciplines. The goal of such an analysis is to investigate alternative causal mechanisms by examining the roles of intermediate variables that lie in the causal paths between the treatment and outcome variables. In this paper we first prove that under a particular version of sequential ignorability assumption, the average causal mediation effect (ACME) is nonparametrically identified. We compare our identification assumption with those proposed in the literature. Some practical implications of our identification result are also discussed. In particular, the popular estimator based on the linear structural equation model (LSEM) can be interpreted as an ACME estimator once additional parametric assumptions are made. We show that these assumptions can easily be relaxed within and outside of the LSEM framework and propose simple nonparametric estimation strategies. Second, and perhaps most importantly, we propose a new sensitivity analysis that can be easily implemented by applied researchers within the LSEM framework. Like the existing identifying assumptions, the proposed sequential ignorability assumption may be too strong in many applied settings. Thus, sensitivity analysis is essential in order to examine the robustness of empirical findings to the possible existence of an unmeasured confounder. Finally, we apply the proposed methods to a randomized experiment from political psychology. We also make easy-to-use software available to implement the proposed methods.

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TL;DR

The paper proves nonparametric identification of the average causal mediation effect (ACME) under a sequential ignorability assumption that does not condition on post-treatment confounders, shows when the popular linear structural equation model (LSEM) product-of-coefficients estimator can be given a causal interpretation, and proposes a sensitivity analysis for unmeasured pre-treatment confounding. It applies the methods to a randomized media-framing experiment and finds the parametric estimates relatively robust to some violations, while nonparametric sensitivity bounds are more sensitive.

Key findings

  • Under Assumption 1 (sequential ignorability), the ACME and average natural direct effects are nonparametrically identified; the LSEM estimator \(\delta(0)=\delta(1)=\beta_2\gamma\) is valid under this assumption plus the no-interaction assumption.
  • The proposed sensitivity analysis uses \(\rho \equiv \mathrm{Corr}(\varepsilon_{i2},\varepsilon_{i3})\); in the media-framing application, the original negative ACME conclusion is maintained unless \(\rho < -0.68\), and the 95% confidence interval covers zero only when \(-0.79 < \rho < -0.49\).
  • In the media-framing experiment (\(n=136\); 67 free-speech frame, 69 public-order frame), parametric estimates were \(\delta(0)=-0.566\) \([-1.081,-0.050]\), \(\delta(1)=-0.451\) \([-0.871,-0.031]\), and total effect \(\tau=-0.540\) \([-1.207,0.127]\); nonparametric estimates were \(\delta(0)=-0.596\) \([-1.168,-0.024]\), \(\delta(1)=-0.374\) \([-0.823,0.074]\), and \(\tau=-0.540\) \([-1.206,0.126]\).
  • A nonparametric sensitivity analysis for binary mediator/outcome found the estimates more sensitive: upper bounds crossed zero at \(\upsilon=0.019\) for \(\delta(1)\) and \(\upsilon=0.022\) for \(\delta(0)\).
  • Monte Carlo simulations with true \(\delta(0)=0.675\) and \(\delta(1)=4.03\) showed nonparametric estimators had smaller bias but larger variance than parametric estimators; 95% CI coverage converged to nominal as sample size increased.

Methods

  • Potential outcomes framework with binary treatment \(T_i\), mediator \(M_i\), outcome \(Y_i\), and pre-treatment covariates \(X_i\); defines ACME \(\delta(t)=E\{Y_i(t,M_i(1))-Y_i(t,M_i(0))\}\).
  • Proves identification under Assumption 1, compares it with Pearl (2001), Robins (2003), Petersen et al. (2006), and Rubin (2004), and derives LSEM implications including interaction and no-interaction cases.
  • Proposes parametric and nonparametric estimators, derives asymptotic variance for the nonparametric plug-in estimator, conducts a Monte Carlo experiment (sample sizes 50, 100, 500; 50,000 iterations), and applies methods to Nelson, Clawson, and Oxley’s (1997) randomized media-framing experiment.

Limitations

  • The paper states Assumption 1 may be too strong in many applied settings, especially when the treatment is randomized but the mediator is not; it cannot condition on observed or unobserved post-treatment confounders.
  • The proposed sensitivity analysis addresses unmeasured pre-treatment confounding but does not address possible post-treatment confounders; the nonparametric sensitivity analysis is developed only for the special case of binary mediator and outcome.
  • The paper states future work is needed on multiple mediators, measurement error in the mediator, and experimental designs that identify causal mediation effects under weaker assumptions; the empirical example also makes evident that subjects’ attitudes were observed rather than randomized.

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Crossref · 10.1214/10-sts321

Cites

44 of 44 references verified0 citation errors · 0 not checkable

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Claims the paper makes

0 of 26 cited sources checked · 0 supported · 0 flagged

Show all 85 claims

  • Probably not a substantial claim. 35
  • It may state two separate claims as one. 29
  • The quote only partly states this claim. 2
56 of 56

  • The authors state this as an assumption.

    Sequential ignorability: the treatment is ignorable given pre-treatment covariates, and the mediator is ignorable given the observed treatment and pre-treatment covariates, with the conditional independence in equation (5) holding without conditioning on observed post-treatment confounders.

    “ASSUMPTION 1 (Sequential ignorability).”
    “Thus, the treatment is first assumed to be ignorable given the pre-treatment covariates, and then the mediator variable is assumed to be ignorable given the observed value of the treatment as well as the pretreatment covariates.”
    “We emphasize that, unlike the standard sequential ignorability assumption in the literature (e.g., Robins, 1999), the conditional independence given in equation (5) of Assumption 1 must hold without conditioning on the observed values of posttreatment confounders.”

  • The authors state this as an assumption.

    The potential outcome depends only on the values of the treatment and mediating variables and not on how they are realized.

    “the above notation implicitly assumes that the potential outcome depends only on the values of the treatment and mediating variables and, in particular, not on how they are realized.”

  • The authors state this as an assumption.

    The no-interaction assumption between the treatment and the ACME holds: δ(1) = δ(0).

    “ASSUMPTION 2 (No-interaction between the Treatment and the ACME). δ(1) = δ(0).”

  • The authors state this as an assumption.

    The sensitivity parameter ρ satisfies -1 < ρ < 1.

    “If we further assume -1 < ρ < 1, then the ACME is identified and given by”

  • The authors state this as an assumption.

    The error terms in equations (12) and (13) can be decomposed as εij = λj Ui + εij for j = 2, 3, where Ui is an unobserved confounder.

    “This alternative version of ρ is based on the following decomposition of the error terms in equations ( 12) and ( 13): ε ij = λ j U i + ε ij for j = 2, 3, where U i is an unobserved confounder”

  • The authors state this as an assumption.

    Sequential ignorability is assumed given Ui and Ti.

    “where U i is an unobserved confounder and the sequential ignorability is assumed given U i and T i .”

  • The authors state this as an assumption.

    Ui has to be a pre-treatment variable so that the resulting estimates can be given a causal interpretation.

    “Again, note that U i has to be a pre-treatment variable so that the resulting estimates can be given a causal interpretation.”

  • The authors state this as an assumption.

    εij is independent of Ui for j = 2, 3.

    “In addition, we assume that ε ij ⊥ ⊥ U i for j = 2, 3.”

  • The authors state this as an assumption.

    The sequential ignorability assumption (Assumption 1) holds.

    “We begin by the nonparametric bounds on the ACME without assuming equation ( 5) of the sequential ignorability assumption.”
    “Clearly, if and only if υ = 0, the sequential ignorability assumption is satisfied.”

  • The authors state this as an assumption.

    There is no post-treatment variable that confounds the relationship between mediator and outcome.

    “Fourth, an important limitation of our framework is that it does not allow the presence of a post-treatment variable that confounds the relationship between mediator and outcome.”

  • The authors state this as an assumption.

    The standard LSEM (linear structural equation model) holds.

    “Finally, under Assumption 1 and the LSEM, we can write E(M i |T i ) = α 2 + β 2 T i , and E(Y i |M i , T i ) = α 3 + β 3 T i + γ M i .”
11 of 11

Citations

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Research works printed but not linked: 3

  1. Pearl, J. (2001). Direct and indirect effects. In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence (J. S. Breese and D. Koller, eds.) 411–420. Morgan Kaufman, San Francisco, CA.
  2. Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2010). Causal mediation analysis using R. In Advances in Social Science Research Using R (H. D. Vinod, ed.). Lecture Notes in Statist. 196 129–154. Springer, New York. · DOI
  3. Imai, K., Keele, L. and Yamamoto, T. (2010). Replication data for: Identification, inference, and sensitivity analysis for causal mediation effects. Available at http://hdl.handle.net/1902.1/14412.

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Metadata

Authors
Imai, K., Keele, L., & Yamamoto, T.
Affiliations
Princeton University (US)
ORCID
Yamamoto, Teppei
Venue
Statistical Science
Publisher
Institute of Mathematical Statistics
Published
2010-02-01
Volume
25
Issue
1
Pages
51-71
ISSN
  • 0883-4237 (ISSN-L)
  • 2168-8745
DOI
10.1214/10-sts321
OpenAlex
W2056621649
Semantic Scholar
305c1929f9b5cd60aede21b30b7f6fb65b31dbd3
Cited by
1,628
References
45
Citations by year
2026: 217 · 2025: 175 · 2024: 177 · 2023: 146 · 2022: 157 · 2021: 133 · 2020: 118 · 2019: 121 · 2018: 76 · 2017: 79 · 2016: 59 · 2015: 54 · 2014: 41 · 2013: 29 · 2012: 24
FWCI
20.03
Type
journal article
Language
en
Open access
Locations
  • Statistical Science (publishedVersion)
  • arXiv (Cornell University) (publishedVersion)
  • Project Euclid (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • National Science Foundation (0918968)
  • University of Pennsylvania
  • University of Wisconsin-Madison
  • Harvard University
  • York University
Topics
  • Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
  • Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
  • Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Modeling and Causal Inference
Keywords
causal mediation analysis, nonparametric identification, sequential ignorability, linear structural equation models, sensitivity analysis, unmeasured confounding, causal mechanisms, intermediate variables, randomized experiments, political psychology

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