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

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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Written from the paper’s full text.

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

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Statistics

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Claims the paper makes

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

  • Findingnone given

    The variance is larger for the nonparametric estimator than for the parametric estimator.

    “As expected, the variance is larger for the nonparametric estimator than the parametric estimator.”
    Quote in the paperNot a substantial claim
  • Findingown data

    The performance of the estimators is quite good in this particular simulation setting.

    “The performance of the estimators turns out to be quite good in this particular setting.”
    Quote in the paperNot a substantial claim
  • Findingown data

    The public order treatment decreased the subjects' tolerance toward the Ku Klux Klan speech in the news clips compared to the free speech frame.

    “Moreover, the public order treatment decreased the subjects' tolerance toward the Ku Klux Klan speech in the news clips compared to the free speech frame.”
    Quote in the paperNot a substantial claim
  • Findingown data

    Being exposed to the public order frame as opposed to the free speech frame significantly increased the subjects' perceived importance of public order.

    “For example, being exposed to the public order frame as opposed to the free speech frame significantly increased the subjects' perceived importance of public order”
    Quote in the paperNot a substantial claim
  • Findingown data

    Even with sample size as small as 50, estimated biases are essentially zero for the nonparametric estimates.

    “Even with sample size as small as 50, estimated biases are essentially zero for the nonparametric estimates.”
    Quote in the paperNot a substantial claim
  • Classified as probably not a substantial claim. 35
  • May combine two claims. 29
  • The quote only partly states this claim. 2

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingown data

    Being exposed to the public order frame as opposed to the free speech frame decreased the subjects' perceived importance of free speech, although the latter effect is not statistically significant.

    “while decreasing the importance of free speech (although the latter effect is not statistically significant).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The parametric estimators are slightly more biased for small sample sizes, but they converge to the true values by the time the sample size reaches 500.

    “The parametric estimators are slightly more biased for the small sample sizes, but they converge to the true values by the time the sample size reaches 500.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    Under the scenario where the unobserved pre-treatment confounder represents subjects' political ideology, the original conclusion about the direction of the ACME is perfectly robust to the violation of sequential ignorability, because the estimated ACME is always negative in the upper right quadrant of each plot.

    “Under this scenario, the original conclusion about the direction of the ACME is perfectly robust to the violation of sequential ignorability, because the estimated ACME is always negative in the upper right quadrant of each plot.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Nonparametric estimators have smaller bias than the parametric estimator, though their variance is much larger.

    “The results indicate that nonparametric estimators have smaller bias than the parametric estimator though its variance is much larger.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    For most of the estimates, the 95% confidence intervals do not contain zero, mirroring the finding from the original study that general attitudes about public order mediated the effect of the media frame.

    “For most of the estimates, the 95% confidence intervals do not contain zero, mirroring the finding from the original study that general attitudes about public order mediated the effect of the media frame.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingcitation

    The nonparametric bounds on the ACME for binary mediator and outcome always contain zero, implying that the sign of the ACME is not identified without an additional assumption even in this special case.

    “These bounds always contain zero, implying that the sign of the ACME is not identified without an additional assumption even in this special case.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    This result holds even after taking into account sampling variability, as the confidence interval covers zero only when -0.79 < ρ < -0.49.

    “This result holds even after we take into account the sampling variability, as the confidence interval covers the value of zero only when -0.79 < ρ < -0.49.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Under the no-interaction assumption, the estimated ACME is between the previous two estimates, and the 95% confidence interval does not contain zero.

    “As expected, the estimated ACME is between the previous two estimates, and the 95% confidence interval does not contain zero.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The no-interaction assumption between the treatment and the ACME holds: δ(1) = δ(0).

  • May combine two claims.

    Findingown data

    The nonparametric sensitivity analysis estimates are rather sensitive to the violation of Assumption 1, in contrast to the parametric sensitivity analyses reported in Section 6.2, where the estimates appeared quite robust.

    “In contrast to the parametric sensitivity analysis reported in Section 6.2, the estimates are shown to be rather sensitive to the violation of Assumption 1.”
    • Also stated in: § APPENDIX E: NONPARAMETRIC SENSITIVITY ANALYSIS · hedged
      “This contrasts with the parametric sensitivity analyses reported in Section 6.2, where the estimates of the ACME appeared quite robust to the violation of Assumption 1.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For both δ(1) and δ(0), the upper bounds sharply increase as the value of υ increases and cross the zero line at small values of υ (0.019 for δ(1) and 0.022 for δ(0)).

    “For both δ(1) and δ(0), the upper bounds sharply increase as we increase the value of υ and cross the zero line at small values of υ [0.019 for δ(1) and 0.022 for δ(0)].”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data

    For both estimators, the coverage probabilities fully converged to their nominal values by the time the sample size reached 5000.

    “(Although not reported in the table, we confirmed that for both estimators the coverage probabilities fully converged to their nominal values by the time the sample size reached 5000.)”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    When the mediator is continuous, the population proportion of those with M_i(1)=M_i(0) can be essentially zero.

    “Another difficulty of this approach is that when the mediator is continuous the population proportion of those with M i (1) = M i (0) can be essentially zero.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationreasoning

    Assumption 2 is related to and implied by Robins' no-interaction assumption given in equation (9), but differs in being written in terms of the ACME rather than controlled direct effects.

    “Although Assumption 2 is related to and implied by Robins' no-interaction assumption given in equation ( 9), the key difference is that Assumption 2 is written in terms of the ACME rather than controlled direct effects.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The no-interaction assumption between the treatment and the ACME holds: δ(1) = δ(0).

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationreasoning

    In general, there is no simple expression for the asymptotic variance of the estimator in equation (19), so one may use a nonparametric or parametric bootstrap to compute uncertainty estimates.

    “Unfortunately, in general, there is no simple expression for the asymptotic variance of this estimator.Thus, one may use a nonparametric bootstrap [or a parametric bootstrap based on the asymptotic distribution of μtm (x) and νtm (x)] to compute uncertainty estimates.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationcitation

    The standard error for the estimated ACME δ(t) = β2γ can be calculated approximately using the Delta method (Sobel, 1982) or exactly via the variance formula of Goodman (1960).

    “The standard error for the estimated ACME, that is, δ(t) = β2 γ , can be calculated either approximately using the Delta method (Sobel, 1982), that is, Var( δ(t)) ≈ β 2 2 Var( γ ) + γ 2 Var( β2 ), or exactly via the variance formula of Goodman (1960)”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites Sobel, 1982

      Not checked in the free run.

    • cites Goodman, 1960

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    Sobel (2008) maintains the assumption that the causal effect of the treatment is entirely through the mediator and assumes the natural direct effect is zero for all units a priori.

    “However, Sobel (2008) maintains, among others, the assumption that the causal effect of the treatment is entirely through the mediator and applies the instrumental variables technique of Angrist, Imbens and Rubin (1996).That is, the natural direct effect is assumed to be zero for all units a priori, that is, ζ i (t) = 0 for all t = 0, 1 and i.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites Sobel (2008)

      Not checked in the free run.

    • cites Angrist, Imbens and Rubin (1996)

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    Testing β1 = 0 is unnecessary because the ACME can be nonzero even when the average total causal effect is zero.

    “Thus, testing β 1 = 0 is unnecessary since the ACME can be nonzero even when the average total causal effect is zero.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationnone given

    Assuming the absence of post-treatment confounders may not be credible in many applied settings.

    “This is an important limitation since assuming the absence of post-treatment confounders may not be credible in many applied settings.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationreasoning

    The proposed sequential ignorability assumption may be too strong in many applied settings.

    “Like the existing identifying assumptions, the proposed sequential ignorability assumption may be too strong in many applied settings.”
    • Also stated in: § 1. INTRODUCTION · hedged
      “Like many identification assumptions, the proposed assumption may be too strong for the typical situations in which causal mediation analysis is employed.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Interpretationnone given

    In experiments where the treatment is randomized but the mediator is not, the ignorability of the treatment assignment holds but the ignorability of the mediator may not.

    “For example, in experiments where the treatment is randomized but the mediator is not, the ignorability of the treatment assignment holds but the ignorability of the mediator may not.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    Under Assumption 1 and the model defined by equations (12) and (15), the average natural direct effects are identified as ζ(t) = β3 + κ(α2 + β2t), and the average total causal effect is τ = β2γ + β3 + κ(α2 + β2).

    “The average natural direct effects are identified as ζ (t) = β 3 + κ(α 2 + β 2 t), and the average total causal effect is equal to τ = β 2 γ + β 3 + κ(α 2 + β 2 ).”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning

    When J = 2, equation (16) reduces to equation (17).

    “Furthermore, when J = 2, this reduces to δ(t) = {Pr(M i = 1|T i = 1, X i ) -Pr(M i = 1|T i = 0, X i )} (17) • {E(Y i |M i = 1, T i = t, X i ) -E(Y i |M i = 0, T i = t, X i )}.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning

    Equation (11) is redundant given equations (12) and (13).

    “We note that equation ( 11) is redundant given equations ( 12) and (13).”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning

    The proposed general estimation strategies enable Imai et al. (2010) to develop an easy-to-use R package, mediation, that implements these methods.

    “Furthermore, the proposed general estimation strategies enable Imai et al. (2010) to develop an easyto-use R package, mediation, that implements these methods and demonstrate its use with an empirical example.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Imai et al. (2010)

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    Under the same assumptions as Theorem 2, the average natural direct effects are identified as ζ(0) = ζ(1) = β3, and the average total causal effect is τ = β3 + β2γ.

    “The theorem implies that under the same set of assumptions, the average natural direct effects are identified as ζ (0) = ζ (1) = β 3 , where the average total causal effect is τ = β 3 + β 2 γ .”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given

    In randomized experiments where the treatment is randomized, equations (4) and (10) are equivalent.

    “In practice, this difference is only a technical matter because, for example, in randomized experiments where the treatment is randomized, equations ( 4) and ( 10) are equivalent.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim. The quote only partly states this claim.

    Interpretationreasoning

    Theorem 3 gives the asymptotic variance of the nonparametric estimator defined in equation (18) given the realized values of the treatment variable.

    “THEOREM 3 (Asymptotic variance of the nonparametric estimator).Suppose that Assumption 1 holds.Then, the variance of the nonparametric estimator defined in equation ( 18) is asymptotically approximated by”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationcitation

    Within Rubin's framework, there exists no obvious definition for the mediation effect unless the direct effect is zero.

    “Within this framework, there exists no obvious definition for the mediation effect unless the direct effect is zero (in this case, the treatment affects the outcome only through the mediator).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Rubin (2004)

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    Under Assumption 1, in the LSEM defined by equations (11), (12) and (13), the ACME is identified and given by δ(0) = δ(1) = β2γ, where the equality between δ(0) and δ(1) is also assumed.

    “THEOREM 2 (Identification under the LSEM).Consider the LSEM defined in equations (11), ( 12) and (13).Under Assumption 1, the ACME is identified and given by δ(0) = δ(1) = β 2 γ, where the equality between δ(0) and δ(1) is also assumed.”
    • Also stated in: § 3.4 Implications for Linear Structural Equation Model · plain
      “Thus, Assumption 1 enables the identification of the ACME under the LSEM.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given

    These omitted variables must causally precede T.

    “Note that these omitted variables must causally precede T i .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given

    The ACME is zero if and only if ρ equals ρ.

    “Third, the ACME is zero if and only if ρ equals ρ.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationcitation

    The proposed sensitivity analysis is developed within the LSEM framework, but some extensions are possible.

    “The proposed sensitivity analysis above is developed within the framework of the LSEM, but some extensions are possible.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Interpretationreasoning

    Although the ACME is nonparametrically identified under Assumption 1, this assumption may be too strong in many applied settings.

    “Although the ACME is nonparametrically identified under Assumption 1, this assumption, like other existing identifying assumptions, may be too strong in many applied settings.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given

    One particular concern is that subjects' pre-existing ideology affects both their attitudes toward public order issues and their tolerance for the Klan within each treatment condition.

    “As we discussed in Section 2, one particular concern is that subjects' pre-existing ideology affects both their attitudes toward public order issues and their tolerance for the Klan within each treatment condition.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    Under the LSEM, if equation (4) holds but equation (5) may not, the sharp bounds on ρ and the ACME are (-1, 1) and (-∞, ∞), respectively.

    “COROLLARY 1 (Bounds on the sensitivity parameter).Consider the LSEM defined in equations (11), ( 12) and (13).Suppose that equation (4) holds but equation (5) may not.Then, the sharp, that is, best possible, bounds on the sensitivity parameter ρ and ACME are given by (-1, 1) and (-∞, ∞), respectively.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The no-interaction assumption only restricts the way the treatment effect is transmitted to the outcome and thus does not affect the estimate of the overall treatment effect.

    “This makes sense since the no-interaction assumption only restricts the way the treatment effect is transmitted to the outcome and thus does not affect the estimate of the overall treatment effect.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The no-interaction assumption between the treatment and the ACME holds: δ(1) = δ(0).

  • Classified as probably not a substantial claim.

    Interpretationcitation

    The influence of the unobserved pre-treatment confounder can be expressed using coefficients of determination R2*M and R2*Y, which represent the proportion of previously unexplained variance in the mediator or outcome explained by the unobserved confounder.

    “We can then express the influence of the unobserved pre-treatment confounder using the following coefficients of determination: R 2 * M ≡ 1 - Var(ε i2 ) Var(ε i2 ) and R 2 * Y ≡ 1 - Var(ε i3 ) Var(ε i3 ) , which represent the proportion of previously unexplained variance (either in the mediator or in the outcome) that is explained by the unobserved confounder (see Imbens, 2003).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Imbens, 2003

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    In the original analysis, Nelson, Clawson and Oxley (1997) found that subjects who viewed the Klan story with the free speech frame were significantly more tolerant of the Klan than those who saw the story with the public order frame.

    “In the original analysis, Nelson, Clawson and Oxley (1997) used a LSEM similar to the one discussed in Section 3.4 and found that subjects who viewed the Klan story with the free speech frame were significantly more tolerant of the Klan than those who saw the story with the public order frame.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Nelson, Clawson and Oxley (1997)

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Interpretationnone given

    The assumptions required for identifying causal mediation effects are unverifiable.

    “We view sensitivity analysis as an essential part of causal mediation analysis because the assumptions required for identifying causal mediation effects are unverifiable and often are not justified in applied settings.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The proof of Theorem 2 implies that if equation (4) holds, εi2 ⊥ Ti and εi3 ⊥ Ti hold but εi2 ⊥ εi3 does not unless equation (5) also holds.

    “The proof of Theorem 2 implies that if equation (4) holds, ε i2 ⊥ ⊥ T i and ε i3 ⊥ ⊥ T i hold but ε i2 ⊥ ⊥ ε i3 does not unless equation ( 5) also holds.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The relationship between ρ and the coefficients of determination is ρ2 = R2*M R2*Y = R2M R2Y / {(1 - R2M)(1 - R2Y)}, or equivalently ρ = sgn(λ2λ3) sqrt(R2*M R2*Y) = sgn(λ2λ3) sqrt(R2M R2Y / {(1 - R2M)(1 - R2Y)}).

    “In either case, it is straightforward to show that the following relationship between ρ and these parameters holds, that is, ρ 2 = R 2 * M R 2 * Y = R 2 M R 2 Y /{(1 -R 2 M )(1 - R 2 Y )} or, equivalently, ρ = sgn(λ 2 λ 3 )R * M R * Y = sgn(λ 2 λ 3 ) R M R Y (1 -R 2 M )(1 -R 2 Y ) , where R * M , R * Y , R M and R Y are in [0, 1].”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    The analyst can examine variation in the estimated ACME with respect to change in these parameters.

    “Then, the analyst can examine variation in the estimated ACME with respect to change in these parameters.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The original researchers found evidence supporting their main hypothesis that subjects' general attitudes mediated the causal effect of the news story frame on tolerance for the Klan.

    “The researchers also found evidence supporting their main hypothesis that subjects' general attitudes mediated the causal effect of the news story frame on tolerance for the Klan.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Nelson, Clawson and Oxley (1997)

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Interpretationnone given

    The assumptions required for identifying causal mediation effects are often not justified in applied settings.

    “We view sensitivity analysis as an essential part of causal mediation analysis because the assumptions required for identifying causal mediation effects are unverifiable and often are not justified in applied settings.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data

    The stark difference between nonparametric and parametric sensitivity analyses illustrates the potential importance of parametric assumptions in causal mediation analysis; a significant part of identification power could be attributed to functional form assumptions as opposed to empirical evidence.

    “Although the direct comparison is difficult because of different parametrization and variable coding, this stark difference illustrates the potential importance of parametric assumptions in causal mediation analysis; a significant part of identification power could in fact be attributed to such functional form assumptions as opposed to empirical evidence.”
    Quote in the paperSubstantial claim

  • May combine two claims. The quote only partly states this claim.

    Interpretationreasoning

    If the treatment is randomized, equation (4) holds, and ρ is given with -1 < ρ < 1, then the ACME is identified and given by the formula in Theorem 4.

    “THEOREM 4 (Identification with a given error correlation).Consider the LSEM defined in equations ( 11), ( 12) and (13).Suppose that equation (4) holds and the correlation between ε i2 and ε i3 , that is, ρ, is given.If we further assume -1 < ρ < 1, then the ACME is identified and given by”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The sensitivity parameter ρ satisfies -1 < ρ < 1.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Imai, Keele and Tingley (2009) show how to conduct sensitivity analysis with probit models when the mediator and/or the outcome are discrete.

    “For example, Imai, Keele and Tingley (2009) show how to conduct sensitivity analysis with probit models when the mediator and/or the outcome are discrete.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Imai, Keele and Tingley (2009)

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationnone given

    This strategy will also work when there is an interaction term between the treatment and mediating variables as in equation (15) and/or when there are observed pre-treatment covariates X.

    “This strategy will also work when there is an interaction term between the treatment and mediating variables as in equation ( 15) and/or when there are observed pre-treatment covariates X i .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationcitation

    A more straightforward and general approach to compute standard errors is to apply the iterative feasible generalized least square algorithm of seemingly unrelated regression (Zellner, 1962) and use the associated asymptotic variance formula.

    “A more straightforward and general approach is to apply the iterative feasible generalized least square algorithm of the seemingly unrelated regression (Zellner, 1962), and use the associated asymptotic variance formula.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Zellner, 1962

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning

    Another interpretation uses R2M and R2Y, the proportion of original variance explained by the unobserved confounder.

    “Another interpretation is based on the proportion of original variance that is explained by the unobserved confounder.In this case, we use the following sensitivity parameters:”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The formal definition of the concept of causal mediation had to await the later works by epidemiologists and statisticians (Robins and Greenland, 1992; Pearl, 2001; Robins, 2003).

    “Indeed, the formal definition of the concept of causal mediation had to await the later works by epidemiologists and statisticians (Robins and Greenland, 1992;Pearl, 2001;Robins, 2003).”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites Robins and Greenland, 1992

      Not checked in the free run.

    • cites Pearl, 2001

      Not checked in the free run.

    • cites Robins, 2003

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationreasoning

    Whether a variable qualifies as a mediator in a given empirical study relies crucially on the investigator's belief in the theory being considered.

    “whether a variable qualifies as a mediator in a given empirical study relies crucially on the investigator's belief in the theory being considered.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationreasoning

    Causal mediation analysis can be uncomfortably far from a completely data-oriented approach to scientific investigations.

    “Thus, causal mediation analysis can be uncomfortably far from a completely data-oriented approach to scientific investigations.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationnone given

    The new identification condition for the ACME is relatively easy to interpret in substantive terms and is weaker than existing results in some situations.

    “First, we provide a new identification condition for the ACME, which is relatively easy to interpret in substantive terms and also weaker than existing results in some situations.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Premisenone given

    Little attention was paid to the conditions under which the popular LSEM estimator can be given a causal interpretation.

    “While psychologists extended this LSEM framework in a number of ways, little attention was paid to the conditions under which their popular estimator can be given a causal interpretation.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisereasoning

    The progress made on the identification of causal mediation effects by these authors has led to the recent development of alternative and more general estimation strategies (e.g., Imai, Keele and Tingley, 2009; VanderWeele, 2009).

    “The progress made on the identification of causal mediation effects by these authors has led to the recent development of alternative and more general estimation strategies (e.g., Imai, Keele and Tingley, 2009;VanderWeele, 2009).”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites Imai, Keele and Tingley, 2009

      Not checked in the free run.

    • cites VanderWeele, 2009

      Not checked in the free run.

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

Cites

Flagged first, then by citations

Research works printed but not linked: 3

  1. Pearl, J. (2001). Direct and indirect effects. In <i>Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence</i> (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 <i>Advances in Social Science Research Using R</i> (H. D. Vinod, ed.). <i>Lecture Notes in Statist.</i> <b>196</b> 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 <a href="http://hdl.handle.net/1902.1/14412">http://hdl.handle.net/1902.1/14412</a>.

Cited by

Sorted by citations

Related

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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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.

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