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Chapter_23

Aaron L9/17/25About 4 minCausal inferenceDeepSeek-R1-0528-2ndresponse_md
### Chapter 23: Causal Mediation

23.1 Mediation Analysis Under Attack
Introduces causal mediation as a framework to decompose treatment effects into direct and indirect pathways. Uses a smoking cessation trial example where treatment (A) (smoking cessation), mediator (M) (hypertension), and outcome (Y) (myocardial infarction) are analyzed. Defines:

  • Pure direct effect: (E[Y^{a=1, M^{a=0}}] - E[Y^{a=0, M^{a=0}}]), representing the effect of (A) on (Y) if (M) is set to its value under no treatment.
  • Total indirect effect: (E[Y^{a=1, M^{a=1}}] - E[Y^{a=1, M^{a=0}}]), representing the effect mediated through (M).
    Both effects are cross-world counterfactuals (involve interventions incompatible in a single world). The mediation formula identifies (E[Y^{a=1, M^{a=0}}]) under Figure 23.1:
    [
    \sum_m E[Y \mid A=1, M=m] \Pr(M=m \mid A=0)
    ]
    This requires unverifiable cross-world independence assumptions under the NPSEM-IE model, which are not needed in the FFRCISTG model used elsewhere in the book.

Technical Point 23.1: Proof of the Mediation Formula
Derives the mediation formula under the NPSEM-IE model, assuming cross-world independence (Y^{a=1,m} \perp!!!\perp M^{a=0}). The proof uses:

  1. Laws of probability
  2. Cross-world independence
  3. Exchangeability and consistency
    The formula is invalid under FFRCISTG, where sharp bounds replace point identification.

23.2 A Defense of Mediation Analysis
Advocates for the policy relevance of pure direct effects by reinterpreting them through separable components of treatment. In the smoking example:

  • (N): Nicotine exposure (affects (M) but not (Y))
  • (O): Non-nicotine components (affects (Y) but not (M))
    The pure direct effect (E[Y^{a=1, M^{a=0}}]) equals (E[Y^{n=0, o=1}]) (effect of nicotine-free cigarettes), identifiable via the same mediation formula under deterministic relationships (A = N = O) (Figure 23.2).

Technical Point 23.2: When the Mediation Formula is the g-Formula
Shows that under Figure 23.3 (expanded DAG with separable components), the g-formula for (E[Y^{n=0, o=1}]) simplifies to the mediation formula despite positivity violations, due to deterministic links between (A), (N), and (O).

23.3 Empirically Verifiable Mediation
Proposes a three-arm randomized trial to test assumptions of separable effects:

  1. Smoking cessation ((A=0))
  2. Continued smoking ((A=1))
  3. Nicotine-free cigarettes ((N=0, O=1))
    Discrepancies between observed (E[Y \mid N=0, O=1]) and the mediation formula refute:
  • No direct effect of nicotine on (Y) (assumption i)
  • No direct effect of non-nicotine components on (M) (assumption ii)
  • No unmeasured (M)-(Y) confounders (assumption iii)

Fine Point 23.1: Empirical Falsification of Separable Effects Assumptions
Details falsification strategies, e.g., if (N) and (Y) are associated given (M) and (O), either assumption (i) or (iii) is violated. An 8-arm trial intervening on (M) can distinguish these.

23.4 An Interventionist Theory of Mediation
Presents a framework avoiding cross-world counterfactuals by focusing on separable effects:

  1. Core idea: Decompose treatment into intervenable components (e.g., (N) and (O)).
  2. Verifiability: Assumptions testable via future trials (e.g., 6-arm trial in Section 23.4).
  3. Mediator interventions: Unnecessary; effects exist if separable components are well-defined.
  4. Equivalence: Under valid assumptions, separable effects equal mediation-based effects.

Technical Point 23.3: Path-Specific Effects and the Front Door Formula
Extends the front door formula to settings with direct effects (Figure 23.6) by introducing separable components ((N), (O)) for path-specific effects. The g-formula under this expansion matches the front door formula.

Technical Points Summary

  • Pure/Total Effects: Require cross-world counterfactuals; identifiable only under strong assumptions.
  • Separable Effects: Policy-relevant, empirically verifiable, and avoid cross-world quantities.
  • g-Formula Equivalence: Mediation and front door formulas emerge as special cases of the g-formula under specific causal stories.

Fine Points Summary

  • Falsification: Three-arm trials test separable effect assumptions; discrepancies indicate violated assumptions.
  • Surrogate Mediators: When interventions on (M) are ill-posed, separable components provide an alternative.

Key Formulas

  • Mediation formula: (\sum_m E[Y \mid A=1, M=m] \Pr(M=m \mid A=0))
  • Pure direct effect: (E[Y^{a=1, M^{a=0}}] - E[Y^{a=0, M^{a=0}}])
  • Total indirect effect: (E[Y^{a=1, M^{a=1}}] - E[Y^{a=1, M^{a=0}}])

Figures

  • Figure 23.1: Basic mediation DAG ((A \rightarrow M \rightarrow Y), no confounders).
  • Figure 23.2: Expanded DAG with separable components (N) and (O) (deterministic links to (A)).
  • Figure 23.3: SWIG for intervention on (N) and (O).
  • Figures 23.4–23.8: Extensions for surrogate mediators and path-specific effects (descriptions only; images not analyzed).