Last updated: 2025-02-13

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Some notes on causal mediation analysis from VanderWeele’s materials.

Traditional Approach

Symbols
  • \(A\): exposure of interest
  • \(M\): potential mediator
  • \(Y\): outcome
  • \(C\): baseline covariates

The Difference Method

M1: a regression of the outcome \(Y\) on the exposure \(A\) and covariates \(C\)

\(E[Y|a, c] = \phi_0 + \phi_1a + \phi'_4c\)

\(\phi_1\) is interpreted as the total effect (TE) of the exposure \(A\) on the outcome \(Y\).


M2: similar but add the mediator \(M\) in the regression as well

\(E[Y|a, m, c] = \theta_0 + \theta_1a + \theta_2m + \theta'_4c\)

\(\theta_1\) is interpreted as the direct effect (DE) of \(A\) on \(Y\) when control has been made for the mediator.

The difference of TE and DE is interpreted as the indirect effect (IE), th effect of \(A\) on \(Y\) explained by the mediator.

\(IE = \phi_1 - \theta_1\)

The Product Method

Once again, M2 is employed.

M3: a regression of the mediator \(M\) on the exposure \(A\) and covariates \(C\)

\(E[M|a, c] = \beta_0 + \beta_1a + \beta'_2c\)

Then, substitute M3 in M2 for \(m\).

\(E_M[E[Y|a, m, c]]\)
\(= \theta_0 + \theta_1a + \theta_2(\beta_0 + \beta_1a + \beta'_2c) + \theta'_4c\)
\(= \theta_0 + (\theta_1 + \theta_2\beta_1)a + (\theta_2\beta'_2c + \theta_2\beta_0 + \theta'_4c)\)

\(DE = \theta_1\)
\(IE = \theta_2\beta_1\)

The interpretation of DE is the same as the difference method. The IE, however, is interpreted as the effect of the exposure on the mediator \(\beta_1\) times the effect of the mediator on the outcome \(\theta_2\).

Limitations
  • Two methods differ in the case of binary outcome.
  • Confounders for the mediator-outcome relationship may exist.
  • Interactions of the exposure-mediator relationship may exist.

Counterfactual Approach

Symbols
  • \(Y_a = Y(a)\): Y when setting \(A\) to \(a\)
  • \(Y_{am} = Y(a, m)\): Y when setting \(A\) to \(a\) and \(M\) to \(m\)
  • \(M_a = M(a)\): M when setting \(M\) to \(m\)

\(TE = Y_1 - Y_0 = Y_{1M_1} - Y_{0M_0} = (Y_{1M_1} - Y_{1M_0}) + (Y_{1M_0} - Y_{0M_0}) = NIE + NDE\)

Controlled direct effect (CDE): the outcome change on average when \(a = 1\) → \(0\) and \(M = m\).

\(CDE(m) = Y_{1m} - Y_{0m}\)

Natural direct effect (NDE): the outcome change on average when \(a = 1\) → \(0\) and the mediator is controlled at the level in the absence of the exposure.

\(NDE = CDE(M_0) = Y_{1M_0} - Y_{0M_0} = Y_{1M(A = 0)} - Y_{0M(A = 0)}\)

Natural indirect effect (NIE): the outcome change on average when \(a = 1\) and the mediator change if \(a = 1\) → \(0\).

\(NIE = Y_{1M_1} - Y_{1M_0} = Y_{1M(A = 1)} - Y_{1M(A = 0)}\)

Proportion Mediation (PM): how much of the effect is mediated.

\(PM = \frac{NIE}{TE}\)

Proportion Eliminated (PE): how much of the effect is eliminated when \(M = m\).

\(PE = \frac{TE - CDE(m)}{TE}\)