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A treatment-induced mediator–outcome confounder breaks the formula

Example

Let $A \sim Bernoulli (1 / 2)$ , $U \sim Bernoulli (1 / 2)$ , and $W \sim Bernoulli (1 / 5)$ be mutually independent. Define the post-treatment variable $Z = A U$ , mediator $M = Z x o r W$ , and outcome $Y = Z + M$ . Thus $Z$ is caused by treatment and causes both $M$ and $Y$ ; it is a treatment-induced mediator–outcome confounder. Treatment and mediator positivity hold for both treatment levels and mediator values.

Facts & Assumptions

Given: The definitions of natural effects and the observed-data functional in the mediation formula.

[F1]

$NDE (a, a^{'}) = E [Y (a, M (a^{'})) - Y (a^{'}, M (a^{'}))]$ and $NIE (a, a^{'}) = E [Y (a, M (a)) - Y (a, M (a^{'}))]$ . Controlled and natural mediation effects

[F2]

The mediation functional averages the observed conditional outcome mean under one treatment over the mediator law under the reference treatment. The natural-effect mediation formula under independent errors

[F3]

The standard formula excludes a treatment-induced post-treatment common cause of mediator and outcome. Assumptions for identifying natural effects

Verification

Given: The full-data construction in the Example and treatment levels $a = 1$ , $a^{'} = 0$ .

1.1F1givenalgebra

Under interventions, $Z (a) = a U$ , $M (a) = a U x o r W$ , and $Y (a, m) = a U + m$ . Hence the true nested means are $E [Y (1, M (0))] = E [U + W] = 7 / 10$ , $E [Y (1, M (1))] = E [U] + E [M (1)] = 1$ , and $E [Y (0, M (0))] = E [W] = 1 / 5$ . By [F1], the true natural effects are $NDE (1, 0) = 1 / 2$ and $NIE (1, 0) = 3 / 10$ .

1.2givenalgebra

At $A = 0$ , $M = W$ and $Y = M$ , so $Q_{0} (1) = 1 / 5$ and $μ_{0} (m) = m$ . At $A = 1$ , $M = U x o r W$ is fair. The joint probabilities $P (U = 1, M = 1 ∣ A = 1) = 2 / 5$ and $P (U = 1, M = 0 ∣ A = 1) = 1 / 10$ give $μ_{1} (1) = 9 / 5$ and $μ_{1} (0) = 1 / 5$ . Thus $Q_{1} (0) = Q_{1} (1) = 1 / 2$ .

2.1F2F3step 1.1step 1.2algebra∎

The observed-data functional gives $\tilde{θ} (1, 0) = (1 / 5) (9 / 5) + (4 / 5) (1 / 5) = 13 / 25$ , $\tilde{θ} (1, 1) = (1 / 2) (9 / 5) + (1 / 2) (1 / 5) = 1$ , and $\tilde{θ} (0, 0) = 1 / 5$ . Its direct and indirect contrasts are therefore $8 / 25$ and $12 / 25$ , not the true $1 / 2$ and $3 / 10$ from step 1.1. The functional is computable, but the identification theorem does not apply because the post-treatment common cause $Z$ violates [F3].

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