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Single-world assumptions do not identify natural effects

Statement refuted

The observed law of $(A, M, Y)$ identifies the natural direct and indirect effects whenever treatment is randomized, consistency and treatment/mediator positivity hold, and the same-world mediator–outcome independences $Y (a, m) ⊥ ⁣ ⁣ ⁣ ⊥ M (a)$ hold for each $a, m$ , even without a cross-world restriction.

Facts & Assumptions

Given: The definitions of the natural effects, the single-world conditions in the statement, and the definition of identifiability.

[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]

Identifiability requires the target to be constant over all full-data laws with the same observed-data law. Identifiability of an estimand

[F3]

The standard formula's additional cross-world restriction is $Y (a, m) ⊥ ⁣ ⁣ ⁣ ⊥ M (a^{'}) ∣ σ (L)$ , which is not one of the single-world assumptions stated above. Assumptions for identifying natural effects

Counterexample

Given: Binary treatment $A$ , mediator $M$ , and outcome $Y$ .

1.1givenconstruct

In model I, let $A, M (0), M (1), V$ be independent fair Bernoulli variables. In model II, let $A, M (1), V$ be independent fair Bernoulli variables and set $M (0) = V$ . In both models define $Y (0, m) = 0$ and $Y (1, m) = 1_{{V = m}}$ for $m \in {0, 1}$ . Potential responses define observed $M = M (A)$ and $Y = Y (A, M (A))$ , so consistency holds; treatment is randomized and both mediator values have probability $1 / 2$ in each treatment arm.

2.1step 1.1algebra

Under $A = 0$ , $Y = 0$ and $M (0)$ is fair in both models. Under $A = 1$ , $M (1)$ is fair and independent of $V$ in both models, so $Y = 1_{{V = M (1)}}$ is fair and independent of $M (1)$ . Thus the conditional observed laws of $(M, Y)$ agree in each treatment arm, and the randomized treatment law is the same; the two models have identical observed laws of $(A, M, Y)$ .

2.2F3step 1.1algebra

The same-world independence holds in both models: for $a = 0$ the outcome response $Y (0, m)$ is constant; for $a = 1$ , $Y (1, m)$ is a function of $V$ and $M (1)$ is independent of $V$ . In model I, $M (0)$ is independent of every $Y (1, m)$ ; in model II, $M (0) = V$ makes $Y (1, m)$ dependent on $M (0)$ , so the cross-world restriction in [F3] differs.

2.3F1step 1.1algebra

In model I, $E [Y (1, M (0))] = 1 / 2$ and $E [Y (1, M (1))] = 1 / 2$ ; in model II, $E [Y (1, M (0))] = 1$ and $E [Y (1, M (1))] = 1 / 2$ . Both models have $E [Y (0, M (0))] = 0$ . By [F1], the natural-effect pairs are $(1 / 2, 0)$ in model I and $(1, - 1 / 2)$ in model II, in the order (NDE, NIE).

3.1F2step 2.1step 2.3∎

The two compatible full-data laws have the same observed law but different natural effects by step 2.3. Therefore the target is not constant on the observational equivalence class in [F2], refuting the stated claim.

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