The formal definition indicates that NDE requires counterfactuals. We ask the following questions in the definition: what will happen had X been set to 1? What will happen had X been set to 0?

In the Berkeley scenario, X represents the "reported" sex, which means a woman can also report her sex as a male in the what-if scenario. In this way, the admission is decided by the reported(fake) sex rather than the actual sex, invoking the counterfactual subscript.

Apply NDE to Berkeley Admissions Paradox

Goal: To reduce the effect of the overcontrolled mediator value, we can use natural direct effect to model the effect. 

Scenario: Berkeley Admissions Paradox

Definition:
$$NDE = P(Y_{M = M_0} = 1 | do(X = 1)) – P(Y_{M = M_0} = 1 | do(X = 0)) $$

University of Michigan - Ann Arbor

The direct effect means that a variable (X) does not go through (without the intervention of) a mediator (M) to produce a causal impact on the outcome (Y).  That is: 

$$X -> Y$$   ( as opposed to   $$X -> M -> Y$$)$$

Direct Effect

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

Natural Direct Effect(NDE)

Scenario: Berkeley Admissions Paradox

Setup:
We include three variables in our definition: 
a treatment $$X$$
an outcome $$Y$$
a mediator $$M$$

Formal Definition:
$$CDE(0) = P(Y = 1 | do(X = 1), do(M = 0)) – P(Y = 1 | do(X = 0), do(M = 0)) $$   


Learn Before

Related

Learn After