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.

University of Michigan - Ann Arbor

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

Natural Direct Effect(NDE)

In 1973, Eugene Hammel, an associate dean at the University of California, noticed a worrisome trend in the university’s admission rates for men and women. According to his data, it is shown that 44 percent of the men who applied to graduate school at Berkeley had been accepted, compared to only 35 percent of the women.
After a careful investigation, Eugene found out that even though for each apartment, the admissions decisions were consistently more favorable to women than to men, the admission rate for women is lower than that of men.
The causal story Eugene discovered is that women were rejected in greater numbers because they applied to harder departments to get into, like history and literature, therefore having a lower acceptance rate.

Berkeley Admissions Paradox

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

The Book of Why

Apply NDE to Berkeley Admissions Paradox

Diagram for Berkeley Admissions Paradox

In the context of Berkeley's admission paradox, Bickel contends that bias is to do with a pattern of association between a particular decision and particular sex of the applicant.  The words "pattern" and "association" are indications that bias belongs to the rung one of the Ladder of Causation, association.  

Bickel's Definition of Bias

In the context of Berkeley's admission paradox, Bickel contends that discrimination is to do with making decisions that are influenced by the sex of the applicant, even if it is immaterial for the admission decision.  The words like "exercise of decision," "influence," and "immaterial" are suggestive of causation, either belonging to the second (intervention)or third (counterfactuals) Ladder of Causation.  

Bickel's Definition of Discrimination

We would like to examine the direct effect of $$X$$ on $$Y$$ without allowing $$M$$ to change. 
In the context of the Berkeley example, X represents the gender, Y represents the admission result, M represents the mediator(department candidates apply to).
We force everybody to apply to the history department by setting $$do(M = 0)$$. People are randomly asked to report their sex as male ($$do(X = 1)$$) or as female ($$do(X = 0)$$), regardless of their actual genders. We obtain CDE through the difference in admission rates between the two groups.
In symbols, $$CDE(0)$$ indicates that we forced the mediator to take on the value zero, i.e. the history department. We could also force everybody to apply to the engineering department: $$do(M = 1)$$. We would then denote the resulting controlled direct effect as $$CDE(1)$$. 


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