In causal models, the total effect represents the complete influence of a treatment variable on an outcome, encompassing both direct and indirect pathways. For linear models, the total effect is simply the sum of direct and indirect effects, calculable via the "sum of products" rule. For non-linear systems, a modified additivity holds involving the Natural Direct Effect (NDE) and Natural Indirect Effect (NIE): $$Total \ Effect (X = 0 \rightarrow X = 1) = NDE (X = 0 \rightarrow X = 1) - NIE (X = 1 \rightarrow X = 0)$$.

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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)

Scenario: Daisy and the kittens

Formal Definition:
$$NIE = P(Y_{M = M_1} = 1 | do(X = 0))–P(Y_{M = M_0} = 1 | do(X = 0))         $$

Natural Indirect Effect(NIE)

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

The Book of Why

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

Total Effect

Daisy, a dog adopted by Dana and his wife, had been having a hard time accepting training/supervision. After several weeks Daisy was still having occasional “accidents” inside the house. 
Dana and his wife by chance observed that bringing foster kittens in the house can stop Daisy from producing "accidents" in the house. According to Dana, the foster kittens remained with them for three weeks, and Daisy the dog did not break her training at all during that period. 
Why would this happen? Is it because of the foster kittens that Dana and his wife accidentally brought into the house?


Senario Setup: Daisy and the kittens

The first P term is the outcome of the Daisy experiment. It represents the probability of successful house training ($$Y = 1$$), given that we set X to 0 (do not introduce other pets) but set the mediator $$M$$ to $$M_1$$ (the value it would have if we had introduced them). The counterfactual $$M_1$$ needs to be computed accordingly: different dogs might have different needs for supervision.
The second P term is the probability of successful house training of the dog under the condition with no other pets. Here we set $$X = 0$$ to reflect that no other pets are present, and $$M = M_0$$ to reflect that we do not introduce the mediator.

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