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: Berkeley Admissions Paradox

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

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


Controlled Direct Effects(CDE)

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   $$X -> M -> Y$$)$$

University of Michigan - Ann Arbor

Mediation analysis is a causal analysis technique used to assess and measure the impact of different "pathways" and "mechanisms" by disentangling and differentiating between direct effect and indirect effect. As such, it serves to distinguish whether an outcome is caused directly in response to the variable of interest (the direct effect), or through mediators (the indirect effect). Assessing the counterfactuals of the situation can be a notable part of the process.

Mediation Analysis

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

The Book of Why

Direct Effect

The indirect effect means that the variable (X) passes through the mediator (M) to cause its effect on the outcome (Y).  Said another way, the variable (X) affects the outcome (Y) by way of the mediator (M).  The indirect effect can be expressed as an amount of mediation (a measured magnitude) by the mediator (M).  

In the context of the causal diagram, the magnitude of the mediation can be observed when the variable (X) will result in the increase on the outcome (Y) while holding the variable (X) constant, while at the same time, when a unit increase in the variable (X) would achieve an increase in the mediator (M).  

Indirect Effect

The so-called "smoking gene" (rs16969968) is a genetic variant that predisposes individuals with the variant to smoking and lung cancer. This genetic variant is also known as "Mr. Big," and "The Big One." In VanderWeele's study referenced in the book, the interaction impact between the smoking gene and smoking was overlooked so the mediation analysis was inaccurate.

"Smoking Gene"

For linear causal models, total effect is the sum of direct and indirect effects, and can be calculated with the "sum of products" rule.

However, for non-linear systems, a modified version of additivity holds true:
$Total ~Effect (X = 0\rightarrow X = 1) \\= NDE (X = 0\rightarrow X = 1) – NIE (X = 1 \rightarrow X = 0)$


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