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.

Apply NIE to Daisy and the kittens

Scenario: Daisy and the kittens

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

University of Michigan - Ann Arbor

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

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

Learn Before

Related

Learn After