The equation for eliminate a confounder via the back door method. The left side of this equation P(Y | do(X)) represent what is the effect of X on Y.  The right side of the equation is an estimate of the query for eliminating the confounder.

P(Y | do(X)) = ∑z P(Y | X, Z = z) P(Z = z)  


University of Michigan - Ann Arbor

For Causal Nodes X,Y:
X and Y will be deconfounded if we block every backdoor, because such paths allow for spurious correlations. 

When eliminating backdoors, we additionally need to ensure that anything closed off from X is not a descendent of X.

Back-Door Criterion

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

The Book of Why

Controlling descents of variable $X$ is like "partially" controlling for variable $X$ itself. Besides the back-door path, we might partly or completely close off the path we are interested in.

Why not controll for a descent of X?

Back Door Adjustment Equation

This diagram has no confounders. B is associated with X because it is in the X->A->Y chain (X->A->B) and it is associated with Y because of the open path of B<-A->Y without having to go through X (where X=0). It also is not a causal path in X->A->Y so it passes the classical epidemilogical definition of a confounder. However, B cannot be a confounder because X is not being impacted by B. 

Game 1 of the Book of Why
$X \rightarrow A \rightarrow Y$ 
              $\searrow B$ 

The causal path here is X -> E-> Y
A, B, C, D can be referred to as "pretreatment" variables. 
There is one possible Backdoor path which is:
X<-A->B<-D->E->Y
However, this path is blocked by B. So we do not have to control for 
anything to meet the backdoor-criterion. If there is confounding because B or C is adjusted for, you can also choose to control for A or D to meet the backdoor-criterion.
Traditional Statistician would control for B and C but this is wrong because 
it opens up noncausal path (by eliminating collider B) which confounds 
X and Y.

Game 2 of the Book of Why
$A \rightarrow B \rightarrow C$ 
  |          $\nwarrow D$
 $\downarrow$          $\swarrow$
$X \rightarrow E \rightarrow Y$ 

This diagram has no confounders. Because A->B<-C is a collider, confounding cannot occur in the backdoor path.

Using the classical epidemiological definition, I will explain through example of why B is not a confounder and why this definition is not completely accurate. Let's say B is seat belt usage. X is smoking and Y is lung disease. We will say A is societal norms towards safety and C is societal norms towards health. 
1. People who smoke and get lung disease also use seat belts
2. People who do not use seatbelts can still smoke and/or get lung disease
3. People who do use seatbelts do not automatically smoke or get lung disease. It is still in relation to it by proxy of societal norms of health and safety but do not cause it.

Game 4 of the Book of Why
$X \leftarrow A \rightarrow B \leftarrow C \rightarrow Y$ 

B is a confounder in the path from X to Y and B has to be adjusted for in order to account for the confounding. Despite A's association with X and B, adjusting for A creates collider bias and is insufficient for resolving the back-door path X <- B -> Y.

(Recreation of the diagram as shown in the book)

Game 3 of the Book of Why

There is a back-door path X <- B <- C -> Y that creates confounding. The confounding will be resolved if only C is adjusted for. However, M-bias will result if only B is adjusted for. Despite that, if A or C is adjusted for in addition to B, then the confounding will also be resolved.

(Recreation of the diagram shown in the book)

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