In Game 4 of *The Book of Why*, represented by the causal diagram $$X \leftarrow A \rightarrow B \leftarrow C \rightarrow Y$$, there are no confounders between $$X$$ and $$Y$$. Because $$A \rightarrow B \leftarrow C$$ forms a collider, the back-door path is already blocked and no confounding can occur. Under the classical epidemiological definition, $$B$$ might incorrectly be identified as a confounder. For example, if $$X$$ is smoking, $$Y$$ is lung disease, $$B$$ is seat belt usage, $$A$$ represents societal safety norms, and $$C$$ represents societal health norms, $$B$$ is associated with both $$X$$ and $$Y$$ without causing them. Adjusting for the collider $$B$$ would actually open the back-door path, introducing M-bias.

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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

In a path that would impact both variables but also contains a collider, M-bias occurs when the variable in the center of the collider is adjusted for. 

The general structure for M-bias is A <- B -> C <- D -> E where adjusting for C would create bias. 

M-bias

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

The Book of Why

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)  


Back Door Adjustment Equation

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)

Game 5 of the Book of Why

In Game 1 of the Book of Why, represented by the causal diagram $$X \rightarrow A \rightarrow Y$$ and $$A \rightarrow B$$, there are no confounders between $$X$$ and $$Y$$. The variable $$B$$ is associated with $$X$$ because it is in the $$X \rightarrow A \rightarrow B$$ chain, and it is associated with $$Y$$ because of the open path $$B \leftarrow A \rightarrow Y$$. Although $$B$$ is correlated with both $$X$$ and $$Y$$ and is not on the causal path between them (thus passing the classical epidemiological definition of a confounder), $$B$$ is not a confounder because it does not causally influence $$X$$.

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

Controlling for a descendant of variable $$X$$ is equivalent to partially controlling for the variable $$X$$ itself. While our goal might be to close off back-door paths, adjusting for a descendant might also partially or completely block the direct causal path of interest that we are trying to measure.

Controlling for a Descendant of X

In Game 2 of *The Book of Why*, the causal path of interest is $$X \rightarrow E \rightarrow Y$$. The variables $$A$$, $$B$$, $$C$$, and $$D$$ are pretreatment variables.

The only potential back-door path is $$X \leftarrow A \rightarrow B \leftarrow D \rightarrow E \rightarrow Y$$. This path is already blocked because $$B$$ acts as a collider ($$A \rightarrow B \leftarrow D$$). Consequently, no variables need to be controlled to satisfy the back-door criterion.

However, if confounding is accidentally introduced by controlling for the collider $$B$$ (or its descendant $$C$$), which opens the back-door path, deconfounding can still be achieved by additionally controlling for either $$A$$ or $$D$$. Traditional statistical approaches that unconditionally control for all observed variables (like $$B$$ and $$C$$) are incorrect here because adjusting for the collider $$B$$ opens a non-causal path, confounding the relationship between $$X$$ and $$Y$$.

Game 2 of the Book of Why

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

Given the definition of M-Bias, we realize that the "Epidemiological Definition of Confounders" doesn't provide everything necessary to define confounders (#TODO Explain why here).

Weakness of the Classical Definition of Confounders

Cinelli, C., Andrew, A., & Pearl, J. (2020). A Crash Course in Good and Bad Controls. TECHNICAL REPORT, 491, 1–9.

Pearl, J. (2009). Causality. Cambridge: Cambridge University Press.

Reference for bad control and good control

Take an example in M-Bias, (A) is exposure, (E) is disease, A is effected by (B) Education, (E) is effected by both (D) age and (A). Here (C) is car type  which is indicative of (B) Eduation and (D) age. Thus (C) associates with A and E. However, it's not a confounder. It has no effect on A or E.

If we make adjustment for  Z will yiled a biased result and it's called Bad control:

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