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

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 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.

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

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 $$Y$$.

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