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 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 $$Y$$.

Game 2 of the Book of Why: $$X \rightarrow E \rightarrow Y$$ and $$X \leftarrow A \rightarrow B \leftarrow D \rightarrow E$$

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.

University of Michigan - Ann Arbor

Pearl notes two ways of eliminating unmeasurable confounders, in the context of:

- RCT
- Observational studies

In each of these methods, we can use the front-door criterion, back-door criterion, or if necessary, do-calculus.

Deconfounding/Eliminating Unmeasurable Confounder Variables

When deconfounding variables $$X$$ and $$Y$$, a back-door path is any path from $$X$$ to $$Y$$ that starts with an arrow pointing into $$X$$. (Notice that if there is a collider on the back-door path, the path is already blocked by the collider.) The path $$A \leftarrow B \leftarrow C \rightarrow D \rightarrow E$$ is a back-door path from $$A$$ to $$E$$. Forks are also back-door paths.

Back-Door Path

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

The Book of Why

Pearl introduces an axiomatic system containing three rules to remove do operators in Chapter 7. 

Rule 1. Irrelevance
Rule 2. Conditional equivalence of *do(x)* and *see(x)*
Rule 3. Lack of causal paths

Generalizing Do-Calculus

A Randomized Controlled Trial (RCT) or a Controlled Experiment is a method to eliminate unmeasurable confounders effect on causal inference by implementing an intervention on a randomly selected group of individuals (treatment/intervened group) while keeping other individuals unchanged (controlled group, placebo, not-intervened group) to observe the changes in outcomes due to the intervention effect. The RCTs are intended to "disarm" confounders (confounding variables).

Randomized Controlled Trial (RCT) = Controlled Experiment

An observational study, as supposed to an experimental study is a type of study that investigators observe the effect of an intervention without assigning it to the subjects at random.  That is, the subjects choose their own treatment/ intervention.  

Observational studies may prove to be less costly that RCT's, given that the investigator need not actively change the behavior of the subjects


Observational Studies

The front-door criterion is a method used in causal inference when the causal effect of a variable $$X$$ on $$Y$$ is mediated by a variable $$M$$ that is shielded from a confounding set $$C$$. If $$C$$ is unmeasurable, we cannot block the back-door path $$X \leftarrow C \rightarrow Y$$. Instead, we use the front-door path $$X \rightarrow M \rightarrow Y$$ to estimate the causal effect.

Front-Door Criterion

In the following causal diagram, which of the following are confounding back-door paths for A to F?

To determine if $$X$$ is ignorable relative to outcome $$Y$$, conditional on a set $$Z$$ of matching variables, we need only test to see if $$Z$$ blocks all the back-door paths between $$X$$ and $$Y$$ and no member of $$Z$$ is a descendant of $$X$$.

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