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

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)

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

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

Back-Door Criterion

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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