In Adam Glynn and Konstantin Kashin's study of the Job Training Partership Act (see causual diagram below), even though the mediator variable "Showed up" was affected by the confounder "Motivation", the researchers used the front-door criterion. 

Through this study they showed that as long as the effect of the confounder (motivation) on the mediator (showed up) is relatively small, it can be treated as absent with minimal bias in the results


Front Door Criterion: Causation from C to M

Derivation of the Front-Door Adjustment Formula from the Rules of Do-Calculus

Given that that there is a smoking gene which predisposes an individual to both smoking and cancer, with tar as a mediator, the the front-door criterion would be satisfied and the pathway Smoking --> Tar --> Cancer could be examined instead. It should be noted that this front-door pathway is valid for two main reasons:
- The confounder cannot be measured (at the time at least)
- The confounder does not interfere with the mediator

Smoking Gene as An Example of Front Door Criterion

To measure the average total effect of $$X$$ on $$Y$$, we need to measure the following two: 1. We can estimate the average causal effect (ACE) of a unit increase in $$X$$ on $$M$$ because the back-door path $$X \leftarrow C \rightarrow Y \leftarrow M$$ is blocked by the collider at $$Y$$. $$ACE_{X \rightarrow M} = P(M|X) - P(M|no X)$$ 2. To measure the ACE of $$M \rightarrow Y$$, we can block the back-door path $$M \leftarrow X \leftarrow C \rightarrow Y$$ by controlling for $$X$$. $$ACE_{M \rightarrow Y} = P(Y|do(M)) - P(Y|do(no M))$$ 3. According to the back-door adjustment formula: $$P(A|do(B)) = \sum_{C} P(A|B, C=c)P(C=c)$$ $$\Rightarrow P(Y|do(X)) = \sum_{M} P(M=m|X) \sum_{X} P(Y|X=x,M=m)P(X=x) \Rightarrow ACE_{X \rightarrow Y} = ACE_{X \rightarrow M} * ACE_{M \rightarrow Y}$$

Applying the Front-Door Criterion to Estimate Average Causal Effect

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.

University of Michigan - Ann Arbor

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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

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

The Book of Why

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 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


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