A conventional measure of a treatment's effectiveness. It averages treatment efficacy over a population and can be found using a randomized control trial (RCT). It measures how much an increase in a single unit of treatment variable X will affect the outcome variable Y. When you have properly eliminated all confounding variables, you can obtain an estimate through running a regression.

Average Causal Effect (ACE)

When we observe a variable W that is irrelevant to Y (possibly conditional on other variables Z), then the probability distribution of Y will not change. 

This assertion of irrelevance translates into a symbolic manipulation:
$P(Y | do(X), Z, W) = P(Y | do(X), Z)$

The stated equation holds provided that the variable set $Z$ blocks all the paths from $W$ to $Y$ after we have deleted all the arrows leading into $X$.


Rule 1. Irrelevance

If a set Z of variables blocks all back-door paths from $X$ to $Y$, then conditional on $Z$, $do(X)$ is equivalent to $see(X)$. We can, therefore, write
$P(Y | do(X), Z) = P(Y | X, Z)$
if $Z$ satisfies the back-door criterion.

Rule 2: Conditional Equivalence of $do(x)$ and $see(x)$

We can remove $do(X)$ from $P(Y| do(X))$ in any case where there are no causal paths from $X$ to $Y$. That is, 
$P(Y | do(X)) = P(Y)$
if there is no path from $X$ to $Y$ with only forward-directed arrows.

We can paraphrase this rule is follows: if we do something that does not affect $Y$, then the probability distribution of $Y$ will not change.

Rule 3. Lack of Causal Paths

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

Rule 1 permits the addition or deletion of observations. 
Rule 2 permits the replacement of an intervention with an observation, or vice versa. 
Rule 3 permits the deletion or addition of interventions. 

All of these permits are issued under appropriate conditions, which have to be verified in any particular case from the causal diagram.


A Simple Syntactic Interpretation for the Rules

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

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

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

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

When the causal effect of a variable $X$ on $Y$ is mediated by a variable $M$, which is shielded from a confounding set $C$, if $C$ is *unmeasurable*, we cannot block the back-door $X \leftarrow C \rightarrow Y$. So, we use the front-door $X \rightarrow M \rightarrow Y$. 

Front-Door Criterion (Unmeasurable confounder of a mediated path)

$\ \ \ \ \ \ \ \ \ \ \ C$

$\ \ \ \ \swarrow \ \ \ \ \ \ \ \ \searrow$

$X \rightarrow M \rightarrow Y$

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