University of Michigan - Ann Arbor

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

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)

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

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

$X \rightarrow M \rightarrow Y$

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

The Book of Why

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

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

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:
$$P(A|do(B)) = \sum_{C} P(A|B, C=c)P(C=c)$$

$$\Rightarrow P(Y|do(X)) = \sum_{M} P(M=m|X) \sum_{X} P(Y|X=x,M=m)P(X=x) \Rightarrow ACE_{X \rightarrow Y} = ACE_{X \rightarrow M} * ACE_{M \rightarrow Y}$$ 

How to use Front-Door Criterion?
     $\swarrow C \searrow$
$X \rightarrow M \rightarrow Y$ 

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

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