A proxy is used as a substitute measure when a variable that is not directly measurable is needed for analysis.  For a variable to be a proxy for another variable, the proxy variable needs to be highly correlated with the variable of interest.  The proxy can be used for any variables such as a mediator, confounder, collider.  

University of Michigan - Ann Arbor

The mediator is a variable that functions as the mechanism by way of transferring an effect of one thing to another.  

Consider the chain junction, where $A -> B -> C$.  $B$ is the mediator that transfers the effect of $A$ to $C$.   In other words, the mediator, $B$, is the variable that explains the causal effect of $A$ on $C$.  $A$ can be an indirect cause of $C$, meaning that $A$ affects $B$, which in turn contributes to the change in $C$.  

Mediator

When a single node is caused by two other nodes we encounter something called "collider-bias" where one condition explains-away another. 

Since A and C both effect the outcome of B, by conditioning on B, A and C become **dependent**.

$A \rightarrow B \leftarrow C$: The Collider

When the junction $A \leftarrow B \rightarrow C$ appears in a Causal Diagram, we say that B is a confounder for both A and C. It might appear that there is a causal relationship between A and C, because there is an observable relationship between A and C when B is varied. However, if we condition on B ("hold B constant") we observe that A and C become conditionally independent. 

$A \leftarrow B \rightarrow C$: The Fork and Confouders

Pearl outlines 3 basic types of junctions in Causal Diagrams which appear commonly and help explain the data. 

The three junctions take the forms:
   1. $A \rightarrow B \rightarrow C$ (Chain)
 2. $A \leftarrow B \rightarrow C$ (Fork)
 3. $A \rightarrow B \leftarrow C$ (Collider)

The 3 Junctions in Causal Diagram

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

The Book of Why

Proxy

We want to ask certain typical questions about a mediator: Does it account for the entire effect? Does Drug B work exclusively through blood pressure or perhaps through other mechanisms as well? 
The answer depends on whether the observed salary disparity is produced directly in response to the applicant’s sex or indirectly, through a mediator such as qualification, over which the employer has no control. All the above questions require a sensitive ability to tease apart total effects, direct effects (which do not pass through a mediator), and indirect effects (which do).

Mediation

A placebo effect is a positive outcome resulting from a simulated treatment, driven largely by an individual's expectation that they will improve. Holding the expectation of recovery can autonomously lead to physiological and psychological benefits, such as reduced stress, lower anxiety and depression, and enhanced immune system functioning, even without an active intervention.

Placebo Effect

Cole, S. R., Platt, R. W., Schisterman, E. F., Chu, H., Westreich, D., Richardson, D., & Poole, C. (2010). Illustrating bias due to conditioning on a collider. $\mathit{International \ Journal \ of \ Epidemiology, 39}$, 417-420. [doi.org/10.1093/ije/dyp334](https://doi.org/10.1093/ije/dyp334)
[ncbi.nlm.nih.gov/pmc/articles/PMC2846442/](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2846442/) 

Illustrating bias due to conditioning on a collider

In a path that would impact both variables but also contains a collider, M-bias occurs when the variable in the center of the collider is adjusted for. 

The general structure for M-bias is A <- B -> C <- D -> E where adjusting for C would create bias. 

M-bias

This a subcategory of a collider bias (Collider is the selection of the data). When two factors individually are usually not severe enough to cause some effect but the combination of the two is. Usually both factors are not thought to be correlated with each other in real life settings but in a certain condition they are. This seems to occur because the resulting effect is being adjusted for by not considering the situations where both factors do not occur. This association is caused based on the conditional collider of the environment the data is selected. 

Berkson's Paradox

The statistical procedure of controlling for everthing that can be measured is misguided. 

For instance, in a collider $A\rightarrow B \leftarrow C$ where the path between $A$ and $C$ is initially independent, controlling for B would make them dependnet due to the explain-away effect. This means that a back-door is opened.

Also, controlling for descendents of the varibles we are interested in should also be avoided. This is like "partially" controlling for the interested variable itself. 

Controlling for everything is misguided

Proposed by Hans Reichenbach, the common cause principal argues against correlation doesn’t imply causation. It states that if there are two variables I and J and they are correlated, then I causes J or J causes I or there is another variable X that precedes and causes both I and J. 

This theory is not true because it doesn't account for how the data is selected and appeals to our human nature to look for casual explanations whenever there is a pattern. This is an example of collider bias.

Common Cause Principal 

In the Book of Why, some of the classic examples of collider bias are discussed:

1. Celebrity's talent and beauty
2. Birth-weight paradox (Simpson's paradox)
3. Admission-rate bias (Berkson's paradox)

List of Collider Bias examples

The classic example of the Confounder effect is the strong association between ice cream sales and crime rate. 

The observed relationship is that as ice cream sales increase, the crime rate increases, however we attribute the increase in ice cream sales and crime rate both to change in season from winter to spring and summer. 

During nice weather, there are more ice cream sales, but also more people outside in close proximity to each other, increasing the potential for crime. 

If we hold the weather constant, we will observe that ice cream sales have no effect on the crime rate.

Example of the Fork and Confounders

When deconfounding $X$ 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 <- B <- C -> D -> E is a back-door path for A to E. Forks are also back-door paths.

Back-door path

When the junction $$A \rightarrow B \rightarrow C$$ appears in a causal diagram, we note that the presence of $$B$$ "screens-off" the value of $$A$$. This means that $$C$$ and $$A$$ are conditionally independent given the value of $$B$$.

Learn Before

Related