Consider the making of a movie-star. We might choose to believe celebrity is a combination of talent and beauty. 

If we find that a certin movie-star is exceptionally talented, this might "explain away" their bad looks. Conversely, if a movie-star is gorgeous, we might attribute their beauty to their success, despite their terrible acting. 

This is collider bias.

Collider Bias Example for Celebrity's Talent and Beauty

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)

University of Michigan - Ann Arbor

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

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

The Book of Why

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 

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.  

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