Given the definition of M-Bias, we realize that the "Epidemiological Definition of Confounders" doesn't provide everything necessary to define confounders (#TODO Explain why here).

Weakness of the Classical Definition of Confounders

Cinelli, C., Andrew, A., & Pearl, J. (2020). A Crash Course in Good and Bad Controls. TECHNICAL REPORT, 491, 1–9.

Pearl, J. (2009). Causality. Cambridge: Cambridge University Press.

Reference for bad control and good control

Take an example in M-Bias, (A) is exposure, (E) is disease, A is effected by (B) Education, (E) is effected by both (D) age and (A). Here (C) is car type  which is indicative of (B) Eduation and (D) age. Thus (C) associates with A and E. However, it's not a confounder. It has no effect on A or E.

If we make adjustment for  Z will yiled a biased result and it's called Bad control:

Bad control in M-Bias

This diagram has no confounders. Because A->B<-C is a collider, confounding cannot occur in the backdoor path.

Using the classical epidemiological definition, I will explain through example of why B is not a confounder and why this definition is not completely accurate. Let's say B is seat belt usage. X is smoking and Y is lung disease. We will say A is societal norms towards safety and C is societal norms towards health. 
1. People who smoke and get lung disease also use seat belts
2. People who do not use seatbelts can still smoke and/or get lung disease
3. People who do use seatbelts do not automatically smoke or get lung disease. It is still in relation to it by proxy of societal norms of health and safety but do not cause it.

Game 4 of the Book of Why
$X \leftarrow A \rightarrow B \leftarrow C \rightarrow Y$ 

There is a back-door path X <- B <- C -> Y that creates confounding. The confounding will be resolved if only C is adjusted for. However, M-bias will result if only B is adjusted for. Despite that, if A or C is adjusted for in addition to B, then the confounding will also be resolved.

(Recreation of the diagram shown in the book)

Game 5 of the Book of Why

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. 

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

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.  

Proxy

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)

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