Explaining Away is a pattern of reasoning by which the presence of one cause of a phenomenon reduces the need to invoke the proposition of alternative causes or patterns in which variables may relate to each other.

Explain-Away Effect

The statistical procedure of controlling for everything 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 dependent due to the explain-away effect. This means that a back-door path is opened. Furthermore, controlling for descendants of the variables we are interested in should also be avoided. This is equivalent to "partially" controlling for the variable of interest itself.

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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

Given two variables X, Y and a confounder Z (i.e. X<-Z->Y). To obtain the relationship between X and Y not affected by Z, we can separately compare X and Y in every groups of Z. Then we can take an average of the effects, weighing each group according to its percentage. This method of compensation is called "adjusting for Z" or "controlling for Z" by statisticians.

Deconfounding/Adjusting/Controlling a Measurable Confounder

Controlling for a descendant of variable $$X$$ is equivalent to partially controlling for the variable $$X$$ itself. While our goal might be to close off back-door paths, adjusting for a descendant might also partially or completely block the direct causal path of interest that we are trying to measure.

Controlling for a Descendant of X

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. *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

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)

List of Collider Bias examples

In the regression model $Y \sim X$, to control for the effect of confounder $Z$ (where $X \leftarrow Z \rightarrow Y$) we can use the regression model $Y \sim X + Z$, i.e., taking the average effect of $X$ on $Y$ across differnt values of $Z$.

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