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.  

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

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**.

University of Michigan - Ann Arbor

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

- Recall bias
- Selection bias
- Collider bias
- Omitted variable bias

Types of Bias

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

The Book of Why

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

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

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 $$B$$.

Chain $$A \rightarrow B \rightarrow C$$: The Screening-Off Effect

Catalogue of Bias Collaboration, Nunan D, Bankhead C, Aronson JK. Selection bias. Catalogue Of Bias 2017: [www.catalogofbias.org/biases/selection-bias/](https://www.catalogofbias.org/biases/selection-bias/) 

Catalog of Bias

Recall bias is a error that occurs when participants do not remember previous events or experiences accurately or omit details: the accuracy and volume of memories may be influenced by subsequent events and experiences.

Recall Bias

This type of bias is also known as "Confounded." It happens when good controls or confounders are not included in the regression model.

Omitted variable bias

Sampling bias is a systematic error that occurs when the method used to select participants results in a sample that fails to accurately represent the broader target population. Because the selected individuals differ in important ways from the overall group, any statistical findings derived from this unrepresentative sample will likely be inaccurate and cannot be safely generalized.

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