- Third type of variable relation
- In a collider there is no association between X and Y, unless we condition on Z. Z opens this path up. 
- Once its open, information can move between X and Y without having direct causal influence on each other. 

University of California, Santa Cruz

- When blocking confounding paths between a predictor and an outcome, we are shutting what's known as the backdoor 
- Causal DAGs allow us to determine which variables to control for (to close the backdoors) and which ones to not control for (to not create new confounds). 
- There are four types of variable relations that merge to create all possible confounding paths; DAGs regardless of their causal complexity are generated from these relations. 

Backdoor 

McElreath, R. (2020). Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd ed.). Chapman and Hall/CRC.

Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd edition)

- The first of the four relations, its one of the most common ones. 
- Conditioning on Z, updating X doesn't change Y. So X and Y are independent of each other, only conditional on Z. 

Fork

- The second type of relation, it is similar to the Fork in the sense conditioning of the middle variable, Z, blocks the path. 
- Conditioning on Z, the path from X to Y becomes blocked. 

Pipe 

Collider 

- Fourth and final variable relation
- Conditioning on the descendent leads to partially conditioning on the parent as the descendent contains some information about the parent. 
- The consequences of this conditioning on the descendent is dependent on the parent
- A common variable relation as we cannot often directly measure a variable, and instead have to rely on its proxy. 

Descendent 

- List the paths connecting the potential cause of interest and the outcome.
- Classify each path as open or closed; paths are open unless a collider is present. 
- Classify each path by whether its a backdoor path or not; backdoor paths have an arrow entering its potential cause of interest.
- Then, open any backdoor paths, and decide which variables to condition on in order to close it. 

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