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.

University of Michigan - Ann Arbor

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

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

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)

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