It is possible for a suspected confounder Z to be a mediator (or on the causal path) between X and Y. Therefore, controlling for Z in such a situation would lead to major misinterpretations as controlling for Z could effectively block the effect of X in the causal diagram. 

Flaw with Classical Epidemiological Definition

Under the classical epidemiological definition, a confounder of $$X$$ and $$Y$$ is any variable $$Z$$ that is associated with $$X$$ and (among the unexposed) with $$Y$$, and that does not lie on the causal path from $$X$$ to $$Y$$. M-bias shows these association-based criteria are insufficient. In the structure $$A \leftarrow B \rightarrow C \leftarrow D \rightarrow E$$ (with exposure $$A$$ and outcome $$E$$), the middle variable $$C$$ is statistically associated with both the exposure and the outcome and does not lie on any causal path between them, so the classical definition would label $$C$$ a confounder. Yet $$C$$ is a collider: adjusting for it opens a spurious path and introduces bias instead of removing it. Because the classical criteria cannot distinguish such colliders from genuine confounders, they fail to capture the causal structure needed to define confounding correctly.

Weakness of the Classical Epidemiological Definition of Confounders

From the text:

A confounder of X and Y is a variable Z that:
1. is associated with X in the population at large
2. associated with Y among the people who have not been exposed to the treatment in X.
3. (Recently) Z should not be on the causal path between X and Y.

University of Michigan - Ann Arbor

Confounders are variables that affect both the outcome and the factor we are trying to determine as the cause.  The confounders are what RCTs try to "disarm."

For example, when measuring the effect of Alcohol use (X) on Lung cancer (Y), Smoking (Z) may be a confounder because those who consume more alcohol also tend to smoke more. 

Confounders in Causal Inference

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

The Book of Why

Sensitivity analysis refers to investigating whether the specified outcome could have resulted from alternative hypotheses and how strong those other hypotheses would need to be in order to explain the observed data.

The resulting relationships can be used to contrast the likelihoods with the original hypothesis. 

Sensitivity Analysis

The following names have been used in different references to refer to "confounders": 
- "Lurking variables" (The Book of Why, p. 18) or "Suspected Lurking third variable" (The Book of Why, p. 138)
- "Common causes" (The Book of Why, p. 48)
- "Background factors" (The Book of Why, p. 48)
- "Mixing (of effects)" (The Book of Why, p. 138)
- "Good controls" (A Crash Course in Good and Bad
Controls, p. 1)

Other Names for "Confounders"

1- Bad controls
2- Good controls (also known as “confounders” or “deconfounders”)
3- Neutral controls

Types of control variables in causal inference

Causal Definition of Confounder

Prior to the advent of do-operators and causal diagrams, statisticians and researchers defined confounders based on surrogate definitions which were often unsatisfactory from a causal point of view. One surrogate definition was declarative, stating that "A confounder is any variable that is correlated with both X and Y." Another definition was procedural, stating that a confounder could be characterized by statistical testing. Either one leads to the existence of "surrogate confounders," which help to reduce, but not eliminate bias when determining relationships between variables. 

Previous Surrogate Definitions of Confounder 

VanderWeele, T. J., & Shpitser, I. (2013). On the definition of a confounder. Annals of statistics, 41(1), 196–220. https://doi.org/10.1214/12-aos1058
Link: https://bit.ly/3fTAeZY

On the definition of a confounder

Deconfounding is defined as mitigating the adverse effect of a confounder in causal inference.  There are multiple methods of deconfounding depending on whether the confounder is:
- Measurable
- Unmeasurable

Deconfounding in Causal Inference

When the junction $$A \leftarrow B \rightarrow C$$ appears in a causal diagram, it is known as a fork, and $$B$$ is considered a common cause or confounder of $$A$$ and $$C$$. It might appear that there is a direct causal relationship between $$A$$ and $$C$$ because there is an observable statistical association between them. However, if we condition on $$B$$ (hold $$B$$ constant), we observe that $$A$$ and $$C$$ become conditionally independent.

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