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 $Z$.

Deconfounding/Adjusting/Controlling a Measurable Confounder in a Regression Model

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.

Controlling for Everything is Misguided

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.

University of Michigan - Ann Arbor

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

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

The Book of Why

Deconfounding/Adjusting/Controlling a Measurable Confounder

Pearl notes two ways of eliminating unmeasurable confounders, in the context of:

- RCT
- Observational studies

In each of these methods, we can use the front-door criterion, back-door criterion, or if necessary, do-calculus.

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