Cinelli, Carlos, et al. “A Crash Course in Good and Bad Controls.” Technical Report -UCLA, 21 Mar. 2020.

https://ftp.cs.ucla.edu/pub/stat_ser/r493.pdf

A Crash Course in Good and Bad
Controls

A variable is neutral to be controlled or adjusted for in regression analysis for causal inference if controlling it does not affect the estimated causal relation between the predictor and the outcome of interest. However, controlling it may affect the precision of the estimation.

Neutral control in causal inference

Bad Controls are those variables when added to the regression equation will impose bias or other disparities between the real effect size to be estimated, and the regression coefficient that represents it. They are defined on the contrary to good controls (i.e., confounders or deconfounders), which should be adjusted (controlled) in a regression model to improve the accuracy of estimating the desired parameters. Bad and Good Controls are necessary to be considered when deciding which variables to control for in regression analysis for causal inference.

Bad control in causal inference

Common causes of X and Y or any mediator between X and Y confound the causal effect of X on Y.

Common Good controls in causal inference

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

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

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.

Classical Epidemiological Definition of a Confounder

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"

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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