The classic example of the Confounder effect is the strong association between ice cream sales and crime rate. 

The observed relationship is that as ice cream sales increase, the crime rate increases, however we attribute the increase in ice cream sales and crime rate both to change in season from winter to spring and summer. 

During nice weather, there are more ice cream sales, but also more people outside in close proximity to each other, increasing the potential for crime. 

If we hold the weather constant, we will observe that ice cream sales have no effect on the crime rate.

Example of the Fork and Confounders

When deconfounding $X$ and $Y$, a back-door path is any path from $X$ to $Y$ that starts with an arrow pointing into $X$. (Notice that if there is a collider on the back-door path, the path is already blocked by the collider.)

The path A <- B <- C -> D -> E is a back-door path for A to E. Forks are also back-door paths.

Back-door path

A proxy is used as a substitute measure when a variable that is not directly measurable is needed for analysis.  For a variable to be a proxy for another variable, the proxy variable needs to be highly correlated with the variable of interest.  The proxy can be used for any variables such as a mediator, confounder, collider.  

Proxy

When the junction $A \leftarrow B \rightarrow C$ appears in a Causal Diagram, we say that B is a confounder for both A and C. It might appear that there is a causal relationship between A and C, because there is an observable relationship between A and C when B is varied. However, if we condition on B ("hold B constant") we observe that A and C become conditionally independent. 

University of Michigan - Ann Arbor

Pearl outlines 3 basic types of junctions in Causal Diagrams which appear commonly and help explain the data. 

The three junctions take the forms:
   1. $A \rightarrow B \rightarrow C$ (Chain)
 2. $A \leftarrow B \rightarrow C$ (Fork)
 3. $A \rightarrow B \leftarrow C$ (Collider)

The 3 Junctions in Causal Diagram

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

$A \leftarrow B \rightarrow C$: The Fork and Confouders

When a single node is caused by two other nodes we encounter something called "collider-bias" where one condition explains-away another. 

Since A and C both effect the outcome of B, by conditioning on B, A and C become **dependent**.

$A \rightarrow B \leftarrow C$: The Collider

When the junction $$A \rightarrow B \rightarrow C$$ appears in a causal diagram, we note that the presence of $$B$$ "screens-off" the value of $$A$$. This means that $$C$$ and $$A$$ are conditionally independent given the value of $$B$$.

Chain $$A \rightarrow B \rightarrow C$$: The Screening-Off Effect

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"

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

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