University of Michigan - Ann Arbor

In conditional probability, when we want to find the likelihood of an occurence given we **know** an *intervention* occured and the value of that intervention, we use the do() operator. 

`do()` operator

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

The following are different ways the do() operator can be used
- P(A | do(B), C)
    Intervening on B, seeing C
    
- P(A | do(B), do(C))
    Multiple interventions
    
- P(A | do(P(B) = P’))
    A change on the distribution of B (not only a point
    mass distribution)

Different uses of do() operator

Pearl, Judea. “On the Interpretation of Do(x).” Technical Report - UCLA, 13 Feb. 2019.

https://ftp.cs.ucla.edu/pub/stat_ser/r486.pdf


On the Interpretation of do(x)


Pearl (1995) uses expression of the form Q = E[Y |do(X = x)] to denote the causal effect of X on Y .  Where do(x) is a minimally invasive intervention that sets the value of X to x and leaves other relationships unchanged.

However, when X is non-manipulable for experimenters (for example, gender/race are non-manipulable), critics raise concern about the interpretation of the casual effect Q. 


Question of do(x) in Structural Causal Modeling (SCM) 


Assuming an observational study guided by model M, we identify and evaluate the Q with the expression Q = E[Y |do(x)] = q(x), where q(x) is a function of x computed from the joint distribution of observed variables in the model. 

One can use this information in three ways:
1. Q represents a limit on pending interventions
2. Q tells us about the effects of feasible interventions
3. Q serves as an auxiliary mathematical operation in the derivation of causal effects of manipulable variables.


The Semantics of Q 

When X is non-manipulable, although direct tests of Q(x) are infeasible, indirect tests are available.We can use back-door adjustment or front-door adjustment to test Q.

For example, if a set of variable z satisfies the back door criterion relative to (X,Y),  Q(x) is identifiable and is given by the formula: $$ P(y|do(x))=\sum_{z} P(y|x,z)P(z)$$. 
Equating the adjustment formulae generated by each of those sets provides a test for M, and hence a test for Q.

Fig.1 shows a front-door model where I is manipulable, X is non-manipulable, and U is an unobserved confounder. X satisfies the front door criterion relative to (I,Y). Randomizing I yields an estimate of P(y|do(I)), which should be equal to the estimand of P(y|do(I)) obtained through the front-door formula. The causal effect of I on Y is identifiable and is given by the formula:  $$Q(x)=\sum_{i} P(y|x, I = i)P(I = i)$$


Testing do(x) Claims

Suppose we have a given patient avoids the flue (A) given the flu shot (FS). 
- $P(A|do(FS))$
    - The probability that a typical patient would avoid the flu if the patient got the shot without choice.
- $ P(A|FS) $
    - The probability a patient avoided the flu given they were observed to take the flue shot. This probability asks the observed frequency of patients who avoided the flu when they *volunteered* to get a flu shot

Conditional probability with do() versus without

Causal Definition of Confounder

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

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. 

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

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

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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