Critical thinking
Article: Pearl (2018)
Confounding and deconfounding: or, slaying the lurking variable

Introduction

Confounding bias occurs when a variable influences both who is selected for the treatment and he outcome of the experiment.
Sometimes the confounders are known. Other times they are merely suspected and act as a ‘lurking third variable’.

If we have measurements of the third variable, then it is very easy to deconfound the true and spurious effects.

Statisticians both over- and underrate the importance of adjusting for possible confounders

• Overrate in the sense that they often control for many more variables than they need to and even for variables that they should not control for
• Underrate in the sense that they are loath to talk about causality at all, even if the controlling has been done correctly.

The chilling fear of confounding

Knowing the set of assumptions that stand behind a given conclusion is not less valuable than attempting to circumvent those assumptions with and RCT, which has complications on its own.

The skillful interrogation of nature: why RCTs work

The one circumstance under which scientists will abandon some of their reticence to talk about causality is when they have conducted a randomized controlled trial (RCT).

Randomization brings two benefits:

• It eliminates confounder bias
• It enables the researcher to quantify his uncertainty

Another ways is, if you know what all the possible counfounders are, to measure and adjust for them.
But, randomization had one great advantage: it servers every incoming link to the randomized variable, including the ones we don’t know about or cannot measure.

RCTs are preferred to observational studies.
But, in some cases, intervention may be physically impossible or unethical.

Provisional causality: causality contingent upon the set of assumptions that our causal diagram advertises.

The principal objective of an RCT is to eliminate confounding.

The new paradigm of confounding

Confounding is not a statistical notion. It stands for the discrepancy between what we want to assess (the causal effect) and what we actually do assess using statistical methods.
If you can’t articulate mathematically what you want to assess, you can’t expect to define what constitutes a discrepancy.

Historically, the concept of ‘confounding’ has evolved around two related conceptions:

• Incomparability
• A lurking third variable.

Both these concepts have resisted formalization.

The do-operator and the back-door criterion

Confounding: anything that makes P(Y|do(X)) differ from P(Y|X).
The do-operator erases all the arrows that come into X, and in this way prevents any information about X from flowing in the noncausal direction.
Randomization has the same effect. So does statistical adjustment, if we pick the right variables to adjust.

Three rules that tell us how to stop the flow of information through any individual junction.

• The chain junction. A → B → C.
  Controlling for B prevents information about A from getting to C or vice versa.
• Fork or confounding junction A ← B → C.
  Controlling for B prevents information about A from getting to C or vice versa.
• A collider. A → B ← C.
  The variables A and C start out independent, so that information about A tells you nothing about C. But if you control for B, then information starts flowing though the ‘pipe’ due to the explain-away effect.

Controlling for descendants (or proxies) of a variable is like ‘partially’ controlling for the variable itself. Controlling for a descendant of a mediator partly closes the pipe, controlling for a descendant of a collider partly opens the pipe.

To deconfound two variables X and Y, we need only block every non-causal path between them without blocking or perturbing any casual paths.
A back-door path: any path from X to Y that starts with an arrow pointing into X.
X and Y will be deconfounded if we block every back-door path.
If we do this by controlling for some set of variables Z, we also need to make sure that no member of Z is a descendant of X on a causal path, otherwise we might partly or completely close off that path.