Confounding

Ice cream sales and drownings rise together, and neither causes the other — summer causes both. A confounder is a common cause that makes two variables move together without either producing the other, and it is the reason a correlation is not an effect. It can be strong enough to reverse a comparison entirely rather than merely inflate it.

Controlling for a confounder is the standard repair and it is not free: condition on the wrong variable and you create the bias you were avoiding. A collider — something both variables cause — is exactly this trap, and conditioning on one opens a path that was closed. Which variables to adjust for is a question about the graph you assumed, not one the data can answer.

Simpson's paradox is the extreme case: an association can hold in every subgroup and reverse in the aggregate, so the direction of an effect depends on which variables you conditioned on. Under a back-door adjustment set Z, p(y|do(x)) = Σ sub _z  p(y|x,z)·p(z) — but the validity of Z is an assumption about the graph, and the formula is silent about whether you chose it correctly.