Three ways variables relate causally. Adjusting for the wrong one destroys the analysis.

Confounding

A confounder is a variable Z that causally affects both X (the treatment or intervention) and Y (the outcome). The classic structure: Z → X and Z → Y. Because Z causes both, X and Y will be correlated even if X has no causal effect on Y whatsoever.

Example: ice cream sales and drowning deaths are correlated. Neither causes the other. Temperature — the confounder — causes both. Any regression of drowning on ice cream sales that omits temperature will find a spurious positive coefficient.

In a causal graph, a confounder creates a back-door path from X to Y: a path that runs through the parents of X rather than through X’s descendants. The back-door criterion identifies the minimal set of confounders to adjust for in order to close all such paths and recover the true causal effect.

Mediation

A mediator is a variable M that lies on a causal path from X to Y: X → M → Y. The mediator transmits — or partially transmits — the effect of X on Y. If you include M in a regression, you block the indirect pathway and your estimate of X’s effect will be attenuated or eliminated — even though X genuinely causes Y.

Example: a training program (X) improves job performance (Y) partly by increasing employee confidence (M). A regression that controls for confidence will underestimate the training effect, because confidence is the mechanism by which training works. The “total effect” of training flows partly through confidence; controlling for confidence estimates only the “direct effect” that bypasses it.

Mediation analysis — decomposing total effects into direct and indirect components — is valuable and legitimate. But it requires an explicit decision to decompose, not an inadvertent inclusion of mediators in a control set.

Colliders: the hidden danger

Of the four causal structures, colliders are the most dangerous and the least intuitive. Confounders and mediators produce biased estimates if you get them wrong. A collider, conditioned on, creates an association that does not exist in the population — and every additional control variable added to a regression is a potential collider.

A collider is a variable C that is caused by both X and Y: X → C ← Y. Unlike confounders, colliders do not create an association between X and Y in the full population. But if you condition on C — by including it in a regression, or by selecting a sample where C has a particular value — you open a spurious path between X and Y.

The intuition: if you know that C occurred, and C is caused by both X and Y, then observing that X did not occur raises the probability that Y must have occurred to explain C. This creates a negative correlation between X and Y within the conditioned sample, even if X and Y are independent unconditionally.

Berkson’s Bias — a collider in clinical data

A hospital study finds that among hospitalised patients, patients with disease A are less likely to have disease B — suggesting A protects against B. But hospitalisation is a collider: you are hospitalised if you have A or B (or both). Conditioning on hospitalised patients opens the A → Hospitalisation ← B path and creates a spurious negative association. In the general population, A and B may be independent.

Moderation

A moderator (also called an effect modifier) is a variable W that changes the magnitude of X’s effect on Y. The causal structure is: W modifies the X → Y relationship. This is represented in a regression as an interaction term: Y = β₀ + β₁X + β₂W + β₃XW + ε.

Example: a marketing campaign (X) increases sales (Y), but the effect is larger among customers under 35 (W=young) than among customers over 55. Age moderates the treatment effect. Unlike confounding — where omitting W biases the average estimate — omitting a moderator averages over heterogeneous effects and may produce a correct average estimate but mislead you about who to target.

In a Bayesian network, moderation is encoded directly in the conditional probability table: P(Y | X, W) will show different conditional distributions for different values of W. The BN naturally represents effect heterogeneity that a regression model requires explicit interaction terms to capture.

The practical rule

The practical rule is disarmingly simple: draw the causal graph before choosing your control set. The graph makes the distinction between confounders, mediators, and colliders unambiguous. Without it, the choice of what to adjust for is a guess — and a wrong guess produces conclusions that are precisely wrong rather than approximately right.

The formal tool is d-separation: given a causal DAG, two variables are independent conditional on a set Z if and only if all paths between them are blocked by Z. A path is blocked if it contains a non-collider that is in Z, or a collider that is not in Z (and has no descendants in Z). The back-door criterion uses d-separation to identify valid adjustment sets.

For practitioners who cannot build a full causal graph, the minimum viable version is a list of variables and a question for each: does this variable cause the treatment, does it cause the outcome, or is it caused by both? That three-way classification distinguishes confounders (causes both), mediators (caused by treatment, causes outcome), and colliders (caused by both) well enough to avoid the most damaging errors.