Normal Distribution
nothing is called 95% until it has been counted
Concept

Adjustment set — where it appears

The covariates a regression conditions on when estimating an effect. Which set is correct is a property of the causal structure rather than of the data, and the set that includes every covariate measured is correct only when every covariate happens to be a common cause.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

The same covariate, three ways round. Three worlds over a treatment, an outcome and a covariate, joined by the same three edges at the same three strengths — 0.90, 0.50 and 0.70 — differing only in which way the two edges touching the covariate point. In the first the covariate causes both and adjusting for it recovers the effect of 0.50 exactly. In the second the treatment causes the covariate, the effect is 1.13, and adjusting returns 0.50 — the direct edge alone, with the part that travels through the covariate deleted. In the third the treatment and the outcome both cause the covariate, the effect is 0.50, and adjusting returns -0.087. The regression that produces those three numbers is one formula, and nothing in the data says which panel it is being run in.

One arithmetic, three decisions

A covariate beside a treatment and an outcome can be a common cause of both, a step on the path between them, or an effect of both. The regression that includes it is the same arithmetic in all three, and it is right in one — returning 0.5000, deleting 0.6300 of the effect, and turning 0.5000 into −0.0872.

collider · Conditioning Two structures in three are made worse. What adjusting for every covariate measured does to the bias in the treatment's estimated effect, against adjusting for none, over 4000 randomly drawn structures of 6 covariates each. Each covariate is independently a common cause with probability 0.25, a cause of the treatment only, a cause of the outcome only, a cause of neither, a step on the causal path, or a common effect. The rule leaves a larger bias on 65.5% of structures, a smaller one on 33.8%, and the same on 0.7%. The share is a property of that population of structures rather than of adjustment, which is why the weights are stated; what does not depend on them is that the rule has no direction — it is not a conservative default that occasionally overcorrects, it is a rule whose error is whatever the structure happens to be.

Adjusting for everything

"Control for every covariate that was measured" leaves a larger bias than controlling for nothing on 65.5% of four thousand randomly drawn structures and a smaller one on 33.8%. Its squared error is 4.110 times that of using no covariate at all, and half of it sits in its worst tenth of structures.

collider · Conditioning A covariate that is prior to everything and still ruins it. A covariate measured before the treatment, caused by neither the treatment nor the outcome, and not a common cause of them. Two unmeasured variables sit behind it: one reaches the treatment, the other reaches the outcome, and both reach the covariate. Every rule of thumb for including a baseline variable is satisfied, and the regression that leaves the covariate out estimates the treatment's effect of 0.50 without bias, while the regression that includes it is off by −0.2000 — because the covariate is a common effect of the two unmeasured causes, and conditioning on a common effect makes its causes dependent. The path it opens runs from the treatment back through the first unmeasured cause, through the covariate, and out through the second to the outcome.

A collider before the treatment

A covariate measured before the treatment, on no causal path, and not a common cause of anything, still biases the estimate by exactly −0.2000 against an effect of 0.5 — while the regression that leaves it out is exact. The bias saturates at 0.3536, and the two paths that make it a collider do not appear in that bound.

collider · Conditioning The covariate the treatment caused, and what it hides. A covariate on the causal path: the treatment causes it and it causes the outcome, so the treatment's total effect of 1.130 runs partly through it. Adjusting for it returns the direct edge alone, 0.500, which is what somebody wanting the total effect should not have asked for. The dashed variable is the second problem: an unmeasured cause of both the covariate and the outcome. It does not touch the treatment, so the unadjusted regression still recovers 1.130 exactly. It does touch the covariate, so once the covariate is conditioned on the treatment and the outcome are linked through it, and the adjusted coefficient lands on 0.050 — neither the total effect nor the direct one.

The variable the treatment caused

Adjusting for a covariate the treatment caused stops estimating the total effect and starts estimating the direct one. When that covariate shares an unmeasured cause with the outcome it estimates neither: the total effect is 1.1300, the direct effect is 0.5000, and the regression returns 0.0500.

collider · Conditioning A proxy removes less than its reliability, always. The share of the confounding bias removed by adjusting for a proxy, against how well the proxy measures the confounder. The diagonal is the answer a reader would guess — a covariate that is 80% signal removes 80% of the problem. The curve is what the arithmetic gives: the reliability, times one minus the squared correlation between the treatment and the confounder, divided by one minus the product of those two. That squared correlation is 0.4475. A reliability of 0.8 removes 68.85% and one of 0.6 removes 45.32%. The two agree only at the ends, and the gap is widest where most applied covariates sit.

Adjusting for a shadow

A covariate that is 80% signal removes 68.85% of the confounding, not 80% — the share is λ(1 − ρ²)/(1 − λρ²) and it is below the reliability everywhere. The residual bias is 0.1084 against an effect of 0.5, and at 25,600 rows it is 17.6 standard errors wide.

collider · Conditioning

Named alongside it

The objects these essays reach for when they reach for this one.

Causal diagramCovariate adjustmentBiasColliderCollider biasLatent variableMediatorPath coefficientUnmeasured confoundingBack door pathClosed formConfounding

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