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

Collider — where it appears

A variable that two others both cause, so the two arrows meet at it. Two causes that are independent in the population become dependent once it is conditioned on, which makes it the one kind of covariate that adjustment creates a bias by including.

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 line is the sample, and the sample is the finding. 900 draws of two independent standard normal causes, with the 453 of them past a threshold of 0.00 marked and the 447 that fall short left pale. In the population the two are independent by construction. Inside the selected sample the correlation is -0.4669 in closed form and -0.5050 counted on these 453 rows, and the least-squares line through them has a slope of -0.545. The mechanism is visible in the picture rather than argued: the threshold removes one corner of the cloud, and a cloud with a corner missing is a cloud whose two coordinates carry information about each other.

The sample is a condition

Two independent standard normals, selected on their sum exceeding its median, read a correlation of exactly −1/(π − 1) = −0.4669 inside the sample. Nothing is measured badly and nothing is missing — and both halves of that split read it, in the same direction, while the population containing both reads zero.

collider · Conditioning Three adjustment rules on four thousand structures, as every covariate gets noisier. Four thousand random structures of six covariates, the treatment's effect fixed at 0.5, every covariate observed with the same reliability. Mean absolute bias of adjusting for nothing stays 0.190; adjusting for everything measured falls from 0.377 with exact covariates to 0.279 at reliability 0.8 and 0.203 at 0.5; adjusting for the common causes rises from 0 to 0.056 and 0.118. Adjusting for everything beats adjusting for nothing on 33.8% at 1, 38.3% at 0.9, 42.0% at 0.8, 45.3% at 0.7, 48.3% at 0.6, 50.8% at 0.5. The common causes never do worse than nothing at any common reliability.

A set of shadows

Measure every covariate of four thousand random structures with reliability 0.8 and adjusting for the common causes still never does worse than adjusting for nothing — the bias left is each confounder's part times one minus its proxy's reliability, scaled alike. Adjusting for everything measured gets better, beating nothing on 42.0% of structures against 33.8% with exact covariates, because noise blunts the bad adjustments too. Spread the reliabilities to 0.75 ± 0.25 and the right set does harm on 7.15% of structures, every one of them with confounders pushing opposite ways.

collider · Conditioning

Named alongside it

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

Causal diagramAdjustment setCovariate adjustmentBiasClosed formCollider biasConfoundingMediatorMonte CarloBack-door pathInstrumental-variableMean squared error

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