Concept

Direct effect — where it appears

The part of a treatment's influence that does not travel through a stated intermediate variable. Estimating it means conditioning on that variable, which is valid only when nothing unmeasured causes both the intermediate and the outcome.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Also named here as total effect — the same set of essays touches all of them, so they are one junction rather than several.

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

What stratifying on a mediator estimates, and what it does not. The direct effect stays at -0.109 throughout, because it is defined with the mediator held fixed. The total effect runs from -0.218 to 0.027 and crosses zero at 1.5. On 15% of the sweep the two have opposite signs, and both are correct answers.

Conditioning on what the treatment caused

When the grouping variable lies on the path from treatment to outcome, the stratified answer is the direct effect and the aggregate is the total effect. Both are correct. Over 15% of a sweep of the indirect path they have opposite signs, and no arithmetic on the table says which question was being asked.

reversal · Simpson

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

Named alongside it

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

Causal diagram Mediator Total effect Adjustment set Collider bias Confounding Covariate adjustment Path coefficient Regression Aggregation Bias Causal effect

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