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

Partial correlation — where it appears

The correlation between two quantities once a third has been removed from both. It is what a regression coefficient is reading when a covariate is included, and a zero in one is the signature that separates causal structures a covariance matrix cannot.

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

One distribution, two effects. Three causal structures fitted to one covariance matrix over a treatment, a covariate and an outcome. Each reproduces it exactly — the largest entry-wise disagreement across all three is 4.4e-16 — so no sample of any size distinguishes them. The regression of the outcome on the treatment and the covariate returns 0.500 in all three, to within 4.4e-16, because that coefficient is a function of the covariance and of nothing else. The effect the three worlds hold is 0.500, 0.848 and 0.848: adjusting is exactly right in the first and off by −0.348 in the other two. The arithmetic cannot see the difference and the difference is the whole question.

The two worlds that look the same

Three causal structures were fitted to one covariance matrix and agree with it to 4.4·10⁻¹⁶. The regression returns 0.5000 under all three; the effect they hold is 0.5000, 0.8481 and 0.8481. What separates structures is a missing edge, and the signature of one is a correlation of exactly zero.

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 diagramAdjustment setAsymptotic biasBiasCausal effectClosed formConditional independenceCovariance matrixCovariate adjustmentEstimation errorFisher transformIdentification

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