Field

Both halves of the dependence at once

One field varies the marginal with the copula held Gaussian and another varies the copula with the marginal held normal, and the natural guess is that the two leaks compound. They do not add in either direction: eleven of twenty cells cancel and nine compound, a mildly skewed covariate under a lower-tail copula leaks 0.002% where adding the two gives 16.6%, and a heavy-tailed symmetric covariate that leaks exactly nothing on its own doubles what an asymmetric copula leaks. A median split's zero survives all thirty combinations at under 10⁻¹⁶.

Two failures that cancel

A mildly skewed covariate under a lower-tail copula leaks 0.002% of an interaction where each failure alone leaks eight and seven per cent. Turn the copula over and the same pair compounds.

The same copula, turned over. A Clayton copula and its reflection, at the same Spearman correlation of 0.40 and the same Kendall tau of 0.275, against the covariate's marginal. With a symmetric covariate the two are the same number to nine decimals — 7.707% apiece — because the leak then depends on how much asymmetry the copula has and not on which way it points. Skew the covariate and they come apart: at a skewness of 2.26 the lower-tail copula leaves 3.431% and the upper-tail one 36.213%, a factor of 10.6. Both halves of the dependence are asymmetries and an asymmetry has a direction; a lower-tail copula concentrates the dependence where a right-skewed marginal is compressed and the two distortions partly undo each other, and an upper-tail one concentrates it where the marginal is stretched.

A symmetry that was not enough

A heavy-tailed symmetric covariate has a skewness of zero and leaks exactly nothing under three copulas. Under the two asymmetric ones it doubles the leak, from 7.707% to 14.229%.

Three copulas that break nothing, and a factor of two between them. The three radially symmetric copulas, at a matched Spearman correlation of 0.40, against the covariate's marginal. All three leave exactly nothing with a symmetric covariate — that is the guarantee, and it holds to twenty decimal places. What they do to a skewed covariate is not the same at all: at a skewness of 2.26 a Frank copula leaves 12.118% where a Gaussian leaves 21.539% and a t on four degrees of freedom leaves 23.640%. A factor of 2.0 between two copulas that are both symmetric, both matched on rank correlation, and both harmless on their own. So the copula matters to the marginal's leak without breaking any symmetry of its own, which is a milder version of the same finding and applies to every trial rather than to the asymmetric ones.

A copula that halves a marginal

Three copulas break nothing on their own and put a factor of two between the same skewed covariate's leaks — 12.118% under a Frank against 23.640% under a t, at the same rank correlation.

The one zero neither half of the dependence can touch. A median split's interaction leak at all 30 combinations of copula and marginal, on a log scale. Every one is under 10⁻¹⁶ and the largest is 1.74e-20, which is the quadrature's own noise rather than a leak. The reason is arithmetic and it is short: a centred median split takes the values ±½, so its square is a quarter identically — for every unit, on every draw, whatever the covariate's scale is and whatever joint law the ranks have. The interaction is then orthogonal to both main effects by construction, and there is nothing for either half of the dependence to break. Both of the fields this one joins report this zero holding under their own variation; running both variations at once is what establishes that it is not two coincidences.

The zero that survives both

A median split's interaction leak is under 10⁻¹⁶ at all thirty combinations of copula and marginal. It is the only guarantee in the collection that neither half of the dependence can touch.

A median with ties

The median split's interaction zero survives every marginal and every copula, because a split that is on for exactly half the units squares to a constant. A covariate measured on a few levels cannot be split exactly in half: on a five-point scale three in eight units sit on the median, and a rule balancing 'above the median' removes 6.39% of the interaction under a Gaussian copula and 10.06% under one of the two Claytons. The size is set by the share of units on the median, which is the formula's own factor. And a fair coin for the tied units restores the zero exactly, on every copula.

Seven products of three splits. The correlation of every product of three centred median splits with every other, at a Spearman rank correlation of 0.4 between each pair of covariates. Under a Gaussian copula a main effect correlates with the other mains and with the triple (0.273), and with no pair at all, because the third moment is 0.000. Under a Frank copula, whose pairs are radially symmetric, the third moment is 0.078 and every other cell fills in: a main correlates 0.082 with the pair it is not in. In both panels a main is exactly uncorrelated with a pair it belongs to, which is the known zero.

The zero that needed two

A rule balancing two median splits removes nothing of their interaction, on every copula, because a centred split squares to a constant. Add a third covariate and the same arithmetic stops protecting the thing it looked as if it protected. The product of three splits is not orthogonal to the mains: at a rank correlation of 0.4 a rule on the mains removes 14.47% of it under a Gaussian copula. And the symmetry that decides the rest belongs to the triple rather than the pair, so Frank's copula, whose pairs are radially symmetric, leaks where Clayton's, whose pairs are not, barely does.

All essays