Field

The other half of the dependence

What a balancing rule can remove of an interaction was measured across six marginals with one joint law of the ranks held fixed. Change the copula instead and the two exact zeros come apart for two different reasons. A median split's zero is arithmetic — a centred median split squares to a quarter identically — and holds under every copula there is. A mean's zero needs the copula to be symmetric under reflection as well as the covariate to be symmetric: it is exactly nothing under a Gaussian, t or Frank copula and 7.71% under a Clayton, at the same rank correlation and with a normal covariate throughout.

One zero is arithmetic and one is a symmetry. What two balancing rules remove of the interaction they are aimed at, on five joint laws of the ranks matched at a Spearman correlation of 0.4, with a normal covariate throughout. A rule holding a median split of each covariate removes exactly nothing of the product of the splits under every one of them, including the two that are not symmetric under reflection — and the reason is not a symmetry at all: a centred median split takes the values ±½, so its square is a quarter identically, and the interaction is orthogonal to both main effects whatever the joint law is. A rule holding the mean of each removes exactly nothing under the three radially symmetric copulas and 7.71% under the two that are not. Bars at the floor are exact zeros; the axis cannot draw 9e-32.

A zero that is arithmetic

A median split's exact zero was explained by a symmetry of the latent normal. It holds under a Clayton copula, which has no such symmetry, because a centred median split squares to a quarter identically.

The mean's zero is the copula's symmetry. Five copulas, each at a Spearman rank correlation of 0.4, with a normal covariate throughout — so nothing here is about the marginal, which is the whole of the earlier field. Horizontally: how far the copula's density is from its own reflection through the centre of the unit square, measured rather than read off the family's name. Vertically: what a rule balancing the mean of each covariate removes of their product. The three copulas at zero on the horizontal axis remove exactly nothing, to thirty decimal places. The two that are not symmetric remove 7.71%. A guarantee that held for six marginals turns out to have needed something the marginals could not have told anybody about.

The symmetry the marginals could not show

A mean's interaction zero needs the covariate to be symmetric and the copula to be symmetric under reflection. Six marginals could only ever test one of those, and the other is broken by the commonest kind of dependence there is.

Which tail the threshold is in. What a rule balancing a threshold at 1 on each covariate's own scale removes of the interaction between the two thresholds, on five copulas matched at a Spearman rank correlation of 0.4 with a normal covariate throughout. This rule never had a zero to lose — the earlier field establishes that under every marginal — so what is left is a size, and the size depends on where the dependence lives. A Clayton copula, whose density piles up in the lower tail, leaves 5.33%; the same copula turned over, so that it piles up in the upper tail where the threshold is, leaves 33.36%. Same rank correlation, same Kendall tau, same marginal, same threshold: 6.26 times the leak, decided by which end of the distribution the dependence and the cut are both in.

Which tail the cut sits in

The same copula and its reflection have the same rank correlation, the same Kendall tau and the same marginals. A balancing rule holding a threshold at a dose leaves 5.33% under one and 33.36% under the other.

A cut walked from the median into each tail. What a rule balancing a threshold on each of two normal covariates removes of the interaction between the two thresholds, with the cut moved from the 1st percentile to the 99th, on five copulas at a Spearman rank correlation of 0.4. Every copula reads exactly zero at the median. The three radially symmetric copulas are mirror images about it: the Gaussian peaks at 19.49% near the 95th percentile and is down to 13.44% at the 99th; the Frank peaks at 13.22% near the 85th and falls to 2.06%; the t is still near its highest, 37.45%, at the 99th. The Clayton climbs to 56.37% at the 1st percentile, in the tail its dependence is in, and never passes 5.39% in the other; its reflection is the same curve turned round.

A cut walked into the tail

Move a balancing rule's threshold off the median and the interaction it leaks rises from exactly zero — and then either turns back to nothing or climbs to a ceiling, depending on one property of the dependence. With both covariates cut, the whole regression collapses to two probabilities, and far out it settles on 2λ/(1 + λ) of the tail-dependence coefficient: 57.25% under a Clayton, 35.57% under a t, and zero under a Gaussian or a Frank however strongly the covariates are related.

A leak read off a cohort, at two cuts. The threshold rule's leak as a cohort of 1,000 units estimates it from its own two-by-two table: the bar is the middle eighty per cent of 600 cohorts' plug-in estimates, the open circle their median and the stroke the true leak, for five copulas at a Spearman rank correlation of 0.4. At the 90th percentile the estimate is centred on the truth and its middle eighty per cent spans about fourteen points: Gaussian 12.8% to 26.4% around 19.2%, t on four 21.5% to 35.7% around 28.9%, Frank 5.4% to 18.7% around 12.2%, Clayton 0.8% to 9.6% around 4.4%, Clayton turned over 35.9% to 49.5% around 43.1%. At the 99th it is not centred on anything: Gaussian 13.1% to 100.0% around 13.7%, t on four 17.5% to 72.9% around 37.4%, Frank 14.4% to 100.0% around 2.2%, Clayton 16.1% to 100.0% around 0.6%, Clayton turned over 37.0% to 75.7% around 56.3%, because on three of the five copulas between two cohorts in five and four in five have nobody past both cuts.

A leak read off a two-by-two table

The share of an interaction a threshold-balancing rule leaves alone needs one number from the data: how often a unit past the cut on one covariate is past it on the other. A cohort of a thousand estimates it to within a dozen points at the 90th percentile. At the 99th most cohorts have an empty cell in their table, and every one of those reads a leak of exactly one hundred per cent — a reading the bootstrap then confirms with an interval of zero width.

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