The symmetry the marginals could not show

A rule balancing the mean of each of two covariates removes exactly none of their product’s interaction. That is one of the two exact zeros this collection has, it is derived by parity, and the field that tested it established that it needs the covariate to be symmetric rather than normal — separated by a heavy-tailed symmetric marginal on which the zero holds to the last digit.

It needs a second thing, and six marginals could not have found it. The copula has to be symmetric under reflection too.

The measurement #

Five copulas, all at a Spearman rank correlation of 0.4, all with a standard normal covariate. Nothing about the marginal varies, so nothing here can be attributed to it.

The mean-balancing rule removes $8.5\times10^{-32}$, $4.4\times10^{-29}$ and $1.5\times10^{-31}$ of the product under a Gaussian copula, a t copula on four degrees of freedom and a Frank copula. It removes 7.707% under a Clayton copula and 7.707% under the same copula turned over.

The three with exact zeros are the three that are radially symmetric — $(U, V)$ distributed as $(1-U, 1-V)$ — and the two without are the two that are not. The correspondence is exact and it is measured rather than looked up: the total absolute difference between each density and its own reflection is below $10^{-15}$ for the first three and 0.136 for the other two.

Why parity needs both halves #

The argument behind the zero is one line and it makes the two conditions visible.

The covariate is an odd function of a latent variable, and the interaction is a product of two of them, which is even. An odd function is orthogonal to an even one when the law is invariant under the joint sign flip — and the joint sign flip is a statement about the whole joint distribution, which factors into what each margin does and what the dependence does.

The marginal has to be odd: that is the parity field’s condition, and a skewed covariate breaks it.

The dependence has to be invariant under flipping both signs: that is radial symmetry of the copula, and a Clayton breaks it. A copula whose density piles up where both variables are small is not the same law as one whose density piles up where both are large, so reflecting through the centre gives a different distribution — and an odd function is no longer orthogonal to an even one under it.

Two conditions, one from each half of the joint law. Either one alone is enough to break the zero, and this collection had only ever varied one of them.

Why this is the condition that matters in practice #

Both conditions can fail, and they do not fail equally often or equally visibly.

A skewed covariate is visible. Skewness is a number a practitioner computes from their own data without thinking about it, and a covariate that is obviously skewed — a cost, a duration, a count — is obviously skewed. The parity field’s warning is one anybody can act on.

Tail dependence is not. Two covariates can have a perfectly ordinary correlation, perfectly symmetric marginals, and still be much more likely to be extreme together at one end than the other. Nothing in a correlation, a scatter plot’s overall shape, or either margin’s summary statistics reveals it, and it is the standard finding wherever anybody looks for it. It is also invisible to the one summary this collection uses everywhere else for a pair of covariates: a correlation is a single number and radial asymmetry is a property of the whole joint law.

So the condition that is harder to check is the one this field is about, and it is broken by the commonest kind of departure there is.

The two Claytons #

The clearest single fact in the table is that a Clayton copula and its reflection give the same leak — 7.707% each, to every digit reported.

That is not an accident and it is a useful check on the mechanism. Reflecting a copula through the centre of the square maps radial asymmetry to radial asymmetry of the same magnitude, and the mean’s zero depends on the magnitude rather than on the direction. So the two copulas are equally far from symmetric and equally far from the zero.

They are not equivalent for everything. A threshold at a value on the covariate’s own scale reports 5.331% under a Clayton and 33.360% under its reflection — a factor of six, from the same rank correlation, the same tau, the same marginal and the same threshold. What separates them is that the threshold sits in the upper tail, so an upper-tail dependence and a lower-tail one are entirely different situations for it.

So the pair of copulas separates two kinds of quantity: those that depend on the amount of radial asymmetry, which is the mean’s zero, and those that depend on which tail, which is everything involving a cut at a fixed value.

The three zeros are zeros by twenty-seven orders of magnitude #

The exact readings are worth putting on one scale before anything is made of the non-zero ones.

The largest of the three is the t copula’s $4.4\times10^{-29}$, and the smallest non-zero leak in the table is 7.707%. The ratio is about $6\times10^{-28}$.

So there is no threshold to argue about and no question of a small number being rounded to zero. The gap between the zeros and the leaks is twenty-seven orders of magnitude, and a quadrature that could not tell them apart would be one that had failed at something far more basic.

The spread among the three zeros — from $8.5\times10^{-32}$ to $4.4\times10^{-29}$, a factor of five hundred — is a fact about the integrands rather than about the mathematics. The t copula’s density has the heaviest tails of the three, so its grid works hardest, and it is duly the least exact zero of the three by a wide margin and still a zero by any standard.

What the table cannot say about how the leak grows #

One limitation is worth naming here rather than leaving to a reader, because it bears directly on what a practitioner does with the finding.

The five copulas supply exactly two radial gaps: below $10^{-15}$ and 0.136. Everything in the correspondence is therefore a presence-against-absence statement, and the only slope available is the one through those two points — 56.7 percentage points of leak per unit of radial gap.

Whether the response is linear is not measured and probably not true. The leak is a squared multiple correlation, and a squared quantity vanishing at zero typically vanishes quadratically, in which case a copula half as far from radial symmetry would leak a quarter as much — about 1.9% rather than the 3.9% a linear reading gives.

That factor of two matters for the practical question, which is not whether an asymmetric copula breaks the zero but how much asymmetry is tolerable. Nothing in this table answers it, and a sweep over the asymmetry rather than over five named copulas would.

The size, and what it is a size of #

7.707% at a rank correlation of 0.4 is not a catastrophe and it is not nothing.

The right comparison is to what the rule is worth when it works. The dictionary field measures a rule holding the means and squares of both covariates removing 64.00% of the same product, so the mean-only rule’s zero is a guarantee of no protection rather than a guarantee of protection — and 7.707% is the amount by which a guarantee of nothing fails to be exactly nothing.

That inverts the usual reading of a broken guarantee, and it is worth sitting with. The exact zero was never good news for a trial. It says that against an outcome depending on the product of two covariates, balancing their means removes none of the risk — and the parity argument is what makes that statement exact rather than approximate.

So a broken zero is an improvement, in the narrow sense that 7.7% of something is removed where none was before. The reason it is still a finding is that the guarantee has stopped being statable: it was a number that held whatever the covariates’ distribution was, and it is now a number that depends on a copula nobody measured.

Where the worst case goes #

The reading a trial acts on is the worst case over the shapes an outcome might have, because an outcome model is not known before the trial.

Under the three symmetric copulas a rule holding a mean and a median split of each covariate — the two things every trial balances — has a worst case of exactly 0.00%. Under a Clayton it is 1.74%.

The rule holding the mean and the square of each does better everywhere, and it moves the other way: 5.34% under the Gaussian, 3.11% under the t, 3.26% under Frank and 4.41% under the two Clayton variants.

Two lessons in one table. The asymmetry that broke the zero also removed the shape the zero was protecting, so the rule every trial runs is better off under a Clayton than under a Gaussian. And holding one more function is worth several times more than either — the square dictionary’s worst case is two to five points above the split dictionary’s on every copula measured.

Which is the same conclusion the dictionary field reaches from inside a Gaussian world, arrived at here by changing the world instead of the rule.

What “matched” means, and that it moves things #

Five copulas cannot be compared until they are matched on something, and there are two conventional somethings.

Matched at a Spearman rank correlation of 0.4, the Clayton’s parameter is 0.759 and its Kendall tau is 0.2749. Matched instead at the Kendall tau the Gaussian copula has there — 0.2730 — its parameter is 0.752 and the mean’s leak moves from 7.707% to 7.668%.

Small, for the mean’s leak. Larger elsewhere: a threshold rule’s share moves by 0.81 points on the t copula, whose two rank statistics disagree most because its dependence is concentrated in tails neither statistic looks at.

Neither matching is more correct, and the exact zeros do not move under either — which is the check that says those zeros are not artefacts of a scale. What is not allowed is a table with the matching left out, and this is the same discipline the block-window field applies to a comparison at a shared length: a number is a number at a setting, and the setting has to be in the sentence.

The condition read as a scale rather than a switch #

The three symmetric copulas give exact zeros and the two asymmetric ones give 7.707%, which looks like a switch. It is a scale, and the reason is worth stating because it decides what a practitioner should worry about.

Radial asymmetry is measured here as the total absolute difference between the copula’s density and its own reflection — a distance between two probability measures on the square, zero for a symmetric copula and 0.136 for these two. A copula slightly off symmetric would have a small distance and a small leak, and the exact zero would be an exact zero only at the boundary case.

Nothing in this field measures the slope of that relationship, because the two asymmetric copulas here have the same distance. What it does establish is the shape of the answer: the zero is not a structural property that either holds or fails, it is the value of a continuous quantity at a symmetric point, and a practitioner whose dependence is nearly symmetric has a leak that is nearly zero.

That is a more useful thing to be told than a switch would be, and it is also what makes the missing test above matter: what has to be measured is a distance, not a yes or no.

What a practitioner should check #

Two things, and the second has no standard tool.

Is the covariate symmetric? A skewness near zero, checked on the sample. The parity field measures what a skewness of 0.3 costs — 11.63% of the product — so the answer has a scale.

Is the dependence symmetric under reflection? This is the harder one. A rank scatter plot with the two tails compared, or a formal test of radial symmetry, and the honest position is that most analyses never ask. What this field supplies is the size of the consequence at one setting: 7.707% at a rank correlation of 0.4 with a Clayton, which is 7.707% of a quantity that was exactly zero.

What the earlier field could and could not have found #

It is worth being precise about why six marginals could not reach this, because the reason is structural rather than a matter of not having tried.

Every marginal in the parity field is a monotone transformation of one latent normal. That is what makes the comparison clean: the dependence is held exactly fixed while the axes are relabelled. And a monotone transformation of each axis leaves the copula unchanged, by definition — the copula is the joint law of the ranks and a monotone map moves no rank.

So the six marginals are six relabellings of one copula. Whatever that copula’s symmetry is contributing, it contributes identically to all six, and no comparison among them can see it. The field’s design, which is exactly right for isolating the marginal’s contribution, makes the copula’s invisible by construction.

That is the general hazard in a well-designed comparison: holding something fixed to isolate one effect also makes the held-fixed thing’s own contribution unmeasurable, and a conclusion phrased as what the zeros need will silently mean what the zeros need, given this copula.

The repair is the one this field applies: run the comparison along the other axis, holding the first fixed instead.

One more reading of the same table #

The five copulas can be read as a sweep over one quantity rather than as five cases, and doing so makes the finding transportable.

Order them by radial gap: the Gaussian, t and Frank at essentially zero, and the two Claytons at 0.136. The mean’s leak follows exactly — zero, zero, zero, 7.707%, 7.707% — with no other property of the five predicting it. Not tail dependence, since the t copula has it in both tails and leaks nothing. Not Kendall’s tau, which is 0.2730, 0.2817, 0.2722, 0.2749 and 0.2749 and has no relation to the leaks at all. Not the family, since a Clayton and its reflection are different families in every ordinary sense and give identical leaks.

One quantity predicts it and the others do not, which is what turns a comparison of five named distributions into a statement about a property. That is worth more than any of the five numbers, and it is the reason the radial gap is measured on each copula rather than the families being labelled symmetric or not.

What is claimed here, and what is not #

This essay takes the second condition behind a mean-balancing rule’s exact interaction zero. The claims are that at a Spearman rank correlation of 0.4 with a standard normal covariate throughout, the rule removes $8.5\times10^{-32}$, $4.4\times10^{-29}$ and $1.5\times10^{-31}$ of the product under the Gaussian, t and Frank copulas and 7.707% under a Clayton and under its reflection; that the three with exact zeros are exactly the three whose density equals its own reflection through the centre of the unit square, measured at below $10^{-15}$ against 0.136; that the worst case of a rule holding a mean and a median split goes from exactly 0.00% to 1.74% between a Gaussian and a Clayton, so the asymmetry improves the guarantee by removing the shape it was protecting; and that matching the copulas on Kendall’s tau instead of Spearman’s rank correlation moves the leak to 7.668% and moves a threshold rule’s share by up to 0.81 points.

What stays out, and is named as a decision: a test for radial symmetry. The essay says the condition should be checked and offers no procedure for checking it. Tests exist in the literature and none has been run here, because running one would mean pricing its own size and power on this collection’s designs — a field rather than a paragraph.

Also out: a copula with asymmetric marginals. Every measurement here holds the covariate normal so that the copula’s contribution is isolated, and every measurement in the parity field holds the copula Gaussian so that the marginal’s is. What happens when both fail at once is not measured, and the natural guess — that the two leaks compound — is a guess.

The boundary against the first essay of this field is that it finds one zero needs no condition at all and this one finds the other needs two.