A zero that rests on a symmetry

What a balancing dictionary buys turns on one line. An interaction between two odd functions is even under the joint sign flip (X, Y) → (−X, −Y); the flip leaves a bivariate normal alone at every correlation; an odd function is orthogonal to an even one. So a rule holding odd functions of each covariate removes exactly nothing of their interaction, whatever the correlation and however many such functions are held.

That result covers the two things every trial balances — a mean and a median split, both odd — and gives them a worst case of exactly zero.

The argument is about a symmetry of the law, and it is worth asking what happens when the law does not have one. Age, dose, income, tumour volume, time to event: none of them is symmetric about anything.

Changing the marginal and keeping the dependence #

The right way to break the symmetry is to change one thing. Let (Z₁, Z₂) be bivariate normal at correlation ρ and let the covariates be X = T(Z₁) and Y = T(Z₂) for a monotone T — a Gaussian copula with whatever marginal T produces.

Everything about the dependence is untouched: the copula is the same, and any statement that is a statement about the joint ranks is unchanged. Everything about the shape of a covariate is not.

Six marginals. The identity, which is normal. A hyperbolic sine, which is heavy-tailed and symmetric. Three Tukey transformations at increasing skewness. And an exponential, which is a covariate with a meaningful zero somewhere other than its median.

The quantity that decides everything is not the skewness but the odd share: how much of the covariate’s squared spread changes sign when the latent normal does. The normal is 100% odd. So is the hyperbolic sine, which is not normal at all. The three skewed ones are 95.7%, 84.9% and 72.2% odd, and the exponential is 82.2%.

The six are chosen to separate three things that could each explain a zero, and the separation is the whole design of the comparison. Normality is what the original argument invoked. Symmetry is what it actually used. Light tails is a third possibility nobody stated, because the normal has both symmetry and light tails and no case distinguished them. The hyperbolic sine has symmetry and heavy tails; the exponential has light tails on one side and no symmetry at all. Between them the three explanations come apart.

The zero, and what breaks it #

A rule balancing the mean of each covariate, against an outcome depending on their product.

On a normal covariate it removes 6.5 × 10⁻³³ of that interaction — zero to machine precision, as the parity argument says. On a hyperbolic sine, at a skewness of zero and tails far heavier than a normal’s, it removes 7.9 × 10⁻³³. The same zero.

On the mildly skewed covariate it removes 11.63%. On the clearly skewed one 26.30%, on the strongly skewed one 29.73%, and on the exponential 28.51%.

It is not normality the zeros needed. It is symmetry. The heavy-tailed symmetric marginal is the case that separates the two, and without it the result would be a result about the normal law and could not be carried anywhere.

Why the odd share is the quantity #

The odd share is not a summary of the marginal. It is the quantity the whole result is a function of, and it is worth showing where it enters.

Write the centred covariate on the latent scale as a sum of its odd and even parts, T(z) − E[T] = a(z) + b(z) with a odd and b even. The interaction is the product T(Z₁)T(Z₂) centred, which expands into four terms: odd×odd, odd×even, even×odd and even×even. Under the joint sign flip the first and last are even and the two mixed terms are odd.

The dictionary holds a(Z₁) + b(Z₁) and a(Z₂) + b(Z₂). Its odd parts are orthogonal to the even terms of the interaction, exactly as before; its even parts are not. So everything the rule removes comes from the b’s, and when b is zero — when T is odd — the removal is zero.

The leak is a function of how much even part there is, and the odd share is one minus that. A skewness of 0.95 corresponds to an even share of 4.3% and a leak of 11.63%; a skewness of 4.75 to an even share of 27.8% and a leak of 29.73%. The two are not proportional — the geometry mixes them through the correlation — and they are monotone together, which is the direction the mechanism predicts.

The first sliver of asymmetry does most of the damage #

Setting the leaks against the even share — one minus the odd share, which is the part of the covariate the symmetry argument has no hold on — says how the guarantee degrades, and it does not degrade gently.

The four non-zero marginals are 4.3%, 15.1%, 17.8% and 27.8% even, and they leak 11.63%, 26.30%, 28.51% and 29.73%.

Read as marginal rates, the leak rises by 2.70 points for each point of even share over the first stretch, then by 1.36, then by 0.82, then by 0.12. A factor of twenty-two between the first rate and the last.

So the response saturates almost immediately. A covariate that is 95.7% odd — mildly skewed, the kind nobody would flag — is already at 39% of the leak the most asymmetric covariate on the table produces.

Which is the number a designer needs #

Turning that round gives the share of the guarantee each covariate still delivers, taking the strongly skewed case as the floor:

• 95.7% odd — delivers 61% of the protection
• 84.9% odd — delivers 12%
• 82.2% odd (the exponential) — delivers 4%
• 72.2% odd — delivers 0%

Being ninety-six per cent odd is worth sixty-one per cent of the zero, and being eighty-five per cent odd is worth twelve. That is the sentence a trial designer should have, and it is a long way from what “nearly symmetric” suggests.

The reason is visible in the algebra the next section sets out. The interaction the rule fails to remove is built from products of the even parts, and the rule’s own span is built from the odd ones; a product of two small even parts is small, but so is the projection available to remove it, and the ratio of the two — which is what a share is — does not go to zero anything like as fast as the even part does.

One more thing the ordering settles. The four are ordered identically by their even share and by their skewness, so this table cannot distinguish the two as explanations. What it can do is rule out skewness’s usual reading: the exponential, at a skewness of 2, sits between two Tukey transforms rather than at the end, and it sits where its even share puts it. A covariate’s asymmetry matters through how much of it changes sign, and skewness is one summary of that among several.

The direction it breaks in #

A guarantee that degrades has to degrade in a stated direction or it is not a mechanism, and this one has two.

With the correlation. At independence nothing leaks under any marginal, because the interaction is then orthogonal to everything in the span for a reason that has nothing to do with parity. As the correlation rises the leak rises: on the mildly skewed covariate it runs 2.51%, 8.35%, 11.63%, 14.81% and 20.39% across correlations of 0.2 to 0.8.

With the skewness. At a fixed correlation of a half the three skewed marginals leak 11.63%, 26.30% and 29.73% at skewnesses of 0.95, 2.26 and 4.75.

Neither is a small effect and neither is a large one at the correlations a trial actually meets. What they are is numbers that depend on the marginal, where the guarantee was a statement that needed no marginal at all.

The direction the guarantee runs in #

There is a reading of the numbers above that is exactly backwards and it is worth heading off.

The zero is a zero of protection: it says the rule removes none of that interaction, which is bad. So a number rising from zero to 26% is the rule protecting better on a skewed covariate than on a normal one.

That is true and it is not the point. What a guarantee is for is being statable in advance, and exactly zero is statable: a trial designer who knows their rule is a list of odd functions knows precisely what it cannot protect against, without knowing anything about their covariate’s distribution. A number between 0.2% and 2.5% that depends on a marginal nobody measured is not a guarantee, it is an unknown that happens to be small — and its being small is what the next essays measure rather than what this one assumes.

The two other zeros #

The zero the means enjoy is one of three the parity argument produces, and the other two behave differently enough to need their own essays.

A dictionary of median splits removes exactly nothing of the product of the median splits, and that zero is 1.08 × 10⁻³⁰ under every marginal here, identical to the last digit. Not small under all six — the same number under all six. That is not a degradation at all, and why it is invariant is the next essay’s subject.

A dictionary of thresholds at a value on the covariate’s own scale removes 22.49%, 20.08%, 19.92%, 17.69%, 15.80% and 4.88% across the six. It never had a zero to lose — including on the normal covariate, where it removes the most of the six — and why is a third essay.

Three rules, one symmetry argument, and three different fates: one that breaks with the marginal, one that does not depend on it at all, and one that never had it. Which of the three a trial is running is decided by a sentence in a protocol.

What is computed, and how #

Every quantity here is an inner product in a bivariate Gaussian expectation, and none of it is simulated.

The route the dictionary field uses is a Hermite series: condition on one variable, expand, and every mixed inner product becomes ρ^j times a one-variable answer. That is exact for polynomials and cut points and it converges slowly for a lognormal tail, so the arithmetic that is right there is the wrong tool here.

What is used instead is nested quadrature. Writing Z₂ = ρZ₁ + √(1 − ρ²)U, every expectation is an integral over z of a function times an integral over u, and both are Gauss–Legendre rules split at the points the integrand jumps. The inner breakpoints move with the outer variable: a threshold on Y at c is a jump of the inner integrand at u = (c − ρz)/√(1 − ρ²), so the split has to be recomputed at every outer node. Splitting at the fixed points instead — the obvious thing — integrates across a discontinuity and loses four digits without any sign that it has.

The check is four hundred thousand draws, and the two routes agree to within a fortieth of a percentage point on the cases where the target has a fourth moment a lognormal tail can supply.

Where the arithmetic was checked against the field it came from #

The quadrature reproduces three numbers the dictionary field computed by a completely different route, which is the strongest confirmation available for either.

A dictionary of squares on a normal covariate removes 64.00% of the covariates’ product at a correlation of a half, which is the corrected closed form 4ρ²/(1 + ρ²)². A rule holding a covariate and its median split of each has a worst case over seven shapes of exactly zero, and one square moves it to 6.84%. And the leak from a threshold at one on a normal covariate is 22.49%.

All three come out of a nested Gauss–Legendre rule that has never heard of a Hermite polynomial, and all three match to the digits quoted. Two routes to a number are two routes only if they can disagree about something, and these two share no arithmetic at all.

What a practitioner would have to know #

The practical form of the result is uncomfortable and it is worth stating as plainly as it deserves.

A trial designer choosing a balancing rule wants to know what it protects against. The parity argument gives an answer that needs nothing about the data: a rule made of odd functions is worth nothing against an interaction between two of them, at any correlation. That is a sentence a protocol can carry.

Replace it with what is true off the normal: a rule made of functions that are odd on the latent scale of a Gaussian copula is worth between nothing and twenty-nine per cent against an interaction between two of them, depending on the marginal of the covariate and on the correlation between the two covariates. That is not a sentence a protocol can carry, and estimating the two quantities it depends on from the trial’s own covariates is a measurement nobody is currently making.

The honest position is that the guarantee has become a calculation rather than a statement, and the calculation needs the marginal. That is what the field’s last essay is about, and its conclusion is the small consolation that all the numbers are small — which is a different kind of reassurance from an exact zero and should not be mistaken for it.

What is claimed here, and what is not #

This essay takes what the exact interaction zeros rest on. The claims are that changing the marginal under a Gaussian copula leaves the dependence untouched and the parity of a covariate not; that the odd share of a covariate on the latent scale falls from 100% to 72.2% across the marginals here; that a rule balancing the means removes 6.5 × 10⁻³³ of the covariates’ product under a normal covariate and 7.9 × 10⁻³³ under a heavy-tailed symmetric one, so the guarantee needs symmetry rather than normality; that it removes 11.63%, 26.30%, 29.73% and 28.51% under the four asymmetric marginals; and that the leak grows with both the correlation and the skewness.

What stays out, and is named as a decision: a copula that is not Gaussian. Everything here keeps the Gaussian dependence and changes the marginals, which is what isolates the effect. A different copula changes the conditional expectations the whole calculation is built on, and the result would be a measurement of two things at once.

Also out: the effect on a real trial’s imbalance. These are exact geometric quantities — shares of a variance removed by a projection — and translating them into what a rule does to a hundred and twenty units is the geometry-against-trials comparison that this collection already runs, at one marginal.

The boundary against the essay that established the parity is that it proves the zeros and this one asks what they need. Neither is a claim that the zeros are wrong; they are exactly right, under exactly the condition they were derived under.

What it does not change #

Two things this collection establishes about balancing rules are untouched by any of it, and saying so bounds the damage.

A rule only sees its span. What a balancing rule removes of a shape is the shape’s squared multiple correlation on the span of what the rule was told to hold, and two bases for the same subspace are the same rule. That is a fact about projections and it holds in any inner-product space, under any marginal, with no symmetry anywhere in the argument.

A correlation is what makes an interaction reachable at all. At independence every main effect is orthogonal to every pure interaction, so a rule holding main effects removes exactly none of one — and that zero survives every marginal here too, which the leak curves show by all starting at the origin.

So the geometry is intact and the parity is not. What changed is a statement about which particular functions land orthogonally, and that statement was the one doing the work in the guarantee.

The checks, and the refusals that make them mean something #

Three claims are gated. The quadrature is required to agree with four hundred thousand draws on five cases, including one with a threshold in it under a skewed marginal, which is where the moving breakpoints are and where getting them wrong costs digits silently. The zeros are required to hold exactly under every symmetric marginal and to fail under every asymmetric one, because a single case of each would leave open that the tails or the particular law were doing the work. And the leak is required to grow with the correlation on every skewed marginal and to stay at machine zero at every correlation on the symmetric ones.

The refusal is the guarantee itself, carried across. The exact worst case, quoted for a covariate that is not symmetric, is refused — with the normal covariate’s exactly-zero worst case printed beside the skewed ones’ 0.74%, 1.83%, 2.24% and 2.49%.