The zero that needed two

A rule that balances a median split of each of two covariates removes nothing of their interaction. That is the fact this line of essays has been testing from every side. The arithmetic behind it is one line: a split centred to plus or minus one squares to one, so the product of the split with the interaction is the other split, whose mean is zero whatever the two covariates are doing together. It held through every copula, through every marginal, through both at once, and failed only when a covariate’s ties stopped the split being a split of exactly half.

That essay left a question it called a short calculation nobody had made. Three covariates give a three-way interaction, which is a product of three centred splits, and each of them still squares to one. The obvious guess is that the zero carries over: the triple is orthogonal to every main effect and every pair, under any dependence, for the same reason.

The guess is half right, and the half it gets wrong is the half it was most sure of.

A table of covariances with one number per row #

Write $z_i = \pm 1$ for the centred median split of covariate $i$, and $T_A$ for the product of the splits in a set $A$. Because each $z_i^2 = 1$, multiplying two products cancels whatever they share:

$$T_A T_B = T_{A \triangle B},$$

the product over the symmetric difference, the covariates that appear in exactly one of the two sets. So the covariance of any two products is not a new quantity. It is

$$\operatorname{Cov}(T_A, T_B) = m(|A \triangle B|) - m(|A|)\, m(|B|),$$

where $m(k)$ is the expected product of $k$ splits, on a dependence that treats its covariates alike. Every question this essay asks is a question about a short list of moments: $m(1)$, which is zero because a median split is balanced; $m(2)$, the quadrant dependence between two covariates, which the field has been calling the joint share in other units; and $m(3)$, $m(4)$ and so on, which nothing so far has needed.

Now the known zero reads differently. A pair against one of its own mains leaves a symmetric difference of size one, and $m(1) = 0$. That is the whole of it. It needs a balanced split and nothing else, which is why no copula could touch it.

The triple against a main leaves a symmetric difference of size two. The covariance is $m(2)$, and $m(2)$ is not zero; it is the very dependence the copula was introduced to supply. So the product of three splits correlates with each main by exactly the quadrant dependence of the other two. A pair against the third main, the one it does not contain, leaves a symmetric difference of size three, and that is $m(3)$, which no calculation on two covariates ever had occasion to compute.

Seven products in one picture #

The figure at the head of this essay draws the whole table for three covariates at a Spearman rank correlation of 0.4 between every pair. There are seven products: three mains, three pairs, one triple. The grey cells are exact zeros and the tinted ones are not.

Under a Gaussian copula, on the left, the table falls into two blocks. The mains and the triple correlate among themselves; the pairs correlate among themselves; nothing crosses. The mains sit at 0.273 with each other, which is $m(2)$, and every main sits at the same 0.273 with the triple. The odd products and the even products have nothing to do with each other, because a Gaussian dependence is unchanged when every covariate’s sign is flipped, and that flip sends an odd product to its own negative. Any expectation an odd product has must therefore be zero.

Frank’s copula, on the right, has the same diagonal structure and most of the same pattern, with values a little larger because a calibrated Frank carries more dependence at the median. But the two blocks are joined. A main now correlates 0.082 with the pair it is not in, and the triple correlates −0.03 with each pair. The cells that stay exactly grey are the six where a main meets a pair containing it — those are the old zero, and they survive here as they survive everywhere.

The picture has been drawn from the moments rather than from a simulation, so the zeros are exact and the grey is not a rounding. Counting 400,000 draws from the Gaussian copula and from Clayton’s gives $m(2)$ within a standard error of the integral and $m(3)$ within one standard error of its value, which is a check on the moments and not a separate estimate of anything.

The triple, against a rule on the mains #

A rule that balances all three splits removes from the triple whatever the three mains can explain of it. With the third moment zero that share has a closed form. Let $q = m(2)$. The three covariances are each $q$; the mains’ own correlation matrix has one on the diagonal and $q$ off it; and solving the regression gives

$$R^2 = \frac{3 q^2}{1 + 2q}.$$

At Spearman 0.4 the Gaussian $q$ is 0.273 and the share is 14.47%. Nothing about this number is a tail effect or a consequence of a skew. It comes from a perfectly balanced split on a perfectly symmetric dependence, at a moderate correlation, and it is not small.

Read across the copulas the share is 14.47% on the Gaussian, 15.23% on the $t$ on four degrees of freedom, 17.48% on Frank and 14.45% on Clayton, either way up. It rises steeply with the correlation: about 1% at 0.1, 4–5% at 0.2, 9–11% at 0.3, 21–25% at 0.5 and between 38% and 45% at 0.7.

Three things are worth reading off this. The first is the direction of the surprise. In the earlier essays a non-zero share was a failure of a guarantee, a leak, in that a rule on the mean happened to remove part of an interaction it was not aimed at and so a report of that interaction’s imbalance would have been misleading. Here the same kind of number is a free good if it is anything: a rule nobody designed to protect a three-way interaction removes about a seventh of one. The second is that the exact zero for two covariates was not evidence of a general orthogonality. It was a coincidence of parity at the smallest case, and the smallest case is the only one the field had examined.

The third is where the $t$ copula sits. The sign of a $t$ variate ignores its chi-square scale, so a median split cannot see the difference between a $t$ copula and a Gaussian one at the same correlation parameter. The two curves differ only because calibrating to one Spearman gives the $t$ a slightly larger correlation, 0.428 against 0.416. A median split is blind to the tails in a stronger sense than the field has so far needed: its whole table depends on the copula through a handful of orthant probabilities.

A symmetry that belongs to the triple #

What decides whether a pair leaks into a third main is $m(3)$, and it is where the surprise is.

Everything in the copula field so far has sorted dependences by radial symmetry: Gaussian, $t$ and Frank are symmetric, Clayton and its reflection are not, and the means’ zero stands or falls on the difference. The natural reading is that $m(3)$ sorts them the same way. A radially symmetric copula has $m(3) = 0$ and an asymmetric one does not.

It does not. Frank’s third moment is 0.078 at Spearman 0.4 and peaks a little above 0.08; Clayton’s is −0.0009 at the same correlation and never exceeds about 0.007 in size anywhere on the range. The copula that the field had always treated as symmetric has a third moment ninety times larger than the one it had always treated as the asymmetric case.

There is no contradiction, and the reason is worth stating carefully. Radial symmetry of a bivariate copula says that the pair looks the same with both signs flipped. Frank’s pair does. But the three-dimensional Frank copula — the exchangeable Archimedean one, which is the natural extension and the one meant whenever this essay says Frank — is a different object, and a symmetric margin says nothing about a symmetric joint. The sign-flip symmetry of three covariates is a property of the triple. The Gaussian and the $t$ have it because flipping all signs of an elliptical vector leaves its law alone. Frank’s trivariate form does not.

Clayton is the opposite case: strong asymmetry that nearly balances at this one quantile. Its moments are available in closed form, since the copula at a corner of one half is $(j\,2^\theta - j + 1)^{-1/\theta}$, and at $\theta = 1$ the third moment is

$$m(3) = 1 - \tfrac{6}{2} + \tfrac{12}{3} - \tfrac{8}{4} = 0$$

exactly. That is a Spearman of about 0.478. Below it the moment is slightly negative and above it slightly positive, which is a sign change the bivariate story gives no hint of. It is a statement about the one quantile a median split reads: move the cut, as the essay on the cut walked into the tail did for pairs, and there is no reason to expect Clayton’s near-balance to survive. That walk has not been made for three.

What this does to a trial is modest in size and specific in kind. A rule holding all three splits removes 0.78% of a pair’s interaction under Frank at Spearman 0.4, and 1.17% at 0.7, through the main it does not contain. Under the Gaussian, the $t$ and Clayton it removes nothing measurable. So the statement “a median-split rule removes nothing of an interaction” has to acquire a condition as soon as a third covariate is present: nothing, of a pair on its own two mains, on any copula; and nothing of a pair on a third main only if the triple is balanced, which is not the condition anybody would have written down.

The same arithmetic in a design book #

Anyone who has read what a fraction of a factorial costs has seen this table before, at its extreme. A half fraction of a $2^3$ design is the set of runs where $z_1 z_2 z_3 = 1$. On those runs $m(3) = 1$, every main is the same column as the pair it is not in, and the design book calls them aliased. That is a statement about exactly the cell this essay has just computed, $\operatorname{Cov}(T_{12}, T_{3}) = m(3)$, at the value of one.

So a population with a third moment of 0.078 is, in a precise and limited sense, a very soft fractional factorial that nobody designed: the words of the defining relation hold on average, by a small amount, with the covariates supplying the fraction instead of a table of runs. And $m(2) \neq 0$ is a softer aliasing still, of a kind a design book has no need for, because a designed factorial has balanced pairs by construction. The covariates of a trial do not, and that is the entire difference between the table of a factorial and the table in the first figure.

It also suggests why the zero looked so general. In a full factorial, every product is orthogonal to every other, because all the moments vanish. A pair of correlated covariates spoils only one of those orthogonalities, the one between the two mains, and the pair is orthogonal to both of them as it would be in the design book. The smallest case is unusually kind. With three covariates the spoiling reaches into the products themselves.

What the rule holds, at more covariates #

The moments go on being available, so the same calculation runs for a product of any number of splits, against a rule on the mains or a rule on the mains and the pairs.

The pattern is parity all the way down. Under the Gaussian an even product has nothing to do with the mains, which is the old zero at two splits and also at four and at six: 0.00% each time, exact. An odd product has nothing to do with the pairs, so holding the pairs as well adds nothing at three or five splits: 14.47% with or without them, 4.98% with or without them. And an even product has a lot to do with the pairs: a rule on the mains and pairs removes 18.69% of the product of four splits and 7.28% of the product of six, against nothing for the mains alone.

So there is a clean way to say what a rule is buying, under any dependence the sign flip leaves alone. A rule on the mains protects the odd products and a rule on the pairs protects the even ones; neither crosses. Frank fills in the crossed entries by small amounts: the product of four splits on the mains goes from 0 to 1.31%, the product of six to 2.00%, and the triple on mains and pairs rises from 17.48% to 17.82%.

The figure is also a warning about reading those shares as a protection worth paying for. A product of six median splits is not an interaction anyone expects, and a share of it is not a clinical benefit. What is usable here is the parity statement, which tells a protocol writer that adding pair products to a rule is worth nothing for an odd-order interaction on a symmetric dependence, however much they cost in the imbalance of the mains, a cost holding more functions at once has already put a price on.

What a protocol should write, and what it should stop writing #

The two-covariate statement was “a median-split rule removes none of the interaction between the covariates it splits”, and it was correct at two. For more than two, the sentence that replaces it has three parts.

A pair stays out of its own mains. That is still arithmetic: it requires balanced splits and nothing else, and a coin for tied units restores it when a scale has ties.

A triple does not. A rule on the mains removes $3q^2 / (1 + 2q)$ of it when the third moment vanishes, with $q$ the quadrant dependence. That is available from a sample’s two-by-two tables without a model, so a protocol can report how much of a three-way interaction its rule is removing simply by reading off the observed $q$, and the number can be substantial: a seventh at a rank correlation of 0.4, about two fifths at 0.7.

The pair’s relation to a third main is an empirical question. It is decided by the expected product of three splits, an ordinary sample average of plus and minus ones, and at any realistic number of units its standard error will dwarf Frank’s 0.078: at a hundred units it is 0.10. So no trial can read it from its own baseline table. What it can do is note that the effect it is bounding is under one per cent even under the worst of these five copulas.

How the numbers were made #

Everything here is computed from the moments $m(k)$, with no grid and no simulation in the reported figures. On the elliptical copulas $m(k)$ is a one-dimensional integral: conditional on a common factor each split is independent with mean $2\Phi(a w) - 1$, so $m(k)$ is that quantity to the power $k$ integrated over the factor. On Clayton and Frank it is the copula’s Archimedean function evaluated at multiples of its inverse at one half, expanded by the binomial theorem, because $z = 1 - 2\cdot \mathbf{1}(U \le \tfrac{1}{2})$. Each copula’s parameter is the one the copula field’s calibration gives for the stated Spearman correlation on a pair, and the shares are regressions among the products using only those moments.

Two routes agree. The closed form $3q^2/(1+2q)$ matches the general solver to a billionth on the Gaussian and the $t$; the triple moment’s value of exactly zero at $\theta = 1$ on Clayton matches the arithmetic above; and 400,000 draws reproduce $m(2)$ and $m(3)$ on the Gaussian and on Clayton within a standard error or two. The calculation also has to reject the guess that motivated it: the claim that the triple keeps the zero is refused on all five copulas, by a margin of more than ten percentage points at Spearman 0.4, and the pair on its own mains is required to come back below $10^{-14}$ at the same time, so a broken group identity would fail in one direction or the other.

Still open: a cut other than the median, and a dependence that is not exchangeable #

Two limits of this essay are the natural next measurements. The first is the cut. All of the above sits at the median, where $m(1)$ vanishes and the group identity does nearly all the work. The cut walked into the tail showed for pairs that a threshold elsewhere leaks by a formula in the joint share. The three-way version has an additional term, since the triple then correlates with its mains through $m(1)$ as well as through $m(2)$, and whether Clayton’s near-balance at the median leaves any trace at the 90th percentile is computable from the same closed form.

The second is exchangeability. Every pair here has the same correlation, and the three-dimensional Archimedean copulas force that. A real baseline table is not like that: two covariates may be strongly related and a third barely. With three different quadrant dependences the triple’s share on the mains is a different function, and which pair dominates is the useful part. A Gaussian with an arbitrary correlation matrix can be computed from the orthant formula; for an asymmetric dependence a vine would be needed, and the third moment there would no longer be a single number.

And the first essay’s smaller question is still unanswered. A coin-broken split is balanced in expectation and not in a sample, so the finite-sample share a rule removes at forty units has not been measured here either.

How far the parity reading goes #

This essay takes products of centred median splits under an exchangeable dependence. Its claims are that every covariance among such products is a function of one symmetric difference; that a pair on its own mains is exactly zero on every copula and a triple on its mains is not, at 14.47% on a Gaussian at Spearman 0.4, 15.23% on a $t$, 17.48% on Frank and 14.45% on Clayton; that whether a pair leaks into a third main is decided by the expected product of three splits, which is zero on the elliptical copulas, 0.078 on Frank and −0.0009 on Clayton with an exact zero at a parameter of one; and that under a sign-flip-symmetric dependence rules on the mains and rules on the pairs protect only odd and even products respectively.

What stays out, by decision rather than oversight: rules actually run in a finite trial, since every number here is a population regression; dependences other than exchangeable; cuts away from the median; and any claim that a three-way interaction exists in a given outcome, which is a clinical question this arithmetic cannot address. The boundary against two failures that cancel is that it held two covariates and let the marginal vary, and this one holds the marginal irrelevant, as a median split does, and adds covariates. The boundary against where the skew puts a threshold is that it moved the cut and this one stays at the only cut where the group identity is exact.