A cut on each side

A rule that balances two threshold indicators also removes part of their interaction, unless both thresholds are medians. A cut walked into the tail measured how much, with the same percentile on each covariate, and found the amount governed by the tail dependence of the copula in the limit. Which tail the threshold is in showed that side matters when the copula is asymmetric, and where the skew puts a threshold showed that a threshold at a fixed value on a skewed scale is simply a cut at a different percentile, moving along that same sweep.

All of those are readings of one line through a surface. A pair of covariates with different skews lands at two different percentiles. So does a protocol that sets its own clinical cut on each scale, and so does a pair in which high on one covariate is the bad direction and low on the other. Those are the ordinary cases, and the shared cut is the special one. The last of those essays asked whether a pair of cuts on opposite sides of the median could make the two departures compound where a single cut makes them cancel, and called it a short calculation. It is, and the answer to that question is yes; but the surface it is read from has a stranger feature than the one it was asked about.

The diagonal was the quiet part #

With the first indicator on for a share $p$ of units, the second for a share $q$, and a joint share $p_{11}$, write $c = p_{11} - pq$. The share of the centred product removed by a rule on the two indicators is a closed expression in those three numbers,

$$\frac{c^2\,\bigl[(1-2p)^2 q(1-q) - 2(1-2p)(1-2q)c + (1-2q)^2 p(1-p)\bigr]}{\bigl[p(1-p)q(1-q) - c^2\bigr]\,\operatorname{Var}(ab)},$$

where $a$ and $b$ are the centred indicators. The cohort essay used the same formula for a single median split against a threshold. Here it is read over a grid of pairs: eleven percentiles on each covariate, 121 cells, five copulas at a Spearman rank correlation of 0.4.

The figure at the head of this essay is the Gaussian and the $t$ on four degrees of freedom. Each has its one exact zero at the double median, as it must: both factors $1 - 2p$ and $1 - 2q$ vanish there and nothing else can. Down the diagonal from that zero are the numbers this line of essays already has: 19% at the 95th percentile on the Gaussian and 34% on the $t$, the heavier tail leaking more, which is the ordering the field has reported since it began.

Now look at the far corners, top right and bottom left. On the Gaussian, a cut at the 5th percentile on one covariate and the 95th on the other removes 54.8% of the product. On the $t$ it removes 1.1%. The two copulas have exchanged order and the gap between them has gone from a factor of less than two in one direction to a factor of about fifty in the other. The same five-copula comparison across the three extreme corners:

Tail dependence had been the organising idea of every table up to here. It does organise the near corner. It does not organise the far one, because the far corner reads a different pair of tails: the discordant corner, where one covariate is extreme upward and the other extreme downward. Nothing in the field has measured that.

One cut held still #

Reading a single row makes the shape easier to see. Fix the first cut at the 95th percentile and sweep the second from the 5th to the 95th.

The Gaussian falls throughout: 54.8% at the 5th, 39.5% at the median, 19.5% at the 95th. The $t$ does the opposite, from 1.1% through 30.9% to 34.5%. They cross between the 60th and 70th percentiles. Frank has an interior maximum of about 30% and falls either side of it. Clayton turned over has a maximum of 66.2% at the median, higher than its value with both cuts at the 95th, and that is the one row where a covariate split at the median carries more leak than the same covariate cut in its tail.

Two practical readings follow, and neither has been available from the diagonal.

The first is that the median split on one covariate does not protect a threshold on the other, which the cohort essay established for one pair, and that on one copula here the median is the worst place to put it. Moving the second cut from the 95th to the median, on Clayton turned over, moves the share from 51.0% to 66.2%. The rule’s unintended removal is a function of both cuts jointly, and neither “make it a median” nor “take it into the tail” is a safe direction by itself.

The second is that the leak is not monotone in either cut, on any copula. On the Gaussian, with the first cut at the 80th, moving the second inward from the 80th to the 70th lowers the share from 14.0% to 12.1%, and moving it further to the 50th raises it to 14.0% again. The reader who remembered the diagonal as a rule — further out, more leak — would predict the opposite at this one step. It is a small dip, but it means the minimum for a given first cut is off the diagonal, at a pair of unequal cuts.

A cell that empties #

Why should the far corner be so different? Write the two indicators out at one such pair: the first is on for 95% of units, and the second for 5%. There are four cells, and the one that matters is the off cell, units past the rare cut on the second covariate and short of the first: high on $Y$ and low on $X$.

Under a positive dependence that cell is thin. Suppose it were empty. Then the second indicator being on implies the first is on, so the product of the two indicators is the second indicator. A product of indicators that equals one of them is not an interaction at all: its centred form is an exact linear combination of the two centred indicators, a rule on those two removes it entirely, and the share is 100%. The same algebra forced a full leak on a table with an empty cell in a sample; here it is the population heading towards one.

So the far corner is asking how fast the off cell empties. Against the size it would have under independence, the Gaussian’s off cell is at 6% at a depth of a twentieth, 1% at a hundredth and essentially nothing at a two-hundredth. Clayton’s is 11%, 3% and 2%. Frank’s falls to about 21% and stays there. And the $t$ on four, with its tail dependence in every corner, does something none of the others can: its off cell is 70% of its independent size at a twentieth, 96% at three hundredths, and then larger than independent, 2.3 times at a hundredth and 4.1 times at a two-hundredth.

That is why the $t$ curve touches zero. At the depth where the off cell passes through its independent size, $c = 0$: the two indicators are uncorrelated at those cuts, and an uncorrelated pair has a product a rule on them cannot touch. On a $t$ copula at a positive rank correlation there is therefore a zero of the threshold’s leak that has nothing to do with medians, near the cuts at 3% and 97%. It is an accident of this correlation, in that it would move with the correlation, and it is a real exact zero of the formula rather than a small number.

Clayton is the instructive middle case. Its off cell does empty against independence, but it climbs only to about 40% and then falls back, to 34.6% at a two-hundredth. Emptying against independence, it turns out, is not the relevant comparison.

Two limits, either side of a third #

What the off cell has to beat is everything else the centred product varies by. With the cuts at $d$ and $1 - d$, the indicators’ own contribution to the product’s variance is of order $d^3$: the rare cell times the square of a small centring constant. The off cell contributes its own mass times a constant near one. So the share heads to one if the off cell shrinks faster than $d^3$ and to zero if it shrinks more slowly.

For a Gaussian copula the rate is known. The off cell is a bivariate normal tail at correlation $-\rho$, which shrinks like $d^{2/(1-\rho)}$ up to a logarithm. So the boundary is at

$$\frac{2}{1 - \rho} = 3, \qquad \text{that is,} \qquad \rho = \tfrac{1}{3},$$

or a Spearman rank correlation of 0.32. The prediction is sharp and a little startling: a Gaussian pair just below that correlation has a far corner that leaks less and less as the cuts move out, and one just above has a far corner that leaks more and more until the rule removes all of the interaction. At the field’s usual Spearman of 0.4, $\rho = 0.416$, which is on the upward side.

The grid cannot reach this; its joint shares are too coarse past about a two-hundredth. But the Gaussian off cell is a single integral of the conditional law, which can be evaluated to a depth of a millionth.

It is what the exponent says. At $\rho = 0.5$ the share is 93.0% at a hundredth and 99.8% at a ten-thousandth. At $\rho = 0.416$ it is 68.3%, 90.5%, and 98.1% at a millionth. At $\rho = 0.3$ it starts by rising, from 21.2% to 24.1%, and then falls through 12.2% to 5.7%. At $\rho = 0.2$ it is 8.4% at a tenth and 0.1% at a millionth. At $\rho = 1/3$ itself it declines very slowly, from 34.5% to 24.9% across four decades, which is the logarithm in the rate showing through where the powers cancel.

This does not rule out Clayton’s turning back as a similar comparison, and the shape of the argument suggests why: its off cell has its own rate at this correlation, and the curve is what a rate slightly slower than the cube looks like. That has not been worked out here, and the measured values stop at a depth of one in two hundred.

Nor is the limit a recommendation. A pair of cuts leaving one unit in a million past each is not a pair anyone uses. What it shows is a qualitative thing about the near range: which way the far corner is heading at the percentiles a protocol does use depends on a correlation threshold, so a reading of 44.6% at a tenth under a Gaussian at Spearman 0.4 and 21.2% at a slightly lower correlation are on curves going in opposite directions, and no table at a single depth would say so.

The question that was asked #

The question the earlier essay left was specifically about compounding. Its reading compares a cell’s leak with what adding two departures would give: the copula’s leak at the field’s baseline cut, the 84th percentile on both covariates, and the cuts’ own leak under a Gaussian copula. Above that sum the failures compound, and below it they cancel.

So yes: opposite-side cuts can compound where a shared cut cancels. Clayton with both cuts at the 95th cancels, at −5.7 points; with both at the 5th it compounds, at +42.6. At the 20th and 80th it compounds, at +8.2. Its reflection compounds at both the 95th, at +14.5, and cancels at the 20th and 80th, at −19.8. In all, Clayton’s surface has 85 cells above the sum and 36 below; its reflection’s has 22 above and 97 below, ignoring cells within half a point. The picture has a simple structure: a block where either cut enters the tail its copula is weak in.

The conclusion for the field’s table is the one an answer that changes drew for the correlation dial and the other dial for the skew. Compounding is not a property of the pair of failures. It is a property of the pair of failures at a stated cut, and the surface now says how much of the answer depends on which cut.

The reading against the $t$ copula’s far corner shows the limit of the frame. There the excess is −59.4 points: an enormous cancellation, by the bookkeeping. But nothing has cancelled. The Gaussian’s far corner is near its off-cell limit and the $t$’s is near a zero, for a reason with no connection to adding departures. The additive baseline is a good question to ask of a table near its diagonal, and it stops describing a mechanism by the time it reaches the discordant corner.

What a protocol can take from it #

A protocol cutting two covariates can read its own position off its own two-by-two table: the counts give $p$, $q$ and $p_{11}$, and the formula gives the share, whatever the copula, without any model. That has always been available. What this surface adds is a warning about how to interpret the answer.

Check the off cell before believing a small leak will stay small under a similar pair. A covariate pair where high on one tends to go with low on the other, in a clinical sense, is the discordant case when the covariates correlate positively. The corner is a fact about which two tails are cut and not about how the indicators are coded, since flipping an indicator’s coding leaves the share exactly where it was. A report that a rule’s leak is small should say which corner it was measured at.

Do not treat the median as neutral for a single covariate. It is neutral for a pair; it is not neutral for one of them.

Do not assume a rank correlation names the leak. At 0.4 the far corner at a twentieth runs from 1.1% to 54.8% across five copulas that all match on that one number, a range far wider than the 2.7% to 51.0% at the near corner. The discordant tail is a separate dependence fact, and one baseline correlation cannot tell these apart.

How the numbers were made #

The surface uses the field’s grid at 240 points per axis and calibrates each copula to the Spearman target as the copula field did. Each cut is snapped to a grid boundary, so the share is exact for the distribution on the grid rather than an interpolation, and the three inputs to the formula are internally consistent. The walk uses a finer grid at 480, and the Gaussian limit uses a Simpson rule on the conditional integral at each depth, with no grid at all.

Two routes agree where they meet: on the diagonal at the 95th the grid matches the copula field’s cut sweep, read through its closed joint share, within a point on every copula, which is the grid’s discretisation at that depth; on the Gaussian at a depth of a tenth the grid’s far corner of 44.4% matches the integral’s 44.6%. And the reading the surface was drawn to replace has to fail: the claim that moving a cut towards the median cannot lower the share is refused on the Gaussian at the 80th and 70th, and the far corner’s two limits are required on either side of one third, with a share below one per cent at 0.2 and above 99% at 0.5.

Still open: Clayton’s rate, and a trial that reads the wrong corner #

Clayton’s own exponent. The off cell’s rate on an Archimedean copula is available from its behaviour at the corner, and the fact that the far corner turns back at this correlation suggests a threshold of its own in the copula’s parameter, the analogue of the Gaussian’s one third. Writing it down would say whether a reading of a Clayton far corner is climbing or falling at any correlation, and whether Frank’s stable ratio near 0.21 implies a limit of zero at all correlations.

A sample of a realistic size. Everything here is the population. A far-corner cell at a twentieth by a twentieth holds about a quarter of one per cent of units under independence, and the discordant count in a trial of a thousand is two or three. The estimated share at that corner must be wildly unstable, and the cohort essay’s interval by inversion should say by how much. That is the sample-size question this surface has not touched.

The surface and its edges #

This essay takes the population share of a product of two threshold indicators, with one cut per covariate. Its claims are that the share over a grid of pairs of percentiles has its only guaranteed zero at the double median; that the far corner at the 5th and 95th percentiles removes 54.8% on a Gaussian copula at Spearman 0.4, 1.1% on a $t$ on four degrees of freedom, 18.2% on Frank and 40.2% on Clayton, reversing the near corner’s order between the first two; that an empty off cell forces a share of one, and that on a Gaussian copula the far corner’s limit is one above a correlation of a third and zero below it; that a $t$ copula has a further exact zero near depth three hundredths at this correlation; and that the compound-or-cancel reading depends on the pair of cuts as well as the pair of failures.

What stays out, as a decision: finite samples, deferred above; a skewed marginal driving the cuts, since a threshold at a value is a cut at a quantile and the marginals’ effect is exactly which cell of this surface they land in; and more than two covariates, where the product of three splits has found the parity argument breaking even at the median. The boundary against the cut walked into the tail is that it walks the diagonal, where the near-corner tail dependence governs, and this one leaves the diagonal for the corner that nothing measured.