Where the skew puts a threshold

The other dial swept the field’s copula-by-marginal table along a sequence of skewness — Tukey’s transform $(e^{gz} - 1)/g$ of a latent normal, from no skew to a skewness of eleven — and found the mean-balancing rule’s cells crossing: two of the four copulas whose departures from the Gaussian compound at low skew cancel at high skew. It read the mean split and left the threshold where it was. A cut at a fixed value on the covariate’s own scale never had a zero, and the essay’s last section said why it still deserved a sequence of its own: on a skewed scale a fixed value sits at a different quantile, so the threshold has a skew dependence of its own, and nobody had made the nine calls per copula that would show it.

They are made here. The finding is that the threshold’s sequence is a curve already measured, read at a different place.

Where the value 1 lands #

On the scale $(e^{gz} - 1)/g$ the value 1 is reached at the latent point $z = \log(1 + g)/g$. At no skew that is $z = 1$, the 84th percentile, which is where every threshold in the copula and compound fields was placed. Along the sequence it moves down.

The value 1 sits at the 84.13th percentile with no skew, the 80.91st at a skewness of 0.95, the 78.33rd at 2.26, the 76.21st at 4.75 and the 74.44th at the top of the sequence, a skewness of 11.16. On a scale stretched to the right, a fixed value is a smaller step from the centre than it was, because the right tail’s values are spread further apart and the bulk is compressed below.

So skewing the covariate does one thing to a threshold rule: it moves where the cut falls on the latent scale, towards the median. And a threshold is a cut at a quantile whatever the scale is — the indicator $1\{X > 1\}$ is $1\{Z > z\}$ for the latent normal — so nothing else about the skewed marginal can reach the rule at all.

The sequence is the cut sweep, read backwards #

A cut walked into the tail showed that with both covariates cut, the share of the interaction a balancing rule removes is a closed function of two probabilities: the share past the cut and the share past it on both covariates. Along the threshold’s sequence the first is the quantile above, and the second is the copula’s own joint tail at that quantile. The skew enters only by choosing the quantile.

Every copula’s leak falls as the sequence rises, because the cut moves from the 84th percentile towards the 74th and every copula’s sweep curve falls towards the median. The Gaussian goes from 16.41% with no skew to 10.07% at the top; the t on four degrees of freedom from 22.18% to 12.11%; the Frank from 13.19% to 10.14%; the reflected Clayton from 33.34% to 17.61%. The Clayton, whose dependence is in the lower tail, barely moves — 5.33% to 4.82%, rising to 5.41% in between — because between the 74th and 84th percentiles its sweep curve is nearly flat, near the gentle summit the copula field found on its upper side.

That accounts for a finding the compound field recorded and explained loosely: a skewed marginal makes the threshold’s leak smaller. It does, and the whole of the reason is the quantile. The compound field’s exponential covariate puts the value 1 at its 63rd percentile, close enough to the median for the leak to fall from 16.42% to 3.58% on that field’s grid; the Tukey sequence moves it less far and the leak falls less.

The one copula that rises before it falls #

The Clayton’s path is the only one that is not monotone, and the cut sweep says why. Its curve on the upper side of the median — the side a threshold at the value 1 is on — rises from zero at the median to a summit of about five and four tenths per cent near the 80th percentile and then falls away towards the tail, because the Clayton has no dependence in its upper tail to sustain a leak there. A cut walking down from the 84th percentile towards the 74th first climbs towards that summit and then passes it: 5.33% with no skew, 5.41% at the second and third steps, 4.82% at the top.

Every other copula’s upper-side curve is still rising at the 84th percentile, so moving the cut towards the median only lowers it. The difference is visible only because the copula field drew the whole curve; on a table with a single cut it would look like the Clayton is uniquely indifferent to skew, when it is a curve read near its turning point.

The exponential, revisited #

The compound field’s sixth marginal, the exponential, is not on the Tukey sequence, and the field’s other dial flagged it as a case that “behaves unlike its neighbours”. For the threshold rule it behaves exactly like them once its quantile is read: the value 1 on an exponential scale is its mean, which is the 63rd percentile of the latent normal, further towards the median than any step of the sequence reaches. Its leak under each copula is the cut sweep read there, below every sequence value.

That is the general statement for any marginal: a threshold’s leak needs the quantile its value falls at and nothing else about the marginal. A heavy-tailed symmetric covariate — the field’s sinh marginal — puts the value 1 at the 81st percentile rather than the 84th, so it too lowers the threshold’s leak slightly, by moving the cut, not by any property of its tails.

Compounding and cancelling, for a threshold #

The field’s question was never only the size. It was whether a skewed covariate under a non-Gaussian copula leaks more or less than its two departures would add to — the copula’s own leak with a symmetric covariate, plus the skewed covariate’s leak under the Gaussian, less the corner where neither departs. For the mean rule the answer changed along the sequence on two copulas of four. For the threshold rule it does not.

The t and the reflected Clayton cancel at every step, and by more as the skew grows: −0.86 and −1.85 points at the first step, −3.73 and −9.39 at the last. The Frank and the Clayton compound at every step: +0.84 and +1.11 at the first, +3.29 and +5.84 at the last. No cell changes sign.

The mean rule’s pattern along the same sequence was nearly the reverse. There the t and the reflected Clayton compounded at low skew and crossed to cancelling, and the Frank and the Clayton cancelled throughout. For the threshold the copulas with dependence in the upper tail cancel and the two without it there compound.

The reason is the slope of each copula’s sweep curve between the 74th and 84th percentiles, measured against the Gaussian’s. Along the sequence every copula’s leak falls, and the additive baseline assumes each falls by as much as the Gaussian does. The t and the reflected Clayton have steep curves there — their leaks are large at the 84th percentile because their dependence lives in that tail, and they lose more of it as the cut moves towards the middle — so they fall by more than the Gaussian and their excess is negative. The Frank and the Clayton have flat curves there and fall by less, so their excess is positive. For a threshold, compounding and cancelling are statements about the slope of a curve the copula field already drew, and a slope that keeps its sign over a range gives a classification that does not change along it.

Two routes to each number #

Every leak on the sequence was computed twice. The quadrant formula takes the share past the cut from the normal law and the joint share from each copula’s own tail — a closed form for the Clayton pair and the Frank, one integral of the conditional law for the Gaussian and the t. The least-squares projection is the field’s own: the threshold function evaluated on the grid of each copula’s density that the whole correlation table was built from, the same assembly the field used for the mean rule’s sequence.

They agree to within 1.14 points on every copula at every step. The grid’s points move in steps rather than smoothly, because the grid can only place a cut on the boundary between two of its cells, a twentieth of a standard deviation apart, and a cut that moves by less than that between steps does not move on the grid at all. The formula has no such resolution limit, which is why the figures above are drawn from it; the grid’s agreement is what says the formula is computing the same quantity the field computed for every other rule.

What this changes about the field’s table #

The field’s copula-by-marginal table and its two dials treat three rules the same way: a mean, a median split and a threshold, each read at every copula and marginal. This essay says the threshold does not belong in a table indexed by marginals at all. Its marginal axis is a quantile axis in disguise, and the copula field’s cut sweep is the threshold’s whole table — every marginal, every copula, every step of any sequence — read at the quantile the threshold’s value lands on.

That is a simplification worth having for a practitioner. A protocol that balances on a clinical threshold does not need to know the covariate’s skewness or fit its marginal. It needs one number, the share of units past the threshold, which a baseline table reports — which tail the cut sits in is then the whole of the copula’s say — and one more, how often a unit past it on one covariate is past it on the other — the conditional exceedance the cut sweep found decides the leak. Skewness is relevant only as one of the things that sets the first number.

The mean rule is different, and the field’s sequence was right to read it. A mean is not a function of one quantile; it weighs the whole marginal, and a skewed marginal changes it in ways that interact with the copula’s shape everywhere, which is why its cells cross and the threshold’s do not.

A worked reading #

Suppose a protocol balances on a laboratory value with a clinical cut-off, the value is right-skewed with a skewness of about two and a quarter, and the cut-off sits where the value 1 sits on the field’s scale. The sequence puts that cut at the 78th percentile of the latent covariate. Under a Gaussian dependence between two such covariates at a rank correlation of 0.4 the rule removes 12.69% of the interaction between the two thresholds; under a t dependence 15.91%; under the reflected Clayton 23.65%.

None of that needed the skewness once the cut’s quantile was known. A baseline table that reports what share of units are past the cut-off — here about a fifth — puts the cut on the copula field’s sweep directly, and the skewness only said where to read. The same table, with the share past the cut-off on both covariates, would settle which of the three dependences the data are closer to at that cut, through the conditional exceedance; and the conditional exceedance is what a practitioner with prior data can estimate without fitting any copula at all.

The reading also says what the skew bought. Had the same protocol cut a symmetric covariate at one standard deviation, the 84th percentile, the Gaussian figure would have been 16.41%. The skewed scale moved the cut six percentiles towards the median and took about four points off the leak, which is a fact about where the cut-off is, not about the skew.

That is a reading a protocol committee can make before the first patient is enrolled: find the share of the target population beyond the cut-off in published reference data, place the cut on the sweep at that quantile, and read off the leak under whichever dependence seems plausible. The skewness never has to be estimated, and a protocol that reports it in place of the share past the cut-off is reporting the less useful of the two numbers.

Two rules, two sequences, one table #

Set beside each other, the field’s two sequences say something about the rules rather than about the dependence. The same five copulas and the same nine steps give the mean rule two cells that cross and the threshold rule none, and the copulas that cancel are different in the two cases.

A practitioner reading the symmetry the marginals could not show and its successors could take away a rule of thumb about which dependences make skewed covariates dangerous. This essay says the rule of thumb is not about the dependence alone. Whether a skewed covariate and an asymmetric copula compound depends on what the balancing rule holds: for a mean, on how the copula reshapes the conditional expectations the mean is built from; for a threshold, on how steeply that copula’s sweep curve falls between two quantiles. The same pair of departures can compound for one rule and cancel for the other, and the Frank does exactly that — cancelling at every step for the mean, compounding at every step for the threshold.

Balancing a skewed covariate asked what a trial should balance when its covariate is skewed. The answer this adds is that a threshold, uniquely among the three rules, can be priced without knowing the skew, from the share of units past it; the price is the cut sweep’s, and it is the same price whatever made the covariate skewed.

Still open: a threshold on each covariate at a different quantile #

Everything here, as in the copula field, places the same threshold on both covariates, so the two indicators share one quantile. A covariate pair with different skews — or the same value on scales of different spread — puts the two cuts at different quantiles, and the quadrant formula generalises to unequal shares on the two sides. Whether the slope argument still classifies the cells, and whether a pair of cuts on opposite sides of the median can make the departures compound where a single cut makes them cancel, is a short calculation on the quadrant formula and has not been made.

The claim, and where it stops #

This essay takes the threshold rule read along the field’s sequence of skewness. The claims are that a threshold at the value 1 on the scale $(e^{gz} - 1)/g$ cuts the latent covariate at $z = \log(1 + g)/g$, from the 84.13th percentile at no skew to the 74.44th at a skewness of 11.16; that the rule’s leak is therefore the cut sweep read at that quantile, falling from 16.41% to 10.07% under a Gaussian copula, from 22.18% to 12.11% under a t and from 33.34% to 17.61% under the reflected Clayton, and barely moving under the Clayton; that against the sum of its two departures the t and the reflected Clayton cancel and the Frank and the Clayton compound at every step, with no cell changing sign; and that the quadrant formula and the field’s grid agree to within 1.14 points at every step. The practical consequence is that a threshold’s leak can be read from the share of units past it and the share past it on both covariates, with the covariate’s skewness entering only through the first, and that the mean rule’s sequence and the threshold’s classify the same copulas differently because they are reading different features of the same dependence.

What stays out, named as a decision: other rank correlations, since the sequence is read at the field’s 0.4; unequal quantiles on the two covariates, deferred above; and a threshold at a value below the median, which the sequence would move towards the median from the other side. The boundary against the parity field’s threshold is that it showed a threshold has no zero under any marginal, and this essay that its size under any marginal is a quantile read off one curve.