A median with ties

The zero that survives both ended the compound field on the one guarantee no asymmetry could touch. A balancing rule that holds a median split of each of two covariates removes exactly nothing of the interaction between the splits, under every marginal and every copula, because a centred median split takes the values plus and minus a half and so squares to a quarter identically. Its last section named what would break it, and the third item was the most ordinary: a covariate that is not continuous. The median of a five-point scale is one of its five values, and the units sitting on it are neither above nor below.

This essay measures that case. The answer turns on a fact the earlier essays proved without needing to state it: the zero needs the split to be balanced, and nothing else.

A split that cannot be halved #

A covariate measured on $k$ ordered levels is modelled here as the latent normal read through a symmetric binomial with $k - 1$ trials, which gives the bell-shaped spread of levels a rounded measurement has: three levels in the proportions one, two and one; five levels in the proportions one, four, six, four, one; and so on. With an even number of levels nothing sits on the median — the split falls between the two middle levels and puts exactly half the units on each side. With an odd number the middle level is the median, and it holds 50.00% of the units at three levels, 37.50% at five, 31.25% at seven, 24.61% at eleven and 14.45% at thirty-one.

A protocol that says “split at the median” then has to say which way the middle level goes. Counted below, the split is on for the units strictly above the median — a quarter of them at three levels, five sixteenths at five. Counted above, it is on for the units at or above — three quarters at three levels. Either way the split is not balanced, and so it is not a median split in the sense the zero needs. It is a threshold at some other quantile of the latent covariate.

What a threshold at another quantile removes #

A cut walked into the tail gave the share of the interaction a rule balancing a threshold removes in closed form, from two probabilities: the share $p$ of units past the threshold, and the share past it on both covariates. It contains the factor $(1 - 2p)^2$, which is zero at $p = \tfrac12$ — the median split’s arithmetic — and otherwise it is not. For a discrete covariate’s split, $1 - 2p$ is exactly the share of units on the median level, whichever way they are counted. So the share of units tied at the median is the quantity the leak is built on.

On every copula, every even number of levels gives exactly zero, as a continuous covariate does. Every odd number gives something. With the middle level counted below, the Gaussian copula’s leak is 10.44% at three levels, 6.39% at five, 4.58% at seven, 2.92% at eleven, 1.53% at twenty-one and 1.04% at thirty-one. The t on four degrees of freedom and the Frank are close to it — 7.31% and 7.02% at five levels — and the Clayton pair is where the direction of the tie matters, which is the next section.

The decline is about as fast as the square of the tied share, which is what the factor in the formula predicts. From three levels to thirty-one the tied share falls by a factor of 3.46 and the Gaussian leak by a factor of 10.0, against the 12.0 the squared factor alone would give; the rest of the formula, which depends on how often both covariates are past the split together, moves a little the other way.

A five-point scale is not an exotic case. Ratings, severity grades, stage classifications and short counts are all measured on a handful of levels, and the trials that balance on them — “above the median severity”, “more than the median number of prior admissions” — are exactly the trials that rely on the median split’s guarantee. At five levels that guarantee is replaced by a leak of six or seven per cent of the interaction, which is the size of a threshold at the 69th percentile, because that is what the split is.

Five copulas at five levels #

At five levels — the scale this question most often arises on — the five copulas give 6.39% for the Gaussian, 7.31% for the t on four degrees of freedom, 7.02% for the Frank, 3.69% for the Clayton and 10.06% for its reflection, with the middle level counted below the split.

Two features of that list come straight from the copula field’s sweep. The Frank removes more than the Gaussian here, although at the 84th-percentile threshold the copula field started from it removes less. A five-level split sits at the 69th percentile, and the sweep of the cut found the Frank above the Gaussian at the 70th and below it from about the 90th: the Frank’s dependence is concentrated near the middle of the scale, which is where a coarse scale’s split falls. And the Clayton pair are three times apart, which is the tail effect, here at a split nobody chose to put in a tail.

So the coarseness of a scale does not only create a leak; it decides where on the copula field’s curves the leak is read. Fine scales put the split near the median, where every copula’s curve starts from zero together; coarse scales put it at the 69th or 75th percentile, where the copulas have already begun to separate.

How many levels before it stops mattering #

The decline is smooth enough to read a rule of thumb off. Under the Gaussian copula at this correlation the leak is still above two per cent at fifteen levels, 2.15%, falls below it by twenty-one, 1.53%, and reaches about one per cent only at thirty-one levels, 1.04%, where the median level still holds 14.45% of units. A scale needs several dozen levels before its median split behaves like a continuous covariate’s, and most measured scales have far fewer.

The same reasoning covers a continuous measurement recorded coarsely. A laboratory value reported to one decimal place, a weight to the nearest kilogram, an age in completed years: each turns a continuous covariate into one with ties, and the share of units on the median value is what matters. A covariate whose recorded median is shared by a tenth of the sample is, for this purpose, a scale of sixty-odd levels and leaks under a per cent; one whose median is shared by a third is a five- or seven-point scale and leaks between four and a half and six and a half per cent. The coin repairs both.

Which way the tie goes #

Under the three radially symmetric copulas it makes no difference whether the middle level is counted above the split or below it: a split in the upper part of the scale and the mirror split in the lower part see the same dependence. Under a Clayton copula, whose dependence is concentrated in the lower tail, it makes a large difference.

At three levels, the Clayton’s leak is 4.91% with the middle level counted below — the split then sits in the upper part of the scale, away from the dependence — and 18.44% with it counted above, which puts the split in the lower part, where the dependence is. At five levels the two are 3.69% and 10.06%; at eleven, 2.05% and 4.01%. The reflected Clayton gives the same pairs the other way round.

So the choice a protocol makes when it writes “above the median” or “at or above the median” — a choice nobody would think of as a modelling decision — moves the leak by a factor of three to four under an asymmetric dependence, with the same covariates, the same rank correlation and the same scale. It is which tail the cut sits in again, arriving through a tie rather than through a dose — and, as a copula that halves a marginal found for a skewed covariate, it is the asymmetric copulas that make a choice of convention into a choice of result.

A coin for the units on the median #

The formula’s factor $(1 - 2p)$ also says how to get the zero back, and the answer does not involve the copula, the marginal or the number of levels. Make the split balanced. The units on the median level are the obstacle; send each one above or below by an independent fair coin, tossed once before the trial is assigned and recorded with the covariate. Then the split is on for exactly half the units in expectation, $p = \tfrac12$, its centred version takes the values plus and minus a half, and its square is a quarter identically.

At five levels the coin-broken split removes exactly 0 of the interaction on all five copulas, against the 3.69% to 10.06% of the two deterministic splits. That is not a measurement of a small number; it is the arithmetic of the zero that is arithmetic, applied to a split constructed to satisfy its one condition. The coin does not know the copula and does not need to.

What the coin costs is that the balanced split is no longer a function of the covariate alone. Two units on the median level are treated differently by the split depending on a coin, so the split carries a little noise that is not about the covariate. For a balancing rule that is harmless — the rule balances whatever it is given, and a split it is given is balanced — and for an analysis it means only that the coin has to be kept with the data. A trial that wants the split for its meaning, rather than for its guarantee, is the trial the copula field’s threshold essay priced; a trial that wants the guarantee can have it back for one coin per tied unit.

How the joint share was computed, and checked #

The leak at a discrete split needs one number the copula field did not: the share of units past the split on both covariates, at a quantile such as the 75th that the copula field’s grid of cells does not land on exactly. For the Clayton pair and the Frank it is a closed form, the copula’s diagonal. For the Gaussian and the t it is one integral of the conditional law of the second covariate given the first, taken on a logarithmic scale where the tail lives.

The integral was checked against drawing. For a three-level covariate split above its median at a Spearman rank correlation of 0.4, the integral gives a joint share of 0.10924, and 400,000 pairs drawn from the Gaussian copula at the same correlation put 0.10998 of themselves past the split on both covariates, with a standard error of 0.00049 — a difference of one and a half standard errors. The formula that turns the two shares into a leak was checked in the copula field against the grid’s own regression, to $10^{-10}$, and is not re-derived here.

What the field’s guarantee now says #

The compound field established three things about a median split’s zero: it survives any marginal, because it is about ranks; it survives any copula, because it is arithmetic; and — this essay’s addition — it survives exactly as long as the split is balanced, which a continuous covariate gives for free and a discrete one does not.

That locates the guarantee precisely. It was never about the median as such. A split at any quantile that puts exactly half the units on each side has the same zero, and a split at the median of a covariate with ties does not have it. The practical form is short: a median split of a measured scale is a threshold, and is priced as one, unless the tied units are split by a coin.

The field’s other two zeros are untouched by ties. A mean’s interaction zero needs a symmetric marginal and a radially symmetric copula, and a symmetric binomial covariate is symmetric, so under a Gaussian copula the mean-balancing rule keeps its zero at every number of levels; the threshold rule never had a zero to lose.

What a protocol should write #

The finding reduces to a sentence a protocol can carry. Where a balancing rule is to hold a median split of a covariate measured on a scale, the protocol should state how units on the median value are assigned to the split, and the choice that keeps the median split’s guarantee is a coin: each tied unit sent above or below with probability a half, by a draw made and recorded before randomisation. Any deterministic choice — “above”, “at or above”, “the nearer half” — is a threshold, and the protocol should then say which tail it puts the split in, because under an asymmetric dependence that is the difference between a leak of four per cent and one of eighteen.

None of this is visible in the trial’s data. A balance table for a coarsely measured covariate shows the split balanced between arms, which it is; what it cannot show is that the split is unbalanced as a covariate, and that the interaction the protocol believed untouched has been partly removed.

Still open: three covariates, and a coin’s variance #

The compound field’s last essay named two other ways the median split’s zero might fail, and this essay answers only one. A three-way interaction of median splits is a product of three centred splits, each squaring to a quarter; whether the same arithmetic makes it orthogonal to every pairwise product and every main effect under an arbitrary three-dimensional copula is a short calculation that has not been made, and the obvious guess — that it does — is a guess.

The coin raises a smaller question of its own. A coin-broken split is balanced in expectation; in a sample of fourteen or forty units the tied units’ coins land unevenly, and the split’s realised share differs from a half by a random amount. The zero is about the population, and the rule balances the realised split, so the finite-sample version — how much of the interaction a rule balancing a coin-broken split removes in a trial of forty — is a simulation that can be run and has not been.

The claim, and where it stops #

This essay takes the median split of a covariate measured on a few levels. The claims are that with an even number of symmetric levels the split is balanced and the rule removes nothing, as for a continuous covariate; that with an odd number the share of units on the median level is the formula’s factor $1 - 2p$, and at a Spearman rank correlation of 0.4 the Gaussian leak is 10.44% at three levels, 6.39% at five and 1.04% at thirty-one, falling roughly as the square of that share; that under a Clayton copula counting the tied level above or below moves the leak from 4.91% to 18.44% at three levels and from 3.69% to 10.06% at five; that a fair coin for the tied units restores the exact zero on every copula; and that the Gaussian joint share at the three-level split, 0.10924 by integral, agrees with 400,000 draws to one and a half standard errors.

What stays out, named as a decision: covariates whose levels are not symmetric, where the median level is not in the middle and a coin would have to be biased to balance the split; three covariates, deferred above; and a covariate with ties away from the median, which change nothing about the median split. The boundary against a cut walked into the tail is that it priced a threshold placed on purpose, and this essay a threshold placed by a scale’s coarseness on a split that was meant to be a median.