Normal Distribution
nothing is called 95% until it has been counted
Concept

Covariate balance — where it appears

How alike the arms of an experiment are on the characteristics recorded before anything was given to anybody. It is what a rule can be judged on before any outcome exists, and it is only a proxy for what actually matters, which is the variance of the treatment estimate.

Named by 88 essays across 21 fields — each of them below, with the objects they name alongside it.

The six best bases of 2 functions, and what each protects. Every cell is R²(g | span B) — the share of the imbalance in that shape a rule balancing that basis removes — computed from exact inner products between Hermite functions and indicators, with nothing simulated. The rows are ordered by their worst cell, which is the number an experimenter who does not know the shape is exposed to. The best row here guarantees 26.8% against every shape in the list, and the worst of the six guarantees 15.1%: the difference between them is entirely which subspace was picked, at the same cost per arrival.

A basis is a subspace

A balancing rule cannot tell one basis from another with the same span, so choosing what to hand it is choosing a subspace — and then what it removes of any outcome shape is a projection, computable exactly, with no trial anywhere in it.

basis · Criterion The expansion that never terminates. The Hermite coefficients of a median split, in magnitude, against the reference j to the power −3/4, anchored at the first one. Every even order is exactly zero because sign is an odd function, and every odd order is not, so no truncation is exact — where a polynomial of degree d is exact at any order past d. Summed, the tail past J falls like 1/√J: sixty orders still leave 6.6% of the variance outside. That statement is what made a cut dictionary's geometry unavailable in closed form, and it is a statement about the function against itself. What it is not is the accuracy of an inner product between two correlated variables, where every term past J carries a factor of ρ^m as well.

A cut is not a polynomial, and it does not have to be

A threshold's expansion never terminates, which is why a balancing dictionary's geometry was closed for powers and taken to draws for cut points. Conditioning on the second variable closes it for both.

splits · Blocking Two covariates make the dictionary an outer product. Four functions of each covariate, and everything a balancing rule may be handed. The margins are the 8 main effects and the block between them is the 16 interactions, which are 66.7% of the dictionary. Every inner product in it is closed form — ⟨f₁g₁, f₂g₂⟩ = ⟨f₁,f₂⟩⟨g₁,g₂⟩ when the covariates are independent — so nothing about the geometry gets harder. What gets harder is the counting: choosing k of 24 is C(24, k), which is 10,626 at four and 735,471 at eight.

A dictionary that is a product

Two covariates make what a balancing rule may read an outer product — eight main effects and sixteen interactions — and every inner product in it is still closed form. What a rule holding all eight main effects removes of a pure interaction is not small. It is zero.

product · Blocking The rule is parity, and it runs both ways. At a correlation of 0.5, four combinations of a dictionary and an outcome shape. The joint sign flip (X, Y) → (−X, −Y) leaves the bivariate normal alone at every correlation, so a function that changes sign under it is orthogonal to one that does not. A product of two odd functions is even; a product of an odd and an even one is odd. So an odd dictionary removes exactly none of the first and something of the second, and an even dictionary does the reverse — which it does, to machine precision, in both of the two rows that should be zero. This is one rule where there had been two: that a median split's square is constant, and that a polynomial dictionary contains the products a correlation generates.

A dictionary that is neither

A rule handed two median splits removes none of their interaction; a rule handed two covariates removes none of their product. Those were two results with two explanations, and they are one result with one — and finding it corrected the number underneath both.

dict · Criterion One margin rises; the other turns over. The two halves of the table's margin across the sweep: a lower-tail copula's own leak with a symmetric covariate, and a covariate skewed at 0.95 under a Gaussian copula. The marginal's leak rises at every step, from 3.727% to 36.056%. The copula's does not: it rises to 9.064% at a Spearman of 0.6 and falls to 8.219% by 0.7. It has to turn over, because at a rank correlation of one the two variables are a deterministic function of each other and there is no interaction left for a split to leak. So the margin of the table turns over before any cell in it does.

A margin that turns over

A skewed covariate's leak grows without limit as the dependence strengthens. A copula's own leak does not — it peaks at a rank correlation of 0.6 and falls. The margin of the table turns over before any cell in it does.

stronger · Adjustment A proposal that moves more, refused more often. The two halves of the trade, both exact, on the 410 admissible assignments of twelve units. The integrated autocorrelation time of an imbalance the rule was never handed falls from 7.30 at one swap to 3.97 at three, and the acceptance rate falls with it, from 58.8% to 40.8%. A rejected proposal costs one evaluation and leaves the chain where it was, so acceptance is not the price of anything and the ranking by acceptance is the reverse of the ranking by cost. Past three the family folds: exchanging k of six from each arm is the complement of exchanging six − k, so k = 5 has the same 36 proposals as k = 1 and k = 6 has 1.

A proposal that moves more than two units

The walk's autocorrelation is a fact about its step size and not about its acceptance rate. Exchanging three units from each arm mixes nearly twice as fast as exchanging one, and is refused a third more often.

blocks · Randomisation The test, checked where the answer is known. A fourteen-unit trial at eight tolerances. At each one the admissible set is enumerated — 1534, 886, 304, 158, 126, 116, 102, 84 assignments — and its components counted, which is only possible because 3432 equal splits of fourteen units can be walked. The dots are the test, which walks none of them: two chains, one started at an assignment and one at its complement, compared on a statistic the rule was not handed. Filled marks are tolerances the enumeration says leave the set in more than one piece. The test fires on every one of them and on none of the others, 0 misses and 0 false alarms.

A test rather than a survey

A thin admissible set falls into an arrangement and its mirror image, and the walk that samples it is uniform on half the reference distribution for ever. That was found by enumerating fourteen units, and enumeration stops at twenty-four.

reach · Randomisation One zero is arithmetic and one is a symmetry. What two balancing rules remove of the interaction they are aimed at, on five joint laws of the ranks matched at a Spearman correlation of 0.4, with a normal covariate throughout. A rule holding a median split of each covariate removes exactly nothing of the product of the splits under every one of them, including the two that are not symmetric under reflection — and the reason is not a symmetry at all: a centred median split takes the values ±½, so its square is a quarter identically, and the interaction is orthogonal to both main effects whatever the joint law is. A rule holding the mean of each removes exactly nothing under the three radially symmetric copulas and 7.71% under the two that are not. Bars at the floor are exact zeros; the axis cannot draw 9e-32.

A zero that is arithmetic

A median split's exact zero was explained by a symmetry of the latent normal. It holds under a Clayton copula, which has no such symmetry, because a centred median split squares to a quarter identically.

copula · Criterion One zero holds and one does not. Three rules, at a correlation of 0.5, against the skewness of the covariate. A rule balancing the mean of each covariate removes exactly nothing of their product when the marginal is symmetric — including the heavy-tailed symmetric one at skewness zero, which is what says the guarantee needs symmetry rather than normality — and removes up to 29.7% when it is not. A rule balancing a median split of each removes exactly nothing of the product of the splits under every marginal here, to 1e-30: both sides are functions of the sign of the latent normal, and a monotone transformation moves neither. A rule balancing a threshold at a value on the covariate's own scale removes between 4.9% and 22.5% — it never had a zero to lose, under any marginal at all.

A zero that rests on a symmetry

A balancing rule removes exactly none of an interaction between two odd functions, at every correlation. The argument needs the joint sign flip to preserve the law, and no real covariate is symmetric about anything.

skew · Criterion Four rules, three shapes, and no ordering that survives. The variance of the unadjusted treatment estimate under each rule, as a fraction of the variance a coin gives, over 450 trials of 200 units each. Against a covariate that enters linearly the rule that reads the number nearly halves it. Against a threshold at 1 it removes about a fifth. Against a quadratic every rule here is at or worse than a coin — they are all optimising a criterion that is one over the variance of an estimate in a model this outcome does not obey, and a constraint that helps nothing still costs something. Nothing in a trial says which column it is in.

Balanced on the wrong function

A rule that reads a covariate's numbers halves the variance of the treatment estimate, if the covariate enters the outcome as a straight line. If it enters as a threshold the rule is worth a fifth of that, and if it enters as a curve every rule here is worse than a coin.

shape · Blocking What each rule leaves behind, at 120 patients. Four allocation rules over the same cohorts and the same seeds, each scored on three imbalances: the number of patients in each arm, the worst of the nine factor levels, and the worst of the 24 cells of the cross-classification. No rule holds all three. Permuted blocks hold the totals exactly and leave the margins near a coin's. Blocks inside every cell hold the cells and let the totals drift, because 24 part-filled blocks do not have to end level. Minimisation holds the margins and the totals and is at 83% of a coin's cell imbalance. Each of the three columns is somebody's definition of a balanced trial.

Balancing what is known in advance

Four allocation rules, three definitions of balance, and no rule that holds more than one of them. Minimisation keeps the worst factor margin near three patients whether the trial has forty or six hundred and forty — and lets the imbalance in the cross-classified cells climb to 86% of a coin's, because the cells are not what it is watching.

covadapt · Assignment The fourth-order expectation, by two routes. Every inner product in an eight-term slice of the dictionary at ρ = 0.5, computed from the linearisation and Mehler's formula and counted from two hundred thousand draws of a correlated pair. The entries that matter are the ones off the main effects: ⟨f(X)u(Y), g(X)v(Y)⟩ is a fourth-order expectation, which the independent-covariate field could not write down. The worst departure is 1.99 standard errors over 36 pairs, measured in each pair's own error because the entries differ in size by two orders of magnitude.

The fourth moment that was missing

Mehler's formula makes the main effects exact at any correlation and stops there, because the interactions need an expectation of four Hermite functions rather than two. A linearisation turns the four into two, and the whole geometry becomes closed again.

joint · Blocking What is left of a probe after the rule has had it. The share of each dictionary function a rule balancing x, x2, x3, cut0 has already taken, on trials of 14 units, averaged over 100 designs. Four of the eight functions are the basis, so their share is exactly one: a randomisation test run on one of them is asking about a quantity the rule forced to zero, and one of them is the default probe of the field this measurement comes from. The four that are not still read 0.919, 0.873, 0.903, 0.832 — between 0.832 and 0.919 of them is inside the span — against closed-form removed shares of 0.000, 0.692, 0.590, 0.692. At 14 units a rule with four functions in it takes most of anything it is shown.

The part the rule already took

A diagnostic that reports on what a balancing rule was not handed is run through a column that is 92% inside the span the rule balanced — because orthogonality in the population is not orthogonality on fourteen units.

aimed · Randomisation Two diagnostics, one answer, two different moments. The two-chain statistic on a covariate probe and on the trial's own difference in arm means, at seven tolerances of a fourteen-unit rule, against the enumerated truth. Both are quiet wherever the set is one set and both fire wherever it is not, at every tolerance — which is what says the outcome probe is the same test rather than a resemblance of it. The difference between them is not accuracy and it is not power. It is when: the covariate probe can be run before a single outcome exists, when a practitioner can still loosen the rule or change the sampler, and it can be run again on a different function if it comes back quiet. The outcome probe runs after the trial, on the one column the trial produced, and what it can do with a positive verdict is repair the p-value rather than the design.

The statistic the p-value is about

The test for whether a balanced-assignment walk reaches its whole set is run on a covariate function chosen before the trial. Run on the difference in arm means it is the same test, and it is about the number the trial publishes.

after · Randomisation Where one rule becomes three. Every arrival in 200 simulated trials is put to all three scores, and the picture is how often they would send that patient to different arms. The range and the pairwise sum are the same rule at two arms and at three — for sorted counts the pairwise sum is twice the range, so the arm that minimises one minimises the other — and they part company at four, where the pairwise sum is 3(d − a) + (c − b) and the range still sees only d − a. The variance disagrees with both from two arms onwards, on 5.4% of arrivals at two and 27.0% at five, because the scores are summed over 3 factors and a sum of squares does not order the candidates the way a sum of absolute values does. All three are called minimisation.

Three arms and three scores

Minimisation balances a trial by keeping the arms' counts even inside every prognostic factor. With two arms there is one way to measure how uneven two counts are. With three there are several, they are all called minimisation, and they send different patients to different arms.

multiarm · Assignment What a mean split leaves, with both halves varying. The share of a mean split's interaction that survives the rule balancing it, at every copula and every marginal, matched at a Spearman correlation of 0.40. The three radially symmetric copulas leave exactly nothing with a symmetric covariate and rise steeply with the skew. The two asymmetric ones start at 7.707% and go opposite ways: the lower-tail copula falls to 0.002% at a skewness of 0.95 — the two failures cancel almost exactly, and a guarantee both fields report as broken is restored — while the upper-tail one climbs to 40.288%. And the heavy-tailed symmetric covariate, which leaks exactly nothing on its own, doubles what the asymmetric copulas leak: 14.229% against 7.707%.

Two failures that cancel

A mildly skewed covariate under a lower-tail copula leaks 0.002% of an interaction where each failure alone leaks eight and seven per cent. Turn the copula over and the same pair compounds.

compound · Adjustment What the rule blocks is not where it splits. How much of the separating direction each probe carries, over 100 designs of 14 units whose admissible set is enumerated and split into two pieces. The deferral this field answers proposed the constraint's active set — which exchanges the tolerance box actually blocks — as a better probe than the design's own leverage, on the ground that leverage is a heuristic and the active set is the quantity. Modelled from the design and the tolerance, it reads 0.4272 against leverage's 0.5395, at 4.43 paired standard errors the wrong way. Counted exactly over the enumerated set — at a cost no trial can pay — it reads 0.3854, worse again. Both beat a random direction at 0.2622, so they are probes; neither beats the two the earlier field already had.

What the rule blocks

A balancing rule breaks the admissible set into pieces by refusing exchanges. Which exchanges it refuses is computable from the design and the tolerance alone, before any assignment exists — and it makes a probe.

blocked · Randomisation A weight that balances, and one that unbalances. The standardised difference between the arms on each covariate, integrated over the population rather than counted in a sample. Unweighted, the arms differ by 0.8310 on the first covariate and 0.6015 on the second, which is what makes the raw difference of arm means 2.7102 against a true average effect of 1.0000. Weighting each unit by one over its own assignment probability removes both differences exactly — -2.78e-17 and -5.69e-19, which is machine precision and not a small number — because the weighted density of the treated arm is the population's own whatever the propensity is. Weighting by a score fitted without the second covariate balances the first to 0.0035 and pushes the second out to 0.7057, further apart than doing nothing.

A score that balances

Weighting each unit by one over its own assignment probability drives the standardised difference between the arms from 0.8310 to 2.8×10⁻¹⁷ — exactly, not nearly. A score fitted without the second covariate leaves that covariate at 0.7057, further apart than doing nothing at all.

weights · Weighting How often a randomised trial reports the reversal, advantage 6 points. Simple randomisation against randomisation stratified by group, 4,000 trials at each size. The simple design reverses on 3.40% of trials at its worst size and 0.50% at 1280 units; the stratified design reverses on none of them, at any size.

The reversal a coin cannot prevent

Randomisation removes Simpson's reversal in expectation, which is not the same as removing it. A correctly randomised trial of eighty units, on a population where the treatment helps in both groups, reports it losing overall on 3.40% of trials — and stratifying the randomisation takes that to zero at every size.

reversal · Simpson A model of the active set, and the active set. Each of the 14 units of one design, at the share of its exchanges the tolerance box blocks — computed from the design's columns and the tolerance under a uniform position in the box, against counted over all 116 admissible assignments. The diagonal is where the two would agree. Over 192 designs they agree about the ordering of the units at a correlation of 0.8141 ± 0.0112, negative on 0.5% of them, and disagree about the level: 0.8442 counted against 0.8170 modelled, a gap of 0.0272 ± 0.0051. An admissible assignment does not sit uniformly in its box, and this is the size of that.

A model and a count

The share of a unit's exchanges a tolerance box refuses can be modelled from the design or counted over the admissible set. They order the units the same way at a correlation of 0.81 and disagree about the level by 0.027.

blocked · Randomisation How far apart the two components are, on each probe. The median separation between the two components of the admissible set — the difference in their mean probe values, over the spread inside a component — over the 100 of 200 designs whose set is enumerated and found split. The separating direction carries 10.565 and needs the enumeration. The fourth power as the earlier fields use it carries 1.543; projected off the span the rule balances, 5.080. The design's own leverage, which uses no dictionary and no outcome, carries 3.836. A random direction in the same subspace carries 0.942, and a direction chosen by looking for concentrated structure carries 0.543 — below random, and the one heuristic here that is worse than not choosing at all.

A probe chosen from the design

The design's own leverage aligns with the separating direction four times better than a random direction in the same subspace. The concentrated direction the argument invites is worse than random.

aimed · Randomisation The probe a trial has is the probe a trial got. What the two-chain test says when it is run on the trial's own difference in arm means, over 24 outcomes on one fourteen-unit set. The set is in 2 mirror components — that is enumerated, not inferred — so every quiet reading is a miss. 29% of them are quiet. The reason is in the enumerated set rather than in the run: how far the two components are apart on a given probe ranges from 0.001 to 4.938 of a within-component spread across these outcomes, a factor of several thousand. Both covariate probes — chosen before any outcome existed, and replaceable if they had been quiet — report the split. An outcome cannot be chosen and cannot be replaced.

A probe nobody chose

On a set that is definitively in two pieces, seven of twenty-four outcomes report nothing at all. Every covariate probe reports it. What separates them is not accuracy — it is that one of them can be chosen and the other is what happened.

after · Randomisation Where the two kinds of cut sit. Six covariates, each a monotone transformation of the same latent normal. The vertical line at zero is where every median split sits, on every one of them, because a monotone map preserves order: the median of the covariate is the image of the median of the latent normal. The marks on the curves are where a threshold at 1 on the covariate's scale falls — 1.000, 0.881, 0.875, 0.783, 0.713, 0.337 — and none of them is at zero. That is the whole of the difference. A function of the sign of the latent normal is odd, and a rule made of odd functions removes exactly nothing of an interaction between two of them; a threshold anywhere else is neither odd nor even and removes something.

A split survives what a mean does not

The two things every trial balances come apart on a skewed covariate. A median split is a function of the sign of the latent normal whatever the marginal is; a mean is not, and its exact zero is gone at a skewness of one.

skew · Criterion The same copula, turned over. A Clayton copula and its reflection, at the same Spearman correlation of 0.40 and the same Kendall tau of 0.275, against the covariate's marginal. With a symmetric covariate the two are the same number to nine decimals — 7.707% apiece — because the leak then depends on how much asymmetry the copula has and not on which way it points. Skew the covariate and they come apart: at a skewness of 2.26 the lower-tail copula leaves 3.431% and the upper-tail one 36.213%, a factor of 10.6. Both halves of the dependence are asymmetries and an asymmetry has a direction; a lower-tail copula concentrates the dependence where a right-skewed marginal is compressed and the two distortions partly undo each other, and an upper-tail one concentrates it where the marginal is stretched.

A symmetry that was not enough

A heavy-tailed symmetric covariate has a skewness of zero and leaks exactly nothing under three copulas. Under the two asymmetric ones it doubles the leak, from 7.707% to 14.229%.

compound · Adjustment A threshold in the tail is a threshold nothing balances. The share of a coin's imbalance in an indicator 1{x > c} that survives a rule which balances the covariate itself. The smooth curve is 1 − ρ² with ρ = φ(c)/√(p(1−p)), a closed form with no trial in it; the points are counted over 500 trials of 200 units at each threshold. At the median the two agree that about a third survives — the removed share is exactly 2/π — and by two standard deviations 86.9% survives. The closed form is exact in the limit and optimistic by a few points at this many units, because the rule balances the sample's mean rather than the population's.

A threshold in the tail

How much of a threshold's imbalance a balanced covariate removes is a correlation, and the correlation is a closed form. At the median it is exactly 2/π — the same 2/π a median split throws away — and two standard deviations out it is an eighth.

shape · Blocking The zero was a fact about independence. What a balancing rule handed every main effect of both covariates removes of a pure interaction, as the covariates are allowed to move together. At ρ = 0 it is exactly nothing — at machine precision, at any number of main effects — which is the independent-covariate result and is correct. It is not small anywhere else: the product of the two covariates loses 64.0% of itself by ρ = 0.5, because h₁h₁ = h₀ + √2·h₂ and Mehler pairs h₂ with h₂ at ρ². Four interactions are drawn and none of them keeps the zero.

A zero that was an assumption

A rule handed every main effect of both covariates removes exactly none of a pure interaction. That is true at machine precision, it is a fact about independence, and it dies as the square of the correlation.

joint · Criterion A trial designed 2:1:1, and what two scores deliver. 500 trials of 180 patients, three arms, a target of 2:1:1. The shaded bars are a minimisation score that divides each arm's count by the share that arm is supposed to receive before measuring the spread; it delivers 49.9% : 25.1% : 25.1%. The others are the same rule with the counts left raw, which delivers 33.4% : 33.3% : 33.3% — the balance it enforces inside every factor level is equality, and equality is what it gets. The marks are the shares that were asked for.

Balancing towards unequal targets

A three-arm trial allocating two to one to one is the ordinary case, and a balancing rule built from raw counts does not know it. It balances the arms towards equality inside every factor level, delivers a third to each arm, and reports that it minimised imbalance.

multiarm · Allocation Every way of splitting 16 units into two halves. All 12,870 assignments, enumerated. The spread of the standardised imbalance is exactly 2/√n = 0.500, whatever the covariate's own distribution, and 33.3% of assignments differ by more than 0.5 standard deviations. Randomisation does not deliver balance; it delivers a known distribution of imbalance.

Randomisation is not balance

A third of all ways to split sixteen units leave the two halves more than half a standard deviation apart on a covariate. What randomisation delivers is not balance but a known reference distribution — and it makes a test exact with no assumption about the data's shape at all.

design · Randomisation Stationary is not the same as convergent. How far each k-swap walk is from uniform after t steps, started at the least balanced admissible assignment of 410. Every one of these chains has a symmetric proposal and rejects by standing still, so every one of them is doubly stochastic and every one preserves the uniform distribution exactly. Only five of the six get there. Exchanging all six units of each arm is a single proposal — the complement — and the admissible set is closed under complement, so the walk takes it every time and oscillates between two assignments for ever: after 160 steps it has visited 1 state and sits 0.9976 from uniform. Its stationary distribution is a fact about the matrix; its limit does not exist.

Stationary is not convergent

A walk that exchanges every unit in each arm preserves the uniform distribution exactly and never gets near it. Every doubly stochastic matrix has the same stationary distribution; only some of them have a limit.

blocks · Randomisation Sheppard's arcsine, by two routes. Corr(sign X, sign Y) as the covariates' correlation runs from zero to one, drawn twice. One route is a sixty-four-node quadrature of the orthant probability over the correlation — the general construction, which works at any pair of cut points; the other is (2/π) arcsin ρ, which is elementary and works only at the median. They agree to 3.3e-16 at every one of 81 correlations, which is what licenses the quadrature everywhere else. The curve is above the diagonal at small ρ and below it at large: two signs agree with probability ½ + arcsin(ρ)/π, so a correlation of 0.5 gives exactly ⅓ and a correlation of 0.8 gives 0.5903.

The arcsine that closes it, and the error that was overstated

Two median splits of a correlated pair agree with probability ½ + arcsin(ρ)/π, exactly. And the truncation the field was avoiding falls geometrically in the correlation, not algebraically in the order.

splits · Routes What a rule gives away by being predictable. Minimisation run at every probability from a coin to fully deterministic, over 500 cohorts of 120 at each. The upper line is the share of assignments an investigator who knows the rule and the enrolled patients can name in advance: 49.7% at p = 0.5, which is a coin and cannot be beaten, and 87.6% at p = 1 — short of everything only where the two arms tie and the rule falls back on a coin. The lower line is what that is worth: an investigator who enrols a patient 0.5 of a standard deviation better than average whenever they predict their favoured arm produces a treatment effect of 0.75 where the truth is zero. Nothing about the randomisation was broken; the bias entered through who was enrolled, which is the one thing an allocation rule cannot control. The dashed line is the closed form 2δ(2g − 1).

The rule that can be guessed

A balancing rule improves as it becomes more deterministic, and a deterministic rule can be worked out in advance from information the person enrolling the patient already has. At full determinism 87.6% of assignments are guessable, and an investigator who acts on the guess produces a treatment effect of three quarters of a standard deviation where the truth is zero.

covadapt · Assignment Three readings, one verdict. Every tolerance of a fourteen-unit trial, with three comparisons on each. The first is between a chain started at an assignment and a chain started at its complement, which is what a mirror split separates. The second is between two chains started at the same assignment on different streams, which nothing about the set can separate — so a large reading there says the run is too short and not that the set is in pieces. The third is the same comparison on the statistic's absolute value, which is symmetric under the complement and therefore blind to the split by construction. The verdict is the pattern rather than any one line: the split is called only where the first fires and the other two do not, which happens at exactly the tolerances the enumeration calls disconnected — 0.8, 0.75, 0.7.

The statistic that changes sign

A test for an unreachable half needs a quantity that tells one half from the other. Every symmetric reading of a mirror pair is identical, and a magnitude is the natural thing to reach for.

reach · Randomisation The mean's zero is the copula's symmetry. Five copulas, each at a Spearman rank correlation of 0.4, with a normal covariate throughout — so nothing here is about the marginal, which is the whole of the earlier field. Horizontally: how far the copula's density is from its own reflection through the centre of the unit square, measured rather than read off the family's name. Vertically: what a rule balancing the mean of each covariate removes of their product. The three copulas at zero on the horizontal axis remove exactly nothing, to thirty decimal places. The two that are not symmetric remove 7.71%. A guarantee that held for six marginals turns out to have needed something the marginals could not have told anybody about.

The symmetry the marginals could not show

A mean's interaction zero needs the covariate to be symmetric and the copula to be symmetric under reflection. Six marginals could only ever test one of those, and the other is broken by the commonest kind of dependence there is.

copula · Criterion What the worst case is worth, one function at a time. The smallest share each dictionary removes, over seven outcome shapes, at a correlation of 0.5. A rule balancing the mean of each covariate has a worst case of exactly zero — against the square, and against both products. Adding a median split to it, which is the second thing every trial balances, leaves the worst case at exactly zero, because a median split is odd and so is a mean. Adding the square instead moves it to 6.8%, and the extra functions after that move it to 7.4%. The worst case is decided by which parities the dictionary contains rather than by how many functions are in it.

What the extra function buys

A rule balancing the mean of each covariate has a worst case of exactly zero. Adding the median split — the other thing every trial balances — leaves it at exactly zero, and one square moves it.

dict · Criterion One curve is a binomial coefficient and the other is a line. The number of subsets a maximin over this dictionary would have to score, against the number the exchange algorithm actually scores. At three functions the walk is 2,024 subsets and is the honest answer; at eight it is 735,471 and the exchange algorithm has looked at 421. The warrant for the second curve is the four sizes where both exist and agree, which is a weak warrant — it says the algorithm has not yet been wrong, not that it cannot be — and it is the only one available past the point the first curve leaves the page.

Where the enumeration stops

A maximin over an eight-function dictionary is a walk over seventy subsets. Over twenty-four it is 735,471 at eight functions, and the exchange algorithm that replaces the walk scores 421. What licenses the second curve is four sizes where both exist and agree, which is a weaker warrant than it looks.

product · Optimum The guarantee, as the basis is allowed more functions. The lower line is the best worst case over the six named shapes for a basis of each size, found by scoring every subset of the dictionary — an exact answer, since the problem is finite. One function guarantees 2.3%, which is nearly nothing; three guarantee 59.0% and the basis that does it is the covariate, its square and its cube, with no indicator in it. The upper line is the same problem with the basis drawn rather than fixed, which is worth 2.09 times as much at two functions and 1.32 at three. The two lines converge because a basis large enough to protect everything has nothing left to randomise over.

Which shapes are worth protecting

Choosing a basis by its worst case is a finite problem with an exact answer. The answer has no tie in it, which a maximin optimum is supposed to have — and the tie comes back, along with twice the guarantee, when the basis is drawn rather than chosen.

basis · Blocking A near-perfect cancellation, at one correlation. A covariate skewed at 0.75 under a lower-tail copula, at each of 7 rank correlations. The earlier field measures this cell at a Spearman of 0.4 and reads 0.002% where adding the two halves' own leaks gives 16.579% — a cancellation so near exact that it is that field's headline. Across the sweep the same cell reads 0.0084%, 0.0182%, 0.0097%, 0.0015%, 0.0864%, 0.4693%, 1.5327%. Its smallest value is at 0.4, in the interior, and by 0.7 it is 1007.40 times larger. The near-zero is where two curves cross, and they cross beside the one correlation that was measured.

The zero that was a crossing

A cell that leaks 0.002% where adding its two halves gives 16.6% is a field's headline. On a finer grid it passes through zero at a rank correlation of 0.38 — two hundredths from where it was measured.

stronger · Adjustment Three copulas that break nothing, and a factor of two between them. The three radially symmetric copulas, at a matched Spearman correlation of 0.40, against the covariate's marginal. All three leave exactly nothing with a symmetric covariate — that is the guarantee, and it holds to twenty decimal places. What they do to a skewed covariate is not the same at all: at a skewness of 2.26 a Frank copula leaves 12.118% where a Gaussian leaves 21.539% and a t on four degrees of freedom leaves 23.640%. A factor of 2.0 between two copulas that are both symmetric, both matched on rank correlation, and both harmless on their own. So the copula matters to the marginal's leak without breaking any symmetry of its own, which is a milder version of the same finding and applies to every trial rather than to the asymmetric ones.

A copula that halves a marginal

Three copulas break nothing on their own and put a factor of two between the same skewed covariate's leaks — 12.118% under a Frank against 23.640% under a t, at the same rank correlation.

compound · Adjustment A rate that does not know how large the trial is. The share of equal splits admitted by a tolerance of 1 coin-spreads on 3 functions, at six trial sizes. The first two are exact — 12,870 and 184,756 splits, walked, averaged over eight draws of the units — and the rest are sampled. From a hundred units on, the rate sits on (2Φ(1) − 1)^3 = 0.3182, which contains no n at all. The two small trials are 29.2% and 27.7% short of it, so the sixteen-unit measurement understates the rate rather than bracketing it. Meanwhile the admissible count — the rate times C(n, n/2) — goes from 2^11.5 to 2^393.7: the exhaustion a small trial runs into is a fact about small trials.

A count that has to be estimated

At sixteen units the admissible assignments can be counted by walking all 12,870 of them. At four hundred there are about 2^393.70, and the share admitted is 0.31885 against a closed form of 0.31818 that has no trial size in it at all. The exhaustion a small trial runs into is a fact about small trials.

product · Randomisation Thinness stays put and reachability does not. The same balancing rule and the same tolerance at five trial sizes. The admitted share barely moves — one admissible assignment in 27, 30, 30, 20, 18 — because the acceptance rate of a rerandomisation is a fact about the basis rather than about the number of units. What does move is the number of single swaps available: 36, 49, 64, 81, 100, growing like a quarter of the square of the trial size. Filled marks are sizes whose admissible set falls into more than one piece. The set is in 4 pieces at 12 units, 2 at 14, and one piece from 16 upwards. The fourteen-unit result is a statement about fourteen units.

A defect that is about size

The admitted share of a rerandomisation barely moves with the number of units. The number of admissible neighbours grows like the square of it, and that is what decides whether the walk can go everywhere.

reach · Randomisation What each probe can see. How far apart the two components of the admissible set are on each probe, over the spread inside a component, on 100 designs whose set is enumerated and split. It is the population quantity a chain is trying to report. The separating direction itself reads 10.5646; the projected fourth power 5.0800, the design's own leverage 3.8362, the modelled active set 1.9529, the counted active set 2.0170 and a random direction in the same subspace 0.9422. The two active-set probes beat the random direction and lose to both of the earlier field's, which is the field's answer to the question that opened it.

A quantity that loses to a heuristic

Leverage is a heuristic about which units a balancing rule has most to say about. The constraint's active set is the thing the rule actually does. As a probe, the heuristic wins by 4.4 paired standard errors.

blocked · Randomisation Four cells change their answer. The four cells of the twenty whose excess changes sign as the dependence strengthens, over 7 recalibrations. Above the line the two failures compound — the cell leaks more than adding the copula's own leak and the marginal's — and below it they cancel. All four start above and end below, and all four are at the two most skewed covariates: skew 0.90 under heavy-tailed, skew 0.95 under heavy-tailed, skew 0.90 under upper tail, skew 0.95 under upper tail. Whether two failures of a dependence compound or cancel is therefore not a property of the pair. It is a property of the pair at a strength of dependence, and a fifth of the table changes its answer inside the range measured here.

An answer that changes

Eleven of twenty cells cancel and nine compound, at one rank correlation. Sweep the correlation and four of the twenty change sides — all four from compounding to cancelling, all four at the most skewed covariates.

stronger · Adjustment The one thing a trial always reports is the one thing that survives. How wrong three p-values are when they are computed over the half of the admissible set a single walk can reach, rather than over all of it, at a fourteen-unit trial where the whole set can be enumerated. The two-sided p-value on the difference in arm means — the number a trial publishes — is wrong by exactly nothing, at every row, to machine precision. That is not luck: the two components are complement pairs and the difference in arm means is exactly negated by the complement, so the distribution of its absolute value is the same on both. A one-sided p-value on the same statistic is out by as much as 0.112, and the largest response observed in the treated arm — a safety reading rather than an effect, and the one statistic here that is not odd under the complement — by as much as 0.172. The defect survived because the commonest thing anybody computes is the one quantity it cannot touch.

Half a reference distribution

A walk that reaches half its admissible set reports the two-sided p-value exactly right, to the last digit, for ever. A one-sided one it puts on the wrong side of five per cent about once in thirty.

after · Reference A cut at a quantile, and a cut at a value. Two rules that read identically in a protocol. One splits each covariate at its median; the other splits it at 1 on the covariate's own scale — a dose, a temperature, a clinical threshold. At a correlation of 0.5 the first removes exactly nothing of the interaction between its own two splits, under every marginal here, because a median split is a function of the sign of the latent normal whatever the marginal is. The second removes what the bars show, and it does so on a normal covariate too: the threshold sits at 1.000 on the latent scale rather than at zero, so it is 59.4% odd and 40.6% even. The exact zero was never about the cut; it was about the cut being at the median.

The cut that is not a quantile

A protocol that says split the covariate at a threshold and one that says split it at the median read the same and are different rules. One has an exact guarantee under every marginal and the other has none under any.

skew · Criterion Whichever dial made the set thin, the crossing is at the same thinness. Each curve is one dictionary, swept over eight tolerances at two hundred units; a point above the line is a set thin enough that walking beats hunting. The curves lie nearly on top of one another, which is the answer to whether the crossing is a fact about the tolerance or about the thinness it produces: the crossings sit between one admissible assignment in 176 and one in 268 for dictionaries of 3 to 6 functions. The mechanism is that a hunt costs exactly 1/p and a walk costs almost the same everywhere — between 82 and 394 evaluations per usable draw across the whole table — so the crossing is wherever 1/p reaches a number that does not move.

The set a dictionary leaves

A rule constrained on six functions at a loose tolerance leaves a set as thin as one constrained on three at a tight one. Both sampling methods cross over at the same thinness, and the tolerance where that happens moves by a factor of three.

dict · Assignment The guarantee that survives a correlation, and the one that does not. What a balancing rule handed both main effects removes of the pure interaction between them, as the covariates become dependent. For median splits it is exactly zero at every correlation, because sign(x)² = 1: the interaction sign(X)sign(Y) is orthogonal to sign(X) and to sign(Y) whatever ρ is. For the product of the raw covariates it is 4ρ²/(1+ρ²)² — 64.00% by ρ = 0.5, rising to all of it at perfect correlation. A cut away from the median sits between them and is not small: 23.01% at a cut of one. The zero is not a fact about interactions. It is a fact about a dictionary whose functions square to a constant, which a polynomial one does not.

The zero that survives a cut

A rule holding both main effects removes half of a pure interaction between correlated powers and exactly none between correlated median splits. The guarantee that a correlation destroyed was never about interactions.

splits · Criterion What a rule reads, against what the outcome uses. The variance of the treatment estimate relative to a coin's, for four things a rule might balance against three shapes the outcome might have, over 350 trials of 200 units. The diagonal is the easy part — a rule that reads the function the outcome uses removes about half the variance. What the table is for is the off-diagonal: reading the covariate alone is worth nothing against a quadratic (0.755), and reading all three is worth nearly as much against every shape as the matching rule is against its own (0.532, 0.493, 0.514).

Three functions of one number

A rule that balances the covariate is exposed to every shape the outcome might have. A rule that balances three functions of it costs two points of variance against the shape the first was built for and takes the worst case from a coin's to about half of it.

shape · Criterion How long a walk has to be given. Every equal split of twelve units is enumerated, the 410 admissible ones are found, the transition matrix is built, and the distance from uniform is computed exactly at each step — no simulation anywhere. The walk is started at the least balanced admissible assignment, which is the state a rejection sampler is least likely to have handed it and the one a burn-in has to cover. It is 0.0849 away after twenty steps and 0.00008 after ninety. A real cost, and a small one, and naming it is what stops it being assumed to be zero.

Walking the admissible set

A rerandomisation test hunts for admissible assignments and throws away the rest. A walk visits them instead — and it is exactly uniform only because it stands still when a proposal fails, which is the step that looks like waste.

joint · Randomisation How often each probe finds a split that is there. The share of 34 designs — every one of them enumerated to be in two components — on which a two-chain test of 800 draws declares the split, by probe. The fourth power as the earlier fields use it finds it on 55.9%, so it misses 44.1% of the sets that have one. The same column projected off the rule's span finds it on 88.2%, and the separating direction itself on 91.2%. The design's own leverage, chosen without any dictionary, gets 79.4%. A random direction in the same subspace gets 44.1%, and the direction chosen for being concentrated gets 38.2% — worse than random, which is what a heuristic that finds the wrong structure looks like from the outside.

What a chosen probe finds

On a chain of eight hundred draws the probe the earlier fields use misses 44% of the sets that are split. Its own residual off the rule's span misses 12%, for one least-squares fit.

aimed · Randomisation The crossing barely moves. Both methods' costs in one unit — assignments evaluated per usable draw — as the tolerance tightens. A hunt costs 1/p and rises without limit: from 2.22 at a tolerance of 1.2 to 357.14 at 0.18. A walk costs its autocorrelation time and barely moves. The two cross at a tolerance of 0.190 at one swap and 0.195 at eight — the whole family of proposal sizes crosses inside a band of about two hundredths, because where the crossing is, the large proposal has already lost its advantage. A multi-swap proposal is worth a factor of 5.65 in the regime where the walk should not be used at all.

Where the gain is, and where the decision is

A bigger proposal is worth a factor of six at a loose tolerance and nothing at a tight one. The tolerances where it helps are the ones where a hunt costs two evaluations a draw, and the crossing barely moves.

blocks · Assignment A class that is a subspace has no guarantee below its own dimension. Each cell is the worst case over every unit-variance function in a class of dimension m, for a rule reading k functions: the smallest squared principal-angle cosine between the two subspaces. Wherever k is less than m the number is zero to machine precision, and that is not a weak guarantee but the absence of one — some direction of the class is orthogonal to the entire basis, and against an outcome in that direction the rule does exactly what a coin does. An experimenter who declines to name the shapes and asks instead to be protected against everything smooth is asking for the cells above the diagonal.

Where the guarantee is exactly zero

An experimenter who declines to name the shapes, and asks instead to be protected against anything in a class, is asking for a number that is not small but zero. Bounding the class is unavoidable, and the two ways of doing it choose different bases.

basis · Randomisation Which tail the threshold is in. What a rule balancing a threshold at 1 on each covariate's own scale removes of the interaction between the two thresholds, on five copulas matched at a Spearman rank correlation of 0.4 with a normal covariate throughout. This rule never had a zero to lose — the earlier field establishes that under every marginal — so what is left is a size, and the size depends on where the dependence lives. A Clayton copula, whose density piles up in the lower tail, leaves 5.33%; the same copula turned over, so that it piles up in the upper tail where the threshold is, leaves 33.36%. Same rank correlation, same Kendall tau, same marginal, same threshold: 6.26 times the leak, decided by which end of the distribution the dependence and the cut are both in.

Which tail the cut sits in

The same copula and its reflection have the same rank correlation, the same Kendall tau and the same marginals. A balancing rule holding a threshold at a dose leaves 5.33% under one and 33.36% under the other.

copula · Criterion A guarantee that stops being a number. The worst case of each dictionary over six outcome shapes, at a correlation of 0.5, against the skewness of the covariate. Under a symmetric marginal every rule made of odd functions has a worst case of exactly zero, and the rule holding a mean and a median split of each covariate — the two things every trial balances — is one of them. Under skew that zero becomes 0.74%, 1.83%, 2.24%, 2.49%: small numbers, each of which depends on a marginal nobody stated. The guarantee has not improved by becoming positive. It has stopped being a guarantee, because it can no longer be written down without the covariate's distribution in it.

Balancing a skewed covariate

The worst case of the rule every trial runs goes from exactly zero to somewhere between a quarter of a per cent and two and a half. Which is small, and is a number that cannot be stated without the covariate's distribution in it.

skew · Criterion Where the set stops being one set. How many of the 3,432 equal splits of fourteen units a balancing rule admits, as the tolerance tightens, with the number of components single swaps leave it in. The set falls from 886 to 84 assignments, and somewhere in that fall it stops being connected: at 0.8 it is in 2 pieces and every assignment's complement is in the other one. Nothing about the rule changes at that point and nothing a chain reports changes either, which is the whole difficulty — the acceptance rate, the stationary distribution and the detailed balance are all in order on both sides of it.

Before the trial and after

The same diagnostic run at two moments answers two different questions. Before, a positive verdict changes the design. After, it changes which number gets reported — and only for the numbers the defect can reach.

after · Assignment One of them is mostly leverage. How much of the design's own leverage direction each active-set probe carries, once both are standardised and projected off the rule's span — which is what a probe is, so it is the comparison that matters. Over 189 designs the modelled active set agrees with leverage at |r| = 0.8359 ± 0.0114 and the counted one at 0.4239 ± 0.0216. So the modelled probe is largely leverage under another name and the counted one is genuinely a different direction — and the counted one is the worse probe, at 0.3854 of alignment against 0.4272. What the active set contains beyond leverage points away from where the set splits.

Counting it exactly does not help

If a modelled active set lost because the model was crude, the exact one would win. It is computed at a cost no trial can pay, and it is worse — so the approximation was never what was costing the probe.

blocked · Randomisation What each analysis does at a true null, by shape. Four analyses of the same trials — 500 of them at each shape, 120 units, assigned by the rule that reads the covariate. Every rejection is false. The unadjusted analysis is the one that moves: 1.60% against a linear outcome, where the design removed a great deal that the standard error still prices, and 5.20% against a quadratic, where it removed nothing and the standard error is right. Adjusting holds the level in all three columns, and so does the design's own reference distribution, which needs to be told the rule and nothing else.

The analysis and the shape

An unadjusted analysis after a rule that read the covariate is too cautious — by a third against a linear outcome, by nothing at all against a quadratic. And an adjustment for the wrong function recovers almost none of the precision the right one would.

shape · Randomisation What the diagnostic says at two hundred units. The same test run 8 times on independent streams, at five tolerances of a two-hundred-unit trial, 40,000 steps each. At the loosest tolerance every run says the same thing — the walk reaches the whole set — and it keeps saying it as the set is thinned. Past a point the runs stop agreeing with each other: at the tightest tolerance here 6 of 8 report that the chains have not mixed and 2 report a split, which is a diagnostic disagreeing with itself rather than a property of the set. That disagreement is the honest answer at this size, and it is one nothing in this collection could give before: an enumeration stops at about twenty-four units.

The diagnostic at two hundred

Pointed at a trial size no enumeration reaches, the test gives three answers rather than one — and past a certain thinness it stops agreeing with itself, which is the honest reading and the one nothing could give before.

reach · Randomisation A thin enough set is not one set. Every admissible set of 14 units this table can enumerate, by how much of the assignment space it admits and how many pieces it falls into under single swaps. A walk is uniform on the piece it starts in and never leaves it. The pieces are not fragments: at 522 admissible assignments the set splits into 3 halves of exactly 520 each, and every assignment's complement is in the other half — no sequence of admissible single swaps takes an assignment to its own mirror image. Two-swap proposals reconnect four of the six disconnected sets here — the two they do not are the thinnest, where a two-unit move rarely lands anywhere admissible either — which makes a bigger proposal a correctness repair rather than the speed dial it was measured as.

The walk that cannot cross

A thin enough admissible set is not one set. It splits into an assignment and its mirror image, no sequence of admissible single swaps joins them, and the walk that samples it is uniform on half the reference distribution for ever.

dict · Randomisation The one zero neither half of the dependence can touch. A median split's interaction leak at all 30 combinations of copula and marginal, on a log scale. Every one is under 10⁻¹⁶ and the largest is 1.74e-20, which is the quadrature's own noise rather than a leak. The reason is arithmetic and it is short: a centred median split takes the values ±½, so its square is a quarter identically — for every unit, on every draw, whatever the covariate's scale is and whatever joint law the ranks have. The interaction is then orthogonal to both main effects by construction, and there is nothing for either half of the dependence to break. Both of the fields this one joins report this zero holding under their own variation; running both variations at once is what establishes that it is not two coincidences.

The zero that survives both

A median split's interaction leak is under 10⁻¹⁶ at all thirty combinations of copula and marginal. It is the only guarantee in the collection that neither half of the dependence can touch.

compound · Adjustment The rate falls geometrically; the count does not fall at all. The share of equal splits of two hundred units that a tolerance of 1 coin-spreads admits, against the number of functions the tolerance is stated for, with the closed form (2Φ(1) − 1)^k drawn beside it. The rate falls by about two thirds with every constraint. The admissible count is that rate times C(200, 100), and it goes from 2^195 to 2^192 — it does not fall in any sense a trial cares about. What the falling rate costs is sampling: 9,878 draws to collect a thousand admissible ones at six constraints, against 1,465 at one.

What a reference distribution costs to sample

A randomisation test on a trial too large to enumerate has to sample its reference distribution, at 1/p attempts per draw and a p-value resolved to 1/(B + 1). Six constraints cost 9,878 attempts per thousand draws, and a thousand draws resolve p to 9.99·10⁻⁴ and not one digit finer.

product · Reference Where the constraints exhaust the randomisation. At 16 units there are 12,870 equal splits, so the ones meeting a stated tolerance can be counted rather than estimated. With each of the first k standardised imbalances required to be within 0.4 of a coin's own spread, the admissible count runs 3874 → 1006 → 314 → 0 → 0 → 0 — and at 4 functions there is no admissible assignment at all. The count is the number of distinct answers a randomisation test can give: at 3 functions its finest attainable p-value is 1 in 314. Balance improves with every constraint and the reference distribution shrinks with it, and the two run out at different rates.

When the constraints run out

Every function added to a basis is a constraint the assignment has to satisfy with the same units. At sixteen units and a stated tolerance the admissible assignments run 3,874, then 1,006, then 314, then none — and the count is exact, because the assignment space is finite.

basis · Allocation A cut walked from the median into each tail. What a rule balancing a threshold on each of two normal covariates removes of the interaction between the two thresholds, with the cut moved from the 1st percentile to the 99th, on five copulas at a Spearman rank correlation of 0.4. Every copula reads exactly zero at the median. The three radially symmetric copulas are mirror images about it: the Gaussian peaks at 19.49% near the 95th percentile and is down to 13.44% at the 99th; the Frank peaks at 13.22% near the 85th and falls to 2.06%; the t is still near its highest, 37.45%, at the 99th. The Clayton climbs to 56.37% at the 1st percentile, in the tail its dependence is in, and never passes 5.39% in the other; its reflection is the same curve turned round.

A cut walked into the tail

Move a balancing rule's threshold off the median and the interaction it leaks rises from exactly zero — and then either turns back to nothing or climbs to a ceiling, depending on one property of the dependence. With both covariates cut, the whole regression collapses to two probabilities, and far out it settles on 2λ/(1 + λ) of the tail-dependence coefficient: 57.25% under a Clayton, 35.57% under a t, and zero under a Gaussian or a Frank however strongly the covariates are related.

copula · Criterion Projection as a dial. Each probe projected a fraction λ of the way off the span of the rule's balanced columns, over the 100 designs of 200 whose admissible set is enumerated to be split; the median separation — the two components' difference in mean probe value over the spread inside a component. the separating direction: 10.565 at 0, 11.500 at 0.25, 11.456 at 0.4, 11.424 at 0.5, 11.240 at 0.6, 10.664 at 0.75, 10.427 at 0.9, 10.273 at 1; the design's leverage: 4.537 at 0, 4.845 at 0.25, 4.924 at 0.4, 4.849 at 0.5, 4.678 at 0.6, 4.417 at 0.75, 4.205 at 0.9, 3.836 at 1; the fourth power: 1.543 at 0, 1.846 at 0.25, 2.094 at 0.4, 2.570 at 0.5, 2.870 at 0.6, 3.686 at 0.75, 4.579 at 0.9, 5.080 at 1.

How far to project

The field's repair for a probe is to project it off the span the balancing rule already pinned. Treated as a dial rather than a switch, that repair is right all the way for the fourth power — its separation rises from 1.543 unprojected to 5.080 fully projected — and wrong for the design's own leverage, which carries 4.537 unprojected, 4.924 at four tenths of the way and only 3.836 projected fully. The leverage probe the field recommended was a third of the way past its best. Even the separating direction itself gains a little from a partial projection and loses a little from a full one.

aimed · Randomisation The table swept along the covariate instead. What a rule holding a mean of each covariate fails to remove of their interaction, at each of 4 copulas, as the covariate is skewed further and the rank correlation is held at 0.4. The sweep runs from a symmetric covariate at g = 0 to a skewness of 11.16, and the three settings the earlier table names — g = 0.3, 0.6 and 0.9 — are on it, where this sweep reproduces that table to the last digit. Every row rises and then falls: the lower-tail copula from 7.707% through 0.0015% and back to 5.587%, the upper-tail one to a maximum of 36.213%. So the quantity a trial is exposed to is not monotone in how skewed its covariate is.

The other dial

The table is swept along the strength of the dependence and never along the shape of the covariate. Swept along the shape at a fixed correlation, the same two copulas cross, the same way — and the near-zero cell turns out to be a minimum in both directions at once.

stronger · Adjustment The pairing recovers most of it and passes nothing. How much of the separating direction each probe carries, over 100 designs of 14 units whose admissible set is enumerated and split in two. The four rows the pairing adds are the dominant direction of what each blocking matrix keeps past its degrees, and the cut that direction's signs induce. Counted, they read 0.5266 and 0.5258 against the counted per-unit share's 0.3854 — most of the gap between that share and the design's own leverage at 0.5395, closed. Modelled, they read 0.4274 and 0.4954 against 0.4272. Nothing built from the active set passes leverage, and the projected fourth power is still ahead of all of them at 0.6583.

A set of pairs, not a vector

The active set is a graph on the units, and every probe built from it so far has been its degree. Read as a graph it recovers 0.1326 of the alignment the summary lost — and draws level with leverage rather than passing it.

blocked · Randomisation A median split of a measured scale. The share of the interaction a rule balancing a split above the median of each covariate removes, when the covariate is measured on k levels of a symmetric binomial, on five copulas at a Spearman rank correlation of 0.4. With an even number of levels nothing sits on the median and the split is exactly balanced: the rule removes nothing, as for a continuous covariate. With an odd number, 3 levels (50.00% on the median): Gaussian 10.44%, Clayton 4.91%, its reflection 18.44%; 5 levels (37.50% on the median): Gaussian 6.39%, Clayton 3.69%, its reflection 10.06%; 7 levels (31.25% on the median): Gaussian 4.58%, Clayton 2.91%, its reflection 6.77%; 9 levels (27.34% on the median): Gaussian 3.57%, Clayton 2.41%, its reflection 5.05%; 11 levels (24.61% on the median): Gaussian 2.92%, Clayton 2.05%, its reflection 4.01%; 15 levels (20.95% on the median): Gaussian 2.15%, Clayton 1.59%, its reflection 2.82%; 21 levels (17.62% on the median): Gaussian 1.53%, Clayton 1.19%, its reflection 1.93%; 31 levels (14.45% on the median): Gaussian 1.04%, Clayton 0.85%, its reflection 1.26%.

A median with ties

The median split's interaction zero survives every marginal and every copula, because a split that is on for exactly half the units squares to a constant. A covariate measured on a few levels cannot be split exactly in half: on a five-point scale three in eight units sit on the median, and a rule balancing 'above the median' removes 6.39% of the interaction under a Gaussian copula and 10.06% under one of the two Claytons. The size is set by the share of units on the median, which is the formula's own factor. And a fair coin for the tied units restores the zero exactly, on every copula.

compound · Adjustment A threshold read along the skew. What a rule balancing a threshold at the value 1 on each covariate's own scale removes of the interaction between the thresholds, against the covariate's skewness, on five copulas at a Spearman rank correlation of 0.4. Gaussian: 16.41% with no skew, 10.07% at a skewness of 11.16; t on four: 22.18% with no skew, 12.11% at a skewness of 11.16; Frank: 13.19% with no skew, 10.14% at a skewness of 11.16; Clayton: 5.33% with no skew, 4.82% at a skewness of 11.16; Clayton turned over: 33.34% with no skew, 17.61% at a skewness of 11.16. Every leak falls, because the value moves towards the median.

Where the skew puts a threshold

The field read the mean-balancing rule along a sequence of skewness and left the threshold rule — a cut at a fixed value on the covariate's own scale — unread. Read, it turns out to be no new curve. Skewing the scale moves the value 1 from the 84th percentile of the latent covariate to the 74th, and the rule's leak is the copula field's cut sweep read at that quantile: it falls on every copula, from 16.41% to 10.07% under a Gaussian. Whether skew and dependence compound or cancel is then a fact about the slope of each copula's curve, and unlike the mean rule's, no cell changes its answer along the sequence.

stronger · Adjustment The same number of assignments, held less tightly. For each basis size the box admits at a tolerance of 0.4, the Mahalanobis rule is given exactly the same number of admissible assignments — the ones with the smallest Mahalanobis distance — so the two differ only in the shape of the region. Each bar is the worst named function's remaining imbalance variance as a share of a coin's, with the mean over the named functions beside it. At one function the regions are the same interval: 0.0523 both ways. At 3 functions and 314 assignments, the box leaves 0.0771 in its worst function and the ellipse 0.1545; on average 0.0633 against 0.0927. The ellipse spends its assignments on a direction no function in the basis names.

What the ellipse spends its assignments on

Rerandomising on Mahalanobis distance never runs out of assignments, where a box of per-function tolerances does at four functions. But at the same number of admissible assignments it holds the worst named function half as tightly, because it spends its constraint on a combination of the basis nobody named.

basis · Allocation A basis drawn rather than chosen. The share of the worst of seven shapes a balancing rule removes, over two covariates and a dictionary of twenty-four functions, for the best fixed basis of each size (lower line) and for a basis drawn at random from the least favourable mixture (upper line), at one to four functions. The fixed basis guarantees 0.0000, 0.0000, 0.4244, 0.6366: nothing at one function or two, because two of the shapes are pure interactions and two of them are main effects of different covariates, and no one or two functions touch all of them. The drawn basis guarantees 0.2042, 0.4084, 0.6133, 0.7920 in expectation over the draw — about a fifth of a shape for every function up to three. The marks above the upper line are the best reply to nature's mixture, 0.2051, 0.4098, 0.6142, 0.7931: the game's value lies between the two.

A basis drawn rather than chosen

On two covariates no fixed basis of one or two functions protects every shape an outcome might have — its worst case is exactly zero. Draw the basis at random from the right mixture and the guarantee is a fifth of a shape for every function. The price is per trial — at three functions, two draws in five leave one shape wholly unprotected.

product · Optimum The best worst cases among twenty-eight pairs. Every pair of two functions from eight — the covariate, its square, cube and fourth power, and cuts at −1, 0, 1 and 2 — held on both covariates, ranked by the smallest share it removes over seven outcome shapes at a correlation of 0.5. The best is the square and a cut at 1 at 7.00%; the mean and the square reach 6.84%. For almost every pair the shape holding the floor is the product of two median splits, which no pair removes more than a small part of.

Two functions a covariate

Every constraint costs admissible assignments at the same rate, so the decision a trial faces is a budget: which two functions of each covariate. Over twenty-eight pairs the worst case never picks the mean. At a correlation of a half it picks the square with a cut at one, beating the mean and the square by sixteen hundredths of a point on a floor held by a shape no pair removes a tenth of — and giving up 41% of a linear outcome to do it. Over the 127 lists of shapes a designer could write, twenty-three different pairs are best somewhere.

dict · Criterion The worst case against the number of functions, at four trial sizes. The largest share of a coin's imbalance the rule leaves in any of the four shapes, against how many functions it reads, at 50, 100, 200, 400 units. At 50 units: 103.5%, 38.6%, 37.2%, 37.0%, 39.8%, 41.7%. At 100 units: 105.9%, 37.8%, 38.9%, 30.3%, 27.0%, 24.6%. At 200 units: 95.9%, 44.2%, 42.8%, 32.9%, 28.4%, 11.7%. At 400 units: 108.5%, 41.8%, 41.3%, 32.1%, 26.1%, 5.8%. At fifty units every basis from two functions up leaves the worst shape between 37.0% and 41.7%; at four hundred, six functions leave it 5.8%.

How many functions a trial can hold

A rule that reads more functions of one covariate protects more shapes, and every function it reads costs the shapes it already held: the imbalance it leaves in a linear outcome falls like one over the trial size, with a constant that runs from about 3 for one function to about 16 for six. At fifty units that cost eats the protection, and the worst case over four shapes sits between 37% and 42% of a coin's from two functions to six. At four hundred it falls from 42% to 6%. By six functions the worst shape, at every size, is one the rule is reading.

shape · Criterion What exact balance on the main effects removes of the interaction, in a sample. The share of the interaction's in-sample variance explained by the eight main effects — what exact balance on them would remove — averaged over 4,000 samples, at 25, 50, 100, 200, 400 units and correlations of 0, 0.3, 0.5. At 0, whose closed form is 0.0%: 62.1%, 40.3%, 24.5%, 14.9%, 8.4%. At 0.3, whose closed form is 30.3%: 69.8%, 54.2%, 44.4%, 38.4%, 34.8%. At 0.5, whose closed form is 64.0%: 80.7%, 73.1%, 69.2%, 66.8%, 65.4%.

The share a sample removes

A rule balancing eight main effects of two independent covariates removes exactly none of their interaction — in the population. A trial is a sample, and in a sample of a hundred units exact balance on the same eight functions removes 24% of the interaction's variance on average, between 10% and 42% from one trial to the next. At twenty-five units it removes 62%. The zero is the centre of nothing: it is the limit the in-sample share falls towards, by about two fifths with every doubling of the trial, carried there by an interaction whose spread grows with the very functions it is regressed on. At a correlation of a half the closed form says 64%; a trial of a hundred gets 69% on average and anything from 53% to 83%.

joint · Criterion What the rule leaves of the product, against what exact balance would. For a rule holding the mean of each covariate, at a correlation of 0.5, the share of a coin's imbalance left in the product of the covariates: by the closed form for exact balance, and in trials of 100 and 400 units, on four marginals. normal: 100.0% exact, 98.6% at 100, 108.4% at 400; clearly skewed: 73.7% exact, 92.9% at 100, 91.5% at 400; strongly skewed: 70.3% exact, 93.9% at 100, 94.4% at 400; exponential: 71.5% exact, 84.1% at 100, 87.3% at 400.

A share at a hundred units

On a skewed covariate the closed form credits a rule that balances the means with removing 26% of the covariates' product and 52% of a square, shares that rest on symmetry having failed. A trial does not collect them. At a hundred units the rule removes 12% of the product and 37% of the square on the clearly skewed covariate, and at four hundred units barely more, because the shares live in the long tail and the tail is a handful of units. On the most skewed covariate the rule removes a fifth of the median split at a hundred units and nothing of it at four hundred: balancing the mean of such a covariate is balancing its largest few values.

skew · Criterion One unit that never changes arm. Every admissible assignment of one design of fourteen units — 192 of them, enumerated, in two components of 96 and 96 — read unit by unit: for each unit, the share of a component's assignments that put it in the treated arm, in the first component (filled) and the second (open). Most units are treated on between a third and two thirds of either component's assignments. One unit, the one at x = 0.28, is treated on 0% of the first component's assignments and 100% of the second's. Asking which arm that unit is in tells the two components apart without error, which no linear probe built from the covariate does.

A ceiling that is one unit

The benchmark every probe of a split admissible set was measured against maximised the wrong thing, so the obvious next step was the direction that maximises the separation itself. On 88 of 100 split designs that direction has no finite value: some combination of units reads the same on every assignment of a component. On 83 of them it is a single unit, sitting in one arm across one component and the other arm across the other — and a chain can see a unit that never moves without enumerating anything.

aimed · Randomisation A leak read off a cohort, at two cuts. The threshold rule's leak as a cohort of 1,000 units estimates it from its own two-by-two table: the bar is the middle eighty per cent of 600 cohorts' plug-in estimates, the open circle their median and the stroke the true leak, for five copulas at a Spearman rank correlation of 0.4. At the 90th percentile the estimate is centred on the truth and its middle eighty per cent spans about fourteen points: Gaussian 12.8% to 26.4% around 19.2%, t on four 21.5% to 35.7% around 28.9%, Frank 5.4% to 18.7% around 12.2%, Clayton 0.8% to 9.6% around 4.4%, Clayton turned over 35.9% to 49.5% around 43.1%. At the 99th it is not centred on anything: Gaussian 13.1% to 100.0% around 13.7%, t on four 17.5% to 72.9% around 37.4%, Frank 14.4% to 100.0% around 2.2%, Clayton 16.1% to 100.0% around 0.6%, Clayton turned over 37.0% to 75.7% around 56.3%, because on three of the five copulas between two cohorts in five and four in five have nobody past both cuts.

A leak read off a two-by-two table

The share of an interaction a threshold-balancing rule leaves alone needs one number from the data: how often a unit past the cut on one covariate is past it on the other. A cohort of a thousand estimates it to within a dozen points at the 90th percentile. At the 99th most cohorts have an empty cell in their table, and every one of those reads a leak of exactly one hundred per cent — a reading the bootstrap then confirms with an interval of zero width.

copula · Criterion When each unit first changed arm. One design of two hundred units, the rule balancing the covariate, its square and a median cut, and a swap-one-pair chain run for 1.6 million steps from a random admissible start, at six tolerances: for each unit, the step at which its arm first differed from the start, the two hundred sorted from first to last. At 1: half the units had moved by step 79, nine in ten by 269, all by 1,073; At 0.6: half the units had moved by step 79, nine in ten by 362, all by 1,073; At 0.4: half the units had moved by step 106, nine in ten by 429, all by 1,593; At 0.3: half the units had moved by step 153, nine in ten by 645, all by 3,878; At 0.22: half the units had moved by step 194, nine in ten by 747, all by 1.1 million; At 0.16: half the units had moved by step 237, nine in ten by 1270, and 1 never moved. The last units to move set the run length a diagnostic built on them needs, and at the tightest tolerance one never does.

A unit that has not moved

At two hundred units the two-chain diagnostic could not settle its own verdict, because its standard error needed a relaxation time the run was too short to estimate. Reading which units moved needs no standard error. Run for 1.6 million steps on five designs, it finds a unit in place in both chains on three designs at the tightest tolerance and one at the next — and every one of them is a unit no swap can move from any admissible assignment, by arithmetic. The admissible set does split at two hundred units, and the split is one extreme unit.

reach · Randomisation How many starts it takes to land in two pieces. Averaged over the split designs at each of three tolerances, the chance that k starting assignments drawn uniformly from the admissible set — what hunting for a start by rejection gives — do not all fall in one component, which is the most any diagnostic built on comparing chains from those starts can detect. At 0.6: 28% with 2, 47% with 4, 63% with 8, 77% with 16, 88% with 32, 95% with 64, 99% with 128. At 0.8: 12% with 2, 21% with 4, 32% with 8, 46% with 16, 62% with 32, 77% with 64, 90% with 128. At 1.0: 4% with 2, 8% with 4, 13% with 8, 21% with 16, 33% with 32, 47% with 64, 64% with 128. The median split design needs 21, 109 and 439 starts at the three tolerances to expose its split with probability 95%. With an even allocation the mirror makes two starts enough whenever the split is a unit.

After the mirror is gone

The two-chain diagnostic starts its second chain at the complement of the first's start, which exists only when the arms are equal. At five treated against ten, enumerated on two hundred designs, a split admissible set is one big piece and some crumbs — the median puts 97.3% of its assignments in one component — so two random starts see the split on 12% of split designs and 95% needs a median of 109 starts. Worse, the rule fixes units in a connected set all the time, so a unit that never moves stops being evidence. A hunt that forces such a unit into the other arm finds 41 of 70 splits; 29 carry no unit that marks them.

after · Randomisation Weights that balance a sample by construction. What three sets of weights leave of the standardised difference between the arms on each covariate, as a root mean square over 1200 samples of 600 units. The true propensity leaves 0.1317 and 0.1186 — a sampling error, since it is right about the population and knows nothing of the draw. A likelihood fit leaves 0.0770 and 0.0657, having absorbed part of the draw's imbalance as a side effect of fitting the treatment. Weights fitted so that each arm's weighted means are the sample's leave 1.4e-14 and 1.2e-14, which is the arithmetic's floor rather than a small number: the largest gap between a weighted arm mean and the sample mean in any draw is 9.8e-14.

A weight fitted to balance

Weights fitted so that each arm's weighted covariate means equal the sample's leave a difference of 1.4×10⁻¹⁴ between the arms and give the estimate a third of the variance of weights fitted by likelihood — 0.011883, within a relative 5.8% of the bound no estimator can beat. In the world where the assignment carries a square nobody named, the same exact balance leaves the square further apart than no weighting at all, and where the outcome carries it too the estimate is wrong by 0.6973 with an interval that covers 1.5%.

weights · Weighting Separation, how far against whether. The median separation each probe carries between the pieces of a split admissible set, over the 100 designs whose set is split. The share refused per unit carries 1.953; the refusal pairing's direction 3.103, and with the saturated pairs dropped 2.917. The depth pairing's direction carries 2.345, and each unit's mean depth 3.836 — exactly the leverage's. A random direction carries 0.942 and the projected fourth power 5.080.

How far, not whether

The blocking matrix records whether a tolerance box refuses an exchange. Dropping the pairs it refuses for certain changes nothing. Recording instead how far each exchange moves the balance vector turns the pairing into its degree plus a part that lies inside the rule's own span — and the degree is the design's leverage, exactly. The heuristic that beat every reading of what the rule blocks was the unclamped pairing's degree all along.

blocked · Randomisation Seven products of three splits. The correlation of every product of three centred median splits with every other, at a Spearman rank correlation of 0.4 between each pair of covariates. Under a Gaussian copula a main effect correlates with the other mains and with the triple (0.273), and with no pair at all, because the third moment is 0.000. Under a Frank copula, whose pairs are radially symmetric, the third moment is 0.078 and every other cell fills in: a main correlates 0.082 with the pair it is not in. In both panels a main is exactly uncorrelated with a pair it belongs to, which is the known zero.

The zero that needed two

A rule balancing two median splits removes nothing of their interaction, on every copula, because a centred split squares to a constant. Add a third covariate and the same arithmetic stops protecting the thing it looked as if it protected. The product of three splits is not orthogonal to the mains: at a rank correlation of 0.4 a rule on the mains removes 14.47% of it under a Gaussian copula. And the symmetry that decides the rest belongs to the triple rather than the pair, so Frank's copula, whose pairs are radially symmetric, leaks where Clayton's, whose pairs are not, barely does.

compound · Adjustment A cut on each covariate, every pair of them. The share of the product of two threshold indicators that a rule balancing the two indicators removes, in per cent, at a Spearman rank correlation of 0.4, for cuts at the fifth to the ninety-fifth percentile on each covariate. On the Gaussian the far corner, a cut at the 5th percentile on one and the 95th on the other, is 54.8% against 19.5% with both at the 95th. On the t copula it is 1.1% against 34.5%. Both surfaces have their only zero at the double median and leave the diagonal lower in places: moving one Gaussian cut from the 80th to the 70th lowers the share from 14.0% to 12.1%.

A cut on each side

Every table of the threshold's leak put one quantile on both covariates, which is the diagonal of a surface nobody had drawn. Off the diagonal the copulas swap places: a cut at the 5th percentile on one covariate and the 95th on the other removes 54.8% of the interaction under a Gaussian copula and 1.1% under a t copula, the reverse of their order at the near corner. Walked further out, the far corner has two opposite limits, and which one a Gaussian reaches is decided by whether its correlation is above or below exactly one third.

stronger · Adjustment Sixty-three subsets of six functions. The largest share of a coin's imbalance left in any of four shapes, measured on fresh trials, for each of the 63 subsets of six functions at 100 units. Subsets of one size spread widely: at four functions from 12.1% to 72.9%. The designer's order of addition gives 1: 94.4%, 2: 41.2%, 3: 39.0%, 4: 31.5%, 5: 27.0%, 6: 22.7%, against the best chosen subset of each size at 1: 83.6%, 2: 41.2%, 3: 25.4%, 4: 12.1%, 5: 16.2%, 6: 22.7%.

The list it was chosen on

Six functions of one covariate looked like 720 designs, one per order of addition. The rule reads a span and not a sequence, so they are 63, and at a hundred units the best four leave 12.1% of a coin's imbalance against the designer's first four at 31.5%. Those best four contain no cube and are, function for function, the four shapes the worst case was scored on. Read on six shapes it was not told about, the same design leaves the cube 77.0% of a coin's imbalance, and the designer's four, which lost by a factor of two and a half, leave it 28.1%.

shape · Criterion Four rules and what each leaves. The share of a coin's imbalance each rule leaves in the true value of each covariate, at 200 units and a reliability of 0.8. one reading of X: 21.4% in X and 100.1% in Z; two readings of X: 12.3% in X and 105.0% in Z; one reading each of X and Z: 21.7% in X and 21.5% in Z; two of X and one of Z: 13.4% in X and 22.0% in Z. The floors in closed form are 20.0% for one reading and 11.1% for two.

A second reading or a second covariate

A rule that balances a recording balances the truth only as far as the recording reaches, and a second reading of the covariate raises that reach. So does a first reading of something else. At a reliability of 0.8 the first reading of a second covariate does more once that covariate matters a third as much as the first, and the threshold is exact: the square root of (1 − λ)/(1 + λ), with no trial size in it. A protocol's list of measurements turns out to be a small knapsack, and it is rarely filled by reading anything twice.

continuous · Assignment What a floor costs a drawn basis. The expected guarantee of a basis drawn from a mixture restricted to bases that each remove at least a stated share of every shape, solved at each distinct floor. 2 functions, of 276 bases: 40.8% at a floor of 0.0%; the best fixed basis floors at 0.0%. 3 functions, of 2024 bases: 61.3% at a floor of 0.0%, 50.0% at a floor of 31.8%, 50.0% at a floor of 34.7%, 49.7% at a floor of 40.5%, 42.4% at a floor of 42.4%; the best fixed basis floors at 42.4%. 4 functions, of 10626 bases: 79.2% at a floor of 0.0%, 77.1% at a floor of 31.8%, 77.1% at a floor of 34.7%, 77.1% at a floor of 35.1%, 77.1% at a floor of 35.6%, 77.1% at a floor of 37.0%, 77.1% at a floor of 38.3%, 77.1% at a floor of 40.5%, 76.5% at a floor of 41.0%, 76.5% at a floor of 42.4%, 76.5% at a floor of 42.8%, 76.5% at a floor of 42.9%, 76.5% at a floor of 43.5%, 76.5% at a floor of 44.7%, 76.5% at a floor of 45.3%, 76.5% at a floor of 46.0%, 76.5% at a floor of 46.8%, 76.5% at a floor of 47.2%, 76.5% at a floor of 50.0%, 76.5% at a floor of 56.0%, 76.5% at a floor of 63.7%; the best fixed basis floors at 63.7%.

A floor for the draw

A basis drawn at random guarantees more on average than any fixed one, at the price of leaving some shape wholly untouched in a large share of trials. Requiring every basis in the draw to keep a floor looks like a smooth trade between the two. It is a staircase, and the first step does nearly all the work. No basis has a floor between zero and 31.83%, so there is nothing to give up until a floor of that size is asked for. And at four functions, seven bases tie at the fixed basis's own 63.66%: a mixture of them keeps that floor exactly and adds thirteen points to the guarantee.

product · Optimum Three regions at the box's own count. Across 24 sets of 16 units, on the sets where a box at a tolerance of 0.4 has any admissible assignment, the share of its own variance left in the worst of the k named functions at the box's count. the ellipse: 0.070, 0.134, 0.196, 0.194, 0.182. a sphere on the standardised functions: 0.062, 0.076, 0.091, 0.086, 0.080. the box: 0.062, 0.072, 0.081, 0.073, 0.069, at 2, 3, 4, 5, 6 functions, on 24, 20, 9, 4, 4 of the sets. The box's mean admissible count is 1045, 354, 115, 32, 13.

A box with a count

The ellipse never runs out of assignments and holds the named functions loosely; the box holds them tightly and can come up empty. Dropping the ellipse's thin directions was meant to give both, and it does not: no cutoff comes within a third of the box. But the box's emptiness was never a property of the box. It belongs to a tolerance fixed in advance. Set the tolerance by a count instead, and at that count the region is the box, on every set of units, with nothing left to run out.

basis · Allocation The two shapes a scale decides between. The share of a coin's imbalance left at 400 units, by marginal and by what the rule reads. normal: linear 1.1% on values and 5.7% on ranks; split 37.4% on values and 25.4% on ranks. skew 2.26: linear 1.5% on values and 28.6% on ranks; split 61.9% on values and 25.0% on ranks. skew 4.75: linear 2.5% on values and 52.2% on ranks; split 95.5% on values and 25.1% on ranks. exponential: linear 1.4% on values and 28.1% on ranks; split 58.7% on values and 25.0% on ranks. A rank's hold on the split does not move with the marginal.

The covariate on its ranks

A skewed covariate spoils every parity argument a balancing rule relies on, and a rule that balances ranks instead of values looks like the way out, since a rank has the same uniform law on any marginal. The measurement says the skew really does belong to the scale for one kind of shape: a rank holds a median split at exactly a quarter of a coin's imbalance whatever the covariate. But it pays with a cap on any shape linear in the value, at nearly half a coin's imbalance on the strongest skew. And a rule holding values and median splits together holds both, beating the rank on the shape a rank gives up and matching it on the one it gains.

skew · Criterion One weighting told the means and one told the second moments, in five worlds. The bias of the fit to balance over 600 samples of 600 units in each world, fitted to the covariates' means and fitted to their means, squares and product. Told the means it is off by -0.0004, -0.0020, 0.0103, 0.6973, 0.2698 in the worlds with no square, a square in the assignment, a square in the outcome, a square in both and a cube in both; told the second moments, by -0.0009, -0.0000, 0.0009, -0.0045, 0.3099. Its interval covers 94.0%, 94.7%, 95.0%, 1.5%, 51.0% and 93.7%, 89.8%, 94.0%, 91.0%, 48.3%.

The moments a balance is told

Weights fitted to balance the covariates' means were wrong by 0.6973 in the world where both the assignment and the outcome carry a square. Told the squares and the product as well, the same construction is off by −0.0045 there and its interval covers 91.0%. The failure moves up a moment rather than away: with a cube in both, the second-moment balance is off by 0.3099 and leaves the cube twice as far apart as no weighting. And where overlap is thin, 37.0% of samples have no such weights at all.

weights · Weighting The bias a tolerance on the second moments leaves where a square drives both the assignment and the outcome. 600 samples a point. Bias at δ = 0, 0.02, 0.05, 0.1, 0.2, 0.5: −0.0074, 0.0500, 0.1353, 0.2782, 0.5473, 0.6983; coverage 89.7%, 89.5%, 72.4%, 26.5%, 1.0%, 1.0%. The tolerance spent in full at the outcome's slope on the square would give 2·δ·√2: 0.0566, 0.1414, 0.2828, 0.5657 at the four smaller values; balancing the means alone gives 0.6973.

A balance allowed a tolerance

Weights that balance the means exactly and the second moments only to within a tenth of a standard deviation were supposed to keep the second-moment repair and recover the samples exact balance loses. Where a square drives both the assignment and the outcome they keep neither half of the repair: the bias is 0.2782 against −0.0074 with exact balance, and the interval covers 26.5%, because the weights spend the whole tolerance — every second-moment gap sits at the edge of its band — and the outcome converts a gap of a tenth into bias at its own slope, 2·δ·√2 = 0.283. At thin overlap the same tolerance does recover samples: 89.0% have weights at all, against 62.0% with exact balance.

weights · Weighting

Named alongside it

The objects these essays reach for when they reach for this one.

Closed formInteractionBasis functionsImbalanceRandomisation testRerandomisationExperimental designProjectionOrthogonalityReference distributionAssignment mechanismExact enumeration

All concepts