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

Orthogonality — where it appears

Zero correlation between two functions, which for a balancing rule means one is exactly invisible while balancing the other. For two independent covariates every main effect is orthogonal to every interaction, so the invisibility is exact rather than a matter of degree.

Named by 28 essays across 14 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 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 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 How much of one search the other has already found. Five pairs of searches on one sample, on a scale whose zero and one are both fixed by construction. Zero is two searches over disjoint sets of independent columns: they remove shares of the residual sum that add, at 0.8 standard errors from exactly additive, and they read 0.004. One is a break search paired with a step column it contains, which reads exactly one on every draw because the step adds nothing at all. Between them: two dictionaries of step columns cut a few rows apart read 0.125, and the pair the earlier field measured — a break and a whitening window, both reading the same residual series — reads 0.762, three quarters of the way to one search containing the other. And below zero, a break paired with a search over independent columns reads -0.306: the joint search finds configurations neither half of it contains, so charging the two separately under-charges.

Two searches that share nothing

Two searches over independent columns remove shares of the residual sum that add exactly. On the scale a chi-square point is quoted on they look super-additive by a fifth of a unit, and none of it is overlap.

apart · Criterion 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 What the second search finds, alone and afterwards. For four of the pairs, what the second search removes on its own and what it removes once the first has already run. The gap between the two is the overlap in absolute terms. Where the searches share nothing the two readings are the same: an independent column removes 0.0261 alone and 0.0260 afterwards. Where one contains the other they are 0.1387 and exactly zero. The pair the earlier field measured sits between: a whitening window removes 0.5033 alone and 0.3120 after a break search has run. This is the earlier field's own reading of its pair, on the share scale rather than in log-likelihood units, and it is the number a rule that runs both searches actually has to charge for.

A search that is already the other

A break search shifts every coefficient after a row, so a step column is one of the directions it can move in. Paired with a dictionary of them it reads exactly one, on every draw, and that fixes the top of the scale.

apart · Criterion 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 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 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 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 Both halves grow; the difference does not. The control pair's two components and their difference, against how much each of its two searches can find, over 1200 draws at each dictionary size. Two disjoint sets of independent columns are additive at every size — the excess stays inside a standard error or two of zero throughout — and it is not because there is nothing there. The overlap grows from 0.000112 at two columns to 0.000870 at ten, a factor of 7.76, and the interaction grows with it, staying within a factor of two of the overlap at every size. Two searches competing for one residual sum share ground and find configurations neither has alone, in almost equal measure, and their difference is what the earlier field's scale calls zero.

What a zero is made of

Two disjoint dictionaries of independent columns read an excess of 0.000116 and are made of an overlap of 0.000583 and an interaction of 0.000467. The control the whole scale is anchored on reads zero because two effects cancel.

separate · Break point 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 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 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 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 Two arms leave one degree of freedom per block unaccounted for. Each point is one run. The one-mean field's identity is (b − 1) + (N − b) = N − 1, and every schedule moves along that line rather than off it. Two arms give the rule N − 2b and the interval b − 1, which come to N − b − 1 — short of the N − 2 two arms leave by exactly one per block, since a block's arm counts absorb one degree of freedom each and only one of the two directions carries the difference. The hollow points add what the block sums are worth, b − 1 more, and land on the total. The missing degrees of freedom are not lost; they are in a place the interval has to be shown it may read.

The degrees of freedom in the sums

One arm partitions N − 1 exactly. Two arms give the rule N − 2b and the interval b − 1, which is short by one per block — and the missing ones are in the block sums, which are correlated with the differences at −0.79 and are usable anyway.

contrast · Blocking 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 The ladder is the same ladder under every law. Each pair's overlap under each of the four laws, over 400 draws apiece. The scale is fixed at both ends by construction: two searches over disjoint sets of independent columns read -0.001, 0.013, -0.002, 0.004, and a break search paired with a step column it already contains reads exactly one under every law. Between them the pair that reads one residual series twice runs 0.763, 0.805, 0.577, 0.752 — lowest under long memory, where a whitening has most to do and the break search has least left to find that the whitening has not taken. The rung that moves most is the pair of step dictionaries, from -0.148 under a moving average to 0.419 under a break; and the pair that is negative is negative under all four.

Three quarters of the way to one search

The pair that started this reads 0.762 on a scale whose one is containment. And the pair that shares nothing but its response reads −0.306, so the sign the earlier field found does not transport at all.

apart · Criterion 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 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 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 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 What one lost run costs a 16-run factorial fitting 11 coefficients. Every run is worth the same: dropping any one multiplies every coefficient's variance by 1.2000, which is 1 + 1/(N − p) with N = 16 and p = 11, and gives every pair of coefficients a correlation of 0.1667 where the complete design had exactly zero.

The run that did not happen

Lose one run from any orthogonal design and every coefficient's variance is multiplied by exactly 1 + 1/(N − p), and every pair of coefficients acquires a correlation of exactly 1/(N − p + 1) where there was none. The price is set by the design's spare capacity and by nothing else, and a saturated design cannot survive it at all.

design · Factorial

Named alongside it

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

Covariate balanceBasis functionsProjectionClosed formInteractionExperimental designHermite polynomialsCorrelationCovariate adjustmentThresholdRerandomisationVariance explained

All concepts