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

Basis — where it appears

The list of functions of a covariate a balancing rule is told to make even between the arms. What it can remove of an outcome shape depends only on the span of that list, though a per-function tolerance depends on the list itself.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

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 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 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 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 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 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 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 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

Named alongside it

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

Covariate balanceClosed formOrthogonalityProjectionCorrelationInteractionRerandomisationExperimental designHermite polynomialsBasis functionsImbalanceRandomisation

All concepts