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

Mahalanobis distance — where it appears

A distance that measures a vector against its own covariance rather than in raw units, so it does not depend on which basis the vector was written in. That invariance is why a rerandomisation rule stated as an ellipsoid accepts the same assignments under any basis for the same span, where a per-function tolerance box does not.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

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 balancing several numbers at once costs each of them. The criterion generalises without a word changing — the covariate imbalance becomes a vector and the correction a quadratic form — so the question is what it is worth rather than whether it can be done. At n = 200 with 200 trials per point, a rule balancing one covariate leaves 12.7% of a coin's imbalance in it; balancing eight leaves 23.2% in each. The assignment has a fixed amount of freedom and every covariate added takes a share of it. The rule degrades rather than failing: at eight covariates it is still four times better balanced than a coin, and the eight are being held simultaneously rather than in turn.

Balancing more than one number

The criterion generalises to several covariates without a word changing, which makes the question what it is worth rather than whether it can be done. Each one added takes a share of the assignment's freedom, and the imbalance left in every one of them rises.

continuous · Blocking 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 point that decides a coefficient, on two covariates correlated at 0.9. A hundred points on two normal covariates with correlation 0.9. Dashed lines mark where the most extreme five per cent of each covariate begin. The point carrying the largest share of the first coefficient's information, 14.2%, is inside the lines on both axes, so neither covariate's plot on its own would show it. The coefficient rests on 26.1 effective points, with a variance inflation of 6.27.

A point ordinary on every axis

In a multiple regression each coefficient rests on its own number of points, n divided by the kurtosis of that covariate's residual after the others, exactly. Correlation between normal covariates inflates a coefficient's variance fivefold and leaves it resting on the same third of the points. What correlation does instead is hide the point that decides the coefficient: at a correlation of 0.9 it stands out on no single covariate's plot in 77.3% of designs of a hundred, and the largest leverage a normal design reaches depends on the number of points and covariates and on nothing else.

spread · Summary 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

Named alongside it

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

Covariate balanceExperimental designRandomisationBasisBasis functionsBlockingExact enumerationMonte CarloReference distributionRerandomisationAcceptance rateAllocation

All concepts