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 2 essays across 2 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

Named alongside it

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

BlockingMonte CarloRandomisationAcceptance rateAllocationBasisCentral limit theoremClosed formCombinatorial searchContinuous covariateCovariate-adaptive randomisationCovariate balance

All concepts