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

Admissible set — where it appears

The set of assignments a randomisation rule could have produced: every allocation of units to arms that passes its balance or restriction criterion. A randomisation test's reference distribution is drawn from it, so its shape — one connected block or several components — decides what the test can detect.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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 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

Named alongside it

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

Covariate balanceEnumerationLeverageMarkov chainRandomisation testFalse-alarmOrthogonal complementProjectionSeparationBasis functionsConnectivityMixing

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