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

Orthogonal complement — where it appears

Every vector at right angles to a given subspace: what is left of a space once that subspace is taken out. Projecting onto it removes exactly the part of a vector the subspace can already explain, and keeps the rest.

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

Also named here as separation — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

Admissible setCovariate balanceEnumerationLeverageMarkov chainProjectionRandomisation testSeparationBasis functionsFalse-alarm

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