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

Allocation rule — where it appears

The mechanism that decides which arm each unit goes to, which is part of the result rather than an administrative detail. It fixes what a randomisation test may condition on, and two rules that produce identical-looking arms can support quite different inferences from them.

Named by 12 essays across 5 fields — each of them below, with the objects they name alongside it.

The six best bases of 2 functions, and what each protects. Every cell is R²(g | span B) — the share of the imbalance in that shape a rule balancing that basis removes — computed from exact inner products between Hermite functions and indicators, with nothing simulated. The rows are ordered by their worst cell, which is the number an experimenter who does not know the shape is exposed to. The best row here guarantees 26.8% against every shape in the list, and the worst of the six guarantees 15.1%: the difference between them is entirely which subspace was picked, at the same cost per arrival.

A basis is a subspace

A balancing rule cannot tell one basis from another with the same span, so choosing what to hand it is choosing a subspace — and then what it removes of any outcome shape is a projection, computable exactly, with no trial anywhere in it.

basis · Criterion Four rules, three shapes, and no ordering that survives. The variance of the unadjusted treatment estimate under each rule, as a fraction of the variance a coin gives, over 450 trials of 200 units each. Against a covariate that enters linearly the rule that reads the number nearly halves it. Against a threshold at 1 it removes about a fifth. Against a quadratic every rule here is at or worse than a coin — they are all optimising a criterion that is one over the variance of an estimate in a model this outcome does not obey, and a constraint that helps nothing still costs something. Nothing in a trial says which column it is in.

Balanced on the wrong function

A rule that reads a covariate's numbers halves the variance of the treatment estimate, if the covariate enters the outcome as a straight line. If it enters as a threshold the rule is worth a fifth of that, and if it enters as a curve every rule here is worse than a coin.

shape · Blocking A threshold in the tail is a threshold nothing balances. The share of a coin's imbalance in an indicator 1{x > c} that survives a rule which balances the covariate itself. The smooth curve is 1 − ρ² with ρ = φ(c)/√(p(1−p)), a closed form with no trial in it; the points are counted over 500 trials of 200 units at each threshold. At the median the two agree that about a third survives — the removed share is exactly 2/π — and by two standard deviations 86.9% survives. The closed form is exact in the limit and optimistic by a few points at this many units, because the rule balances the sample's mean rather than the population's.

A threshold in the tail

How much of a threshold's imbalance a balanced covariate removes is a correlation, and the correlation is a closed form. At the median it is exactly 2/π — the same 2/π a median split throws away — and two standard deviations out it is an eighth.

shape · Blocking The guarantee, as the basis is allowed more functions. The lower line is the best worst case over the six named shapes for a basis of each size, found by scoring every subset of the dictionary — an exact answer, since the problem is finite. One function guarantees 2.3%, which is nearly nothing; three guarantee 59.0% and the basis that does it is the covariate, its square and its cube, with no indicator in it. The upper line is the same problem with the basis drawn rather than fixed, which is worth 2.09 times as much at two functions and 1.32 at three. The two lines converge because a basis large enough to protect everything has nothing left to randomise over.

Which shapes are worth protecting

Choosing a basis by its worst case is a finite problem with an exact answer. The answer has no tie in it, which a maximin optimum is supposed to have — and the tie comes back, along with twice the guarantee, when the basis is drawn rather than chosen.

basis · Blocking What a rule reads, against what the outcome uses. The variance of the treatment estimate relative to a coin's, for four things a rule might balance against three shapes the outcome might have, over 350 trials of 200 units. The diagonal is the easy part — a rule that reads the function the outcome uses removes about half the variance. What the table is for is the off-diagonal: reading the covariate alone is worth nothing against a quadratic (0.755), and reading all three is worth nearly as much against every shape as the matching rule is against its own (0.532, 0.493, 0.514).

Three functions of one number

A rule that balances the covariate is exposed to every shape the outcome might have. A rule that balances three functions of it costs two points of variance against the shape the first was built for and takes the worst case from a coin's to about half of it.

shape · Criterion What each analysis does at a true null, by shape. Four analyses of the same trials — 500 of them at each shape, 120 units, assigned by the rule that reads the covariate. Every rejection is false. The unadjusted analysis is the one that moves: 1.60% against a linear outcome, where the design removed a great deal that the standard error still prices, and 5.20% against a quadratic, where it removed nothing and the standard error is right. Adjusting holds the level in all three columns, and so does the design's own reference distribution, which needs to be told the rule and nothing else.

The analysis and the shape

An unadjusted analysis after a rule that read the covariate is too cautious — by a third against a linear outcome, by nothing at all against a quadratic. And an adjustment for the wrong function recovers almost none of the precision the right one would.

shape · Randomisation Where the constraints exhaust the randomisation. At 16 units there are 12,870 equal splits, so the ones meeting a stated tolerance can be counted rather than estimated. With each of the first k standardised imbalances required to be within 0.4 of a coin's own spread, the admissible count runs 3874 → 1006 → 314 → 0 → 0 → 0 — and at 4 functions there is no admissible assignment at all. The count is the number of distinct answers a randomisation test can give: at 3 functions its finest attainable p-value is 1 in 314. Balance improves with every constraint and the reference distribution shrinks with it, and the two run out at different rates.

When the constraints run out

Every function added to a basis is a constraint the assignment has to satisfy with the same units. At sixteen units and a stated tolerance the admissible assignments run 3,874, then 1,006, then 314, then none — and the count is exact, because the assignment space is finite.

basis · Allocation What the guess is worth, when it is worth anything. The variance cost of an even split relative to the variance-minimising one for a risk difference, against the first arm's proportion, with the second at 0.3. The cost is a pure number: it does not depend on the trial's size. It is exactly zero at 0.3 and at 0.70, where the two arms have the same p(1 − p); it is 0.19% at a half and 4.36% at a tenth. Across the whole range from a tenth to nine tenths it never exceeds 4.36%, which is what the variance-minimising rule is worth here — and what it is worth is the reason it is safe to use with a guess.

The arm whose variance is its answer

With a binary outcome the allocation rule is a function of the proportions the trial exists to estimate. It costs at most 4.36% of variance to ignore it anywhere between a tenth and nine tenths, because √(p(1−p)) stays within a factor of two of its peak across 98% of the unit interval.

allocation · Allocation The worst case against the number of functions, at four trial sizes. The largest share of a coin's imbalance the rule leaves in any of the four shapes, against how many functions it reads, at 50, 100, 200, 400 units. At 50 units: 103.5%, 38.6%, 37.2%, 37.0%, 39.8%, 41.7%. At 100 units: 105.9%, 37.8%, 38.9%, 30.3%, 27.0%, 24.6%. At 200 units: 95.9%, 44.2%, 42.8%, 32.9%, 28.4%, 11.7%. At 400 units: 108.5%, 41.8%, 41.3%, 32.1%, 26.1%, 5.8%. At fifty units every basis from two functions up leaves the worst shape between 37.0% and 41.7%; at four hundred, six functions leave it 5.8%.

How many functions a trial can hold

A rule that reads more functions of one covariate protects more shapes, and every function it reads costs the shapes it already held: the imbalance it leaves in a linear outcome falls like one over the trial size, with a constant that runs from about 3 for one function to about 16 for six. At fifty units that cost eats the protection, and the worst case over four shapes sits between 37% and 42% of a coin's from two functions to six. At four hundred it falls from 42% to 6%. By six functions the worst shape, at every size, is one the rule is reading.

shape · Criterion What the rule leaves of the product, against what exact balance would. For a rule holding the mean of each covariate, at a correlation of 0.5, the share of a coin's imbalance left in the product of the covariates: by the closed form for exact balance, and in trials of 100 and 400 units, on four marginals. normal: 100.0% exact, 98.6% at 100, 108.4% at 400; clearly skewed: 73.7% exact, 92.9% at 100, 91.5% at 400; strongly skewed: 70.3% exact, 93.9% at 100, 94.4% at 400; exponential: 71.5% exact, 84.1% at 100, 87.3% at 400.

A share at a hundred units

On a skewed covariate the closed form credits a rule that balances the means with removing 26% of the covariates' product and 52% of a square, shares that rest on symmetry having failed. A trial does not collect them. At a hundred units the rule removes 12% of the product and 37% of the square on the clearly skewed covariate, and at four hundred units barely more, because the shares live in the long tail and the tail is a handful of units. On the most skewed covariate the rule removes a fifth of the median split at a hundred units and nothing of it at four hundred: balancing the mean of such a covariate is balancing its largest few values.

skew · Criterion Three contrasts on one dataset, three different splits. The variance-minimising allocation for each of three ways of reporting the same two-arm comparison, against the first arm's proportion, with the second at 0.1. A risk difference wants the arm with the larger p(1 − p) to get more units; a log odds ratio wants it to get fewer, and the two curves are exact reflections of each other in the half line. A log risk ratio wants something else again. At a first-arm proportion of 0.6 they ask for 62.0%, 21.4% and 38.0% of the units. A trial reporting more than one of them cannot be optimal for either.

Two contrasts, one split

A risk difference wants 62.0% of the units in the first arm, a log risk ratio wants 21.4% and a log odds ratio wants 38.0% — on one dataset, with one pair of proportions. The difference's rule and the odds ratio's are exact reflections of each other, so no split can be near-optimal for both.

allocation · Allocation The guess rate through a trial. Guess rates in each tenth of a trial of 150 patients on three arms with the rule taking its preferred arm with probability 0.8, over 1500 trials. knowing everything: 65.9%, 72.2%, 72.0%, 72.8%, 72.4%, 72.2%, 72.5%, 72.7%, 72.7%, 72.4%. a joint table learned in earlier trials: 53.4%, 55.4%, 54.8%, 55.1%, 54.3%, 53.8%, 54.9%, 54.8%, 54.1%, 54.8%. the centre's totals: 45.4%, 50.3%, 50.2%, 50.8%, 49.8%, 50.2%, 50.9%, 50.6%, 50.4%, 50.7%. following whichever has been right: 46.5%, 50.2%, 49.3%, 49.6%, 48.9%, 48.7%, 49.3%, 49.0%, 49.0%, 48.7%. a joint table learned in this trial: 36.4%, 43.8%, 45.0%, 46.2%, 46.7%, 46.8%, 48.1%, 48.5%, 47.9%, 48.9%. the centre's own factors, scored: 46.8%, 51.4%, 49.7%, 49.9%, 48.4%, 48.6%, 48.5%, 47.7%, 47.6%, 46.7%. Chance is 33.3%.

The guesser who learns

An investigator who watches a three-arm minimisation can tally which of several partial readings has been right so far and follow the leader. It does not help. Following the leader names 48.9% of assignments where the best single reading names 49.9%, and a table of what the three readings say together, learned inside the trial, starts at 36% and has not caught up by a hundred and fifty patients. Learned over earlier trials, that table reaches 54.5%. The full guesser's 71.8% stays out of reach of all of them, because the gap to it is information rather than practice.

multiarm · Assignment

Named alongside it

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

Covariate balanceTreatment effectEfficiencyImbalanceThresholdVariance reductionBasis functionsDesign criterionModel misspecificationClosed formMaximin designMonte Carlo

All concepts