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

Permuted blocks — where it appears

An allocation rule that fills fixed-size blocks with equal numbers of each arm in random order, holding the arm totals level throughout. The block length is a dial between how level the arms stay and how predictable the next assignment is to somebody who has been counting.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

What each rule leaves behind, at 120 patients. Four allocation rules over the same cohorts and the same seeds, each scored on three imbalances: the number of patients in each arm, the worst of the nine factor levels, and the worst of the 24 cells of the cross-classification. No rule holds all three. Permuted blocks hold the totals exactly and leave the margins near a coin's. Blocks inside every cell hold the cells and let the totals drift, because 24 part-filled blocks do not have to end level. Minimisation holds the margins and the totals and is at 83% of a coin's cell imbalance. Each of the three columns is somebody's definition of a balanced trial.

Balancing what is known in advance

Four allocation rules, three definitions of balance, and no rule that holds more than one of them. Minimisation keeps the worst factor margin near three patients whether the trial has forty or six hundred and forty — and lets the imbalance in the cross-classified cells climb to 86% of a coin's, because the cells are not what it is watching.

covadapt · Assignment Where one rule becomes three. Every arrival in 200 simulated trials is put to all three scores, and the picture is how often they would send that patient to different arms. The range and the pairwise sum are the same rule at two arms and at three — for sorted counts the pairwise sum is twice the range, so the arm that minimises one minimises the other — and they part company at four, where the pairwise sum is 3(d − a) + (c − b) and the range still sees only d − a. The variance disagrees with both from two arms onwards, on 5.4% of arrivals at two and 27.0% at five, because the scores are summed over 3 factors and a sum of squares does not order the candidates the way a sum of absolute values does. All three are called minimisation.

Three arms and three scores

Minimisation balances a trial by keeping the arms' counts even inside every prognostic factor. With two arms there is one way to measure how uneven two counts are. With three there are several, they are all called minimisation, and they send different patients to different arms.

multiarm · Assignment A trial designed 2:1:1, and what two scores deliver. 500 trials of 180 patients, three arms, a target of 2:1:1. The shaded bars are a minimisation score that divides each arm's count by the share that arm is supposed to receive before measuring the spread; it delivers 49.9% : 25.1% : 25.1%. The others are the same rule with the counts left raw, which delivers 33.4% : 33.3% : 33.3% — the balance it enforces inside every factor level is equality, and equality is what it gets. The marks are the shares that were asked for.

Balancing towards unequal targets

A three-arm trial allocating two to one to one is the ordinary case, and a balancing rule built from raw counts does not know it. It balances the arms towards equality inside every factor level, delivers a third to each arm, and reports that it minimised imbalance.

multiarm · Allocation What a rule gives away by being predictable. Minimisation run at every probability from a coin to fully deterministic, over 500 cohorts of 120 at each. The upper line is the share of assignments an investigator who knows the rule and the enrolled patients can name in advance: 49.7% at p = 0.5, which is a coin and cannot be beaten, and 87.6% at p = 1 — short of everything only where the two arms tie and the rule falls back on a coin. The lower line is what that is worth: an investigator who enrols a patient 0.5 of a standard deviation better than average whenever they predict their favoured arm produces a treatment effect of 0.75 where the truth is zero. Nothing about the randomisation was broken; the bias entered through who was enrolled, which is the one thing an allocation rule cannot control. The dashed line is the closed form 2δ(2g − 1).

The rule that can be guessed

A balancing rule improves as it becomes more deterministic, and a deterministic rule can be worked out in advance from information the person enrolling the patient already has. At full determinism 87.6% of assignments are guessable, and an investigator who acts on the guess produces a treatment effect of three quarters of a standard deviation where the truth is zero.

covadapt · Assignment Four analyses of the same trials, with no treatment effect at all. 700 trials of 120 patients allocated by minimisation at p = 0.8, with the prognostic factors carrying a real effect on the outcome and no treatment effect — every rejection below is a false one. Two statistics, the plain difference and the same after adjusting for the balanced factors, each read against two reference distributions: a t table, and the set of allocations the rule could have produced from these covariates. The unadjusted comparison rejects 0.6% where it claims 5% — conservative, which is a loss of power rather than an error, and nothing on the output says so. Adjusting puts it back at 5.4%. Both re-randomised versions are at their nominal level by construction, whatever statistic goes into them.

The analysis has to know the rule

A trial balanced by minimisation and analysed by comparing the two arms' means rejects a true null 0.6% of the time where it claims 5%, and at full determinism 0.0%. That is not an error anybody complains about — it is a test that has stopped working, paid for by a balance the analysis then refused to use.

covadapt · Assignment What a guesser gets, and what a guesser gets for nothing. 600 trials of 150 patients under a fully deterministic rule, with an investigator who knows the rule, the factors and every assignment so far. The guess rate falls with the number of arms — 87.8% at 2, 86.2% at 3, 81.0% at 4 — which reads like a trial getting safer and is not: what a guesser can trade on is the excess over the 50%, 33%, 25% they would get by naming an arm at random, and that goes the other way, from 1.76× chance at two arms to 3.24× at 4. The gap a guesser manufactures between the best and worst arm under a true null is 0.759, 0.733, 0.711 standard deviations — nearly unchanged.

Guessing one arm in three

A balancing rule is guessable because it is balancing. With three arms the next assignment is worked out less often than with two — and by more, relative to what a guesser gets for nothing, and the damage they can do is almost unchanged.

multiarm · Assignment

Named alongside it

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

Covariate-adaptive randomisationMinimisationStudy designCovariate balanceRandomisationAllocationExperimental designMonte CarloPredictabilityStratified randomisationBlindingConfounding

All concepts