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

Balancing on what was recorded first

The adaptive field's rules read outcomes and every one of them broke something. These read baseline covariates, and each holds one imbalance flat while the others grow behind it exactly as a coin's do. What follows is the opposite of the outcome-adaptive story at every step: the unadjusted analysis is too cautious rather than too eager, and the exact test that cost nineteen points of power there costs almost nothing here.
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.

 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.

 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.

 The allocations this trial could have made, and the ones it could not. One 120-patient trial allocated by minimisation at p = 1, re-randomised 399 times. No outcome is redrawn anywhere in this figure: each re-randomisation runs the rule again over the same patients in the same order with the same recorded factors, so what is drawn is the set of experiments that could have happened. The bars are that set; the outline is what shuffling the labels gives, which is the reference distribution of a coin and is what every off-the-shelf permutation routine assumes. The coin's is wider — its 5% point is 1.95 against the rule's 1.09 — because a coin's allocations are less balanced and a less balanced allocation gives a larger statistic. Reading this trial against it makes the test conservative rather than anti-conservative, which is the opposite error from the outcome-adaptive case and for the same structural reason.

The reference the covariates supply

Hold the outcomes fixed, re-run the rule that assigned them, count. The same construction cost nineteen points of power in the adaptive field, because its rule chased outcomes and its critical value depended on a rate nobody has. Here the rule reads only what was recorded before anything happened, and the same unadjusted statistic goes from 20.3% power to 55.0% by being read against the right distribution.

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