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.

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).

400 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 0.8% 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 5.5%. Both re-randomised versions are at their nominal level by construction, whatever statistic goes into them.

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.