20 adaptive trials, 45% against 25%

Each line is one trial allocating patients one at a time by the arm's own posterior. The average final share on the better arm is 84.7%, with a standard deviation of 10.3 points across these 20 trials. The rule does not deliver a fixed advantage: it delivers one that depends on how the first few patients came out.

Both arms have a success rate of 30% and it rises by the same amount over the course of the trial, so there is no difference anywhere to find. A fair coin finds none: 5.4% at no drift and 5.6% at 0.5, the nominal rate throughout, and that is what randomisation buys. The adaptive rule starts at 8.4% with no drift at all — the allocation is a function of the outcomes, so the two are not independent — and reaches 75% at a drift of 0.5, because the early patients went to whichever arm looked good early and the late ones went to the other. Comparing within 8 blocks of arrival time removes the part the drift caused, bringing 75% down to 1.3%, and leaves the part the adaptation caused: 7.6% at no drift.

8,000 trials in which every arm has an effect of exactly 0. The arm carried forward is reported by its first stage at +0.187 above the truth — it was chosen for being ahead — and by its second stage at -0.0032, which is unbiased by construction because the selection could not see it. The combination that gets published is at 0.123, exactly the share of the first stage's bias the first stage contributes. Root mean squared error: 0.240, 0.261, 0.182 — the unbiased estimate is the least accurate of the three.

8,000 trials with no effect in any arm. Stage one runs 8 arms at 60 each, the best is carried forward, and stage two adds 30 more to it and to the control. The histogram is where the final statistic lands and the curve is the standard normal it is being read against — shifted right, because the arm was chosen for being ahead. 10.5% of these trials clear 1.96 against a claimed 5%, and the value that actually holds the rate for this design is 2.313.

The same 60 observations, estimated two ways. Keeping the arms separate gives 0.995, which is σ. Pooling them without separating the arms — the price of staying blind to the comparison — gives 1.114, against the identity √(1 + Δ²/4σ²) = 1.118. The sample size is proportional to the variance, so a blinded design at this effect asks for 25% more units than it needs, and it does so systematically rather than by chance.