Nuisance — the series · Normal Distribution

The cheap repair needs a number nobody has. The obvious alternative to re-randomising is to simulate the design under its null once and use the critical value that comes out — which is what the arm-dropping design does, where the critical value has to be solved for and is 2.313. It does not transfer here. The rule chases outcomes, so how imbalanced the allocation gets depends on how often anything succeeds, and the critical value moves from 1.668 at a success rate of 0.05 to 2.718 at 0.8. Calibrated at 0.3 and used at 0.8 the test's real size is 12.4%; used at 0.05 it is 0.12%. The randomisation test needs none of this, because it conditions on the outcomes that happened rather than on a rate they were supposed to come from.

What the exactness buys

Against a z test calibrated to reject exactly 5% of true nulls on this design, the randomisation test loses nineteen points of power. What it buys is that the calibration needs the success rate — which moves the critical value from 1.668 to 2.718 and is the quantity the trial was run to find out.

part 1 · exact

What the exactness costs, and the dial it is bought with. The median half-width of the interval each rule reports, at a requirement of 0.4 and a first look after 5 observations. The flat line is the interval a practitioner writes at the purely sequential rule's stopping time, which covers 91.72% rather than 95%. The curve is the blinded rule, which covers its nominal level at every block size: it reads b − 1 degrees of freedom where the other reads n − 1, and pays for the exactness in width. The best block size is 3, at 0.4712. Larger blocks give the stopping rule a better estimate and the interval a worse one, and the two costs go opposite ways, which is what puts the minimum in the middle.

What the blindfold costs

The exactly-covering rule pays for it in the width of the interval, and the block size is a dial between two costs that run in opposite directions. And on an interval whose width was fixed in advance, the same repair buys nothing at all.

part 2 · blind

The overshoot is the last block size and nothing else. A run stops at a multiple of its own block sizes and cannot land between them, so it ends past its own target by about half a block. Fixed sizes overshoot by 1.5, 2.2, 3.1, 4.7, 8.5 observations as the size goes 2, 3, 5, 8, 16. Every schedule here ends in blocks of two and every one of them lands where blocks of two land — 1.62, 1.32, 1.37 against 1.48 — while having spent most of the run inside blocks four and eight times larger. That is the one thing on this page a schedule genuinely takes from both ends.

What a schedule actually buys

Big blocks early and small blocks late is the right instinct and it does not take both ends of the trade, because there are not two ends to take. What it does take is the overshoot — about four per cent of the observations — and a steadier stopping point.

part 3 · pace

Which weights are the inverse variances

There is an exact estimator when the two arms share a variance and another when every block has the same two counts, and between them they cover every trial anybody designs on purpose. In the corner where neither holds, both cover 98.45% instead of 95%, and the only estimator at its level is the one with no theorem behind it.

part 4 · contrast

Exact in the corner, where nothing was. Coverage of a nominal 95% interval on five designs, at a required half-width of 0.3. The first four are the two-arm field's own and the fifth is its corner — two variances, block sizes that swing by eight, and an allocation that alternates between five to one and one to five — where neither of that field's two conditions holds. The effective-size weights over-cover there at 98.40%; the weights h_b(λ) = (1/m_A + λ/m_B)⁻¹ cover at 94.84%, and at 94.84% when λ is estimated from the within-arm contrasts rather than known. Nothing here is supposed to move.

Weights that need only a ratio

A fixed-width interval about a difference is exact under either of two conditions and under neither in the corner. It is exact there too, and the only thing it needs is how much larger one arm's variance is than the other's.

part 5 · corner

A wrong weight costs width; a random weight costs level. Five weightings on a trial whose variance ratio drifts by a factor of 20.1 between the first block and the last, over 4000 runs. The rule that knows every λ_b covers at 95.1% and sets the width. One ratio for the whole trial is wrong for every block and costs nothing in level — 94.8% — while being 20% wider; equal weights are calibrated by an identity and 22% wider. The ratio estimated inside each block is the only rule aimed at the quantity that actually varies, and it is the only one that misses the level, at 92.0%: a weight computed from a handful of degrees of freedom is mostly noise, and noise in a weight is not a wrong weight. Modelling the drift across blocks recovers the oracle's width at 94.8%.

A ratio that changes between blocks

A wrong weight costs width and a random weight costs level. The rule aimed at the quantity that actually varies is the only one that misses its own coverage, and the rule that models it across blocks recovers the whole of what knowing it is worth.

part 6 · blocks

An exact test rejecting a true hypothesis a fifth of the time. How often each analysis reports an effect when the average treatment effect is exactly zero and the effect varies between units, at 150 units with 25% treated. The permutation test on the difference in means reads 4.20% where the effect is constant — where the two nulls coincide and its exactness applies — and 22.93% where the effect varies with a standard deviation of 3. The same test on the studentised difference reads 6.27% there, and the ordinary large-sample t, which makes no exactness claim at all, reads 6.60%.

The null the exactness is for

A permutation test is exact under the hypothesis that the treatment changed nothing for anybody. Under the hypothesis it changed nothing on average, with a quarter of the units treated and the effect varying between them, it rejects a true null 22.93% of the time.

part 7 · exact

One statistic that is right under both hypotheses. Rejection rates for both statistics under both nulls, at 25% of 150 units treated, with the weak-null readings taken at an effect spread of 3. The difference in means is exact under the sharp null and rejects 22.93% of true weak nulls. The studentised difference is exact under the sharp null — 4.07% — and reads 6.27% under the weak one. The repair is a change of statistic inside the same construction: the same re-randomisations, the same fixed outcomes, a different number compared across them.

A statistic that is exact twice

Dividing the difference in means by its own separate-variance standard error before permuting takes the rejection rate under a true weak null from 20.47% to 6.07%, keeps the exactness under the sharp null at 4.07%, and costs 0.8 points of power against a real effect. At an even split it changes nothing at all, in every draw.

part 8 · exact