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

Inverse variance weighting — where it appears

Weighting each observation or block by one over its variance. It minimises the variance of the weighted mean and makes a weighted least squares decomposition exact, and for a difference of two arms the weights depend on the ratio of the two arm variances and on nothing else.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

The stopping rule costs more than the weighting does. Coverage over 2000 runs of a trial whose variance ratio drifts by a factor of twenty, at three ways of deciding when to stop. Twelve blocks fixed in advance is the top line and reproduces what a trial of fixed length delivers. Stopping when the reported interval is short enough is the bottom line, and it costs between 3.0% and 5.5% of coverage — including for the rule that is told every block's true ratio, which is what says the shortfall belongs to the stopping and not to the weights. Stopping on a width predicted from the within-arm sums of squares is the middle line, and it is back at the fixed-length values. The standard error on each point is 0.49%.

A width the trial has to stop for

The weighting that covers at 94.9% on twelve blocks covers at 91.5% when the trial stops as soon as its interval is short enough — and so does the rule that is told every block's true variance ratio. The shortfall is the stopping, not the weights.

stop · Stopping 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.

corner · Nuisance How wrong the ratio is allowed to be. λ enters only through the weights, so misstating it leaves the estimate unbiased and moves two things — the interval's calibration and its efficiency — both of which are closed forms of the design. Coverage stays at its level over a factor of two in either direction (94.93% at half the truth, 94.27% at twice it) and starts to go at a factor of five. An estimate on hundreds of within-arm degrees of freedom is never wrong by anything like that, which is what makes the feasible rule usable rather than merely definable.

Blinded, and still exact

The one number the exact interval needs is a ratio of within-arm spreads, which is a contrast and contains no mean — so a rule forbidden to look at the effect may compute it, on more degrees of freedom than the interval itself has.

corner · Width Two promises, and no rule here keeps both. A fixed-width procedure promises two things: that the interval covers at its nominal rate, and that it is no wider than the width asked for. Over 1500 runs of the modelled weighting, a rule that stops when the interval it will report is short enough keeps the width — only 2.0% of runs come out wider than 0.34 — and covers at 91.13% against a nominal 95%. A rule that stops on a width predicted from the within-arm sums of squares covers at 94.80% and comes out wider than promised on 42.3% of runs. The two promises are in conflict because keeping the second one exactly requires conditioning on the very quantity that has to be independent of the stopping time for the first.

Stopping on the arms

The width a trial will report is predictable from quantities the interval is not about. A rule that stops on the prediction covers at 94.5% where one that stops on the interval covers at 91.5, and it costs two blocks and half of the width promise.

stop · Width The weights may not read the block they weight. A weighted least squares decomposition needs weights that are constants, or at least independent of the differences they multiply. One λ̂ pooled across the trial is estimated on hundreds of degrees of freedom and is effectively a constant; a λ̂ estimated inside each block is estimated on that block's own two or three, and is correlated with the difference it weights. Coverage falls from 94.68% to 82.76% — and the interval gets wider while doing it, 0.5163 against 0.3024, which is the signature of weights that are noise.

The condition that cannot be dropped

The weights may not read the block they weight. Estimate the variance ratio inside each block rather than across the trial and the coverage falls to 83% — on an interval that is at the same time seventy per cent wider.

corner · Allocation The fixed-width trial's coverage when the outcomes are not normal, for both stopping rules. normal: stopping on the arms 94.05% after 18.1 blocks, on the report 89.95%; log-normal, skewness 0.95: stopping on the arms 94.70% after 18.5 blocks, on the report 90.80%; log-normal, skewness 2.26: stopping on the arms 94.15% after 19.3 blocks, on the report 90.25%; log-normal, skewness 4.75: stopping on the arms 94.45% after 18.7 blocks, on the report 90.50%; t, five degrees of freedom: stopping on the arms 94.35% after 18.3 blocks, on the report 90.30%; skewness 4.75, arm A only: stopping on the arms 93.80% after 26.0 blocks, on the report 89.90%; skewness 4.75, arm B only: stopping on the arms 93.60% after 14.2 blocks, on the report 89.95%; equal variances, normal: stopping on the arms 94.75% after 11.4 blocks, on the report 90.90%; equal variances, skewness 4.75: stopping on the arms 94.05% after 11.1 blocks, on the report 92.00%.

A width rule on skewed outcomes

The blinded fixed-width rule rests on a within-arm spread being independent of the arm means, which only normal samples guarantee. On outcomes with a skewness of 4.75 the independence fails and the overall coverage barely notices — 93.60% to 94.70% across every shape counted, against 94.05% on normal outcomes. What skew moves is the runs that stop by twelve blocks, which cover about 90% with the skew in one arm, and the trial's length: a variance ratio corrected on normal theory lengthens it from 18.1 blocks to 26.0 with the skew in the first arm and shortens it to 14.2 with the skew in the second.

stop · Width One pooled ratio, two periods. The calibration factor of weights built from one variance ratio pooled across the trial — the factor by which the interval's own scale estimate overstates the estimate's variance — when arm B's ratio is F times higher in one kind of period than the other, closed form on the corner's block compositions. With the small A-heavy blocks the noisier, the factor rises: 1.00 at F = 1, 1.50 at F = 1.5, 2.07 at F = 2, 3.12 at F = 3, 3.91 at F = 4, 4.86 at F = 6, 5.44 at F = 8. With the large B-heavy blocks the noisier, it falls: 1.00 at F = 1, 0.74 at F = 1.5, 0.65 at F = 2, 0.58 at F = 3, 0.55 at F = 4, 0.53 at F = 6, 0.53 at F = 8. A ratio estimated per period gives weights that are the inverse variances again, and its factor is exactly 1 at every F.

A ratio that changes with the period

The corner's exact interval needs one variance ratio, and the corner's own alternating allocation makes two kinds of period that need not share one. When they differ the pooled ratio is precise and wrong: at a fourfold difference the interval covers 100.0% one way round and 85.1% the other. A ratio estimated per period is exact again, blind to the effect, and costs under a tenth of the width — and a ratio that merely wanders at random costs width and no coverage at all.

corner · Nuisance A lean, its reflection, and none. The mean error of each weighting's estimated difference at a fixed eighteen blocks, 2,000 trials a shape, with two standard errors. normal: the modelled ratio, every block 0.0031, the modelled ratio, each block left out 0.0028, the modelled ratio, cross-fitted halves 0.0023, the true ratios 0.0037; skewness 0.95: the modelled ratio, every block 0.0092, the modelled ratio, each block left out -0.0134, the modelled ratio, cross-fitted halves -0.0014, the true ratios 0.0002; skewness 2.26: the modelled ratio, every block 0.0166, the modelled ratio, each block left out -0.0298, the modelled ratio, cross-fitted halves -0.0042, the true ratios -0.0029; skewness 4.75: the modelled ratio, every block 0.0255, the modelled ratio, each block left out -0.0455, the modelled ratio, cross-fitted halves -0.0055, the true ratios -0.0051.

A weighting that does not read its own spreads

On skewed outcomes a block's spreads move with its means, so a weighting fitted to every block's spreads leans: +0.0255 at a skewness of 4.75. The obvious repair — fit each block's weight with that block left out — does not remove the lean. It reverses it and nearly doubles it, to −0.0455, because a weighted mean divides by the sum of the weights and every other block's weight still reads the one left out. Splitting the trial into two halves and weighting each by the other's fit does remove it: −0.0055, indistinguishable from the weighting told the true ratios, for a spread about a tenth wider.

stop · Width What each stopping rule spends. The average number of observations the corner design takes before it stops, with the stopping rule reading pooled within-arm spreads (the upper bar) and each kind of period's own spreads (the lower), at a promised half-width of 0.3, 3,000 trials each. one ratio: 1,656 and 1,649; 2-fold, small blocks noisier: 1,020 and 1,015; 2-fold, large blocks noisier: 2,976 and 2,614; 4-fold, small blocks noisier: 751 and 669; 4-fold, large blocks noisier: 5,642 and 3,368; 8-fold, small blocks noisier: 711 and 490; 8-fold, large blocks noisier: 10,989 and 3,019. With one ratio the two rules are the same rule. With the large blocks noisier the pooled rule over-spends by a factor that grows with the difference, to 3.6 times at eightfold; with the small blocks noisier it over-spends less.

A stopping rule that reads the periods

A fixed-width trial in the corner stops when its estimated precision reaches the promise, and it estimated that precision from spreads pooled across a trial whose two kinds of period need not share a variance ratio. Read each kind of period's own spreads instead and the trial stops on 3,019 observations where it took 10,989, at an eightfold difference, with the interval it reports still covering 95.1%. The stopping time reads nothing but within-arm spreads, per period now as well as pooled, and that is all the exactness needed.

corner · Nuisance

Named alongside it

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

CoverageFixed-width intervalVariance ratioBlindingEfficiencyWeighted least squaresDegrees of freedomEstimated varianceInterval widthBlockCalibrationStopping rule

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