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

Contrast — where it appears

A weighted combination of observations whose weights sum to zero, so that it says nothing about the overall level. Contrasts and means are independent for normal data, which is what lets a stopping rule read one while the interval reads the other.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The construction survives a difference of two weighted means. Coverage of δ̂ ± t√(S_D²/H) on b − 1 degrees of freedom, over 900 runs at a requirement of 0.3, where δ̂ is the block differences weighted by h_b = (1/m_A + 1/m_B)⁻¹ and H is their total. The theorem the one-mean field rests on goes through with h_b in place of the block size, and the reason is that the weights a weighted least squares decomposition needs are the inverse variances — which is exactly what h_b is. The stopping rule reads only within-arm within-block contrasts, so it is a function of nothing the interval reports, whatever it does with the block sizes. Each bar is within 2.9% of the level it claims.

A width promised for a difference

The exact fixed-width interval was built for one mean. Two arms make the target 42.7 units of effective size and each unit costs four observations, so the same promise about a difference costs 169.4 rather than 42.7 — and the theorem survives untouched with the harmonic size in place of the block size.

contrast · Width 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 Three sets of weights, five designs, and no estimator that is exact everywhere. Coverage of the same interval under three weightings. h_b is the inverse variance when the arms share a variance or the allocation is constant; equal weights are right when every block has the same two counts; the estimated precision weights are right in the limit and exact nowhere, because the decomposition needs the weights to be the constants they are only estimating. In the corner — two variances, changing sizes, changing allocation — the two exact estimators are the ones that miss, at 98.45% and 95.65%, and the one with no theorem behind it is at 95.05%. That is the whole statement: there is an exact estimator under either condition, and none under both.

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.

contrast · Nuisance Two arms leave one degree of freedom per block unaccounted for. Each point is one run. The one-mean field's identity is (b − 1) + (N − b) = N − 1, and every schedule moves along that line rather than off it. Two arms give the rule N − 2b and the interval b − 1, which come to N − b − 1 — short of the N − 2 two arms leave by exactly one per block, since a block's arm counts absorb one degree of freedom each and only one of the two directions carries the difference. The hollow points add what the block sums are worth, b − 1 more, and land on the total. The missing degrees of freedom are not lost; they are in a place the interval has to be shown it may read.

The degrees of freedom in the sums

One arm partitions N − 1 exactly. Two arms give the rule N − 2b and the interval b − 1, which is short by one per block — and the missing ones are in the block sums, which are correlated with the differences at −0.79 and are usable anyway.

contrast · Blocking Free until the sums stop seeing what the differences see. Coverage with and without the block sums pooled into the interval's variance estimate. With one effect and one level they are free. With an effect that varies between blocks they are still free, because a block's sum picks that variation up exactly as its difference does. With a level that varies they make the interval 37% wider and conservative. And where the effect falls as the level rises — a ceiling, and not an exotic thing to suppose — the sums carry none of the between-block variation while the differences carry all of it, the pooled estimate is short, and the interval that uses it covers 88.75% on a width 20% narrower than the honest one.

What a two-arm rule may not pool

A spread computed "within the block" without the arm label carries a share of the effect, so the trial runs 173 observations at a null and 282 at an effect of 1.5. The stopping rule is reading the thing it exists to measure, and the phrase that produced it is one word long.

contrast · Allocation

Named alongside it

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

CoverageDegrees of freedomFixed-width intervalBlindingTwo-sampleBlockConfidence intervalAllocationEffective sample sizeHarmonic meanCalibrationChi squared

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