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

Calibration — where it appears

Whether an interval's own estimate of its own width is right on average. For fixed weights it is a closed factor in the design and the block variances, equal to one in exactly two cases — weights proportional to the inverse variances, and weights that are all equal.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 charges, and only one of them is a test. What each of three thresholds does to the same decision, under AR(1) at 0.8, against the size of a genuine break in the mean at row 60. A chi-square on the 5 coefficients a split adds — 11.07 — declares a break on 73.6% of samples that have none: it is not a test at all. The break search's own 95% point, 69.6, carried into a rule that also chooses its window, fires on 0.0% of null samples and on 0.0% of samples with the largest break measured — the natural way of combining two published corrections does not lose a little power, it switches the test off. The calibrated charge, 27.2, holds 5.6% at no break and reaches 29.2% at the largest.

The charge that is not a sum

Charging two searches what each costs on its own is conservative, and conservative here means the test never fires. At the largest break measured it declares nothing, on every draw, while a calibrated threshold reaches 29%.

twice · Break point 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

Named alongside it

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

CoverageDegrees of freedomEfficiencyFixed-width intervalInverse variance weightingVariance ratioBlindingBlockContrastEstimated varianceInterval widthNuisance parameter

All concepts