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.

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.

The corner leaves a choice rather than an answer. Equal weights assume nothing about the variance ratio, are correctly scaled at any ratio, and are 118% wider. The weights h_b(λ) assume the ratio, are exact, and are the narrowest interval available. The effective sizes assume one of two conditions that do not hold here, and are both mis-scaled and wide — over-covering at 98.40%, which reads as caution and is not. The first two are a trade; the third is not on the frontier at all.

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.

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.