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 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.

Two thousand runs of each rule, the modelled weighting, a promise of 0.34. Reading its report: 4–8 blocks, 22.3% of runs, 78.2%; 9–12 blocks, 16.6% of runs, 90.4%; 13–16 blocks, 18.4% of runs, 96.2%; 17–20 blocks, 17.4% of runs, 96.0%; 21–28 blocks, 17.9% of runs, 96.4%; 29–36 blocks, 7.4% of runs, 99.3% — 91.45% overall. Reading the arms: 4–8 blocks, 0.0%, none; 9–12 blocks, 0.9%, 94.4%; 13–16 blocks, 30.4%, 95.6%; 17–20 blocks, 50.0%, 93.9%; 21–28 blocks, 18.0%, 94.4%; 29–36 blocks, 0.7%, 92.3% — 94.50% overall.

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%.