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

Alpha spending — where it appears

A rule giving how much of the error rate may have been spent by each fraction of the information collected. Each look's boundary is solved from what the rule allows by then, so the looks may fall wherever they happen, provided their timing is not chosen from the data.

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

Four boundaries for 5 looks, all spending 5% in total. test at 0.05 every look: 1.96, 1.96, 1.96, 1.96, 1.96. Pocock — a constant, higher boundary: 2.41, 2.41, 2.41, 2.41, 2.41. O'Brien–Fleming — strict early, nearly nominal at the end: 4.55, 3.22, 2.63, 2.27, 2.03. Bonferroni across looks: 2.58, 2.58, 2.58, 2.58, 2.58. Every one except the first spends the same total error rate; they differ in when they spend it.

Spending the error rate

The repair for interim testing is to spend 5% across the looks rather than at each one. The boundaries are solvable rather than quotable, and a trial that can stop early uses 298 observations where a fixed design uses 400 — at a cost of half a point of power.

sequential · Stopping The chance of crossing later from each interim z, under six schedules with O'Brien–Fleming-type spending boundaries. Exact. At an interim |z| of 1.0: end only 3.64%, +0.6 3.41%, +0.75 3.20%, +0.9 3.41%, every 0.125 3.00%, every 0.05 2.88%. At 2.5: end only 38.30%, +0.6 45.80%, +0.75 45.36%, +0.9 41.70%, every 0.125 50.08%, every 0.05 53.24%. The heavy line is the largest of the six at each z.

A look the trend asked for

Under an O'Brien–Fleming-type spending function, every schedule of looks fixed in advance spends exactly 5.0000%. A committee that adds a look at three quarters of the trial whenever the interim z is 1.5 or more spends 5.2323% — 5.315% counted over a hundred thousand trials — and the most a committee choosing among six schedules could spend is 5.4390%.

sequential · Stopping

Named alongside it

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

Error rateGroup-sequential designO'Brien–FlemingThe Pocock boundaryClosed formInterim analysisMonte CarloOptional stoppingp-valueStagewise orderingStatistical powerStopping rule

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