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

Selective inference — where it appears

Inference about a result that was reported because it looked interesting, computed from the distribution the result has given that it was selected. It conditions on the selection rule, so its intervals stay honest after the choice rather than before it.

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

Thirty winners of twenty analyses, effect 2 standard errors in one of them, with two intervals each. Each row is the largest of twenty statistics correlated at 0.6, reported because it cleared the family-wise threshold of 2.846. The thin grey bar is the ordinary interval, z ± 1.96; the heavy bar is the interval conditional on the statistic having won; the tick is the true mean of the analysis that won. Of these thirty the ordinary interval misses 1 and the conditional one 2; over 4,468 reported winners they cover 76.0% and 94.9%.

An interval for the winner

The largest of twenty correlated analyses, reported because it cleared the family-wise threshold, carries an ordinary 95% interval that covers its true mean 75.96% of the time at an effect of two standard errors and never at all when there is no effect. An interval built on the fact that it won covers 94.90% — and it is 2.48 times as wide, excludes zero for only 14.10% of the winners, and has no lower limit at all for 14.32% of them. It restores the promise by declining to say most of what the ordinary interval said.

paths · Forking How often a winner's interval covers when it is conditioned on the analyses admitted to rather than the twenty run, effect 2. The largest of twenty analyses reported when it clears the family-wise threshold for the k admitted. At k = 1, 2, 5, 10, 20: correlation 0.6, conditioned on the admitted family 95.3%, 94.9%, 94.9%, 95.1%, 95.2%; the ordinary interval 67.7%, 71.1%, 75.2%, 76.5%, 76.1%; correlation 0, conditioned on the admitted family 94.4%, 95.6%, 95.3%, 94.8%, 95.6%; the ordinary interval 47.2%, 52.0%, 58.9%, 62.4%, 64.7%.

An interval for the analyses admitted to

An analyst runs twenty correlated analyses, admits to one, and reports the winner with an interval conditioned on having cleared 1.96 — the selection as written down rather than as it happened. The interval covers 95.3% of the time, against 95.6% for one conditioned on all twenty, because a winner's truncation point is set by the threshold far more than by its unseen rivals: hiding nineteen analyses moves it by 0.061 on average. What the hiding changes is how often there is a winner at all. With no effect anywhere, a family of twenty produces a reported winner 35.5% of the time against 4.8% when all twenty are admitted, and one whose interval excludes zero 1.48% of the time against 0.13% — the interval keeps its promise and the report does not.

paths · Forking

Named alongside it

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

Confidence intervalCoverageFamilywise error rateThe garden of forking pathsTruncated normalThe winner's curseMedian-unbiased estimateSensitivity analysis

All concepts