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

Permutation — where it appears

A reassignment of the observed labels among the same units, and the set of them is the reference distribution a randomisation test reads its statistic against. Under a null that fixes each unit's outcome every permutation is equally likely, so a p-value is a share of a finite set rather than an approximation to one.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Two diagnostics, one answer, two different moments. The two-chain statistic on a covariate probe and on the trial's own difference in arm means, at seven tolerances of a fourteen-unit rule, against the enumerated truth. Both are quiet wherever the set is one set and both fire wherever it is not, at every tolerance — which is what says the outcome probe is the same test rather than a resemblance of it. The difference between them is not accuracy and it is not power. It is when: the covariate probe can be run before a single outcome exists, when a practitioner can still loosen the rule or change the sampler, and it can be run again on a different function if it comes back quiet. The outcome probe runs after the trial, on the one column the trial produced, and what it can do with a positive verdict is repair the p-value rather than the design.

The statistic the p-value is about

The test for whether a balanced-assignment walk reaches its whole set is run on a covariate function chosen before the trial. Run on the difference in arm means it is the same test, and it is about the number the trial publishes.

after · Randomisation What 20 analyses of one dataset are worth. The threshold giving a 5% family-wise error rate, read back as a number of independent analyses. At no correlation it is 20.05; at 0.6 it is 11.37; at 0.95 it is 2.58. Bonferroni divides by 20 throughout.

How many analyses there really were

Bonferroni divides by twenty because twenty analyses were run. Twenty analyses of one dataset are worth 11.37 independent ones at a correlation of 0.6 and 2.58 at 0.95, and the threshold that controls exactly the same error rate is measurable rather than assumed.

paths · Forking The one thing a trial always reports is the one thing that survives. How wrong three p-values are when they are computed over the half of the admissible set a single walk can reach, rather than over all of it, at a fourteen-unit trial where the whole set can be enumerated. The two-sided p-value on the difference in arm means — the number a trial publishes — is wrong by exactly nothing, at every row, to machine precision. That is not luck: the two components are complement pairs and the difference in arm means is exactly negated by the complement, so the distribution of its absolute value is the same on both. A one-sided p-value on the same statistic is out by as much as 0.112, and the largest response observed in the treated arm — a safety reading rather than an effect, and the one statistic here that is not odd under the complement — by as much as 0.172. The defect survived because the commonest thing anybody computes is the one quantity it cannot touch.

Half a reference distribution

A walk that reaches half its admissible set reports the two-sided p-value exactly right, to the last digit, for ever. A one-sided one it puts on the wrong side of five per cent about once in thirty.

after · Reference

Named alongside it

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

Assignment mechanismConnected componentCovariate balanceExact enumerationImbalanceMarkov chain Monte Carlop-valueRandomisation testReference distributionSharp nullTreatment effectBonferroni

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