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

Dichotomisation — where it appears

Replacing a measured quantity by which side of a cut-off it falls on. A cut at the median keeps 2/π of what a normal quantity carries, and the cost in sample size depends on what the quantity was being used for.

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

How much of a normal outcome's information survives cutting it into two, by where the cut is. For a small shift, a cut at the mean keeps 63.7% of the information, so the trial needs 1.57 times the sample. A cut at the top tenth keeps 34.2% and needs 2.92 times; a cut two standard deviations out keeps 13.1%.

An outcome cut in two

Replacing a measured outcome with whether it crossed a threshold keeps 63.7% of the information when the cut is at the mean, 34.2% at the top tenth and 13.1% two standard deviations out. A trial that needs 63 patients per arm on the measured outcome needs 102 cut at the mean and 185 cut at one and a half standard deviations. The responder rates that result read as a share of patients who respond — 50.0% against 69.1% — when every patient moved by the same amount; and a cut chosen after looking turns a 5% test into a 17.7% one.

planned · Power The sample a trial needs for the same power, as a share of the unadjusted analysis's, by how strongly a baseline covariate predicts the outcome. Adjusting for the covariate as measured needs one minus the squared correlation of the unadjusted sample; cut at the median, one minus 2/π of the squared correlation. At a correlation of 0.7 those are 0.510 and 0.688; cut in three, 0.611; the change from baseline, 0.600. The change score needs fewer patients than the median-cut adjustment once the correlation passes 0.624.

A baseline cut in two

Adjusting a trial's result for a baseline measurement that predicts the outcome shrinks the sample it needs to 1 − ρ² of the unadjusted one. Adjusting for whether that measurement was above its median shrinks it only to 1 − (2/π)ρ² — the cut keeps 63.7% of what the covariate could remove, exactly the fraction a cut outcome keeps. In patients the loss grows with the covariate's strength: a cut costs 12% more patients at a correlation of 0.5, 35% at 0.7 and 2.55 times as many at 0.9, where a simple change from baseline would have done better than the cut adjustment.

planned · Power

Named alongside it

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

Sample sizeStatistical powerAnalysis of covarianceBaseline adjustmentChange scoreCorrelationEffect sizeThe garden of forking pathsInformationRandomisationResponder analysisTwo-sample test

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