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

Change score — where it appears

The difference between a measurement taken after treatment and the same measurement before it, compared between groups. It uses the baseline without estimating how strongly it predicts, which makes it less efficient than adjusting for the baseline and more efficient than ignoring or cutting it.

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

Two groups, a baseline and a follow-up, and nothing happening in between — baseline reliability 0.6. 600 units in two pre-existing groups whose true means are 1.00 apart, read once at baseline and once at follow-up, with no change for anybody. The two groups' mean changes are −0.075 and −0.032, so the change-score analysis reports a group difference of 0.043. The regression of follow-up on baseline and group reports 0.409, against a closed form of (1 − λ) × 1.00 = 0.400: at any one baseline reading the two groups' lines sit that far apart, because each group's units regress towards their own group's mean. The pooled slope in this sample is 0.614, the baseline's reliability.

Two analyses of one baseline

Two groups read at baseline and again at follow-up, with no change for anybody. Subtracting the baseline reports a group difference of −0.0014 and adjusting for it reports 0.4008 — and each analysis is exactly right about one reason the groups started apart and wrong by 0.40 about the other.

paradox · Rtm 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.

RandomisationAnalysis of covarianceAssignment mechanismAttenuationBaseline adjustmentClosed formConfoundingCorrelationCovariate adjustmentDichotomisationLord's paradoxMeasurement error

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