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What partial pooling does to one group, to the set, and to a ranking

Partial pooling minimises the total squared error across groups, and three uses of its estimates are not the total. One group: with each group's standard error equal to the population's spread, every group truly more than 1.73 widths from the centre — 8.33% of a normal population — is estimated worse than by its own mean, and among eight groups with the spread estimated the truly most extreme one is worse off in 61.6% of datasets; capping each group's shift at one standard error improves the total, the truly extreme group and the observed extreme group at once. The set: posterior means spread 0.707 as widely as the truth and count 0.234% of groups beyond two widths against a true 2.28%. The ranking: in a league table of a hundred groups of unequal size, raw means fill 62.1% of the top ten with small groups, posterior means 13.4%, and no ranking recovers more than 5.47 of the true ten.

Which groups partial pooling serves, standard error 1 population width. Pooling's expected squared error for a group, divided by its own mean's, against how far the group truly sits from the centre. It is ×0.25 at the centre and crosses ×1 at 1.732 population widths, beyond which 8.33% of a normal population lies; capping the shift at one standard error holds every group under ×2.

A group from the population's own tail

Partial pooling halves the total squared error when a group's own standard error equals the spread between groups. Every group whose true effect sits more than 1.73 population widths from the centre — 8.33% of a perfectly normal population — does worse than it would have with its own mean, and its loss grows without bound. Among eight groups with the spread estimated, the most extreme is worse off in 61.6% of datasets. Capping the shift at one standard error keeps the total at 0.528 of the unpooled error and holds every group under twice it.

How the truth, the raw means, the posterior means and the constrained estimates spread, standard error 1. Beyond two population widths above the centre lie 2.28% of the true values, 7.86% of the raw means, 0.234% of the posterior means, and 2.28% of the constrained estimates.

Estimates that are too alike

Posterior means give each group its least-error estimate, and as a set they are too alike: with each group's standard error equal to the population's spread, they spread 0.707 as widely as the truth. Beyond two population widths lie 2.28% of the true effects, 7.86% of the groups' own means, and 0.234% of the posterior means — a tenth of the truth. Rescaling the estimates to the right spread counts the tail exactly and costs 17% more squared error; summing each group's posterior chance of being beyond the line counts it without changing any estimate.

How many of a league table's top ten are small groups, ranked four ways. A hundred groups with sizes from 4 to 400, of which 36% have twenty units or fewer. Small groups make up 36.3% of the true top ten, 62.1% of the top ten by raw means, 13.4% by posterior means and 22.9% by the posterior chance of being in the top ten. The three rankings recover 4.43, 5.38 and 5.47 of the true top ten.

A league table of a hundred

A hundred groups with sizes from 4 to 400, and a top ten to publish. Ranked by their own means, small groups fill 62.1% of the top ten against their 36.3% share of the true top ten. Ranked by posterior means they fill 13.4%. The ranking built from each group's chance of being in the top ten recovers 5.47 of the true ten, the best of three and barely half; and the group ranked first could hold any rank from 1 to 31.

The small groups one centre protects

In a league table of a hundred groups where larger groups do better on average, pooling every group towards one centre overstates the small groups by half a population width — 0.501 — because it pulls them towards an average they are not part of. The top ten barely notices. The bottom ten, the list that flags poor performers, holds 3.19 of the true bottom ten when pooled to one centre and 4.28 when the groups are ranked by their own noisy means. Pooling towards a centre that rises with size restores 4.35 and puts the small groups back at the bottom where 80.6% of the true bottom ten sit.

How far each estimate of the small groups sits from their truth, when the size relation is flattening below the median size. The small groups' average true effect is −0.259. Their estimates are off on average by −0.012 (own means), +0.276 (one fitted centre), −0.173 (fitted line in log size), −0.025 (fitted quadratic in log size), −0.003 (the true relation).

The relation the table has to estimate

Pooling a league table's groups towards a centre that rises with size repaired the small groups' bias — when the size relation was known. Fitted from the table itself, a straight line in log size loses almost nothing: the small groups' error is 0.750 against 0.719 with the true relation. When the relation bends below the median size, the fitted line extrapolates the large groups' slope into the region where it is barely observed and misplaces the small groups by −0.173 if the relation flattens and +0.264 if it steepens — smaller than one centre's error, of either sign. A quadratic in log size removes nearly all of it, −0.025 and +0.014, and costs 0.025 in error when the relation was straight.

How often seven 95% intervals cover the small groups of a league table whose size relation is steepening below the median size. Groups of twenty patients or fewer, 600 tables of a hundred groups. Coverage: one centre, plugged in 58.0% (average width 2.81); one centre, its uncertainty added 58.5% (average width 2.83); a fitted line, plugged in 83.1% (average width 1.58); a fitted line, its uncertainty added 88.4% (average width 1.77); a fitted quadratic, plugged in 85.5% (average width 1.53); a fitted quadratic, its uncertainty added 93.7% (average width 1.92); the true relation and spread 94.8% (average width 1.55).

The interval a small group is given

A league table's 95% interval for a small group, pooled towards a size relation fitted from the table, covers 92.8% of the time when the relation is straight and 83.1% when it steepens below the median — and 77.4% for the groups of eight patients or fewer. Adding the fitted centre's own uncertainty, weighted by how much the group borrows from it, brings the line to 88.4% and a fitted quadratic to 93.7%, and the quadratic's interval covers 94.6% of the smallest groups at every shape tried. One centre, plugged in, covers the smallest groups 35.2% of the time when the relation steepens, and no added variance repairs a centre in the wrong place.

Every group's 90% rank interval in one league table whose size relation is steepening below the median size, with its true rank. A hundred groups from four to four hundred patients, ranked from their posterior under a fitted quadratic size relation whose uncertainty is drawn once per table. The groups of twenty patients or fewer have rank intervals 30.1 places wide on average, and 34 of their 36 true ranks fall inside; the groups of over a hundred, 32.1 places, with 27 of 30 inside.

A ranking with its uncertainty

A league table's reader wants a rank, and a rank interval inherits a fitted size relation's errors differently from an effect interval. When the relation steepens below the median size, a fitted line's 95% interval for a small group's effect covers 83.1% of the time; its 90% interval for the same group's rank covers 95.6%, because the line's error lifts the small groups into the crowded middle of the table, where rank intervals are wide. The error surfaces instead in the large groups' ranks, 87.6%, and in width: 36.9 places against the 27.3 the true relation needs. A quadratic's uncertainty has to be drawn once per table, since it moves every small group together — added group by group it over-covers at 92.8%.

How often a league table's own data choose a bent centre over a straight one. Over 400 tables of a hundred groups each. The quadratic's curvature coefficient is significant at 5% on 5.3% of tables whose relation is straight, 14.5% of tables whose relation is flattening below the median, 29.3% of tables whose relation is steepening below the median; its average AIC weight is 0.374, 0.456, 0.558.

A centre chosen by its own table

A league table that picks its centre's shape from its own data — a line unless a curvature test says otherwise — picks the line on 85.5% of tables whose relation flattens below the median, because the bend lives where the groups are smallest and noisiest. Its bottom ten then flags like the line's: 2.61 innocent small groups a table against 2.19 for a quadratic told the shape. Always fitting the quadratic costs 0.07 of a correctly flagged group when the relation is straight and matches the told answer when it bends, and a flag set by probability carries the choice through a mixture of the two.

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