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

Between group variance — where it appears

The variance of the true group means around the overall mean in a hierarchical model, usually written τ². It decides how far each group's estimate is pulled towards the others, and with few groups it is the least well estimated quantity in the whole model.

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

Sixteen groups from two clusters: the true effects, the observed means and the pooled estimates, with the kurtosis of each. One dataset of 16 groups of four observations each, effects drawn from two clusters with total spread 2, noise 3. The fitted spread is 1.701, so every estimate is pulled 43.7% of the way to the centre -0.213. Kurtosis: true effects 1.203, observed means 2.348, pooled estimates 2.348 — the last two identical, because one is an affine image of the other.

The shape the estimates keep

Partial pooling was suspected of destroying the evidence of its own misfit: compressing a two-cluster population of group effects towards one centre before any test could see the clusters. With groups of equal size it destroys nothing, because every estimate moves the same fraction of the way to the same place and the estimates keep the means' shape exactly — except when the fit pools completely, as it does on 29.9% of eight-group datasets at a spread of one. The evidence was never there: a shape check has 6.5% power at eight groups. With groups of unequal size the pooled estimates see the clusters better than the raw means do.

hierarchical · Pooling Coverage of the population mean from two units, by the true spread between units. 4,000 two-unit studies at each true spread, ten observations a unit, within-unit standard deviation 1.2. exact, on one degree of freedom: 94.97%, 95.50%, 94.95%, 94.83%, 95.15%, 95.20%, 94.67%. spread supplied as 1: 100.00%, 100.00%, 99.92%, 95.53%, 81.85%, 70.03%, 51.55%. half-normal, scale 1: 100.00%, 100.00%, 99.78%, 94.63%, 86.25%, 79.70%, 68.38%. half-Cauchy, scale 1: 100.00%, 100.00%, 100.00%, 98.98%, 95.50%, 92.38%, 88.00%. inverse-gamma(0.001, 0.001) on the variance: 100.00%, 100.00%, 100.00%, 99.15%, 97.05%, 95.28%, 92.65% — at spreads 0, 0.25, 0.5, 1, 1.5, 2, 3. The exact interval holds 95% at every spread; every integrated one over-covers when the spread is small and under-covers when it is large, and where it crosses 95% is set by the prior's tail.

A prior on the spread between two sites

Integrate the between-site spread of a two-site study under a half-normal prior of scale 1 and the interval for the overall mean is 3.01 wide and covers 94.63% when the true spread is 1 — the supplied interval again, under another name. At a true spread of 3 it covers 68.38%. Every prior tried covers 95% averaged over its own draws and none covers it at every spread; each holds up to about its own scale, because two sites move the posterior median of the spread by less than a third of what the truth moves.

multilevel · Levels

Named alongside it

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

Hierarchical modelMonte CarloCoverageCredible intervalDegrees of freedomEmpirical BayesKurtosisMixture distributionModel diagnosticsModel misspecificationPartial poolingPosterior distribution

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