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

Random-effects — where it appears

Treating group-level parameters as draws from a distribution rather than as separate unknowns, so that groups borrow strength from each other. The variance of that distribution is estimated, and estimating it as zero collapses the model to complete pooling in a way worth noticing.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

Eight groups, τ = 1 against a within-group spread of 3. Each row is a group. The hollow circle is the group's own mean, the filled one is the estimate after pooling, and the small mark is the truth the data was generated from. The group of 3 moves 75% of the way to the population mean of 0.10; the group of 40 moves 18%.

Eight groups, one population

Eight hospitals are neither one hospital nor eight unrelated problems. The two obvious answers cost 2.23 and 1.15 in squared error; the estimate between them costs 0.88, and the weight it uses is not a matter of taste.

hierarchical · Pooling Six groups of 10, each fitting its own slope, then borrowing. Each faint line is one group's own least-squares slope through its own centre; each solid line is that slope after pooling towards the population slope of 0.79. Every group has the same 10 observations. The group whose x values span 0.4 has a slope standard error of 1.86 and moves 91% of the way in; the group spanning 2.0 has a standard error of 0.37 and moves 28%.

The slope that borrows

Pooling a mean makes it look as though how much a group borrows depends on how much data it has. Pool a slope instead and the illusion breaks — ten groups with ten observations each can borrow anything from 28% to 91%, decided entirely by where those ten observations were placed.

multilevel · Levels 16 groups shrunk towards a fitted line, at γ = 1.2. Hollow circles are the groups' own values, filled ones the estimates after pooling, and the diagonal is the line fitted through them with each group weighted by how well it is measured — slope 1.21, intercept 0.20. The horizontal rule is where the same 16 groups would have been shrunk to with no covariate. The spread left to borrow against is 0.64 with the covariate against 1.28 without, so every group is pulled further in than it would otherwise have been.

Borrowing towards a line

A group shrunk towards the average of all groups is being compared with groups it has nothing in common with. Fit a group-level predictor and it is shrunk towards what the predictor says a group like it should be — which halves the spread left to borrow against and takes a quarter off the squared error.

multilevel · Levels Two arms leave one degree of freedom per block unaccounted for. Each point is one run. The one-mean field's identity is (b − 1) + (N − b) = N − 1, and every schedule moves along that line rather than off it. Two arms give the rule N − 2b and the interval b − 1, which come to N − b − 1 — short of the N − 2 two arms leave by exactly one per block, since a block's arm counts absorb one degree of freedom each and only one of the two directions carries the difference. The hollow points add what the block sums are worth, b − 1 more, and land on the total. The missing degrees of freedom are not lost; they are in a place the interval has to be shown it may read.

The degrees of freedom in the sums

One arm partitions N − 1 exactly. Two arms give the rule N − 2b and the interval b − 1, which is short by one per block — and the missing ones are in the block sums, which are correlated with the differences at −0.79 and are usable anyway.

contrast · Blocking What 240 observations are worth, by which question is asked. 8 rows and 10 columns with 3 observations in each cell — 240 in all, each belonging to one row and one column, neither nested in the other. the overall mean: variance 0.1703 against a naive 0.0060, a design effect of 28.4 and 8.5 effective observations; a difference between two rows: variance 2.0682 against a naive 0.0960, a design effect of 21.5 and 11.1 effective observations; a difference between two columns: variance 1.0845 against a naive 0.1200, a design effect of 9.0 and 26.6 effective observations.

Two groupings that cross

Pupils belong to a school and to a neighbourhood, and neither is nested in the other. There is then no design effect: the overall mean is worth 8.5 independent observations out of 240, a row difference 11.1 and a column difference 26.6, and which grouping matters depends on the question rather than on the study.

multilevel · Levels What a variance estimated from K units is worth. The between-unit mean square is a scaled chi-square on K − 1 degrees of freedom, so the estimator's whole distribution is decided by the number of units. At two units its interquartile range spans a factor of 13.03 and its ten-to-ninety range a factor of 171.3, and it comes out exactly zero on 26.7% of studies. The closed form and 3,000 simulated studies agree to 0.051 at every quantile.

A level with two units

A variance estimated from two units is a scaled chi-square on one degree of freedom. Its interquartile range spans a factor of thirteen, its ten-to-ninety range a factor of a hundred and seventy-one, and it comes out exactly zero on 26.7% of studies — so the design effect it decides runs from 1.00 to 7.01 against a truth of 4.69.

multilevel · Levels Twelve groups from two clusters, τ = 1. Every group's truth is in one of two clusters, and the population's total spread is exactly τ = 1, so the analysis recovers τ̂ = 1.515 and every shrinkage weight is what it would be for a single normal population. The estimates are pulled towards the grand mean, which is the middle of the gap — a place 0 of the 12 truths are and 6 of the estimates end up.

One population, or two

Group effects from two clusters rather than one bell, with the same total spread. The analysis recovers the same population spread, uses the same weight for every group, and reports nothing unusual — while 46% of its estimates land in a region holding 6.6% of the truths.

hierarchical · Pooling Each interval covers one question and not the other. Coverage of each interval for the overall mean, scored against both estimands, over 20,000 two-site studies of 10 observations apiece. The fixed-effect interval covers the mean of the two sites in hand 96.37% of the time and the population mean 54.77%. The random-effects interval covers the population mean 94.96% — exactly its level, from one degree of freedom — and over-covers the two sites in hand at 98.25%. Both are correct; they are answers to different questions printed in the same place.

What a two-unit study should report

The fixed-effect interval covers the mean of the two sites in hand 96.37% of the time and the population mean 54.77%. The random-effects interval covers the population mean 94.96% — exactly its level, from one degree of freedom — and is 11.6 times wider.

multilevel · Levels

Named alongside it

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

Hierarchical modelPartial poolingShrinkageStudy designVariance componentsDegrees of freedomEffective sample sizeCorrelationCoverageLeast squaresMethod of momentsSample size

All concepts