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%.

Every point on this curve is a group of 10 observations. What changes along the axis is only where those 10 x values are placed. A group whose x values span 0.2 has a slope standard error of 3.72 and moves 97% of the way to the population slope; one spanning 2.9 has a standard error of 0.25 and moves 15%. The half-way point is at a spread of 1.25, where the slope's own standard error equals τ.

Each row is a group. The hollow circle is its own proportion, the filled one is the estimate after pooling, and the vertical rule is the pooled population proportion of 31.3%. One group saw no events at all, and its raw proportion of zero becomes 21.2% — an estimate the group's own data cannot produce and the population's can. The arrows are not the same length, and none of the groups differs in size.

For every possible count out of 10: the raw proportion, the estimate from shrinking on the log-odds scale with the ½-corrected standard error, and the posterior mean under the same two-level model computed with the binomial likelihood itself. The two pooled routes differ by at most 0.0815 between 1 and 9 events, and by 0.0334 at zero — where the raw estimate is 0% and both pooled routes are not.

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.

Each study has 400 observations arranged as 20 clusters of 20. The lower points are the counted coverage of the usual interval, which treats them as 400 independent observations; the curve through them is 2Φ(1.96/√deff) − 1 with deff = 1 + 19ρ, computed before any data was drawn. At ρ = 0.81 the interval covers 36% rather than 95%. The upper points treat the cluster as the unit and need no variance components at all.

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.

240 observations arranged in 8 rows and 10 columns. As the column grouping's spread grows from 0.1 to 3, the column contrast falls from 210.2 effective observations to 1.6 and the row contrast stays near 11.3, because the column effects cancel out of a row difference exactly.

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.

the variance itself: 0.000 at the tenth percentile of the variance estimate, 0.390 at the median and 2.900 at the ninetieth, against a true 1.000; the intraclass correlation: 0.000 at the tenth percentile of the variance estimate, 0.213 at the median and 0.668 at the ninetieth, against a true 0.410; the design effect: 1.000 at the tenth percentile of the variance estimate, 2.918 at the median and 7.014 at the ninetieth, against a true 4.689; effective observations: 20.000 at the tenth percentile of the variance estimate, 6.854 at the median and 2.851 at the ninetieth, against a true 4.266. Each is a deterministic function of the same estimate, so the spread is the estimate's spread carried through.

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.