Compute per-site empirical-Bayes shrinkage proportions

Compute the classical empirical-Bayes shrinkage weight per site, \(S_j = \sigma_\tau^2 / (\sigma_\tau^2 + \widehat{se}_j^2)\) — the fraction by which a partial-pooling estimator pulls each site's estimate toward the grand mean. Group D diagnostic (downstream shrinkage — Dr. Chen's question 4: "How much does empirical Bayes pool toward the mean?").

Usage

compute_shrinkage(se2_j, sigma_tau, monotone = FALSE)

Arguments

se2_j

Numeric vector (length \(\ge 1\), strictly positive). Per-site sampling variances \(\widehat{se}_j^2\).

sigma_tau

Numeric scalar (\(\ge 0\)). Between-site SD on the response scale. sigma_tau = 0 returns zeros (no heterogeneity to shrink toward).

monotone

Logical scalar. If TRUE, return sort(shrinkage) for plotting purposes. Default FALSE preserves site-index alignment.

Value

A numeric vector of length length(se2_j), each entry in [0, 1). Sorted ascending when monotone = TRUE.

Details

\(S_j\) ranges over [0, 1). \(S_j \to 1\) means "the prior dominates" — site \(j\) has so much sampling noise that partial-pooling collapses its estimate toward the overall mean. \(S_j \to 0\) means "the data dominate" — the site identifies its own latent effect cleanly. Sites with intermediate \(S_j\) are where empirical-Bayes pooling does real work. \(\sigma_\tau = 0\) returns zero for every site (no heterogeneity to recover).

Reading guide. For diagnostic plots — funnel- or caterpillar-style traces of shrinkage against site precision — pass monotone = TRUE so the returned vector is sorted ascending and renders as a smooth curve. For computing means or downstream summaries, leave monotone = FALSE so each \(S_j\) aligns with its site index.

For the four-question diagnostic walkthrough see the A3 · Diagnostics in practice vignette.

References

Lee, J., Che, J., Rabe-Hesketh, S., Feller, A., & Miratrix, L. (2025). Improving the estimation of site-specific effects and their distribution in multisite trials. Journal of Educational and Behavioral Statistics, 50(5), 731–764. doi:10.3102/10769986241254286 .

See also

compute_I for the geometric-mean aggregate \(\widehat{I}\) that uses the same shrinkage kernel; mean_shrinkage for the across-site mean of \(S_j\); feasibility_index for \(\sum_j S_j\) (Efron) or \(\sum_j (1 - S_j)\) (Morris); scenario_audit for the quality gates that consume these diagnostics; the A3 · Diagnostics in practice vignette.

Other family-diagnostics: bhattacharyya_coef(), compute_I(), compute_kappa(), default_thresholds(), feasibility_index(), heterogeneity_ratio(), informativeness(), ks_distance(), mean_shrinkage(), realized_rank_corr(), realized_rank_corr_marginal(), scenario_audit()

Examples

# Two sites, the second twice as noisy as the first.
compute_shrinkage(c(0.05, 0.10), sigma_tau = 0.20)
#> [1] 0.4444444 0.2857143
# First site keeps more of its own information; second is shrunk harder.

# Sorted view — useful for a per-site shrinkage curve plot.
compute_shrinkage(c(0.10, 0.04, 0.07), sigma_tau = 0.20, monotone = TRUE)
#> [1] 0.2857143 0.3636364 0.5000000

# No heterogeneity collapses every site to zero shrinkage.
compute_shrinkage(c(0.05, 0.10), sigma_tau = 0)
#> [1] 0 0