A small area's rate is mostly noise, so ranking areas by their raw rates puts the smallest areas at both ends of the table by construction. This borrows strength across areas: each rate is pulled toward the overall one by an amount that depends on how little information the area carries.
Value
A data frame with the raw sir, the shrunk eb, the
shrinkage applied (zero means untouched, one means replaced by
the overall rate), and the fitted prior's nu and alpha.
Details
The Clayton-Kaldor construction: the area-specific relative risks are
taken to come from a gamma prior, whose two parameters are estimated
from the observed and expected counts by the method of moments, and the
posterior mean (O + nu) / (E + alpha) is reported. Where the
expected count is large the data dominate and the estimate barely moves;
where it is small the prior does, which is the intended behaviour and
not a defect.
When the between-area variance estimate comes out at or below zero there
is no evidence of any real variation between areas, and every estimate
collapses to the overall rate. That is reported through shrinkage
rather than hidden.
References
Clayton, D. and Kaldor, J. (1987). Empirical Bayes estimates of age-standardized relative risks for use in disease mapping. Biometrics 43(3), 671-681.
Lawson, A. B. Using R for Bayesian Spatial and Spatio-Temporal Health Modeling. Chapman and Hall/CRC, which cites Clayton and Kaldor as the empirical-Bayes approximation in the development of Bayesian disease mapping.
Examples
# Three areas, one of them tiny. The tiny area's raw ratio is
# extreme; its shrunk one is not.
eb_rates(observed = c(30, 45, 2), expected = c(25, 50, 0.5),
area = c("North", "South", "Tiny"))
#> area observed expected sir eb shrinkage nu alpha
#> 1 North 30 25.0 1.2 1.1073869 0.5141392 26.98066 26.45506
#> 2 South 45 50.0 0.9 0.9414767 0.3460211 26.98066 26.45506
#> 3 Tiny 2 0.5 4.0 1.0751472 0.9814506 26.98066 26.45506