Empirical-Bayes smoothing: a rate computed over a small denominator is pulled
toward the overall rate in proportion to how little information stands behind
it, while a rate over a large denominator is left essentially alone. Standard
practice in disease mapping, and the statistical counterpart to
value_by_alpha_map()'s visual answer.
Usage
smooth_rates(
data,
numerator,
denominator,
method = c("eb", "none"),
suffix = "_smoothed"
)Value
data with <numerator>_rate and <numerator>_smoothed columns
added, plus <numerator>_shrinkage – the weight given to the country's own
rate, between 0 (fully shrunk to the global rate) and 1 (untouched).
The model
A Poisson-gamma model: counts \(y_i \sim \mathrm{Poisson}(d_i \theta_i)\)
with \(\theta_i \sim \mathrm{Gamma}\), whose mean and variance are estimated
from the data by the method of moments. The posterior mean is
\(w_i r_i + (1 - w_i)\bar{r}\) with \(w_i = d_i / (d_i + \alpha)\), so the
shrinkage weight is exactly the "how much do we believe this country"
quantity that rate_check() flags. Where the between-country variance is
estimated as non-positive (rates no more dispersed than Poisson noise alone),
every rate shrinks fully to the global mean, which is the right answer:
the data contain no evidence of real between-country variation.
Examples
d <- data.frame(
iso3c = c("CHN", "IND", "TUV", "NRU"),
cases = c(50000, 42000, 3, 1),
pop = c(1.41e9, 1.39e9, 11000, 12000)
)
smooth_rates(d, cases, pop)
#> # A tibble: 4 × 6
#> iso3c cases pop cases_rate cases_smoothed cases_shrinkage
#> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 CHN 50000 1410000000 0.0000355 0.0000355 0.997
#> 2 IND 42000 1390000000 0.0000302 0.0000302 0.997
#> 3 TUV 3 11000 0.000273 0.0000334 0.00236
#> 4 NRU 1 12000 0.0000833 0.0000330 0.00257