A small area's rate is mostly noise, so ranking areas by their raw ratios
puts the smallest areas at both ends of the table by construction. This
function borrows strength across areas: each area's ratio of observed to
expected is pulled toward the overall ratio by an amount that depends on
how little information the area carries. Use the shrunk estimates for
maps and rankings; use stdinc for each area's own evidence.
Arguments
- obs
Numeric vector of observed counts (non-negative, no
NA).- expd
Numeric vector of expected counts (positive), the same length, for example from
expcnts.- area
Optional vector of area labels, added as the first column. Default
NULL.
Value
A data frame with one row per area: area (when given),
observed, expected, the raw ratio sir, the shrunk
estimate eb, the shrinkage applied (zero means untouched,
one means replaced by the overall rate), and the fitted prior's nu
(shape) and alpha (rate), the same on every row.
Details
The area relative risks \(\theta_a\) are taken to come from a gamma prior with shape \(\nu\) and rate \(\alpha\). Its mean is the overall ratio \(m = \sum O_a / \sum E_a\), and its variance is estimated by the method of moments, $$s^2 = \frac{\sum_a E_a (\theta_a - m)^2}{\sum_a E_a} - \frac{m}{\bar E},$$ with \(\theta_a = O_a / E_a\) and \(\bar E\) the mean expected count; the subtracted term is the spread that Poisson sampling alone would produce. Then \(\alpha = m / s^2\), \(\nu = m^2 / s^2\), and the posterior mean \((O_a + \nu)/(E_a + \alpha)\) is reported, with shrinkage \(\alpha / (E_a + \alpha)\). 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
(all ones) and NA prior parameters rather than hidden.
Errors: lengths that differ; missing, negative or infinite counts; missing, infinite or non-positive expected counts; fewer than two areas.
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.
ebrates(obs = c(30, 45, 2), expd = c(25, 50, 0.5),
area = c("North", "South", "Tiny"))
#> ╭── Empirical Bayes rates ────────────────────────────────────────────────╮
#> │ Areas 3 │
#> │ Overall ratio 1.02 │
#> │ Prior shape (nu) 26.98 │
#> │ Prior rate (alpha) 26.46 │
#> │ Method gamma-Poisson, moments (Clayton and Kaldor 1987) │
#> │ │
#> │ area observed expected sir eb shrinkage │
#> │ ----- -------- -------- ----- ----- --------- │
#> │ North 30 25.00 1.200 1.107 0.5141 │
#> │ South 45 50.00 0.900 0.941 0.3460 │
#> │ Tiny 2 0.50 4.000 1.075 0.9815 │
#> │ │
#> │ Each raw ratio (sir) is pulled toward the overall ratio; shrinkage is │
#> │ the share of the pull (0 untouched, 1 replaced), largest for areas │
#> │ with small expected counts. Use eb for maps and rankings, stdinc() │
#> │ for each area's own evidence. │
#> ╰─────────────────────────────────────────────────────────────────────────╯
# no variation beyond chance: every estimate is the overall ratio
ebrates(c(10, 11, 9), c(10, 10, 10))
#> ╭── Empirical Bayes rates ────────────────────────────────────────────────╮
#> │ Areas 3 │
#> │ Overall ratio 1 │
#> │ Prior shape (nu) none │
#> │ Prior rate (alpha) none │
#> │ Method gamma-Poisson, moments (Clayton and Kaldor 1987) │
#> │ │
#> │ observed expected sir eb shrinkage │
#> │ -------- -------- ----- -- --------- │
#> │ 10 10 1.000 1 1 │
#> │ 11 10 1.100 1 1 │
#> │ 9 10 0.900 1 1 │
#> │ │
#> │ The areas vary no more than chance would make them, so every estimate │
#> │ is the overall ratio (shrinkage 1). │
#> ╰─────────────────────────────────────────────────────────────────────────╯
# one area has nothing to borrow from
try(ebrates(5, 4))
#> Error : shrinkage borrows strength across areas, so it needs at least two