Moran's I for rates, not fooled by small denominators

Moran's I on raw rates mistakes the noise of small denominators for clustering: a few small states with extreme rates beside one another look like a hot spot. The empirical-Bayes index (Assuncao & Reis 1999) standardises each rate by how much variation its denominator alone would produce before testing for autocorrelation.

Usage

eb_morans_i(data, numerator, denominator, weights = NULL, n_perm = 999)

Arguments

data

A country-level frame with iso3c.

numerator, denominator

The counts and their population at risk (unquoted).

weights

A country_weights() object; NULL (default) is k-nearest neighbours (k = 5), as for morans_i().

n_perm

Permutations for the pseudo p-value (default 999; 0 skips it).

Value

A one-row tibble like morans_i()'s: i, expected, n, n_excluded, n_links, p_value (one-sided, for positive autocorrelation) and the list-column excluded.

The statistic

With rates \(p_i = y_i / x_i\), the global rate \(b = \sum y / \sum x\) and the method-of-moments between-country variance \(a\) (as in smooth_rates()), each rate is standardised as \(z_i = (p_i - b) / \sqrt{a + b / x_i}\) and Moran's I is computed on \(z\). It agrees with spdep::EBImoran.mc() on the same weights.

References

Assuncao, R. M. & Reis, E. A. (1999). A new proposal to adjust Moran's I for population density. Statistics in Medicine 18(16), 2147-2162. doi:10.1002/(SICI)1097-0258(19990830)18:16<2147::AID-SIM179>3.0.CO;2-I

See also

morans_i(), smooth_rates(), rate_funnel()

Examples

snap <- countryatlas::world_snapshot$countries
snap$births <- snap$population * 0.02
eb_morans_i(snap, births, population, n_perm = 0)
#> # A tibble: 1 × 7
#>         i expected     n n_excluded n_links p_value excluded 
#>     <dbl>    <dbl> <int>      <int>   <int>   <dbl> <list>   
#> 1 -0.0267 -0.00467   215          1     979      NA <chr [1]>