Local Moran's I (LISA)

Local Indicators of Spatial Association (Anselin 1995): one Moran statistic per country, plus the cluster type it belongs to. Where morans_i() answers "is there clustering anywhere", this answers "where, and of what kind".

Usage

local_morans(
  data,
  value,
  weights = NULL,
  n_perm = 9999,
  alpha = 0.05,
  p_adjust = c("fdr", "bonferroni", "holm", "none")
)

Arguments

data: A country-level frame with iso3c and the value column.
value: The value column (unquoted).
weights: A country_weights() object. NULL (default) uses country_weights("knn", k = 5): every country's five nearest neighbours, islands included. country_weights("contiguity") gives the land-border default of earlier versions, which leaves every island out.
n_perm: Permutations for the pseudo-p-value (default 9999; use 0 to skip the test, which leaves p_value as NA). The smallest p-value a permutation test can give is \(1/(n_{perm}+1)\), and false-discovery control across about 190 countries needs a finer floor than 999 permutations give; see the section below.
alpha: Significance threshold for the cluster label (default 0.05).
p_adjust: How to adjust the per-country p-values for testing about 190 countries at once: "fdr" (default; Benjamini-Hochberg false discovery rate), "bonferroni", "holm" or "none". See the section below.

Value

A tibble, one row per country: iso3c, value, lag (the neighbour average), ii (the local statistic), p_value, p_adjusted and cluster ("High-High", "Low-Low", "High-Low", "Low-High" or "Not significant", judged on p_adjusted). The method is recorded as the "countryatlas_p_adjust" attribute.

One test per country

A local statistic runs one test per country, about 190 of them, so at alpha = 0.05 roughly ten countries are expected to look significant when nothing is going on at all. p_adjust = "fdr" controls the false discovery rate instead – the share of flagged countries that are false alarms – which is the standard remedy for exactly this setting (Caldas de Castro & Singer 2006) and what GeoDa offers. "bonferroni" and "holm" control the chance of any false alarm, which is stricter; "none" reproduces the unadjusted labels of earlier versions. p_value itself is never adjusted, so it still matches spdep.

Adjustment needs resolution. A permutation p-value cannot go below \(1/(n_{perm}+1)\), and Benjamini-Hochberg flags the most significant of 190 tests only when its p-value is below \(0.05/190 \approx 0.00026\). With 999 permutations the floor, 0.001, is above that, so nothing short of several countries all reaching the floor can be flagged, and an empty map would say "no clusters" when it means "not enough permutations". Hence the default of 9,999, which takes about a second for the whole world.

p_value is a two-sided pseudo-p from conditional permutation: \((1 + \#\{|I_i^{*}| \ge |I_i|\}) / (n_{perm} + 1)\), so it is never exactly zero and its floor is \(1/(n_{perm}+1)\) – with the default 999 permutations, 0.001. Two-sided because a local statistic is interesting at both ends: a country surrounded by unlike neighbours is as much a finding as one surrounded by like ones. cluster is "Not significant" wherever p_value > alpha, and everywhere when n_perm = 0 leaves it NA. Set a seed beforehand for a reproducible p_value.

References

Anselin, L. (1995). Local Indicators of Spatial Association – LISA. Geographical Analysis 27(2), 93-115. doi:10.1111/j.1538-4632.1995.tb00338.x

Caldas de Castro, M. & Singer, B. H. (2006). Controlling the false discovery rate: a new application to account for multiple and dependent tests in local statistics of spatial association. Geographical Analysis 38(2), 180-208. doi:10.1111/j.0016-7363.2006.00682.x

Examples

# \donttest{
snap <- countryatlas::world_snapshot$countries
set.seed(1)
local_morans(snap, gdp_per_capita, weights = country_weights("knn", k = 5),
             n_perm = 99)
#> # A tibble: 199 × 7
#>    iso3c  value     lag       ii p_value p_adjusted cluster        
#>    <chr>  <dbl>   <dbl>    <dbl>   <dbl>      <dbl> <fct>          
#>  1 ABW   33939.  14923. -0.0605     0.86      0.940 Not significant
#>  2 AFG     374.   3453.  0.333      0.14      0.353 Not significant
#>  3 AGO    2799.   2911.  0.298      0.1       0.353 Not significant
#>  4 ALB    6549.  10175.  0.113      0.34      0.588 Not significant
#>  5 AND   41224. 102279.  2.71       0.01      0.353 Not significant
#>  6 ARE   41605.  30978.  0.433      0.2       0.433 Not significant
#>  7 ARG   12774.  10839.  0.0452     0.6       0.786 Not significant
#>  8 ARM    5378.   8014.  0.161      0.45      0.678 Not significant
#>  9 ATG   18305.  28266.  0.00927    0.63      0.799 Not significant
#> 10 AUS   61486.   1841. -0.941      0.13      0.353 Not significant
#> # ℹ 189 more rows
# }