Permutation test for empirical Bayes index

An empirical Bayes index modification of Moran's I for testing for spatial autocorrelation in a rate, typically the number of observed cases in a population at risk. The index value is tested by using nsim random permutations of the index for the given spatial weighting scheme, to establish the rank of the observed statistic in relation to the nsim simulated values.

Usage

EBImoran.mc(n, x, listw, nsim, zero.policy = attr(listw, "zero.policy"), 
 alternative = "greater", spChk=NULL, return_boot=FALSE,
 subtract_mean_in_numerator=TRUE)

Arguments

n: a numeric vector of counts of cases the same length as the neighbours list in listw
x: a numeric vector of populations at risk the same length as the neighbours list in listw
listw: a listw object created for example by nb2listw
nsim: number of permutations
zero.policy: default attr(listw, "zero.policy") as set when listw was created, if attribute not set, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA
alternative: a character string specifying the alternative hypothesis, must be one of "greater" (default), "two.sided", or "less"
spChk: should the data vector names be checked against the spatial objects for identity integrity, TRUE, or FALSE, default NULL to use get.spChkOption()
return_boot: return an object of class boot from the equivalent permutation bootstrap rather than an object of class htest
subtract_mean_in_numerator: default TRUE, if TRUE subtract mean of z in numerator of EBI equation on p. 2157 in reference (consulted with Renato Assunção 2016-02-19); until February 2016 the default was FALSE agreeing with the printed paper.

Details

The statistic used is (m is the number of observations): $$EBI = \frac{m}{\sum_{i=1}^{m}\sum_{j=1}^{m}w_{ij}} \frac{\sum_{i=1}^{m}\sum_{j=1}^{m}w_{ij}z_i z_j}{\sum_{i=1}^{m}(z_i - \bar{z})^2} $$ where: $$z_i = \frac{p_i - b}{\sqrt{v_i}}$$ and: $$p_i = n_i / x_i$$ $$v_i = a + (b / x_i)$$ $$b = \sum_{i=1}^{m} n_i / \sum_{i=1}^{m} x_i $$ $$a = s^2 - b / (\sum_{i=1}^{m} x_i / m)$$ $$s^2 = \sum_{i=1}^{m} x_i (p_i - b)^2 / \sum_{i=1}^{m} x_i $$

Value

A list with class htest and mc.sim containing the following components:

statistic: the value of the observed Moran's I.
parameter: the rank of the observed Moran's I.
p.value: the pseudo p-value of the test.
alternative: a character string describing the alternative hypothesis.
method: a character string giving the method used.
data.name: a character string giving the name(s) of the data, and the number of simulations.
res: nsim simulated values of statistic, final value is observed statistic
z: a numerical vector of Empirical Bayes indices as z above

References

Assunção RM, Reis EA 1999 A new proposal to adjust Moran's I for population density. Statistics in Medicine 18, pp. 2147–2162; Bivand RS, Wong DWS 2018 Comparing implementations of global and local indicators of spatial association. TEST, 27(3), 716–748 doi:10.1007/s11749-018-0599-x

Author

Roger Bivand Roger.Bivand@nhh.no

Examples

nc.sids <- st_read(system.file("shapes/sids.gpkg", package="spData")[1], quiet=TRUE)
rn <- as.character(nc.sids$FIPS)
ncCC89_nb <- read.gal(system.file("weights/ncCC89.gal", package="spData")[1],
 region.id=rn)
#> Warning: neighbour object has 3 sub-graphs
EBImoran.mc(nc.sids$SID74, nc.sids$BIR74,
 nb2listw(ncCC89_nb, style="B", zero.policy=TRUE), nsim=999,
 alternative="two.sided", zero.policy=TRUE)
#> The default for subtract_mean_in_numerator set TRUE from February 2016
#> 
#> 	Monte-Carlo simulation of Empirical Bayes Index (mean subtracted)
#> 
#> data:  cases: nc.sids$SID74, risk population: nc.sids$BIR74
#> weights: nb2listw(ncCC89_nb, style = "B", zero.policy = TRUE)
#> number of simulations + 1: 1000
#> 
#> statistic = 0.25789, observed rank = 999, p-value = 0.002
#> alternative hypothesis: two.sided
#> 
sids.p <- nc.sids$SID74 / nc.sids$BIR74
moran.mc(sids.p, nb2listw(ncCC89_nb, style="B", zero.policy=TRUE),
 nsim=999, alternative="two.sided", zero.policy=TRUE)
#> 
#> 	Monte-Carlo simulation of Moran I
#> 
#> data:  sids.p 
#> weights: nb2listw(ncCC89_nb, style = "B", zero.policy = TRUE)  
#> number of simulations + 1: 1000 
#> 
#> statistic = 0.20904, observed rank = 997, p-value = 0.006
#> alternative hypothesis: two.sided
#>