A permutation test for Moran's I statistic calculated by using nsim random permutations of x for the given spatial weighting scheme, to establish the rank of the observed statistic in relation to the nsim simulated values.

moran.mc(x, listw, nsim, zero.policy=NULL, alternative="greater",
na.action=na.fail, spChk=NULL, return_boot=FALSE, adjust.n=TRUE)

## Arguments

x

a numeric vector 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 NULL, 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), or "less".

na.action

a function (default na.fail), can also be na.omit or na.exclude - in these cases the weights list will be subsetted to remove NAs in the data. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations. Note that only weights lists created without using the glist argument to nb2listw may be subsetted. na.pass is not permitted because it is meaningless in a permutation test.

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

default TRUE, if FALSE the number of observations is not adjusted for no-neighbour observations, if TRUE, the number of observations is adjusted

## 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

## References

Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion, p. 63-5.

## Author

Roger Bivand Roger.Bivand@nhh.no

moran, moran.test

## Examples

data(oldcol)
colw <- nb2listw(COL.nb, style="W")
nsim <- 99
set.seed(1234)
sim1 <- moran.mc(COL.OLD$CRIME, listw=colw, nsim=nsim) sim1 #> #> Monte-Carlo simulation of Moran I #> #> data: COL.OLD$CRIME
#> weights: colw
#> number of simulations + 1: 100
#>
#> statistic = 0.51095, observed rank = 100, p-value = 0.01
#> alternative hypothesis: greater
#>
mean(sim1$res[1:nsim]) #> [1] -0.003196117 var(sim1$res[1:nsim])
#> [1] 0.008543554
summary(sim1$res[1:nsim]) #> Min. 1st Qu. Median Mean 3rd Qu. Max. #> -0.180085 -0.073721 -0.009528 -0.003196 0.063865 0.206779 colold.lags <- nblag(COL.nb, 3) set.seed(1234) sim2 <- moran.mc(COL.OLD$CRIME, nb2listw(colold.lags[[2]],
style="W"), nsim=nsim)
summary(sim2$res[1:nsim]) #> Min. 1st Qu. Median Mean 3rd Qu. Max. #> -0.15905 -0.07503 -0.01535 -0.02060 0.03305 0.13023 sim3 <- moran.mc(COL.OLD$CRIME, nb2listw(colold.lags[[3]],
style="W"), nsim=nsim)
summary(sim3\$res[1:nsim])
#>      Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
#> -0.192986 -0.055468 -0.024154 -0.026327  0.004769  0.129996