Permutation test for Moran's I statistic

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.

Usage

moran.mc(x, listw, nsim, zero.policy=attr(listw, "zero.policy"),
 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 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".
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
adjust.n: 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

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$res[1:nsim])
#>      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
#> -0.192986 -0.055468 -0.024154 -0.026327  0.004769  0.129996 
crime <- COL.OLD$CRIME
is.na(crime) <- sample(1:length(crime), 10)
try(moran.mc(crime, nb2listw(COL.nb, style="W"), nsim=99,
 na.action=na.fail))
#> Error in na.fail.default(x) : missing values in object
moran.mc(crime, nb2listw(COL.nb, style="W"), nsim=99, zero.policy=TRUE,
 na.action=na.omit)
#> 
#> 	Monte-Carlo simulation of Moran I
#> 
#> data:  crime 
#> weights: nb2listw(COL.nb, style = "W") 
#> omitted: 5, 7, 9, 15, 19, 33, 40, 45, 47, 48 
#> number of simulations + 1: 100 
#> 
#> statistic = 0.42342, observed rank = 100, p-value = 0.01
#> alternative hypothesis: greater
#> 
moran.mc(crime, nb2listw(COL.nb, style="W"), nsim=99, zero.policy=TRUE,
 return_boot=TRUE, na.action=na.omit)
#> NA observations omitted: 5, 7, 9, 15, 19, 33, 40, 45, 47, 48
#> 
#> DATA PERMUTATION
#> 
#> 
#> Call:
#> boot(data = x, statistic = moran_boot, R = nsim, sim = "permutation", 
#>     listw = listw, n = n, S0 = S0, zero.policy = zero.policy, 
#>     parallel = parallel, ncpus = ncpus, cl = cl)
#> 
#> 
#> Bootstrap Statistics :
#>      original     bias    std. error
#> t1* 0.4234221 -0.4429891   0.1293935
moran.mc(crime, nb2listw(COL.nb, style="W"), nsim=99, zero.policy=TRUE,
 na.action=na.exclude)
#> 
#> 	Monte-Carlo simulation of Moran I
#> 
#> data:  crime 
#> weights: nb2listw(COL.nb, style = "W") 
#> omitted: 5, 7, 9, 15, 19, 33, 40, 45, 47, 48 
#> number of simulations + 1: 100 
#> 
#> statistic = 0.42342, observed rank = 100, p-value = 0.01
#> alternative hypothesis: greater
#> 
moran.mc(crime, nb2listw(COL.nb, style="W"), nsim=99, zero.policy=TRUE,
 return_boot=TRUE, na.action=na.exclude)
#> NA observations omitted: 5, 7, 9, 15, 19, 33, 40, 45, 47, 48
#> 
#> DATA PERMUTATION
#> 
#> 
#> Call:
#> boot(data = x, statistic = moran_boot, R = nsim, sim = "permutation", 
#>     listw = listw, n = n, S0 = S0, zero.policy = zero.policy, 
#>     parallel = parallel, ncpus = ncpus, cl = cl)
#> 
#> 
#> Bootstrap Statistics :
#>      original     bias    std. error
#> t1* 0.4234221 -0.4384413   0.1243937
try(moran.mc(crime, nb2listw(COL.nb, style="W"), nsim=99, na.action=na.pass))
#> Error in moran.mc(crime, nb2listw(COL.nb, style = "W"), nsim = 99, na.action = na.pass) : 
#>   na.pass not permitted