Permutation test for Geary's C statistic

A permutation test for Geary's C 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

geary.mc(x, listw, nsim, zero.policy=attr(listw, "zero.policy"), alternative="greater",
 spChk=NULL, adjust.n=TRUE, return_boot=FALSE, na.action=na.fail, scale=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), or "less"; this reversal corresponds to that on geary.test described in the section on the output statistic value, based on Cliff and Ord 1973, p. 21 (changed 2011-04-11, thanks to Daniel Garavito).
spChk: should the data vector names be checked against the spatial objects for identity integrity, TRUE, or FALSE, default NULL to use get.spChkOption()
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
return_boot: return an object of class boot from the equivalent permutation bootstrap rather than an object of class htest
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.
scale: default TRUE, may be FALSE to revert changes made to accommodate localC in November 2021 (see #151)

Value

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

statistic: the value of the observed Geary's C.
parameter: the rank of the observed Geary's C.
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)
set.seed(1)
sim1 <- geary.mc(COL.OLD$CRIME, nb2listw(COL.nb, style="W"),
 nsim=99, alternative="less")
sim1
#> 
#> 	Monte-Carlo simulation of Geary C
#> 
#> data:  COL.OLD$CRIME 
#> weights: nb2listw(COL.nb, style = "W")  
#> number of simulations + 1: 100 
#> 
#> statistic = 0.52987, observed rank = 1, p-value = 0.99
#> alternative hypothesis: less
#> 
mean(sim1$res)
#> [1] 1.002705
var(sim1$res)
#> [1] 0.01043616
summary(sim1$res)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  0.5299  0.9362  1.0141  1.0027  1.0673  1.2459 
colold.lags <- nblag(COL.nb, 3)
sim2 <- geary.mc(COL.OLD$CRIME, nb2listw(colold.lags[[2]],
 style="W"), nsim=99)
sim2
#> 
#> 	Monte-Carlo simulation of Geary C
#> 
#> data:  COL.OLD$CRIME 
#> weights: nb2listw(colold.lags[[2]], style = "W")  
#> number of simulations + 1: 100 
#> 
#> statistic = 0.81129, observed rank = 1, p-value = 0.01
#> alternative hypothesis: greater
#> 
summary(sim2$res)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  0.8113  0.9500  1.0173  1.0147  1.0731  1.1962 
sim3 <- geary.mc(COL.OLD$CRIME, nb2listw(colold.lags[[3]],
 style="W"), nsim=99)
sim3
#> 
#> 	Monte-Carlo simulation of Geary C
#> 
#> data:  COL.OLD$CRIME 
#> weights: nb2listw(colold.lags[[3]], style = "W")  
#> number of simulations + 1: 100 
#> 
#> statistic = 1.1303, observed rank = 89, p-value = 0.89
#> alternative hypothesis: greater
#> 
summary(sim3$res)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  0.7903  0.9609  0.9989  1.0148  1.0656  1.2726 
crime <- COL.OLD$CRIME
is.na(crime) <- sample(1:length(crime), 10)
try(geary.mc(crime, nb2listw(COL.nb, style="W"), nsim=99,
 na.action=na.fail))
#> Error in na.fail.default(x) : missing values in object
geary.mc(crime, nb2listw(COL.nb, style="W"), nsim=99, zero.policy=TRUE,
 na.action=na.omit)
#> Warning: subsetting caused increase in subgraph count
#> 
#> 	Monte-Carlo simulation of Geary C
#> 
#> data:  crime 
#> weights: nb2listw(COL.nb, style = "W") 
#> omitted: 14, 16, 17, 21, 24, 31, 32, 42, 44, 47 
#> number of simulations + 1: 100 
#> 
#> statistic = 0.49368, observed rank = 1, p-value = 0.01
#> alternative hypothesis: greater
#> 
geary.mc(crime, nb2listw(COL.nb, style="W"), nsim=99, zero.policy=TRUE,
 return_boot=TRUE, na.action=na.omit)
#> Warning: subsetting caused increase in subgraph count
#> NA observations omitted: 14, 16, 17, 21, 24, 31, 32, 42, 44, 47
#> 
#> DATA PERMUTATION
#> 
#> 
#> Call:
#> boot(data = x, statistic = geary_boot, R = nsim, sim = "permutation", 
#>     listw = listw, n = n, n1 = wc$n1, S0 = wc$S0, zero.policy = zero.policy, 
#>     scale = scale, parallel = parallel, ncpus = ncpus, cl = cl)
#> 
#> 
#> Bootstrap Statistics :
#>      original    bias    std. error
#> t1* 0.4936767 0.4891213   0.1517218
geary.mc(crime, nb2listw(COL.nb, style="W"), nsim=99, zero.policy=TRUE,
 na.action=na.exclude)
#> Warning: subsetting caused increase in subgraph count
#> 
#> 	Monte-Carlo simulation of Geary C
#> 
#> data:  crime 
#> weights: nb2listw(COL.nb, style = "W") 
#> omitted: 14, 16, 17, 21, 24, 31, 32, 42, 44, 47 
#> number of simulations + 1: 100 
#> 
#> statistic = 0.49368, observed rank = 1, p-value = 0.01
#> alternative hypothesis: greater
#> 
geary.mc(crime, nb2listw(COL.nb, style="W"), nsim=99, zero.policy=TRUE,
 return_boot=TRUE, na.action=na.exclude)
#> Warning: subsetting caused increase in subgraph count
#> NA observations omitted: 14, 16, 17, 21, 24, 31, 32, 42, 44, 47
#> 
#> DATA PERMUTATION
#> 
#> 
#> Call:
#> boot(data = x, statistic = geary_boot, R = nsim, sim = "permutation", 
#>     listw = listw, n = n, n1 = wc$n1, S0 = wc$S0, zero.policy = zero.policy, 
#>     scale = scale, parallel = parallel, ncpus = ncpus, cl = cl)
#> 
#> 
#> Bootstrap Statistics :
#>      original    bias    std. error
#> t1* 0.4936767 0.4570379   0.1307276
try(geary.mc(crime, nb2listw(COL.nb, style="W"), nsim=99, na.action=na.pass))
#> Error in geary.mc(crime, nb2listw(COL.nb, style = "W"), nsim = 99, na.action = na.pass) : 
#>   na.pass not permitted