Mantel-Hubert spatial general cross product statistic

A permutation test for the spatial general cross product statistic with Moran (\(C_{ij} = z_i z_j\)), Geary (\(C_{ij} = (z_i - z_j)^2\)), and Sokal (\(C_{ij} = |z_i - z_j|\)) criteria, for \(z_i = (x_i - \bar{x}) / \sigma_{x}\). plot.mc.sim is a helper function to plot the outcomes of the permutation test.

Usage

sp.mantel.mc(var, listw, nsim, type = "moran", zero.policy = attr(listw, "zero.policy"),
 alternative = "greater", spChk=NULL, return_boot=FALSE)
# S3 method for class 'mc.sim'
plot(x, xlim, xlab, main, sub, ..., ptype="density")

Arguments

var: 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
type: "moran", "geary" or "sokal" criteria for similarity
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
x: the object to be plotted
xlim: the range of the x axis
xlab: a title for the x axis
main: an overall title for the plot
sub: a sub title for the plot
ptype: either "density" or "hist"
...: further arguments passed through

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.
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.
p.value: the pseudo p-value of the test.
res: nsim simulated values of statistic, final value is observed statistic
estimate: the mean and variance of the simulated distribution.

References

Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion, p. 22-24, Haining, R. 1990 Spatial data analysis in the social and environmental sciences, Cambridge: Cambridge University Press, p. 230–1. The function has been checked against general matrix code posted to the r-help list by Ben Bolker on 1 May 2001; another mantel() function is in the vegan package.

Author

Roger Bivand Roger.Bivand@nhh.no

Examples

data(oldcol)
sim1 <- sp.mantel.mc(COL.OLD$CRIME, nb2listw(COL.nb),
 nsim=99, type="geary", alternative="two.sided")
sim1
#> 
#> 	Mantel permutation test for geary measure
#> 
#> data:  COL.OLD$CRIME 
#> weights: nb2listw(COL.nb) 
#> number of simulations + 1: 100 
#> 
#> statistic = 51.927, observed rank = 1, p-value = 0.02
#> alternative hypothesis: two.sided
#> sample estimates:
#> mean of permutations   sd of permutations 
#>            98.852997             8.934135 
#> 
plot(sim1)

sp.mantel.mc(COL.OLD$CRIME, nb2listw(COL.nb), nsim=99,
 type="sokal", alternative="two.sided")
#> 
#> 	Mantel permutation test for sokal measure
#> 
#> data:  COL.OLD$CRIME 
#> weights: nb2listw(COL.nb) 
#> number of simulations + 1: 100 
#> 
#> statistic = 36.695, observed rank = 1, p-value = 0.02
#> alternative hypothesis: two.sided
#> sample estimates:
#> mean of permutations   sd of permutations 
#>            56.418584             3.227064 
#> 
sp.mantel.mc(COL.OLD$CRIME, nb2listw(COL.nb), nsim=99,
 type="moran")
#> 
#> 	Mantel permutation test for moran measure
#> 
#> data:  COL.OLD$CRIME 
#> weights: nb2listw(COL.nb) 
#> number of simulations + 1: 100 
#> 
#> statistic = 24.526, observed rank = 100, p-value = 0.01
#> alternative hypothesis: greater
#> sample estimates:
#> mean of permutations   sd of permutations 
#>           -0.3401381            4.5579226 
#>