Lee's L test for spatial autocorrelation

Lee's L test for spatial autocorrelation using a spatial weights matrix in weights list form. The assumptions underlying the test are sensitive to the form of the graph of neighbour relationships and other factors, and results may be checked against those of lee.mc permutations.

Usage

lee.test(x, y, listw, zero.policy=attr(listw, "zero.policy"),
 alternative="greater", na.action=na.fail, spChk=NULL)

Arguments

x: a numeric vector the same length as the neighbours list in listw
y: a numeric vector the same length as the neighbours list in listw
listw: a listw object created for example by nb2listw

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), less or two.sided.

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. If na.pass is used, zero is substituted for NA values in calculating the spatial lag
spChk: should the data vector names be checked against the spatial objects for identity integrity, TRUE, or FALSE, default NULL to use get.spChkOption()

Value

A list with class htest containing the following components:

statistic: the value of the standard deviate of Lee's L.
p.value: the p-value of the test.
estimate: the value of the observed Lee's L, its expectation and variance under the method assumption.
alternative: a character string describing the alternative hypothesis.
method: a character string giving the assumption used for calculating the standard deviate.
data.name: a character string giving the name(s) of the data.

Note

See Lee (2004) for details on how the asymptotic expectation and variance of Lee's L is computed. In particular, check Lee (2004), table 1, page 1690.

This test may fail for large datasets as the computation of the asymptotic expectation and variance requires the use of dense matrices.

References

Lee (2004). A generalized significance testing method for global measures of spatial association: an extension of the Mantel test. Environment and Planning A 2004, volume 36, pages 1687 - 1703

Author

Roger Bivand and Virgilio GÃ³mez-Rubio Virgilio.Gomez@uclm.es

Examples

data(oldcol)
col.W <- nb2listw(COL.nb, style="W")
crime <- COL.OLD$CRIME

lee.test(crime, crime, col.W, zero.policy=TRUE)
#> 
#> 	Lee's L statistic randomisation
#> 
#> data:  crime ,  crime 
#> weights: col.W  
#> 
#> Lee's L statistic standard deviate = 5.2343, p-value = 8.279e-08
#> alternative hypothesis: greater
#> sample estimates:
#> Lee's L statistic       Expectation          Variance 
#>       0.547064219       0.239417989       0.003454459 
#> 

#Test with missing values
x<-crime
y<-crime
x[1:5]<-NA
y[3:7]<-NA

lee.test(x, y, col.W, zero.policy=TRUE, na.action=na.omit)
#> 
#> 	Lee's L statistic randomisation
#> 
#> data:  x ,  y 
#> weights: col.W 
#> omitted: 1, 2, 3, 4, 5, 6, 7 
#> 
#> Lee's L statistic standard deviate = 6.6873, p-value = 1.137e-11
#> alternative hypothesis: greater
#> sample estimates:
#> Lee's L statistic       Expectation          Variance 
#>       0.706469726       0.260143244       0.004454563 
#> 
#  lee.test(x, y, col.W, zero.policy=TRUE)#Stops with an error



data(boston, package="spData")
lw<-nb2listw(boston.soi)

x<-boston.c$CMEDV
y<-boston.c$CRIM

lee.test(x, y, lw, zero.policy=TRUE, alternative="less")
#> 
#> 	Lee's L statistic randomisation
#> 
#> data:  x ,  y 
#> weights: lw  
#> 
#> Lee's L statistic standard deviate = -11.54, p-value < 2.2e-16
#> alternative hypothesis: less
#> sample estimates:
#> Lee's L statistic       Expectation          Variance 
#>      -0.326297206      -0.105040316       0.000367637 
#> 

#Test with missing values
x[1:5]<-NA
y[3:7]<-NA

lee.test(x, y, lw, zero.policy=TRUE, alternative="less", na.action=na.omit)
#> 
#> 	Lee's L statistic randomisation
#> 
#> data:  x ,  y 
#> weights: lw 
#> omitted: 1, 2, 3, 4, 5, 6, 7 
#> 
#> Lee's L statistic standard deviate = -11.284, p-value < 2.2e-16
#> alternative hypothesis: less
#> sample estimates:
#> Lee's L statistic       Expectation          Variance 
#>     -0.3244748280     -0.1053154332      0.0003772109 
#>