Geary's C test for spatial autocorrelation

Geary's 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 geary.mc permutations.

Usage

geary.test(x, listw, randomisation=TRUE, zero.policy=attr(listw, "zero.policy"),
    alternative="greater", spChk=NULL, adjust.n=TRUE, 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
randomisation: variance of I calculated under the assumption of randomisation, if FALSE normality
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".
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
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.
scale: default TRUE, may be FALSE to revert changes made to accommodate localC in November 2021 (see #151)

Value

A list with class htest containing the following components:

statistic: the value of the standard deviate of Geary's C, in the order given in Cliff and Ord 1973, p. 21, which is (EC - C) / sqrt(VC), that is with the sign reversed with respect to the more usual (C - EC) / sqrt(VC); this means that the “greater” alternative for the Geary C test corresponds to the “greater” alternative for Moran's I test.
p.value: the p-value of the test.
estimate: the value of the observed Geary's C, 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.

References

Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion, p. 21, Cliff, A. D., Ord, J. K. 1973 Spatial Autocorrelation, Pion, pp. 15-16, 21; Bivand RS, Wong DWS 2018 Comparing implementations of global and local indicators of spatial association. TEST, 27(3), 716–748 doi:10.1007/s11749-018-0599-x

Author

Roger Bivand Roger.Bivand@nhh.no

Note

The derivation of the test (Cliff and Ord, 1981, p. 18) assumes that the weights matrix is symmetric. For inherently non-symmetric matrices, such as k-nearest neighbour matrices, listw2U() can be used to make the matrix symmetric. In non-symmetric weights matrix cases, the variance of the test statistic may be negative (thanks to Franz Munoz I for a well documented bug report). Geary's C is affected by non-symmetric weights under normality much more than Moran's I. From 0.4-35, the sign of the standard deviate of C is changed to match Cliff and Ord (1973, p. 21).

Examples

data(oldcol)
geary.test(COL.OLD$CRIME, nb2listw(COL.nb, style="W"))
#> 
#> 	Geary C test under randomisation
#> 
#> data:  COL.OLD$CRIME 
#> weights: nb2listw(COL.nb, style = "W")   
#> 
#> Geary C statistic standard deviate = 4.7605, p-value = 9.655e-07
#> alternative hypothesis: Expectation greater than statistic
#> sample estimates:
#> Geary C statistic       Expectation          Variance 
#>        0.52986993        1.00000000        0.00975278 
#> 
geary.test(COL.OLD$CRIME, nb2listw(COL.nb, style="W"),
 randomisation=FALSE)
#> 
#> 	Geary C test under normality
#> 
#> data:  COL.OLD$CRIME 
#> weights: nb2listw(COL.nb, style = "W")   
#> 
#> Geary C statistic standard deviate = 4.6388, p-value = 1.752e-06
#> alternative hypothesis: Expectation greater than statistic
#> sample estimates:
#> Geary C statistic       Expectation          Variance 
#>        0.52986993        1.00000000        0.01027137 
#> 
colold.lags <- nblag(COL.nb, 3)
geary.test(COL.OLD$CRIME, nb2listw(colold.lags[[2]],
 style="W"))
#> 
#> 	Geary C test under randomisation
#> 
#> data:  COL.OLD$CRIME 
#> weights: nb2listw(colold.lags[[2]], style = "W")   
#> 
#> Geary C statistic standard deviate = 2.2896, p-value = 0.01102
#> alternative hypothesis: Expectation greater than statistic
#> sample estimates:
#> Geary C statistic       Expectation          Variance 
#>       0.811285136       1.000000000       0.006793327 
#> 
geary.test(COL.OLD$CRIME, nb2listw(colold.lags[[3]],
 style="W"), alternative="greater")
#> 
#> 	Geary C test under randomisation
#> 
#> data:  COL.OLD$CRIME 
#> weights: nb2listw(colold.lags[[3]], style = "W")   
#> 
#> Geary C statistic standard deviate = -1.5667, p-value = 0.9414
#> alternative hypothesis: Expectation greater than statistic
#> sample estimates:
#> Geary C statistic       Expectation          Variance 
#>       1.130277918       1.000000000       0.006914551 
#> 
print(is.symmetric.nb(COL.nb))
#> [1] TRUE
coords.OLD <- cbind(COL.OLD$X, COL.OLD$Y)
COL.k4.nb <- knn2nb(knearneigh(coords.OLD, 4))
print(is.symmetric.nb(COL.k4.nb))
#> [1] FALSE
geary.test(COL.OLD$CRIME, nb2listw(COL.k4.nb, style="W"))
#> 
#> 	Geary C test under randomisation
#> 
#> data:  COL.OLD$CRIME 
#> weights: nb2listw(COL.k4.nb, style = "W")   
#> 
#> Geary C statistic standard deviate = 6.4415, p-value = 5.916e-11
#> alternative hypothesis: Expectation greater than statistic
#> sample estimates:
#> Geary C statistic       Expectation          Variance 
#>       0.399254423       1.000000000       0.008697812 
#> 
geary.test(COL.OLD$CRIME, nb2listw(COL.k4.nb, style="W"),
 randomisation=FALSE)
#> 
#> 	Geary C test under normality
#> 
#> data:  COL.OLD$CRIME 
#> weights: nb2listw(COL.k4.nb, style = "W")   
#> 
#> Geary C statistic standard deviate = 6.2873, p-value = 1.615e-10
#> alternative hypothesis: Expectation greater than statistic
#> sample estimates:
#> Geary C statistic       Expectation          Variance 
#>       0.399254423       1.000000000       0.009129529 
#> 
cat("Note non-symmetric weights matrix - use listw2U()\n")
#> Note non-symmetric weights matrix - use listw2U()
geary.test(COL.OLD$CRIME, listw2U(nb2listw(COL.k4.nb,
 style="W")))
#> 
#> 	Geary C test under randomisation
#> 
#> data:  COL.OLD$CRIME 
#> weights: listw2U(nb2listw(COL.k4.nb, style = "W"))   
#> 
#> Geary C statistic standard deviate = 6.4415, p-value = 5.916e-11
#> alternative hypothesis: Expectation greater than statistic
#> sample estimates:
#> Geary C statistic       Expectation          Variance 
#>       0.399254423       1.000000000       0.008697812 
#> 
geary.test(COL.OLD$CRIME, listw2U(nb2listw(COL.k4.nb,
 style="W")), randomisation=FALSE)
#> 
#> 	Geary C test under normality
#> 
#> data:  COL.OLD$CRIME 
#> weights: listw2U(nb2listw(COL.k4.nb, style = "W"))   
#> 
#> Geary C statistic standard deviate = 6.2873, p-value = 1.615e-10
#> alternative hypothesis: Expectation greater than statistic
#> sample estimates:
#> Geary C statistic       Expectation          Variance 
#>       0.399254423       1.000000000       0.009129529 
#> 
crime <- COL.OLD$CRIME
is.na(crime) <- sample(1:length(crime), 10)
try(geary.test(crime, nb2listw(COL.nb, style="W"), na.action=na.fail))
#> Error in na.fail.default(x) : missing values in object
geary.test(crime, nb2listw(COL.nb, style="W"), zero.policy=TRUE,
 na.action=na.omit)
#> 
#> 	Geary C test under randomisation
#> 
#> data:  crime 
#> weights: nb2listw(COL.nb, style = "W") 
#> omitted: 3, 4, 10, 20, 21, 23, 27, 29, 31, 38  
#> 
#> Geary C statistic standard deviate = 4.2726, p-value = 9.661e-06
#> alternative hypothesis: Expectation greater than statistic
#> sample estimates:
#> Geary C statistic       Expectation          Variance 
#>        0.45199742        1.00000000        0.01645071 
#> 
geary.test(crime, nb2listw(COL.nb, style="W"), zero.policy=TRUE,
 na.action=na.exclude)
#> 
#> 	Geary C test under randomisation
#> 
#> data:  crime 
#> weights: nb2listw(COL.nb, style = "W") 
#> omitted: 3, 4, 10, 20, 21, 23, 27, 29, 31, 38  
#> 
#> Geary C statistic standard deviate = 4.2726, p-value = 9.661e-06
#> alternative hypothesis: Expectation greater than statistic
#> sample estimates:
#> Geary C statistic       Expectation          Variance 
#>        0.45199742        1.00000000        0.01645071 
#> 
try(geary.test(crime, nb2listw(COL.nb, style="W"), na.action=na.pass))
#> Error in geary.test(crime, nb2listw(COL.nb, style = "W"), na.action = na.pass) : 
#>   na.pass not permitted