Moran'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 moran.mc permutations.

moran.test(x, listw, randomisation=TRUE, zero.policy=NULL,
 alternative="greater", rank = FALSE, na.action=na.fail, spChk=NULL,
 adjust.n=TRUE, drop.EI2=FALSE)

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 NULL, 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.

rank

logical value - default FALSE for continuous variables, if TRUE, uses the adaptation of Moran's I for ranks suggested by Cliff and Ord (1981, p. 46)

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()

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

drop.EI2

default FALSE, if TRUE, emulate CrimeStat <= 4.02

Value

A list with class htest containing the following components:

statistic

the value of the standard deviate of Moran's I.

p.value

the p-value of the test.

estimate

the value of the observed Moran's I, 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

Var(I) is taken from Cliff and Ord (1969, p. 28), and Goodchild's CATMOG 47 (1986), see also Upton & Fingleton (1985) p. 171; it agrees with SpaceStat, see Tutorial workbook Chapter 22; VI is the second crude moment minus the square of the first crude moment. 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.

References

Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion, p. 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

See also

Examples

data(oldcol)
coords.OLD <- cbind(COL.OLD$X, COL.OLD$Y)
moran.test(COL.OLD$CRIME, nb2listw(COL.nb, style="W"))
#> 
#> 	Moran I test under randomisation
#> 
#> data:  COL.OLD$CRIME  
#> weights: nb2listw(COL.nb, style = "W")    
#> 
#> Moran I statistic standard deviate = 5.6341, p-value = 8.797e-09
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>       0.510951264      -0.020833333       0.008908762 
#> 
moran.test(COL.OLD$CRIME, nb2listw(COL.nb, style="B"))
#> 
#> 	Moran I test under randomisation
#> 
#> data:  COL.OLD$CRIME  
#> weights: nb2listw(COL.nb, style = "B")    
#> 
#> Moran I statistic standard deviate = 6.2116, p-value = 2.622e-10
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>        0.52063815       -0.02083333        0.00759872 
#> 
moran.test(COL.OLD$CRIME, nb2listw(COL.nb, style="C"))
#> 
#> 	Moran I test under randomisation
#> 
#> data:  COL.OLD$CRIME  
#> weights: nb2listw(COL.nb, style = "C")    
#> 
#> Moran I statistic standard deviate = 6.2116, p-value = 2.622e-10
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>        0.52063815       -0.02083333        0.00759872 
#> 
moran.test(COL.OLD$CRIME, nb2listw(COL.nb, style="S"))
#> 
#> 	Moran I test under randomisation
#> 
#> data:  COL.OLD$CRIME  
#> weights: nb2listw(COL.nb, style = "S")    
#> 
#> Moran I statistic standard deviate = 5.9786, p-value = 1.125e-09
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>       0.512957470      -0.020833333       0.007971504 
#> 
moran.test(COL.OLD$CRIME, nb2listw(COL.nb, style="W"),
 randomisation=FALSE)
#> 
#> 	Moran I test under normality
#> 
#> data:  COL.OLD$CRIME  
#> weights: nb2listw(COL.nb, style = "W")    
#> 
#> Moran I statistic standard deviate = 5.6754, p-value = 6.92e-09
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>       0.510951264      -0.020833333       0.008779831 
#> 
colold.lags <- nblag(COL.nb, 3)
moran.test(COL.OLD$CRIME, nb2listw(colold.lags[[2]],
 style="W"))
#> 
#> 	Moran I test under randomisation
#> 
#> data:  COL.OLD$CRIME  
#> weights: nb2listw(colold.lags[[2]], style = "W")    
#> 
#> Moran I statistic standard deviate = 2.6076, p-value = 0.004559
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>       0.168485742      -0.020833333       0.005271314 
#> 
moran.test(COL.OLD$CRIME, nb2listw(colold.lags[[3]],
 style="W"))
#> 
#> 	Moran I test under randomisation
#> 
#> data:  COL.OLD$CRIME  
#> weights: nb2listw(colold.lags[[3]], style = "W")    
#> 
#> Moran I statistic standard deviate = -1.7896, p-value = 0.9632
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>      -0.138930745      -0.020833333       0.004354683 
#> 
print(is.symmetric.nb(COL.nb))
#> [1] TRUE
COL.k4.nb <- knn2nb(knearneigh(coords.OLD, 4))
print(is.symmetric.nb(COL.k4.nb))
#> [1] FALSE
moran.test(COL.OLD$CRIME, nb2listw(COL.k4.nb, style="W"))
#> 
#> 	Moran I test under randomisation
#> 
#> data:  COL.OLD$CRIME  
#> weights: nb2listw(COL.k4.nb, style = "W")    
#> 
#> Moran I statistic standard deviate = 7.2183, p-value = 2.632e-13
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>       0.624933667      -0.020833333       0.008003503 
#> 
moran.test(COL.OLD$CRIME, nb2listw(COL.k4.nb, style="W"),
 randomisation=FALSE)
#> 
#> 	Moran I test under normality
#> 
#> data:  COL.OLD$CRIME  
#> weights: nb2listw(COL.k4.nb, style = "W")    
#> 
#> Moran I statistic standard deviate = 7.2711, p-value = 1.782e-13
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>       0.624933667      -0.020833333       0.007887613 
#> 
cat("Note: non-symmetric weights matrix, use listw2U()")
#> Note: non-symmetric weights matrix, use listw2U()
moran.test(COL.OLD$CRIME, listw2U(nb2listw(COL.k4.nb,
 style="W")))
#> 
#> 	Moran I test under randomisation
#> 
#> data:  COL.OLD$CRIME  
#> weights: listw2U(nb2listw(COL.k4.nb, style = "W"))    
#> 
#> Moran I statistic standard deviate = 7.2183, p-value = 2.632e-13
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>       0.624933667      -0.020833333       0.008003503 
#> 
moran.test(COL.OLD$CRIME, listw2U(nb2listw(COL.k4.nb,
 style="W")), randomisation=FALSE)
#> 
#> 	Moran I test under normality
#> 
#> data:  COL.OLD$CRIME  
#> weights: listw2U(nb2listw(COL.k4.nb, style = "W"))    
#> 
#> Moran I statistic standard deviate = 7.2711, p-value = 1.782e-13
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>       0.624933667      -0.020833333       0.007887613 
#> 
ranks <- rank(COL.OLD$CRIME)
names(ranks) <- rownames(COL.OLD)
moran.test(ranks, nb2listw(COL.nb, style="W"), rank=TRUE)
#> 
#> 	Moran I test under randomisation
#> 
#> data:  ranks using rank correction 
#> weights: nb2listw(COL.nb, style = "W")    
#> 
#> Moran I statistic standard deviate = 6.3815, p-value = 8.766e-11
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>       0.584333495      -0.020833333       0.008992923 
#> 
crime <- COL.OLD$CRIME
is.na(crime) <- sample(1:length(crime), 10)
res <- try(moran.test(crime, nb2listw(COL.nb, style="W"),
 na.action=na.fail))
#> Error in na.fail.default(x) : missing values in object
res
#> [1] "Error in na.fail.default(x) : missing values in object\n"
#> attr(,"class")
#> [1] "try-error"
#> attr(,"condition")
#> <simpleError in na.fail.default(x): missing values in object>
moran.test(crime, nb2listw(COL.nb, style="W"), zero.policy=TRUE,
 na.action=na.omit)
#> 
#> 	Moran I test under randomisation
#> 
#> data:  crime  
#> weights: nb2listw(COL.nb, style = "W") 
#> omitted: 5, 7, 9, 15, 19, 33, 40, 45, 47, 48   
#> 
#> Moran I statistic standard deviate = 3.8759, p-value = 5.311e-05
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>        0.42342213       -0.02631579        0.01346387 
#> 
moran.test(crime, nb2listw(COL.nb, style="W"), zero.policy=TRUE,
 na.action=na.exclude)
#> 
#> 	Moran I test under randomisation
#> 
#> data:  crime  
#> weights: nb2listw(COL.nb, style = "W") 
#> omitted: 5, 7, 9, 15, 19, 33, 40, 45, 47, 48   
#> 
#> Moran I statistic standard deviate = 3.8759, p-value = 5.311e-05
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>        0.42342213       -0.02631579        0.01346387 
#> 
moran.test(crime, nb2listw(COL.nb, style="W"), na.action=na.pass)
#> Warning: NAs in lagged values
#> 
#> 	Moran I test under randomisation
#> 
#> data:  crime  
#> weights: nb2listw(COL.nb, style = "W")    
#> 
#> Moran I statistic standard deviate = 1.3275, p-value = 0.09216
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic       Expectation          Variance 
#>       0.103248466      -0.020833333       0.008736122 
#>