moran.test: Moran's I test for spatial autocorrelation
In spdep: Spatial Dependence: Weighting Schemes, Statistics and Models

Description Usage Arguments Value Note Author(s) References See Also Examples

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.

1 2	moran.test(x, listw, randomisation=TRUE, zero.policy=NULL, alternative="greater", rank = FALSE, na.action=na.fail, spChk=NULL, adjust.n=TRUE)

`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

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.

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.

Roger Bivand Roger.Bivand@nhh.no

Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion, p. 21.

moran, moran.mc, listw2U

data(oldcol)
coords.OLD <- cbind(COL.OLD$X, COL.OLD$Y)
moran.test(COL.OLD$CRIME, nb2listw(COL.nb, style="W"))
moran.test(COL.OLD$CRIME, nb2listw(COL.nb, style="B"))
moran.test(COL.OLD$CRIME, nb2listw(COL.nb, style="C"))
moran.test(COL.OLD$CRIME, nb2listw(COL.nb, style="S"))
moran.test(COL.OLD$CRIME, nb2listw(COL.nb, style="W"),
 randomisation=FALSE)
colold.lags <- nblag(COL.nb, 3)
moran.test(COL.OLD$CRIME, nb2listw(colold.lags[[2]],
 style="W"))
moran.test(COL.OLD$CRIME, nb2listw(colold.lags[[3]],
 style="W"))
print(is.symmetric.nb(COL.nb))
COL.k4.nb <- knn2nb(knearneigh(coords.OLD, 4))
print(is.symmetric.nb(COL.k4.nb))
moran.test(COL.OLD$CRIME, nb2listw(COL.k4.nb, style="W"))
moran.test(COL.OLD$CRIME, nb2listw(COL.k4.nb, style="W"),
 randomisation=FALSE)
cat("Note: non-symmetric weights matrix, use listw2U()")
moran.test(COL.OLD$CRIME, listw2U(nb2listw(COL.k4.nb,
 style="W")))
moran.test(COL.OLD$CRIME, listw2U(nb2listw(COL.k4.nb,
 style="W")), randomisation=FALSE)
ranks <- rank(COL.OLD$CRIME)
names(ranks) <- rownames(COL.OLD)
moran.test(ranks, nb2listw(COL.nb, style="W"), rank=TRUE)
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))
res
moran.test(crime, nb2listw(COL.nb, style="W"), zero.policy=TRUE,
 na.action=na.omit)
moran.test(crime, nb2listw(COL.nb, style="W"), zero.policy=TRUE,
 na.action=na.exclude)
moran.test(crime, nb2listw(COL.nb, style="W"), na.action=na.pass)

Loading required package: sp
Loading required package: Matrix

	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 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 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 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 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 


	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 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 

[1] TRUE
[1] FALSE

	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 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 

Note: non-symmetric weights matrix, use listw2U()
	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 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 


	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 

Error in na.fail.default(x) : missing values in object
[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 I test under randomisation

data:  crime  
weights: nb2listw(COL.nb, style = "W") 
omitted: 9, 10, 14, 16, 21, 32, 37, 38, 45, 48 

Moran I statistic standard deviate = 3.7094, p-value = 0.0001039
alternative hypothesis: greater
sample estimates:
Moran I statistic       Expectation          Variance 
       0.40185281       -0.02702703        0.01336779 


	Moran I test under randomisation

data:  crime  
weights: nb2listw(COL.nb, style = "W") 
omitted: 9, 10, 14, 16, 21, 32, 37, 38, 45, 48 

Moran I statistic standard deviate = 3.7094, p-value = 0.0001039
alternative hypothesis: greater
sample estimates:
Moran I statistic       Expectation          Variance 
       0.40185281       -0.02702703        0.01336779 


	Moran I test under randomisation

data:  crime  
weights: nb2listw(COL.nb, style = "W")  

Moran I statistic standard deviate = 1.8785, p-value = 0.03016
alternative hypothesis: greater
sample estimates:
Moran I statistic       Expectation          Variance 
      0.154373808      -0.020833333       0.008699084 

Warning message:
In lag.listw(listw, z, zero.policy = zero.policy, NAOK = NAOK) :
  NAs in lagged values