Moran's I test for spatial autocorrelation
moran.test.RdMoran'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.
Usage
moran.test(x, listw, randomisation=TRUE, zero.policy=attr(listw, "zero.policy"),
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
listwobject created for example bynb2listw- randomisation
variance of I calculated under the assumption of randomisation, if FALSE normality
- zero.policy
default
attr(listw, "zero.policy")as set whenlistwwas 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.
- 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 bena.omitorna.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 tonb2listwmay be subsetted. Ifna.passis 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
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)
#> Warning: subsetting caused increase in subgraph count
#>
#> Moran I test under randomisation
#>
#> data: crime
#> weights: nb2listw(COL.nb, style = "W")
#> omitted: 7, 10, 12, 13, 14, 18, 26, 41, 42, 47
#> n reduced by no-neighbour observations
#>
#> Moran I statistic standard deviate = 3.9826, p-value = 3.408e-05
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic Expectation Variance
#> 0.43709247 -0.02702703 0.01358066
#>
moran.test(crime, nb2listw(COL.nb, style="W"), zero.policy=TRUE,
na.action=na.exclude)
#> Warning: subsetting caused increase in subgraph count
#>
#> Moran I test under randomisation
#>
#> data: crime
#> weights: nb2listw(COL.nb, style = "W")
#> omitted: 7, 10, 12, 13, 14, 18, 26, 41, 42, 47
#> n reduced by no-neighbour observations
#>
#> Moran I statistic standard deviate = 3.9826, p-value = 3.408e-05
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic Expectation Variance
#> 0.43709247 -0.02702703 0.01358066
#>
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 = 2.4925, p-value = 0.006342
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic Expectation Variance
#> 0.212782897 -0.020833333 0.008784908
#>
columbus <- st_read(system.file("shapes/columbus.gpkg", package="spData")[1], quiet=TRUE)
col_geoms <- st_geometry(columbus)
col_geoms[1] <- st_buffer(col_geoms[1], dist=-0.05)
st_geometry(columbus) <- col_geoms
(nb1 <- poly2nb(columbus))
#> Warning: some observations have no neighbours;
#> if this seems unexpected, try increasing the snap argument.
#> Warning: neighbour object has 2 sub-graphs;
#> if this sub-graph count seems unexpected, try increasing the snap argument.
#> Neighbour list object:
#> Number of regions: 49
#> Number of nonzero links: 232
#> Percentage nonzero weights: 9.662641
#> Average number of links: 4.734694
#> 1 region with no links:
#> 1
#> 2 disjoint connected subgraphs
try(lw <- nb2listw(nb1, style="W"))
#> Error in nb2listw(nb1, style = "W") :
#> Empty neighbour sets found (zero.policy: FALSE)
(lw <- nb2listw(nb1, style="W", zero.policy=TRUE))
#> Characteristics of weights list object:
#> Neighbour list object:
#> Number of regions: 49
#> Number of nonzero links: 232
#> Percentage nonzero weights: 9.662641
#> Average number of links: 4.734694
#> 1 region with no links:
#> 1
#> 2 disjoint connected subgraphs
#>
#> Weights style: W
#> Weights constants summary:
#> n nn S0 S1 S2
#> W 48 2304 48 22.23035 199.8188
moran.test(COL.OLD$CRIME, lw)
#>
#> Moran I test under randomisation
#>
#> data: COL.OLD$CRIME
#> weights: lw
#> n reduced by no-neighbour observations
#>
#> Moran I statistic standard deviate = 2.7707, p-value = 0.002797
#> alternative hypothesis: greater
#> sample estimates:
#> Moran I statistic Expectation Variance
#> 0.23901261 -0.02127660 0.00882535
#>