BB, BW and Jtot join count statistic for k-coloured factors

A function for tallying join counts between same-colour and different colour spatial objects, where neighbour relations are defined by a weights list. Given the global counts in each colour, expected counts and variances are calculated under non-free sampling, and a z-value reported. Since multiple tests are reported, no p-values are given, allowing the user to adjust the significance level applied. Jtot is the count of all different-colour joins.

joincount.multi(fx, listw, zero.policy = attr(listw, "zero.policy"),
 spChk = NULL, adjust.n=TRUE)
# S3 method for jcmulti
print(x, ...)

Arguments

fx: a factor of the same length as the neighbours and weights objects 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
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 consistently (up to and including spdep 0.3-28 the adjustment was inconsistent - thanks to Tomoki NAKAYA for a careful bug report)
spChk: should the data vector names be checked against the spatial objects for identity integrity, TRUE, or FALSE, default NULL to use get.spChkOption()
x: object to be printed
...: arguments to be passed through for printing

Value

A matrix with class jcmulti with row and column names for observed and expected counts, variance, and z-value.

References

Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion, p. 20; Upton, G., Fingleton, B. 1985 Spatial data analysis by example: point pattern and qualitative data, Wiley, pp. 158--170.

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.

Examples

columbus <- st_read(system.file("shapes/columbus.gpkg", package="spData")[1], quiet=TRUE)
HICRIME <- cut(columbus$CRIME, breaks=c(0,35,80), labels=c("low","high"))
(nb <- poly2nb(columbus))
#> Neighbour list object:
#> Number of regions: 49 
#> Number of nonzero links: 236 
#> Percentage nonzero weights: 9.829238 
#> Average number of links: 4.816327 
lw <- nb2listw(nb, style="B")
joincount.multi(HICRIME, lw)
#>           Joincount Expected Variance z-value
#> low:low      35.000   30.102   19.247  1.1164
#> high:high    54.000   27.694   18.219  6.1630
#> high:low     29.000   60.204   26.630 -6.0468
#> Jtot         29.000   60.204   26.630 -6.0468
col_geoms <- st_geometry(columbus)
col_geoms[21] <- st_buffer(col_geoms[21], dist=-0.05)
st_geometry(columbus) <- col_geoms
(nb <- poly2nb(columbus))
#> Neighbour list object:
#> Number of regions: 49 
#> Number of nonzero links: 230 
#> Percentage nonzero weights: 9.579342 
#> Average number of links: 4.693878 
#> 1 region with no links:
#> 21
#> 3 disjoint connected subgraphs
lw <- nb2listw(nb, style="B", zero.policy=TRUE)
joincount.multi(HICRIME, lw)
#>           Joincount Expected Variance z-value
#> low:low      35.000   30.585   19.350  1.0036
#> high:high    52.000   28.138   18.342  5.5716
#> high:low     28.000   61.170   25.882 -6.5200
#> Jtot         28.000   61.170   33.190 -5.7577
# \dontrun{
data(oldcol)
HICRIME <- cut(COL.OLD$CRIME, breaks=c(0,35,80), labels=c("low","high"))
names(HICRIME) <- rownames(COL.OLD)
joincount.multi(HICRIME, nb2listw(COL.nb, style="B"))
#>           Joincount Expected Variance z-value
#> low:low      34.000   29.592   18.895  1.0141
#> high:high    54.000   27.224   17.888  6.3307
#> high:low     28.000   59.184   26.233 -6.0884
#> Jtot         28.000   59.184   26.233 -6.0884
data(hopkins, package="spData")
image(1:32, 1:32, hopkins[5:36,36:5], breaks=c(-0.5, 3.5, 20),
 col=c("white", "black"))
box()

hopkins.rook.nb <- cell2nb(32, 32, type="rook")
unlist(spweights.constants(nb2listw(hopkins.rook.nb, style="B")))
#>       n      n1      n2      n3      nn      S0      S1      S2 
#>    1024    1023    1022    1021 1048576    3968    7936   61984 
hopkins.queen.nb <- cell2nb(32, 32, type="queen")
hopkins.bishop.nb <- diffnb(hopkins.rook.nb, hopkins.queen.nb, verbose=FALSE)
hopkins4 <- hopkins[5:36,36:5]
hopkins4[which(hopkins4 > 3, arr.ind=TRUE)] <- 4
hopkins4.f <- factor(hopkins4)
table(hopkins4.f)
#> hopkins4.f
#>   0   1   2   3   4 
#> 657 215  98  30  24 
joincount.multi(hopkins4.f, nb2listw(hopkins.rook.nb, style="B"))
#>       Joincount   Expected   Variance z-value
#> 0:0   864.00000  816.27273  116.05233  4.4304
#> 1:1    94.00000   87.14015   55.25216  0.9229
#> 2:2    18.00000   18.00379   14.81562 -0.0010
#> 3:3     2.00000    1.64773    1.55539  0.2825
#> 4:4     5.00000    1.04545    0.99845  3.9576
#> 1:0   503.00000  535.05682  227.76750 -2.1241
#> 2:0   213.00000  243.88636   97.21769 -3.1325
#> 2:1    99.00000   79.81061   59.01930  2.4978
#> 3:0    61.00000   74.65909   28.58592 -2.5547
#> 3:1    28.00000   24.43182   18.99976  0.8186
#> 3:2    15.00000   11.13636    9.82411  1.2327
#> 4:0    40.00000   59.72727   22.78583 -4.1327
#> 4:1    23.00000   19.54545   15.26564  0.8842
#> 4:2    14.00000    8.90909    7.90051  1.8112
#> 4:3     5.00000    2.72727    2.58616  1.4133
#> Jtot 1001.00000 1059.89015  273.78610 -3.5591
cat("replicates Upton & Fingleton table 3.4 (p. 166)\n")
#> replicates Upton & Fingleton table 3.4 (p. 166)
joincount.multi(hopkins4.f, nb2listw(hopkins.bishop.nb, style="B"))
#>       Joincount   Expected   Variance z-value
#> 0:0   823.00000  790.76420  144.44877  2.6821
#> 1:1   101.00000   84.41702   55.98143  2.2164
#> 2:2    19.00000   17.44117   14.61542  0.4077
#> 3:3     3.00000    1.59624    1.51444  1.1407
#> 4:4     3.00000    1.01278    0.97111  2.0166
#> 1:0   497.00000  518.33629  234.93545 -1.3920
#> 2:0   216.00000  236.26491  104.42142 -1.9831
#> 2:1    81.00000   77.31652   58.70829  0.4807
#> 3:0    58.00000   72.32599   31.49151 -2.5529
#> 3:1    21.00000   23.66832   18.85316 -0.6145
#> 3:2    17.00000   10.78835    9.62487  2.0022
#> 4:0    48.00000   57.86080   25.15973 -1.9659
#> 4:1    21.00000   18.93466   15.14473  0.5307
#> 4:2    10.00000    8.63068    7.73708  0.4923
#> 4:3     4.00000    2.64205    2.51686  0.8560
#> Jtot  973.00000 1026.76858  284.51030 -3.1877
cat("replicates Upton & Fingleton table 3.6 (p. 168)\n")
#> replicates Upton & Fingleton table 3.6 (p. 168)
joincount.multi(hopkins4.f, nb2listw(hopkins.queen.nb, style="B"))
#>      Joincount  Expected  Variance z-value
#> 0:0  1687.0000 1607.0369  303.8034  4.5877
#> 1:1   195.0000  171.5572  114.2057  2.1936
#> 2:2    37.0000   35.4450   29.6821  0.2854
#> 3:3     5.0000    3.2440    3.0687  1.0024
#> 4:4     8.0000    2.0582    1.9674  4.2361
#> 1:0  1000.0000 1053.3931  480.6959 -2.4353
#> 2:0   429.0000  480.1513  215.0360 -3.4882
#> 2:1   180.0000  157.1271  119.3987  2.0932
#> 3:0   119.0000  146.9851   65.1029 -3.4684
#> 3:1    49.0000   48.1001   38.3268  0.1454
#> 3:2    32.0000   21.9247   19.5237  2.2802
#> 4:0    88.0000  117.5881   52.0312 -4.1019
#> 4:1    44.0000   38.4801   30.7868  0.9948
#> 4:2    24.0000   17.5398   15.6933  1.6308
#> 4:3     9.0000    5.3693    5.0994  1.6078
#> Jtot 1974.0000 2086.6587  582.8326 -4.6665
cat("replicates Upton & Fingleton table 3.7 (p. 169)\n")
#> replicates Upton & Fingleton table 3.7 (p. 169)
# }