BB join count statistic for k-coloured factors
joincount.test.RdThe BB join count test for spatial autocorrelation using a spatial
weights matrix in weights list form for testing whether same-colour joins
occur more frequently than would be expected if the zones were labelled
in a spatially random way. 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 joincount.mc
permutations.
Arguments
- fx
a factor of the same length as the neighbours and weights objects in listw
- listw
a
listwobject created for example bynb2listw- 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".
- sampling
default “nonfree”, may be “free”
- 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 list with class jclist of lists with class htest for each of the k colours containing the following components:
- statistic
the value of the standard deviate of the join count statistic.
- p.value
the p-value of the test.
- estimate
the value of the observed statistic, its expectation and variance under non-free sampling.
- alternative
a character string describing the alternative hypothesis.
- method
a character string giving the method used.
- data.name
a character string giving the name(s) of the data.
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
data(oldcol)
HICRIME <- cut(COL.OLD$CRIME, breaks=c(0,35,80), labels=c("low","high"))
names(HICRIME) <- rownames(COL.OLD)
joincount.test(HICRIME, nb2listw(COL.nb, style="B"))
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: nb2listw(COL.nb, style = "B")
#>
#> Std. deviate for low = 1.0141, p-value = 0.1553
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 34.00000 29.59184 18.89550
#>
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: nb2listw(COL.nb, style = "B")
#>
#> Std. deviate for high = 6.3307, p-value = 1.22e-10
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 54.00000 27.22449 17.88838
#>
joincount.test(HICRIME, nb2listw(COL.nb, style="B"), sampling="free")
#>
#> Join count test under free sampling
#>
#> data: HICRIME
#> weights: nb2listw(COL.nb, style = "B")
#>
#> Std. deviate for low = 0.3993, p-value = 0.3448
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 34.00000 30.19575 90.76809
#>
#>
#> Join count test under free sampling
#>
#> data: HICRIME
#> weights: nb2listw(COL.nb, style = "B")
#>
#> Std. deviate for high = 2.8518, p-value = 0.002173
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 54.0000 27.8284 84.2198
#>
joincount.test(HICRIME, nb2listw(COL.nb, style="C"))
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: nb2listw(COL.nb, style = "C")
#>
#> Std. deviate for low = 1.0141, p-value = 0.1553
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 7.1810345 6.2500000 0.8428969
#>
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: nb2listw(COL.nb, style = "C")
#>
#> Std. deviate for high = 6.3307, p-value = 1.22e-10
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 11.4051724 5.7500000 0.7979712
#>
joincount.test(HICRIME, nb2listw(COL.nb, style="S"))
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: nb2listw(COL.nb, style = "S")
#>
#> Std. deviate for low = 2.5786, p-value = 0.00496
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 8.2425673 6.2500000 0.5971141
#>
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: nb2listw(COL.nb, style = "S")
#>
#> Std. deviate for high = 6.1736, p-value = 3.337e-10
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 10.4249914 5.7500000 0.5734265
#>
joincount.test(HICRIME, nb2listw(COL.nb, style="W"))
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: nb2listw(COL.nb, style = "W")
#>
#> Std. deviate for low = 4.6675, p-value = 1.524e-06
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 9.5190476 6.2500000 0.4905378
#>
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: nb2listw(COL.nb, style = "W")
#>
#> Std. deviate for high = 5.1205, p-value = 1.523e-07
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 9.2920635 5.7500000 0.4784979
#>
by(card(COL.nb), HICRIME, summary)
#> HICRIME: low
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 2.00 2.00 4.00 3.84 4.00 10.00
#> ------------------------------------------------------------
#> HICRIME: high
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 3.000 4.750 6.000 5.667 7.000 9.000
print(is.symmetric.nb(COL.nb))
#> [1] TRUE
coords.OLD <- cbind(COL.OLD$X, COL.OLD$Y)
COL.k4.nb <- knn2nb(knearneigh(coords.OLD, 4))
print(is.symmetric.nb(COL.k4.nb))
#> [1] FALSE
joincount.test(HICRIME, nb2listw(COL.k4.nb, style="B"))
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: nb2listw(COL.k4.nb, style = "B")
#>
#> Std. deviate for low = 4.3698, p-value = 6.217e-06
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 36.500000 25.000000 6.925749
#>
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: nb2listw(COL.k4.nb, style = "B")
#>
#> Std. deviate for high = 6.7293, p-value = 8.523e-12
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 40.500000 23.000000 6.762918
#>
cat("Note non-symmetric weights matrix - use listw2U()\n")
#> Note non-symmetric weights matrix - use listw2U()
joincount.test(HICRIME, listw2U(nb2listw(COL.k4.nb, style="B")))
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: listw2U(nb2listw(COL.k4.nb, style = "B"))
#>
#> Std. deviate for low = 4.3698, p-value = 6.217e-06
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 36.500000 25.000000 6.925749
#>
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: listw2U(nb2listw(COL.k4.nb, style = "B"))
#>
#> Std. deviate for high = 6.7293, p-value = 8.523e-12
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 40.500000 23.000000 6.762918
#>
columbus <- st_read(system.file("shapes/columbus.gpkg", package="spData")[1], quiet=TRUE)
HICRIME <- cut(columbus$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.test(HICRIME, lw)
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: lw
#>
#> Std. deviate for low = 1.1164, p-value = 0.1321
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 35.00000 30.10204 19.24708
#>
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: lw
#>
#> Std. deviate for high = 6.163, p-value = 3.57e-10
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 54.00000 27.69388 18.21936
#>
col_geoms <- st_geometry(columbus)
col_geoms[21] <- st_buffer(col_geoms[21], dist=-0.05)
st_geometry(columbus) <- col_geoms
(nb <- poly2nb(columbus))
#> Warning: some observations have no neighbours;
#> if this seems unexpected, try increasing the snap argument.
#> Warning: neighbour object has 3 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: 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.test(HICRIME, lw)
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: lw
#>
#> Std. deviate for low = 1.0036, p-value = 0.1578
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 35.00000 30.58511 19.35032
#>
#>
#> Join count test under nonfree sampling
#>
#> data: HICRIME
#> weights: lw
#>
#> Std. deviate for high = 5.5716, p-value = 1.262e-08
#> alternative hypothesis: greater
#> sample estimates:
#> Same colour statistic Expectation Variance
#> 52.00000 28.13830 18.34187
#>