The 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.

joincount.test(fx, listw, zero.policy=NULL, alternative="greater",
# S3 method for jclist
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 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".

sampling

default “nonfree”, may be “free”

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.

## References

Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion, pp. 19-20.

## 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.

joincount.mc, joincount.multi, listw2U

## 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
#>