library(spdep)
library(spatialreg)
library(sf)
library(ggplot2)

# 1 Introduction

This guide show the functionalities of the spqdep package to test spatial dependence on qualitative dataset.

## 1.1 Datasets

Two data sets will be used as examples in this guide:

• provinces_spain: The division of Spain into provinces. It is a multypolygon geometry with isolated provinces (islands without neighbouring provinces). See by example Paez et al. 2021.

• FastFood.sf: The data set used as example in . It is a geometry of points.

The package is install like usual and the dataset can be loaded using the next code

library(spqdep)
data("provinces_spain", package = "spqdep")
data("FastFood.sf", package = "spqdep")

## 1.2 Data Generating Process (DGP)

Additional to the two dataset available in the spqdep package. The user can generate structured spatial processes using the function. The DGP generate with this function defined in .

The next code show how to generate a random process on a set of random points localized in a square 1x1. In this case, the connectivity criteria is based on the 4 near neighborhood.

set.seed(123)
N <- 100
cx <- runif(N)
cy <- runif(N)
coor <- cbind(cx,cy)
p <- c(1/6,3/6,2/6)
rho = 0.5
listw <- spdep::nb2listw(knn2nb(knearneigh(coor, k = 4)))
fx <- dgp.spq(list = listw, p = p, rho = rho)

The next plot show the qualitative spatial process defined.

ggplot(data.frame(fx = fx, cx = cx, cy = cy), aes(x = cx, y = cy, color = fx)) +
geom_point(size = 6) +
theme_bw()

# 2 Q-test

• The Q-test is based on m-surroundings

• Before to apply the Q-test it is necessary define a set of the m-surroundings

• The function generate a set of m-surrounding.

• The user can tuning several parameters to obtain a congruent set of m-surroundings.

## 2.1 m-surroundings

• m.surround() is the function to generate m-surroundings.

• The output of this function is a object of the class m_surr

• Using the method the user can explore the coherence of m-surroundings.

By example. the next code obtain m-surroundings with length m = 3 and degree of overlapping r = 1:

m = 3
r = 1
mh <- m.surround(x = cbind(cx,cy), m = m, r = r)
class(mh)
## [1] "m_surr" "list"

## 2.2 Methods for the m_surr class

The spqdep have three methods that can be apply to this class: , and

• list the m-surroundings
print(mh)
##       [,1] [,2] [,3]
##  [1,]    1   19   17
##  [2,]    2   65   53
##  [3,]    3   42   77
##  [4,]    4   11   26
##  [5,]    6   18   85
##  [6,]    7   79   93
##  [7,]    8   21   31
##  [8,]   13   58   68
##  [9,]   15   98   74
## [10,]   17   76   96
## [11,]   20   87   24
## [12,]   22   82   92
## [13,]   25    9   61
## [14,]   26   14    7
## [15,]   27   72   13
## [16,]   31    5   20
## [17,]   37   73   89
## [18,]   38   30   15
## [19,]   41   91   62
## [20,]   46   47   49
## [21,]   48   43   55
## [22,]   49   29   60
## [23,]   52   86    3
## [24,]   53   97   69
## [25,]   55   28   27
## [26,]   56   64   41
## [27,]   57   35   63
## [28,]   60   44   70
## [29,]   61   78   94
## [30,]   62   54   46
## [31,]   63   83   75
## [32,]   67   50   37
## [33,]   68   84    2
## [34,]   69   59   22
## [35,]   70   12   99
## [36,]   71   34   67
## [37,]   74   51    6
## [38,]   75   36  100
## [39,]   77   88    8
## [40,]   85   80   57
## [41,]   89   32    4
## [42,]   90   45   38
## [43,]   92   23   25
## [44,]   93   39   95
## [45,]   94   33   71
## [46,]   95   81   90
## [47,]   96   40   56
## [48,]   99   10   48
## [49,]  100   66   52
• generate a summary of some characteristics of the m-surroundings
summary(mh)
##
## Characteristics of m-surrounding:
##
## Number of m-surrounding (R): 49
## Length of m-surrounding (m): 3
## Number no-symbolized observations: 1
##
## List of no-symbolized observations:
## 16
##
## List of the degree overlaping:
##     There are 2 m-surrounding that have intersection with 1 m-surrounding
##     There are 47 m-surrounding that have intersection with 2 m-surrounding
## Mean degree of overlaping: 1.9592
• show the spatial structure of the m-surroundings
plot(mh, type = 1)

• With the argument control the user can tuning some characteristics of the m-surroundings.

By example, with control argument, the user can ‘prune’ non-coherent m-surroundings.

control <- list (dtmaxknn = 10)
mh.prune <- m.surround(x = coor, m = m, r = r, control = control)
plot(mh.prune)

## 2.3 The Q-test

• The function obtain the Q-test for a spatial process develop in .

The user must select the longitude of the m-surroundings (m) and the overlapping degree (r). In the next code example, the Q-test is obtain for the DGP spatial process (fx) obtain with the . The coordinates coor must be included as argument.

q.test <- Q.test(fx = fx, coor = coor, m = 3, r = 1)
• The output is a list with the result for symbols based on permutations (standard) and combinations (equivalent).

• The output of this function is an object of the spqtest class.

## 2.4 Distribution of Q-test

• The asymptotic distribution is the default distribution to obtain the significance of Q-test .

• Alternatively, the Monte Carlo method can be used to obtain the significance of the test. The paper describe this approach.

q.test.mc <- Q.test(fx = fx, coor = coor, m = 3, r = 1, distr = "mc")
summary(q.test.mc)
Qualitative Dependence Test (Q)
Distribution: mc. Distance: Euclidean
Q p.value k N m R n R/n 5k^m
V1 - standard-permutations
41.51 0.12900 3 100 3 100 27 3.70 135
V1 - equivalent-combinations
32.80 0.03600 3 100 3 100 10 10.00 135

## 2.5 Methods for the spqtest class

A summary can be apply to an object of the spqtest class:

summary(q.test)
Qualitative Dependence Test (Q)
Distribution: asymptotic. Distance: Euclidean
Q df p.value k N m r R n R/n 5k^m
V1 - standard-permutations
30.10 26 0.26354 3 100 3 1 49 27 1.81 135
V1 - equivalent-combinations
16.73 9 0.05318 3 100 3 1 49 10 4.90 135

The histogram of the number of symbols is obtain appling the plot method.

plot(q.test)
## [[1]]

##
## [[2]]

## 2.6 The Q-test using a sf object

• A sf object or a data frame can be used as input of the function:
# Case 3: With a sf object with isolated areas
sf_use_s2(FALSE)
## Spherical geometry (s2) switched off
provinces_spain$Male2Female <- factor(provinces_spain$Male2Female > 100)
levels(provinces_spain$Male2Female) = c("men","woman") f1 <- ~ Male2Female q.test.sf <- Q.test(formula = f1, data = provinces_spain, m = 3, r = 1) • The method show the histogram of the number of symbols plot(q.test.sf) ## [[1]] ## ## [[2]] # 3 Maps comparison. The QMap-test • The function obtain the test for maps comparison publish in ## 3.1 The QMap test The next code generate two qualitative spatial process with different levels of spatial dependence and the Q-Map is apply. p <- c(1/6,3/6,2/6) rho = 0.5 QY1 <- dgp.spq(p = p, listw = listw, rho = rho) rho = 0.8 QY2 <- dgp.spq(p = p, listw = listw, rho = rho) dt = data.frame(QY1,QY2) m = 3 r = 1 formula <- ~ QY1 + QY2 control <- list(dtmaxknn = 10) qmap <- Q.map.test(formula = formula, data = dt, coor = coor, m = m, r = r, type ="combinations", control = control) ## Warning in Q.map.test(formula = formula, data = dt, coor = coor, m = m, : The ## ratio between the number of symbolized observations and the number of symbols is ## lower than 5. • The output of id an object of the classes qmap and htest ## 3.2 Methods for qmap class • The qmap object is a list with two elements. Each element is an object of the class htext print(qmap[[1]]) ## ## Q-Map test of Equivalence for qualitative data. ## ## Symbols type: combinations ## ## Ratio Symbolized observations/Num symbols = 4.6 ## ## data: QY1 and QY2 ## QE = 148.71, df = 9, p-value < 2.2e-16 ## alternative hypothesis: two.sided • The method obtains the distribution of symbols with the confidence intervals specified by the user. plot(qmap, ci=.6) # 4 Runs tests The runs test have global and local versions ## 4.1 Global Runs test • The function sp.runs.test obtain the spatial runs test. listw <- knearneigh(coor, k = 3) srq <- sp.runs.test(fx = fx, listw = listw) • The output of this function is a object of the classes sprunstest and htest ## 4.2 Methods for spruntest class • The spqdep has two methods for this class y print(srq) ## ## Runs test of spatial dependence for qualitative data. Asymptotic ## version ## ## data: mxf ## sp.runs test = -0.81833, p-value = 0.4132 ## alternative hypothesis: two.sided ## sample estimates: ## Total runs Mean total runs Variance total runs ## 276.0000 285.5152 135.1986 plot(srq) ## 4.3 The local Runs test • The function local.sp.runs.test obtain the local test based on runs. ## 4.4 Asymptotic version • Asymptotic version lsrq <- local.sp.runs.test(fx = fx, listw = listw, alternative = "less") • The method list the statistic of each observation (point or region) print(lsrq) ## runs.i E.i Std.i z.value p.value ## 1 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 2 3 2.855152 0.8722689 0.1660594 0.56594492 ## 3 3 2.855152 0.8722689 0.1660594 0.56594492 ## 4 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 5 4 2.855152 0.8722689 1.3124950 0.90532341 ## 6 3 2.855152 0.8722689 0.1660594 0.56594492 ## 7 3 2.855152 0.8722689 0.1660594 0.56594492 ## 8 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 9 3 2.855152 0.8722689 0.1660594 0.56594492 ## 10 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 11 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 12 3 2.855152 0.8722689 0.1660594 0.56594492 ## 13 4 2.855152 0.8722689 1.3124950 0.90532341 ## 14 1 2.855152 0.8722689 -2.1268116 0.01671787 ## 15 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 16 3 2.855152 0.8722689 0.1660594 0.56594492 ## 17 3 2.855152 0.8722689 0.1660594 0.56594492 ## 18 4 2.855152 0.8722689 1.3124950 0.90532341 ## 19 3 2.855152 0.8722689 0.1660594 0.56594492 ## 20 3 2.855152 0.8722689 0.1660594 0.56594492 ## 21 3 2.855152 0.8722689 0.1660594 0.56594492 ## 22 4 2.855152 0.8722689 1.3124950 0.90532341 ## 23 4 2.855152 0.8722689 1.3124950 0.90532341 ## 24 3 2.855152 0.8722689 0.1660594 0.56594492 ## 25 4 2.855152 0.8722689 1.3124950 0.90532341 ## 26 1 2.855152 0.8722689 -2.1268116 0.01671787 ## 27 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 28 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 29 3 2.855152 0.8722689 0.1660594 0.56594492 ## 30 3 2.855152 0.8722689 0.1660594 0.56594492 ## 31 3 2.855152 0.8722689 0.1660594 0.56594492 ## 32 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 33 4 2.855152 0.8722689 1.3124950 0.90532341 ## 34 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 35 3 2.855152 0.8722689 0.1660594 0.56594492 ## 36 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 37 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 38 3 2.855152 0.8722689 0.1660594 0.56594492 ## 39 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 40 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 41 3 2.855152 0.8722689 0.1660594 0.56594492 ## 42 3 2.855152 0.8722689 0.1660594 0.56594492 ## 43 1 2.855152 0.8722689 -2.1268116 0.01671787 ## 44 3 2.855152 0.8722689 0.1660594 0.56594492 ## 45 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 46 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 47 4 2.855152 0.8722689 1.3124950 0.90532341 ## 48 1 2.855152 0.8722689 -2.1268116 0.01671787 ## 49 3 2.855152 0.8722689 0.1660594 0.56594492 ## 50 3 2.855152 0.8722689 0.1660594 0.56594492 ## 51 4 2.855152 0.8722689 1.3124950 0.90532341 ## 52 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 53 3 2.855152 0.8722689 0.1660594 0.56594492 ## 54 3 2.855152 0.8722689 0.1660594 0.56594492 ## 55 4 2.855152 0.8722689 1.3124950 0.90532341 ## 56 4 2.855152 0.8722689 1.3124950 0.90532341 ## 57 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 58 3 2.855152 0.8722689 0.1660594 0.56594492 ## 59 4 2.855152 0.8722689 1.3124950 0.90532341 ## 60 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 61 3 2.855152 0.8722689 0.1660594 0.56594492 ## 62 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 63 4 2.855152 0.8722689 1.3124950 0.90532341 ## 64 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 65 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 66 3 2.855152 0.8722689 0.1660594 0.56594492 ## 67 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 68 4 2.855152 0.8722689 1.3124950 0.90532341 ## 69 3 2.855152 0.8722689 0.1660594 0.56594492 ## 70 4 2.855152 0.8722689 1.3124950 0.90532341 ## 71 3 2.855152 0.8722689 0.1660594 0.56594492 ## 72 4 2.855152 0.8722689 1.3124950 0.90532341 ## 73 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 74 3 2.855152 0.8722689 0.1660594 0.56594492 ## 75 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 76 3 2.855152 0.8722689 0.1660594 0.56594492 ## 77 3 2.855152 0.8722689 0.1660594 0.56594492 ## 78 3 2.855152 0.8722689 0.1660594 0.56594492 ## 79 3 2.855152 0.8722689 0.1660594 0.56594492 ## 80 3 2.855152 0.8722689 0.1660594 0.56594492 ## 81 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 82 3 2.855152 0.8722689 0.1660594 0.56594492 ## 83 3 2.855152 0.8722689 0.1660594 0.56594492 ## 84 3 2.855152 0.8722689 0.1660594 0.56594492 ## 85 3 2.855152 0.8722689 0.1660594 0.56594492 ## 86 3 2.855152 0.8722689 0.1660594 0.56594492 ## 87 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 88 3 2.855152 0.8722689 0.1660594 0.56594492 ## 89 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 90 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 91 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 92 3 2.855152 0.8722689 0.1660594 0.56594492 ## 93 3 2.855152 0.8722689 0.1660594 0.56594492 ## 94 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 95 3 2.855152 0.8722689 0.1660594 0.56594492 ## 96 3 2.855152 0.8722689 0.1660594 0.56594492 ## 97 4 2.855152 0.8722689 1.3124950 0.90532341 ## 98 3 2.855152 0.8722689 0.1660594 0.56594492 ## 99 2 2.855152 0.8722689 -0.9803761 0.16345026 ## 100 2 2.855152 0.8722689 -0.9803761 0.16345026 • The method identify the localization with values of local test significant. plot(lsrq, sig = 0.05) ## 4.5 Monte Carlo local runs test • The Monte Carlo distribution ot the local test using a sf object data("provinces_spain") listw <- spdep::poly2nb(as(provinces_spain,"Spatial"), queen = FALSE) ## although coordinates are longitude/latitude, st_intersects assumes that they are planar provinces_spain$Male2Female <- factor(provinces_spain$Male2Female > 100) levels(provinces_spain$Male2Female) = c("men","woman")
plot(provinces_spain["Male2Female"])

formula <- ~ Male2Female
# Boots Version
lsrq <- local.sp.runs.test(formula = formula, data = provinces_spain, listw = listw, distr ="bootstrap", nsim = 199)
plot(lsrq, sf = provinces_spain, sig = 0.10)

# 5 The scan test

• Two of the scan tests to identify clusters can be apply to test spatial structure in qualitative spatial processes.

• The scan test don’t need pre-define the classical W conectivity matrix.

• See

• The scan tests contrasts the null of independence of a spatial qualitative process and give additional information indicating one (or perhaps more) spatial cluster(s).

• The scan tests don’t have asymptotic distribution. The significance is obtained by permutational resampling.

• The output of the scan function is an object of the classes scantest and htest

## 5.1 Scan bernoulli

• For qualitative spatial process with two categories the bernoulli scan test is obtain with the next code
formula <- ~ Male2Female
scan.spain <- spqdep::scan.test(formula = formula, data = provinces_spain,
case="men", nsim = 99, distr = "bernoulli")
print(scan.spain)
##
##  Scan test. Distribution: bernoulli
##
## data:  Male2Female
## scan-loglik = 6.0359, p-value = 0.07
## alternative hypothesis: High
## sample estimates:
##
## Total observations in the MLC =   17.00
## Expected cases in the MLC =       11.84
## Observed cases in the MLC =       16.00

### 5.1.1 Scan test with flexible windows

• The flexible windows is an option. Note that is slow. Consider maximum nv of 12.
listw <- spdep::poly2nb(provinces_spain, queen = FALSE)
## although coordinates are longitude/latitude, st_intersects assumes that they are planar
scan.spain <- spqdep::scan.test(formula = formula, data = provinces_spain,
case="men", nsim = 99, windows = "flexible",
listw = listw, nv = 6, distr = "bernoulli")
## +++++end
print(scan.spain)
##
##  Scan test. Distribution: bernoulli
##
## data:  Male2Female
## scan-loglik = 1.9464, p-value < 2.2e-16
## alternative hypothesis: High
## sample estimates:
##
## Total observations in the MLC =     6.00
## Expected cases in the MLC =       370.74
## Observed cases in the MLC =         6.00

## 5.2 scan multinomial

• In case of a spatial process with three or more categories
data(FastFood.sf)
formula <- ~ Type
scan.fastfood <- scan.test(formula = formula, data = FastFood.sf, nsim = 99, distr = "multinomial", windows = "elliptic",
nv = 50)
print(scan.fastfood)
##
##  Scan test. Distribution: multinomial
##
## data:  Type
## scan-loglik = 15.506, p-value < 2.2e-16
## sample estimates:
##                  H     P     S Sum
## cases.expect 13.48 14.86 14.66  43
## cases.observ 16.00  1.00 26.00  43

## 5.3 Methods for scan test

• Two method can be used with scantest objects: and
summary(scan.fastfood)
##
## Summary of data:
## Distribution....................: multinomial
## Number of locations.............: 877
## Total number of cases...........: 877
## Names of cathegories...........: H P S
## Total cases per category........: 275 303 299
## Percent cases per category......: 0.31 0.35 0.34
##
## Scan statistic:
## Total cases in the MLC.........: 43
## Names of cathegories...........: H P S
## Observed cases in the MLC......: 13.48 14.86 14.66
## Expected cases in the MLC......: 16 1 26
## Value of statistic (loglik ratio)....: 15.5058
## p-value........................: 0
##
## IDs of cluster detect:
## Location IDs included.....:  68 849 152 499 630 763 827 765 617 600 607 48 58 588 743 843 74 122 750 115 645 61 226 796 876 699 610 597 596 721 751 53 186 659 778 63 106 229 585 738 612 131 208
##
##
## Secondary Scan statistic. Number 1
## Total cases in secondary cluster......:  16
## Names of cathegories.................: H P S
## Percent per category total...........: 0.31 0.35 0.34
## Percent per category inside cluster..: 0.25 0.5 0.25
## Value of statisitic (loglik ratio)....: 11.5285
## p-value.........................: 0.1
## Location IDs included..................:  677 311 781 128 108 436 551 576 21 374 319 717 561 6 629 547
##
##
## Secondary Scan statistic. Number 2
## Total cases in secondary cluster......:  7
## Names of cathegories.................: H P S
## Percent per category total...........: 0.31 0.35 0.34
## Percent per category inside cluster..: 0.14 0.29 0.57
## Value of statisitic (loglik ratio)....: 7.0038
## p-value.........................: 0.96
## Location IDs included..................:  158 709 335 801 749 545 856
##
##
## Secondary Scan statistic. Number 3
## Total cases in secondary cluster......:  17
## Names of cathegories.................: H P S
## Percent per category total...........: 0.31 0.35 0.34
## Percent per category inside cluster..: 0.35 0.35 0.29
## Value of statisitic (loglik ratio)....: 6.8747
## p-value.........................: 0.98
## Location IDs included..................:  521 782 162 643 297 220 267 265 104 530 312 523 783 157 531 848 680
##
##
## Secondary Scan statistic. Number 4
## Total cases in secondary cluster......:  17
## Names of cathegories.................: H P S
## Percent per category total...........: 0.31 0.35 0.34
## Percent per category inside cluster..: 0.47 0.29 0.24
## Value of statisitic (loglik ratio)....: 6.7961
## p-value.........................: 0.98
## Location IDs included..................:  190 837 555 711 646 216 17 390 742 563 307 4 353 197 254 192 66
##
##
## Secondary Scan statistic. Number 5
## Total cases in secondary cluster......:  10
## Names of cathegories.................: H P S
## Percent per category total...........: 0.31 0.35 0.34
## Percent per category inside cluster..: 0.4 0.2 0.4
## Value of statisitic (loglik ratio)....: 6.5951
## p-value.........................: 0.98
## Location IDs included..................:  78 365 668 228 170 857 306 708 651 187
plot(scan.spain, sf = provinces_spain)

plot(scan.fastfood, sf = FastFood.sf)

# 6 Similarity test

The Farber et al. (2014) paper develop the similarity test

## 6.1 Simiarity test

The function calculates the similarity test for both asymptotic distribution and permutational resampling.

coor <- st_coordinates(st_centroid(FastFood.sf))
listw <- spdep::knearneigh(coor, k = 4)
formula <- ~ Type
similarity <- similarity.test(formula = formula, data = FastFood.sf, listw = listw)
print(similarity)
##
##  Similarity test of spatial dependence for qualitative data.
##  Distribution: asymptotic
##
## data:  Type
## Similarity-test = -5.4476, p-value = 5.105e-08
## alternative hypothesis: two.sided

# 7 join-count tests

• The functions of the spdep R-package have been wrapped for Bernoulli and Multinomial distributions. Asymptotic or Monte Carlo distributions (permutations) can be used to evaluate the signification of the tests.

## 7.1 Asyntotic distribution

provinces_spain$Older <- cut(provinces_spain$Older, breaks = c(-Inf,19,22.5,Inf))
levels(provinces_spain\$Older) = c("low","middle","high")
f1 <- ~ Older + Male2Female
jc1 <- jc.test(formula = f1, data = provinces_spain, distr = "asymptotic", alternative = "greater", zero.policy = TRUE)
## although coordinates are longitude/latitude, st_intersects assumes that they are planar
summary(jc1)
JoinCount Spatial Tests (asymptotic)
pairs z-value pvalue Joincount Expected Variance
Older - multinomial - alternative: greater - Join count test under nonfree sampling
high:high −0.56 0.71077 14 15.71 9.48
low:low −0.93 0.82392 7 9.34 6.35
middle:middle −0.88 0.81070 13 15.71 9.48
low:high −0.47 0.68162 24 25.88 15.78
middle:high −1.21 0.88738 28 33.27 18.88
middle:low −0.22 0.58727 25 25.88 15.78
Jtot −1.29 0.90190 77 85.02 38.52
Male2Female - binomial - alternative: greater - Join count test under nonfree sampling
men-men −2.17 0.98504 60 68.39 14.92
woman-woman 2.11 0.01740 13 8.01 5.59
men-woman −2.47 0.99333 38 49.39 21.19

## 7.2 Monte Carlo distribution

jc1 <- jc.test(formula = f1, data = provinces_spain, distr = "mc", alternative = "greater", zero.policy = TRUE)
## although coordinates are longitude/latitude, st_intersects assumes that they are planar
summary(jc1)
JoinCount Spatial Tests (Monte Carlo)
pairs pvalue Joincount Expected Variance
Older - multinomial - alternative: greater - Monte-Carlo simulation of join-count statistic (nonfree sampling)
high:high 0.50900 14 13.77 10.80
low:low 0.68200 7 8.33 6.31
middle:middle 0.53800 13 13.86 10.40
low:high 0.40700 24 22.89 16.40
middle:high 0.68300 28 29.55 18.26
middle:low 0.25300 25 22.60 16.01
Jtot 0.38900 77 75.03 22.51
Male2Female - binomial - alternative: greater - Monte-Carlo simulation of join-count statistic
men-men 0.52150 60 60.34 23.29
woman-woman 0.01750 13 7.10 5.95

# 8 References

Farber, S., Marin, M. R., & Páez, A. (2015). Testing for spatial independence using similarity relations. Geographical Analysis, 47(2), 97-120.

Paez, A., Lopez, F. A., Menezes, T., Cavalcanti, R., & Pitta, M. G. D. R. (2021). A spatio‐temporal analysis of the environmental correlates of COVID‐19 incidence in Spain. Geographical analysis, 53(3), 397-421.

Kanaroglou, Pavlos. 2016. Spatial Analysis in Health Geography. Routledge. https://doi.org/10.4324/9781315610252.
López, Fernando A, and Antonio Páez. 2012. “Distribution-Free Inference for q (m) Based on Permutational Bootstrapping: An Application to the Spatial Co-Location Pattern of Firms in Madrid.” Estadı́stica Española 54 (177): 135–56.
Pebesma, Edzer. 2018. Simple Features for R: Standardized Support for Spatial Vector Data.” The R Journal 10 (1): 439–46. https://doi.org/10.32614/RJ-2018-009.
Ruiz, Manuel, Fernando López, and Antonio Páez. 2010. “Testing for Spatial Association of Qualitative Data Using Symbolic Dynamics.” Journal of Geographical Systems 12 (3): 281–309. https://doi.org/10.1007/s10109-009-0100-1.
———. 2012. “Comparison of Thematic Maps Using Symbolic Entropy.” International Journal of Geographical Information Science 26 (3): 413–39. https://doi.org/10.1080/13658816.2011.586327.
———. 2021. “A Test for Global and Local Homogeneity of Categorical Data Based on Spatial Runs.” Working Paper.