# Chapter 15 Measures of spatial autocorrelation

When analysing areal data, it has long been recognised that, if present, spatial autocorrelation changes how we may infer, relative to the default position of independent observations. In the presence of spatial autocorrelation, we can predict the values of observation $$i$$ from the values observed at $$j \in N_i$$, the set of its proximate neighbours. Early results (Moran 1948; Geary 1954), entered into research practice gradually, for example the social sciences (Duncan, Cuzzort, and Duncan 1961). These results were then collated and extended to yield a set of basic tools of analysis (Cliff and Ord 1973, 1981).

Cliff and Ord (1973) generalised and extended the expression of the spatial weights matrix representation as part of the framework for establishing the distribution theory for join count, Moran’s $$I$$ and Geary’s $$C$$ statistics. This development of what have become known as global measures, returning a single value of autocorrelation for the total study area, has been supplemented by local measures returning values for each areal unit (Getis and Ord 1992; Anselin 1995).

## 15.1 Measures and process mis-specification

It is not and has never been the case that Tobler’s first law of geography: “Everything is related to everything else, but near things are more related than distant things” always holds absolutely. This is and has always been an oversimplification, disguising the underlying entitation, support and other mis-specification problems. Are the units of observation appropriate for the scale of the underlying spatial process? Could the spatial patterning of the variable of interest for the chosen entitation be accounted for by another variable?

Tobler (1970) was published in the same special issue of Economic Geography as Olsson (1970), but Olsson does grasp the important point that spatial autocorrelation is not inherent in spatial phenomena, but often is engendered by inappropriate entitation, by omitted variables and/or inappropriate functional form. The key quote from Olsson is on p. 228:

The existence of such autocorrelations makes it tempting to agree with Tobler (1970, 236 [the original refers to the pagination of a conference paper]) that ‘everything is related to everything else, but near things are more related than distant things.’ On the other hand, the fact that the autocorrelations seem to hide systematic specification errors suggests that the elevation of this statement to the status of ‘the first law of geography’ is at best premature. At worst, the statement may represent the spatial variant of the post hoc fallacy, which would mean that coincidence has been mistaken for a causal relation.

The status of the “first law” is very similar to the belief that John Snow induced the cause of cholera as water-borne from a map. It may be a good way of selling GIS, but it is inaccurate; Snow had a strong working hypothesis prior to visiting Soho, and the map was prepared after the Broad street pump was disabled as documentation that the hypothesis held (Brody et al. 2000).

Measures of spatial autocorrelation unfortunately pick up other mis-specifications in the way that we model data (Schabenberger and Gotway 2005; McMillen 2003). For reference, Moran’s $$I$$ is given as (Cliff and Ord 1981, 17):

$I = \frac{n \sum_{(2)} w_{ij} z_i z_j}{S_0 \sum_{i=1}^{n} z_i^2}$ where $$x_i, i=1, \ldots, n$$ are $$n$$ observations on the numeric variable of interest, $$z_i = x_i - \bar{x}$$, $$\bar{x} = \sum_{i=1}^{n} x_i / n$$, $$\sum_{(2)} = \stackrel{\sum_{i=1}^{n} \sum_{j=1}^{n}}{i \neq j}$$, $$w_{ij}$$ are the spatial weights, and $$S_0 = \sum_{(2)} w_{ij}$$. First we test a random variable using the Moran test, here under the normality assumption (argument randomisation=FALSE, default TRUE). Inference is made on the statistic $$Z(I) = \frac{I - E(I)}{\sqrt{\mathrm{Var}(I)}}$$, the z-value compared with the Normal distribution for $$E(I)$$ and $$\mathrm{Var}(I)$$ for the chosen assumptions; this x does not show spatial autocorrelation with these spatial weights:

glance_htest <- function(ht) c(ht$estimate, "Std deviate" = unname(ht$statistic),
"Analytical randomisation" = unname(mtr$estimate[3])) # Permutation bootstrap Analytical randomisation # 0.000144 0.000140 Geary’s global $$C$$ is implemented in geary.test() largely following the same argument structure as moran.test(). The Getis-Ord $$G$$ test includes extra arguments to accommodate differences between implementations, as Bivand and Wong (2018) found multiple divergences from the original definitions, often to omit no-neighbour observations generated when using distance band neighbours. It is given by (Getis and Ord 1992, 194). For $$G^*$$, the $$\sum_{(2)}$$ constraint is relaxed by including $$i$$ as a neighbour of itself (thereby also removing the no-neighbour problem, because all observations have at least one neighbour). Finally, the empirical Bayes Moran’s $$I$$ takes account of the denominator in assessing spatial autocorrelation in rates data (Assunção and Reis 1999). Until now, we have considered the proportion of valid votes cast in relation to the numbers entitled to vote by spatial entity, but using EBImoran.mc() we can try to accommodate uncertainty in extreme rates in entities with small numbers entitled to vote. There is, however, little impact on the outcome in this case. Global measures of spatial autocorrelation using spatial weights objects based on graphs of neighbours are, as we have seen, rather blunt tools, which for interpretation depend critically on a reasoned mean model of the variable in question. If the mean model is just the intercept, the global measures will respond to all kinds of mis-specification, not only spatial autocorrelation. The choice of entities for aggregation of data will typically be a key source of mis-specification. ## 15.3 Local measures Building on insights from the weaknesses of global measures, local indicators of spatial association began to appear in the first half of the 1990s (Anselin 1995; Getis and Ord 1992, 1996). In addition, the Moran plot was introduced, plotting the values of the variable of interest against their spatially lagged values, typically using row-standardised weights to make the axes more directly comparable (Anselin 1996). The moran.plot() function also returns an influence measures object used to label observations exerting more than propotional influence on the slope of the line representing global Moran’s $$I$$. In figure 15.1, we can see that there are many spatial entities exerting such influence. These pairs of observed and lagged observed values make up in aggregate the global measure, but can also be explored in detail. The quadrants of the Moran plot also show low-low pairs in the lower left quadrant, high-high in the upper right quadrant, and fewer low-high and high-low pairs in the upper left and lower right quadrants. z %>% moran.plot(listw = lw_q_W, labels = pol_pres15$TERYT, cex = 1, pch = ".",
xlab = "I round turnout", ylab = "lagged turnout") -> infl_W

If we extract the hat value influence measure from the returned object, Figure 15.2 suggests that some edge entities exert more than proportional influence (perhaps because of row standardisation), as do entities in or near larger urban areas.

pol_pres15$hat_value <- infl_W$hat
tm_shape(pol_pres15) + tm_fill("hat_value")

### 15.3.1 Local Moran’s $$I_i$$

Bivand and Wong (2018) discuss issues impacting the use of local indicators, such as local Moran’s $$I_i$$ and local Getis-Ord $$G_i$$. Some issues affect the calculation of the local indicators, others inference from their values. Because $$n$$ statistics may be being calculated from the same number of observations, there are multiple comparison problems that need to be addressed. Although the apparent detection of hotspots from values of local indicators has been quite widely adopted, it remains fraught with difficulty because adjustment of the inferential basis to accommodate multiple comparisons is not often chosen, and as in the global case, mis-specification also remains a source of confusion. Further, interpreting local spatial autocorrelation in the presence of global spatial autocorrelation is challenging (Ord and Getis 2001; Tiefelsdorf 2002; Bivand, Müller, and Reder 2009).

args(localmoran)
# function (x, listw, zero.policy = NULL, na.action = na.fail,
#     conditional = FALSE, alternative = "greater", p.adjust.method = "none",
#     mlvar = TRUE, spChk = NULL, adjust.x = FALSE)
# NULL

The mlvar= and adjust.x= arguments to localmoran() are discussed in Bivand and Wong (2018), and permit matching with other implementations. The p.adjust.method= argument uses an untested speculation implemented in p.adjustSP() that adjustment should only take into account the cardinality of the neighbour set of each observation when adjusting for multiple comparisons; using stats::p.adjust() is preferable.

Taking "two.sided" p-values, we obtain:

z %>%
localmoran(listw = lw_q_W, alternative = "two.sided") -> locm

The $$I_i$$ local indicators when summed and divided by the sum of the spatial weights equal global Moran’s $$I$$, showing the possible presence of positive and negative local spatial autocorrelation:

all.equal(sum(locm[,1])/Szero(lw_q_W), unname(moran.test(z, lw_q_W)$estimate[1])) # [1] TRUE Using stats::p.adjust() to adjust for multiple comparisons, we see that almost 29% of the 2495 local measures have p-values < 0.05 if no adjustment is applied, but only 12% using Bonferroni adjustment to control the familywise error rate, with two other choices shown: fdr is the Benjamini and Hochberg (1995) false discovery rate and BY (Benjamini and Yekutieli 2001), another false discovery rate adjustment: pva <- function(pv) cbind("none" = pv, "bonferroni" = p.adjust(pv, "bonferroni"), "fdr" = p.adjust(pv, "fdr"), "BY" = p.adjust(pv, "BY")) locm %>% subset(select = "Pr(z != 0)", drop = TRUE) %>% pva() -> pvsp f <- function(x) sum(x < 0.05) apply(pvsp, 2, f) # none bonferroni fdr BY # 715 297 576 424 In the global measure case, bootstrap permutations may be used as an alternative to analytical methods for possible inference, where both the theoretical development of the analytical variance of the measure, and the permutation scheme, shuffle all of the observed values. In the local case, conditional permutation may be used, fixing the value at observation $$i$$ and randomly sampling from the remaining $$n-1$$ values to find randomised values at neighbours, and is provided as localmoran_perm(), which may use multiple nodes to sample in parallel if provided, and permits the setting of a seed for the random number generator across the compute nodes: library(parallel) set.coresOption(ifelse(detectCores() == 1, 1, detectCores()-1L)) # NULL system.time(z %>% localmoran_perm(listw = lw_q_W, nsim = 499, alternative = "two.sided", iseed = 1) -> locm_p) # user system elapsed # 0.629 1.513 0.695 The outcome is that almost 32% of observations have two sided p-values < 0.05 without multiple comparison adjustment, and under 3% with Bonferroni adjustment. locm_p %>% subset(select = "Pr(z != 0)", drop = TRUE) %>% pva() -> pvsp apply(pvsp, 2, f) # none bonferroni fdr BY # 797 76 463 161 We can see what is happening by tabulating counts of the standard deviate of local Moran’s $$I$$, where the two-sided $$\alpha=0.05$$ bounds would be $$0.025$$ and $$0.975$$, but Bonferroni adjustment is close to $$0.00001$$ and $$0.99999$$. Without adjustment, almost 800 observations are significant, with Bonferroni adjustment, only almost 70 in the conditional permutation case: brks <- qnorm(c(0, 0.00001, 0.0001, 0.001, 0.01, 0.025, 0.5, 0.975, 0.99, 0.999, 0.9999, 0.99999, 1)) (locm_p %>% subset(select = Z.Ii, drop = TRUE) %>% cut(brks) %>% table()-> tab) # . # (-Inf,-4.26] (-4.26,-3.72] (-3.72,-3.09] (-3.09,-2.33] (-2.33,-1.96] # 0 0 1 4 5 # (-1.96,0] (0,1.96] (1.96,2.33] (2.33,3.09] (3.09,3.72] # 459 1239 195 316 145 # (3.72,4.26] (4.26, Inf] # 55 76 sum(tab[c(1:5, 8:12)]) # [1] 797 sum(tab[c(1, 12)]) # [1] 76 In an important clarification, Sauer et al. (2021) show that the comparison of standard deviates for local Moran’s $$I_i$$ based on analytical formulae and conditional permutation in Bivand and Wong (2018) was based on a misunderstanding. Sokal, Oden, and Thomson (1998) provide alternative analytical formulae for standard deviates of local Moran’s $$I_i$$ based either on total or conditional permutation, but the analytical formulae used in Bivand and Wong (2018), based on earlier practice, only use total permutation, and consequently do not match the simulation conditional permutations. Thanks to a timely pull request, localmoran() now has a conditional= argument using alternative formulae from the appendix of Sokal, Oden, and Thomson (1998): z %>% localmoran(listw = lw_q_W, conditional = TRUE, alternative = "two.sided") -> locm_c yielding standard deviates that correspond closely to those from simulation: locm_c %>% subset(select = "Pr(z != 0)", drop = TRUE) %>% pva() -> pvsp apply(pvsp, 2, f) # none bonferroni fdr BY # 789 69 468 156 pol_pres15$locm_Z <- locm[, "Z.Ii"]
pol_pres15$locm_c_Z <- locm_c[, "Z.Ii"] pol_pres15$locm_p_Z <- locm_p[, "Z.Ii"]
tm_shape(pol_pres15) + tm_fill(c("locm_Z", "locm_c_Z", "locm_p_Z"), breaks = brks,
midpoint = 0, title = "Standard deviates of\nLocal Moran's I") +
tm_facets(free.scales = FALSE, ncol = 2) + tm_layout(panel.labels = c("Analytical total",
"Analytical conditional", "Conditional permutation"))

Figure 15.3 shows that conditional permutation scales back the proportion of standard deviate values taking extreme values, especially positive values. The analytical total standard deviates of local Moran’s $$I$$ should probably not be used since alternatives are available, not least thanks to the clarification by Sauer et al. (2021).

In presenting local Moran’s $$I$$, use is often made of “hotspot” maps. Because $$I_i$$ takes high values both for strong positive autocorrelation of low and high values of the input variable, it is hard to show where “clusters” of similar neighbours with low or high values of the input variable occur. The quadrants of the Moran plot are used, by creating a categorical quadrant variable interacting the input variable and its spatial lag split at their means. The quadrant categories are then set to NA if, for the chosen standard deviate and adjustment, $$I_i$$ would be considered insignificant. Here, for the Bonferroni adjusted conditional analytical standard deviates, 14 observations belong to “Low-Low clusters”, and 55 to “High-High clusters”:

quadr <- interaction(
cut(infl_W$x, c(-Inf, mean(infl_W$x), Inf), labels=c("Low X", "High X")),
cut(infl_W$wx, c(-Inf, mean(infl_W$wx), Inf), labels=c("Low WX", "High WX")),
sep=" : ")
pol_pres15$hs_an_q <- quadr is.na(pol_pres15$hs_an_q) <- !(pol_pres15$locm_Z < brks[6] | pol_pres15$locm_Z > brks[8])
b <- table(pol_pres15$hs_an_q) pol_pres15$hs_ac_q <- quadr
is.na(pol_pres15$hs_ac_q) <- !(pol_pres15$locm_c_Z < brks[2] |
pol_pres15$locm_c_Z > brks[12]) c <- table(pol_pres15$hs_ac_q)
pol_pres15$hs_cp_q <- quadr is.na(pol_pres15$hs_cp_q) <- !(pol_pres15$locm_p_Z < brks[2] | pol_pres15$locm_p_Z > brks[12])
d <- table(pol_pres15$hs_cp_q) t(rbind("Moran plot quadrants" = a, "Unadjusted analytical total" = b, "Bonferroni analytical cond." = c, "Bonferroni cond. perm." = d)) # Moran plot quadrants Unadjusted analytical total # Low X : Low WX 1040 370 # High X : Low WX 264 3 # Low X : High WX 213 0 # High X : High WX 978 342 # Bonferroni analytical cond. Bonferroni cond. perm. # Low X : Low WX 14 18 # High X : Low WX 0 0 # Low X : High WX 0 0 # High X : High WX 55 58 tm_shape(pol_pres15) + tm_fill(c("hs_an_q", "hs_ac_q", "hs_cp_q"), colorNA="grey95", textNA="Not significant", title="Turnout hotspot status\nLocal Moran's I") + tm_facets(free.scales=FALSE, ncol=2) + tm_layout(panel.labels= c("Unadjusted analytical total", "Bonferroni analytical cond.", "Cond. perm. with Bonferroni")) Figure 15.4 shows the impact of using analytical or conditional permutation standard deviates, and no or Bonferroni adjustment, reducing the counts of observations in “Low-Low clusters” from 370 to 14, and “High-High clusters” from 342 to 54; the “High-High clusters” are metropolitan areas. Tiefelsdorf (2002) argues that standard approaches to the calculation of the standard deviates of local Moran’s $$I_i$$ should be supplemented by numerical estimates, and shows that Saddlepoint approximations are a computationally efficient way of achieving this goal. The localmoran.sad() function takes a fitted linear model as its first argument, so we first fit a null (intercept only) model, but use case weights because the numbers entitled to vote vary greatly between observations: lm(z ~ 1, weights = pol_pres15$I_entitled_to_vote) -> lm_null

Saddlepoint approximation is much more computationally intensive than conditional permutation:

system.time(lm_null %>% localmoran.sad(nb = nb_q, style = "W", alternative = "two.sided") %>%
summary() -> locm_sad_null)
#    user  system elapsed
#  10.550   0.084  10.643

However, standard approaches do not permit richer mean models with covariates or case weights. Next we add the categorical variable distinguishing between rural, urban and other types of observational unit:

lm(z ~ Types, weights=pol_pres15$I_entitled_to_vote) -> lm_types system.time(lm_types %>% localmoran.sad(nb = nb_q, style = "W", alternative = "two.sided") %>% summary() -> locm_sad_types) # user system elapsed # 11.4 0.0 11.4 To conclude, we add the spatially lagged categories, although the spatial lag of a categorical variable, represented by dummies in the model matrix, is not well defined: spatialreg::lmSLX(z ~ Types, listw = lw_q_W, weights = pol_pres15$I_entitled_to_vote) -> lm_Dtypes
system.time(lm_Dtypes %>% localmoran.sad(nb = nb_q, style = "W", alternative = "two.sided") %>%
summary() -> locm_sad_Dtypes)
#    user  system elapsed
#  11.904   0.004  11.908
pol_pres15$locm_sad_null_Z <- locm_sad_null[, "Saddlepoint"] pol_pres15$locm_sad_types_Z <- locm_sad_types[, "Saddlepoint"]
pol_pres15$locm_sad_Dtypes_Z <- locm_sad_Dtypes[, "Saddlepoint"] tm_shape(pol_pres15) + tm_fill(c("locm_sad_null_Z", "locm_sad_types_Z", "locm_sad_Dtypes_Z", "locm_c_Z"), breaks = brks, midpoint=0, title="Standard deviates of\nLocal Moran's I") + tm_facets(free.scales = FALSE, ncol = 2) + tm_layout(panel.labels = c( "Saddlepoint weighted null", "Saddlepoint weighted types", "Saddlepoint weighted Durbin types", "Analytical conditional")) Figure 15.5 includes the analytical conditional standard deviates for comparison (lower right panel), but in general it can be seen that the Saddlepoint approximation standard deviates lie closer to zero, possibly because some of the mis-specification in the mean model has been removed by using richer versions, and possibly because the approximation approach is inherently local, relating regression residual values at $$i$$ to those of its neighbours. It is also possible to use Saddlepoint approximation where the global spatial process has been incorporated, removing the conflation of global and local spatial autocorrelation in standard approaches. The same can also be accomplished using exact methods (Bivand, Müller, and Reder 2009). ### 15.3.2 Local Getis-Ord $$G_i$$ The local Getis-Ord $$G$$ measure is reported as a standard deviate, and may also take the $$G^*_i$$ form where self-neighbours are inserted into the neighbour object using include.self(). The observed and expected values of local $$G$$ with their analytical variances may also be returned if return_internals=TRUE. system.time(z %>% localG(lw_q_W) -> locG) # user system elapsed # 0.027 0.000 0.036 system.time(z %>% localG_perm(lw_q_W, nsim = 499, iseed = 1) -> locG_p) # user system elapsed # 0.519 1.317 0.627 Once again we face the problem of multiple comparisons, with the count of areal unit p-values < 0.05 being reduced by an order of magnitude when employing Bonferroni correction: locG %>% c() %>% abs() %>% pnorm(lower.tail = FALSE) %>% (function(x) x*2)() %>% pva() -> pvsp apply(pvsp, 2, f) # none bonferroni fdr BY # 789 69 468 156 library(ggplot2) p1 <- ggplot(data.frame(Zi=locm_c[,4], Zi_perm=locm_p[,4])) + geom_point(aes(x=Zi, y=Zi_perm), alpha=0.2) + xlab("Analytical conditional") + ylab("Bootstrap conditional") + coord_fixed() + ggtitle("Local Moran's I") p2 <- ggplot(data.frame(Zi=c(locG), Zi_perm=c(locG_p))) + geom_point(aes(x=Zi, y=Zi_perm), alpha=0.2) + xlab("Analytical conditional") + ylab("Bootstrap conditional") + coord_fixed() + ggtitle("Local G") gridExtra::grid.arrange(p1, p2, nrow=1) Figure 15.6 shows that, when using analytical conditional standard deviates, the values from analytical and bootstrap estimates coincide for both $$I_i$$ and $$G_i$$. In both cases, one may argue that the bootstrap approach is superfluous in exploratory spatial data analysis. pol_pres15$locG_Z <- c(locG)
pol_pres15$hs_G <- cut(c(locG), c(-Inf, brks[2], brks[12], Inf), labels = c("Low", "Not significant", "High")) table(pol_pres15$hs_G)
#
#             Low Not significant            High
#              14            2426              55
m1 <- tm_shape(pol_pres15) + tm_fill(c("locG_Z"), midpoint = 0, title = "Standard\ndeviate")
m2 <- tm_shape(pol_pres15) + tm_fill(c("hs_G"),
title="Local G Bonferroni\nadjusted hotspot status")
tmap_arrange(m1, m2, nrow=1)

As can be seen from Figure 15.6, we do not need to contrast the two estimation methods, and showing the mapped standard deviate is as informative as the “hotspot” status for the chosen adjustment (Figure 15.7). In the case of $$G_i$$, the values taken by the measure reflect the values of the input variable, so a “High cluster” is found for observations with high values of the input variable, here high turnout in metropolitan areas.

Very recently, Geoda has been wrapped for R as rgeoda (Li and Anselin 2021), and will provide very similar functionalities for the exploration of spatial autocorrelation in areal data as spdep. The active objects are kept as pointers to a compiled code workspace; using compiled code for all operations (as in Geoda itself) makes rgeoda perform fast, but leaves less flexible when modifications or enhancements are desired.

The contiguity neighbours it constructs are the same as those found by poly2nb(), as almost are the $$I_i$$ measures. The difference is as established by Bivand and Wong (2018), that localmoran() calculates the input variable variance divinding by $$n$$, but Geoda uses $$(n-1)$$, as can be reproduced by setting mlvar=FALSE:

library(rgeoda)
system.time(Geoda_w <- queen_weights(pol_pres15))
# here
#    user  system elapsed
#   0.105   0.000   0.114
summary(Geoda_w)
#                      name               value
# 1 number of observations:                2495
# 2          is symmetric:                 TRUE
# 3               sparsity: 0.00228786229774178
# 4        # min neighbors:                   1
# 5        # max neighbors:                  13
# 6       # mean neighbors:    5.70821643286573
# 7     # median neighbors:                   6
# 8           has isolates:               FALSE
system.time(lisa <- local_moran(Geoda_w, pol_pres15["I_turnout"],
cpu_threads = ifelse(parallel::detectCores() == 1, 1, parallel::detectCores()-1L),
permutations = 499, seed = 1))
#    user  system elapsed
#   0.273   0.008   0.071
all.equal(card(nb_q), lisa_num_nbrs(lisa), check.attributes = FALSE)
# [1] TRUE
all.equal(lisa_values(lisa), localmoran(pol_pres15\$I_turnout,
listw=lw_q_W, mlvar = FALSE)[,1], check.attributes = FALSE)
# [1] TRUE

## 15.4 Exercises

1. Why are join-count measures on a chessboard so different between rook and queen neighbours?
2. Please repeat the simulation shown in section 15.1 using the chessboard polygons and the row-standardized queen contiguity neighbours. Why is it important to understand that spatial autocorrelation usually signals (unavoidable) mis-specification in our data?
3. Do we need to use conditional permutation in inference for local measures of spatial autocorrelation?
4. Why is false discovery rate adjustment recommended for local measures of spatial autocorrelation?
5. Compare the local Moran’s $$I_i$$ standard deviate values for the simulated data from exercise 15.2 for the analytical conditional approach, and Saddlepoint approximation. Consider the advantages and disadvantages of the Saddlepoint approximation approach.

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