“The Problem of Spatial Autocorrelation:” forty years on

Introduction

Cliff and Ord (1969), published forty years ago, marked a turning point in the treatment of spatial autocorrelation in quantitative geography. It provided the framework needed by any applied researcher to attempt an implementation for a different system, possibly using a different programming language. In this spirit, here we examine how spatial weights have been represented in implementations and may be reproduced, how the tabulated results in the paper may be reproduced, and how they may be extended to cover simulation.

One of the major assertions of Cliff and Ord (1969) is that their statistic advances the measurement of spatial autocorrelation with respect to Moran (1950) and Geary (1954) because a more general specification of spatial weights could be used. This more general form has implications both for the preparation of the weights themselves, and for the calculation of the measures. We will look at spatial weights first, before moving on to consider the measures presented in the paper and some of their subsequent developments. Before doing this, we will put together a data set matching that used in Cliff and Ord (1969). They provide tabulated data for the counties of the Irish Republic, but omit Dublin from analyses. A shapefile included in this package, kindly made available by Michael Tiefelsdorf, is used as a starting point:

library(spdep)
eire <- as(sf::st_read(system.file("shapes/eire.gpkg", package="spData")[1]), "Spatial")
row.names(eire) <- as.character(eire$names)
#proj4string(eire) <- CRS("+proj=utm +zone=30 +ellps=airy +units=km")

class(eire)

## [1] "SpatialPolygonsDataFrame"
## attr(,"package")
## [1] "sp"

names(eire)

##  [1] "A"       "towns"   "pale"    "size"    "ROADACC" "OWNCONS" "POPCHG" 
##  [8] "RETSALE" "INCOME"  "names"

and read into a SpatialPolygonsDataFrame — classes used for handling spatial data in are fully described in Roger S. Bivand, Pebesma, and Gómez-Rubio (2008). To this we need to add the data tabulated in the paper in Table 2,¹ p. 40, here in the form of a text file with added rainfall values from Table 9, p. 49:

fn <- system.file("etc/misc/geary_eire.txt", package="spdep")[1]
ge <- read.table(fn, header=TRUE)
names(ge)

##  [1] "serlet"        "county"        "pagval2_10"    "pagval10_50"  
##  [5] "pagval50p"     "cowspacre"     "ocattlepacre"  "pigspacre"    
##  [9] "sheeppacre"    "townvillp"     "carspcap"      "radiopcap"    
## [13] "retailpcap"    "psinglem30_34" "rainfall"

Since we assigned the county names as feature identifiers when reading the shapefiles, we do the same with the extra data, and combine the objects:

row.names(ge) <- as.character(ge$county)
all.equal(row.names(ge), row.names(eire))

## [1] TRUE

eire_ge <- cbind(eire, ge)

Finally, we need to drop the Dublin county omitted in the analyses conducted in Cliff and Ord (1969):

eire_ge1 <- eire_ge[!(row.names(eire_ge) %in% "Dublin"),]
length(row.names(eire_ge1))

## [1] 25

To double-check our data, let us calculate the sample Beta coefficients, using the formulae given in the paper for sample moments:

skewness <- function(z) {z <- scale(z, scale=FALSE); ((sum(z^3)/length(z))^2)/((sum(z^2)/length(z))^3)}
kurtosis <- function(z) {z <- scale(z, scale=FALSE); (sum(z^4)/length(z))/((sum(z^2)/length(z))^2)}

These differ somewhat from the ways in which skewness and kurtosis are computed in modern statistical software, see for example Joanes and Gill (1998). However, for our purposes, they let us reproduce Table 3, p. 42:

print(sapply(as(eire_ge1, "data.frame")[13:24], skewness), digits=3)

##    pagval2_10   pagval10_50     pagval50p     cowspacre  ocattlepacre 
##      1.675429      1.294978      0.000382      1.682094      0.086267 
##     pigspacre    sheeppacre     townvillp      carspcap     radiopcap 
##      1.138387      1.842362      0.472748      0.011111      0.342805 
##    retailpcap psinglem30_34 
##      0.002765      0.068169

print(sapply(as(eire_ge1, "data.frame")[13:24], kurtosis), digits=4)

##    pagval2_10   pagval10_50     pagval50p     cowspacre  ocattlepacre 
##         3.790         4.331         1.508         4.294         2.985 
##     pigspacre    sheeppacre     townvillp      carspcap     radiopcap 
##         3.754         4.527         2.619         1.865         2.730 
##    retailpcap psinglem30_34 
##         2.188         2.034

print(sapply(as(eire_ge1, "data.frame")[c(13,16,18,19)], function(x) skewness(log(x))), digits=3)

## pagval2_10  cowspacre  pigspacre sheeppacre 
##    0.68801    0.17875    0.00767    0.04184

print(sapply(as(eire_ge1, "data.frame")[c(13,16,18,19)], function(x) kurtosis(log(x))), digits=4)

## pagval2_10  cowspacre  pigspacre sheeppacre 
##      2.883      2.799      2.212      2.421

Using the tabulated value of 23.6 for the percentage of agricultural holdings above 50 in 1950 in Waterford, the skewness and kurtosis cannot be reproduced, but by comparison with the irishdata dataset in , it turns out that the value should rather be 26.6, which yields the tabulated skewness and kurtosis values.

Before going on, the variables considered are presented in Table vars.

Spatial weights

As a basis for comparison, we will first read the unstandardised weighting matrix given in Table A1, p. 54, of the paper, reading a file corrected for the misprint giving O rather than D as a neighbour of V:

fn <- system.file("etc/misc/unstand_sn.txt", package="spdep")[1]
unstand <- read.table(fn, header=TRUE)
summary(unstand)

##      from                to                weight        
##  Length:110         Length:110         Min.   :0.000600  
##  Class :character   Class :character   1st Qu.:0.003225  
##  Mode  :character   Mode  :character   Median :0.007550  
##                                        Mean   :0.007705  
##                                        3rd Qu.:0.010225  
##                                        Max.   :0.032400

In the file, the counties are represented by their serial letters, so ordering and conversion to integer index representation is required to reach a representation similar to that of the SpatialStats module for spatial neighbours:

class(unstand) <- c("spatial.neighbour", class(unstand))
of <- ordered(unstand$from)
attr(unstand, "region.id") <- levels(of)
unstand$from <- as.integer(of)
unstand$to <- as.integer(ordered(unstand$to))
attr(unstand, "n") <- length(unique(unstand$from))

Having done this, we can change its representation to a listw object, assigning an appropriate style (generalised binary) for unstandardised values:

lw_unstand <- sn2listw(unstand)
lw_unstand$style <- "B"
lw_unstand

## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 25 
## Number of nonzero links: 110 
## Percentage nonzero weights: 17.6 
## Average number of links: 4.4 
## 
## Weights style: B 
## Weights constants summary:
##    n  nn     S0         S1        S2
## B 25 625 0.8476 0.01871808 0.1229232

Note that the values of S0, S1, and S2 correspond closely with those given on page 42 of the paper, 0.84688672, 0.01869986 and 0.12267319. The discrepancies appear to be due to rounding in the printed table of weights.

The contiguous neighbours represented in this object ought to match those found using poly2nb. However, we see that the reproduced contiguities have a smaller link count:

nb <- poly2nb(eire_ge1)
nb

## Neighbour list object:
## Number of regions: 25 
## Number of nonzero links: 108 
## Percentage nonzero weights: 17.28 
## Average number of links: 4.32

The missing link is between Clare and Kerry, perhaps by the Tarbert–Killimer ferry, but the counties are not contiguous, as Figure plot_nb shows:

xx <- diffnb(nb, lw_unstand$neighbours, verbose=TRUE)

## Neighbour difference for region id: Clare in relation to id: Kerry 
## Neighbour difference for region id: Kerry in relation to id: Clare

plot(eire_ge1, border="grey60")
plot(nb, coordinates(eire_ge1), add=TRUE, pch=".", lwd=2)
plot(xx, coordinates(eire_ge1), add=TRUE, pch=".", lwd=2, col=3)

County boundaries and contiguities

An attempt has also been made to reproduce the generalised weights for 25 Irish counties reported by Cliff and Ord (1969), after Dublin is omitted. Reproducing the inverse distance component d_ij⁻¹ of the generalised weights d_ij⁻¹β_i(j) is eased by the statement in Cliff and Ord (1973, 55) that the points chosen to represent the counties were their “geographic centres,” so not very different from the centroids yielded by applying a chosen computational geometry function. The distance metric is not given, and may have been in kilometers or miles — both were tried, but the results were not sensitive to the difference as it applies equally across the weights; miles are used here. Computing the proportion of shared distance measure β_i(j) is harder, because it requires the availability of the full topology of the input polygons. Roger S. Bivand, Pebesma, and Gómez-Rubio (2008, 244) show how to employ the vect2neigh function (written by Markus Neteler) in the package when using GRASS GIS vector handling to create a full topology from spaghetti vector data and to extract border segment lengths; a similar approach also is mentioned there using ArcGIS coverages for the same purpose. GRASS was used to create the topology, and next the proportion of shared distance measure was calculated.

library(terra)
v_eire_ge1 <-vect(eire_ge1)
SG <- rasterize(v_eire_ge1, rast(nrows=70, ncols=50, extent=ext(v_eire_ge1)), field="county")
library(rgrass)
grass_home <- "/home/rsb/topics/grass/g820/grass82"
initGRASS(grass_home, home=tempdir(), SG=SG, override=TRUE)
write_VECT(v_eire_ge1, "eire", flags=c("o", "overwrite"))
res <- vect2neigh("eire", ID="serlet")

res$length <- res$length*1000
attr(res, "external") <- attr(res, "external")*1000
attr(res, "total") <- attr(res, "total")*1000
grass_borders <- sn2listw(res)
raw_borders <- grass_borders$weights
int_tot <- attr(res, "total") - attr(res, "external")
prop_borders <- lapply(1:length(int_tot), function(i) raw_borders[[i]]/int_tot[i])
dlist <- nbdists(grass_borders$neighbours, coordinates(eire_ge1))
inv_dlist <- lapply(dlist, function(x) 1/(x/1.609344))
combo_km <- lapply(1:length(inv_dlist), function(i) inv_dlist[[i]]*prop_borders[[i]])
combo_km_lw <- nb2listw(grass_borders$neighbours, glist=combo_km, style="B")
summary(combo_km_lw)

## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 25 
## Number of nonzero links: 108 
## Percentage nonzero weights: 17.28 
## Average number of links: 4.32 
## Link number distribution:
## 
## 1 2 3 4 5 6 7 8 
## 1 2 5 5 8 1 2 1 
## 1 least connected region:
## E with 1 link
## 1 most connected region:
## V with 8 links
## 
## Weights style: B 
## Weights constants summary:
##    n  nn        S0       S1.5        S2
## B 25 625 0.9083144 0.02191845 0.1427746

To compare, we need to remove the Tarbert–Killimer ferry link manually, and view the summary of the original weights, as well as a correlation coefficient between these and the reconstructed weights. Naturally, unless the boundary coordinates used here are identical with those in the original analysis, presumably measured by hand, small differences will occur.

red_lw_unstand <- lw_unstand
Clare <- which(attr(lw_unstand, "region.id") == "C")
Kerry <- which(attr(lw_unstand, "region.id") == "H")
Kerry_in_Clare <- which(lw_unstand$neighbours[[Clare]] == Kerry)
Clare_in_Kerry <- which(lw_unstand$neighbours[[Kerry]] == Clare)
red_lw_unstand$neighbours[[Clare]] <- red_lw_unstand$neighbours[[Clare]][-Kerry_in_Clare]
red_lw_unstand$neighbours[[Kerry]] <- red_lw_unstand$neighbours[[Kerry]][-Clare_in_Kerry]
red_lw_unstand$weights[[Clare]] <- red_lw_unstand$weights[[Clare]][-Kerry_in_Clare]
red_lw_unstand$weights[[Kerry]] <- red_lw_unstand$weights[[Kerry]][-Clare_in_Kerry]
summary(red_lw_unstand)

## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 25 
## Number of nonzero links: 108 
## Percentage nonzero weights: 17.28 
## Average number of links: 4.32 
## Link number distribution:
## 
## 1 2 3 4 5 6 7 8 
## 1 2 5 5 8 1 2 1 
## 1 least connected region:
## E with 1 link
## 1 most connected region:
## V with 8 links
## 
## Weights style: B 
## Weights constants summary:
##    n  nn     S0         S1        S2
## B 25 625 0.8437 0.01870287 0.1222501

cor(unlist(red_lw_unstand$weights), unlist(combo_km_lw$weights))

## [1] 0.969543

Even though the differences in the general weights, for identical contiguities, are so small, the consequences for the measure of spatial autocorrelation are substantial, Here we use the fifth variable, other cattle per 1000 acres crops and pasture (1952), and see that the reconstructed weights seem to “reveal” more autocorrelation than the original weights.

flatten <- function(x, digits=3, statistic="I") {
  res <- c(format(x$estimate, digits=digits),
    format(x$statistic, digits=digits),
    format.pval(x$p.value, digits=digits))
  res <- matrix(res, ncol=length(res))
  colnames(res) <- paste(c("", "E", "V", "SD_", "P_"), "I", sep="")
  rownames(res) <- deparse(substitute(x))
  res
}
`reconstructed weights` <- moran.test(eire_ge1$ocattlepacre, combo_km_lw)
`original weights` <- moran.test(eire_ge1$ocattlepacre, red_lw_unstand)
print(rbind(flatten(`reconstructed weights`), flatten(`original weights`)), quote=FALSE)

##                       I       EI      VI      SD_I P_I    
## reconstructed weights  0.3203 -0.0417  0.0225 2.41 0.00792
## original weights       0.2779 -0.0417  0.0223 2.14 0.0161

Measures of spatial autocorrelation

Our targets for reproduction are Tables 4 and 5 in Cliff and Ord (1969, 43–44), the first containing standard deviates under normality and randomisation for the original Moran measure with binary weights, the original Geary measure with binary weights, the proposed measure with unstandardised general weights, and the proposed measure with row-standardised general weights. In addition, four variables were log-transformed on the basis of the skewness and kurtosis results presented above. We carry out the transformation of these variables, and generate additional binary and row-standardised general spatial weights objects — note that the weights constants for the row-standardised general weights agree with those given on p. 42 in the paper, after allowing for small differences due to rounding in the weights values displayed in the paper (p. 54):

eire_ge1$ln_pagval2_10 <- log(eire_ge1$pagval2_10)
eire_ge1$ln_cowspacre <- log(eire_ge1$cowspacre)
eire_ge1$ln_pigspacre <- log(eire_ge1$pigspacre)
eire_ge1$ln_sheeppacre <- log(eire_ge1$sheeppacre)
vars <- c("pagval2_10", "ln_pagval2_10", "pagval10_50", "pagval50p",
 "cowspacre", "ln_cowspacre", "ocattlepacre", "pigspacre",
 "ln_pigspacre", "sheeppacre", "ln_sheeppacre", "townvillp",
 "carspcap", "radiopcap", "retailpcap", "psinglem30_34")
nb_B <- nb2listw(lw_unstand$neighbours, style="B")
nb_B

## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 25 
## Number of nonzero links: 110 
## Percentage nonzero weights: 17.6 
## Average number of links: 4.4 
## 
## Weights style: B 
## Weights constants summary:
##    n  nn  S0  S1   S2
## B 25 625 110 220 2176

lw_std <- nb2listw(lw_unstand$neighbours, glist=lw_unstand$weights, style="W")
lw_std

## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 25 
## Number of nonzero links: 110 
## Percentage nonzero weights: 17.6 
## Average number of links: 4.4 
## 
## Weights style: W 
## Weights constants summary:
##    n  nn S0       S1       S2
## W 25 625 25 15.84089 103.6197

The standard representation of the measures is:

$$I = \frac{n}{\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}} \frac{\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}(x_i-\bar{x})(x_j-\bar{x})}{\sum_{i=1}^{n}(x_i - \bar{x})^2}$$

for Moran’s I — in the paper termed the proposed statistic, and for Geary’s C:

$$C = \frac{(n-1)}{2\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}} \frac{\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}(x_i-x_j)^2}{\sum_{i=1}^{n}(x_i - \bar{x})^2}$$

where x_i, i = 1, …, n are n observations on the numeric variable of interest, and w_ij are the spatial weights. In order to reproduce the standard deviates given in the paper, it is sufficient to apply moran.test to the variables with three different spatial weights objects, and two different values of the randomisation= argument. In addition, geary.test is applied to a single spatial weights objects, and two different values of the randomisation= argument.

system.time({
MoranN <- lapply(vars, function(x) moran.test(eire_ge1[[x]], listw=nb_B, randomisation=FALSE))
MoranR <- lapply(vars, function(x) moran.test(eire_ge1[[x]], listw=nb_B, randomisation=TRUE))
GearyN <- lapply(vars, function(x) geary.test(eire_ge1[[x]], listw=nb_B, randomisation=FALSE))
GearyR <- lapply(vars, function(x) geary.test(eire_ge1[[x]], listw=nb_B, randomisation=TRUE))
Prop_unstdN  <- lapply(vars, function(x) moran.test(eire_ge1[[x]], listw=lw_unstand, randomisation=FALSE))
Prop_unstdR  <- lapply(vars, function(x) moran.test(eire_ge1[[x]], listw=lw_unstand, randomisation=TRUE))
Prop_stdN  <- lapply(vars, function(x) moran.test(eire_ge1[[x]], listw=lw_std, randomisation=FALSE))
Prop_stdR  <- lapply(vars, function(x) moran.test(eire_ge1[[x]], listw=lw_std, randomisation=TRUE))
})

##    user  system elapsed 
##   0.112   0.000   0.112

res <- sapply(c("MoranN", "MoranR", "GearyN", "GearyR", "Prop_unstdN", "Prop_unstdR", "Prop_stdN", "Prop_stdR"), function(x) sapply(get(x), "[[", "statistic"))
rownames(res) <- vars
ores <- res[,c(1,2,5:8)]

In order to conduct 8 different tests on 16 variables, we use lapply on the list of variables in the specified order, then sapply on a list of output objects by name to generate a table in the same row and column order as the original (we save a copy of six columns for comparison with bootstrap results below):

print(formatC(res, format="f", digits=4), quote=FALSE)

##               MoranN MoranR GearyN  GearyR  Prop_unstdN Prop_unstdR Prop_stdN Prop_stdR
## pagval2_10    3.7851 3.8779 4.3142  3.9016  3.3307      3.4159      3.9276    4.0292   
## ln_pagval2_10 4.0965 4.1074 4.0841  4.0343  3.5795      3.5894      4.1278    4.1393   
## pagval10_50   1.0899 1.1316 2.7511  2.3760  1.3348      1.3882      1.5127    1.5738   
## pagval50p     5.2011 5.0555 4.0178  4.7194  4.6604      4.5247      4.8823    4.7387   
## cowspacre     5.1969 5.3907 4.3531  3.7709  4.1379      4.2991      4.7274    4.9135   
## ln_cowspacre  5.2420 5.2455 3.9211  3.9085  4.2007      4.2037      4.6532    4.6566   
## ocattlepacre  0.5565 0.5593 -0.1707 -0.1668 2.1366      2.1478      1.9219    1.9320   
## pigspacre     2.4807 2.5393 2.2928  2.0802  2.8312      2.9010      3.1908    3.2703   
## ln_pigspacre  2.3015 2.2724 2.0520  2.1893  2.5171      2.4839      2.8460    2.8081   
## sheeppacre    1.0188 1.0630 0.8689  0.7387  1.7398      1.8187      1.4792    1.5470   
## ln_sheeppacre 1.2930 1.2827 1.5156  1.5767  2.3708      2.3511      2.0374    2.0203   
## townvillp     2.2759 2.2681 2.5475  2.5902  1.2148      1.2104      1.6275    1.6216   
## carspcap      4.4927 4.4015 3.2247  3.5992  3.8897      3.8075      4.1826    4.0934   
## radiopcap     0.3156 0.3153 1.2294  1.2348  0.5915      0.5909      0.7857    0.7849   
## retailpcap    3.4985 3.4524 3.1303  3.3497  2.9291      2.8888      3.0346    2.9926   
## psinglem30_34 2.7349 2.6895 2.3382  2.5519  2.7541      2.7065      2.7078    2.6605

The values of the standard deviates agree with those in Table 4 in the original paper, with the exception of those for the proposed statistic with standardised weights under normality for all untransformed variables. We can see what has happened by substituting the weights constants for the standardised weights with those for unstandardised weights:

wc_unstd <- spweights.constants(lw_unstand)
wrong_N_sqVI <- sqrt((wc_unstd$nn*wc_unstd$S1 - wc_unstd$n*wc_unstd$S2 + 3*wc_unstd$S0*wc_unstd$S0)/((wc_unstd$nn-1)*wc_unstd$S0*wc_unstd$S0)-((-1/(wc_unstd$n-1))^2))
raw_data <- grep("^ln_", vars, invert=TRUE)
I <- sapply(Prop_stdN, function(x) x$estimate[1])[raw_data]
EI <- sapply(Prop_stdN, function(x) x$estimate[2])[raw_data]
res <- (I - EI)/wrong_N_sqVI
names(res) <- vars[raw_data]
print(formatC(res, format="f", digits=4), quote=FALSE)

##    pagval2_10   pagval10_50     pagval50p     cowspacre  ocattlepacre     pigspacre 
##        3.8836        1.4957        4.8276        4.6744        1.9003        3.1550 
##    sheeppacre     townvillp      carspcap     radiopcap    retailpcap psinglem30_34 
##        1.4627        1.6093        4.1357        0.7769        3.0006        2.6774

Next, let us look at Table 5 in the original paper. Here we only tabulate the values of the measures themselves, and, since the expectation is constant for each measure, the square root of the variance of the measure under randomisation — extracting values calculated above:

res <- lapply(c("MoranR", "GearyR", "Prop_unstdR", "Prop_stdR"), function(x) sapply(get(x), function(y) c(y$estimate[1], sqrt(y$estimate[3]))))
res <- t(do.call("rbind", res))
colnames(res) <- c("I", "sigma_I", "C", "sigma_C", "unstd_r", "sigma_r", "std_r", "sigma_r")
rownames(res) <- vars
print(formatC(res, format="f", digits=4), quote=FALSE)

##               I       sigma_I C      sigma_C unstd_r sigma_r std_r  sigma_r
## pagval2_10    0.4074  0.1158  0.3477 0.1672  0.4559  0.1456  0.5384 0.1440 
## ln_pagval2_10 0.4444  0.1183  0.3825 0.1531  0.4930  0.1490  0.5680 0.1473 
## pagval10_50   0.0877  0.1143  0.5840 0.1751  0.1577  0.1436  0.1818 0.1420 
## pagval50p     0.5754  0.1221  0.3925 0.1287  0.6545  0.1539  0.6795 0.1522 
## cowspacre     0.5749  0.1144  0.3418 0.1745  0.5764  0.1438  0.6566 0.1421 
## ln_cowspacre  0.5803  0.1186  0.4071 0.1517  0.5858  0.1493  0.6456 0.1476 
## ocattlepacre  0.0244  0.1181  1.0258 0.1547  0.2775  0.1486  0.2422 0.1469 
## pigspacre     0.2527  0.1159  0.6533 0.1667  0.3812  0.1458  0.4296 0.1441 
## ln_pigspacre  0.2314  0.1202  0.6897 0.1417  0.3343  0.1514  0.3787 0.1497 
## sheeppacre    0.0792  0.1137  0.8686 0.1778  0.2182  0.1429  0.1768 0.1412 
## ln_sheeppacre 0.1117  0.1196  0.7708 0.1453  0.3125  0.1506  0.2593 0.1489 
## townvillp     0.2284  0.1191  0.6148 0.1487  0.1398  0.1499  0.1987 0.1482 
## carspcap      0.4914  0.1211  0.5124 0.1355  0.5394  0.1526  0.5761 0.1509 
## radiopcap     -0.0042 0.1188  0.8141 0.1505  0.0467  0.1495  0.0744 0.1478 
## retailpcap    0.3734  0.1202  0.5267 0.1413  0.3959  0.1515  0.4066 0.1498 
## psinglem30_34 0.2828  0.1207  0.6465 0.1385  0.3697  0.1520  0.3583 0.1503

The values are as follows, and match the original with the exception of those for the initial version of Moran’s I in the first two columns. If we write a function implementing equations 3 and 4:

$$I = \frac{\sum_{i=1}^{n}\sum_{j=i+1}^{n}w_{ij}(x_i-\bar{x})(x_j-\bar{x})}{\sum_{i=1}^{n}(x_i - \bar{x})^2}$$

where crucially the inner summation is over i + 1…n, not 1…n, we can reproduce the values of the measure shown in the original Table 5:

oMoranf <- function(x, nb) {
  z <- scale(x, scale=FALSE)
  n <- length(z)
  glist <- lapply(1:n, function(i) {ii <- nb[[i]]; ifelse(ii > i, 1, 0)})
  lw <- nb2listw(nb, glist=glist, style="B")
  wz <- lag(lw, z)
  I <- (sum(z*wz)/sum(z*z))
  I
}
res <- sapply(vars, function(x) oMoranf(eire_ge1[[x]], nb=lw_unstand$neighbours))
print(formatC(res, format="f", digits=4), quote=FALSE)

##    pagval2_10 ln_pagval2_10   pagval10_50     pagval50p     cowspacre  ln_cowspacre 
##        0.8964        0.9776        0.1928        1.2660        1.2649        1.2766 
##  ocattlepacre     pigspacre  ln_pigspacre    sheeppacre ln_sheeppacre     townvillp 
##        0.0536        0.5559        0.5091        0.1743        0.2458        0.5024 
##      carspcap     radiopcap    retailpcap psinglem30_34 
##        1.0811       -0.0093        0.8215        0.6222

The variance term given in equation 7 in the original paper is for the case of normality, not randomisation; the reference on p. 28 to equation 38 on p. 26 does not permit the reproduction of the values in the second column of Table 5. The variance equation given as equation 1.35 by Cliff and Ord (1973, 9) does not do so either, so for the time being it is not possible to say how the tabulated values were computed. Note that since the standard deviances are reproduced correctly, and can be reproduced from the second column values using the measure and its expectance, it is just a matter of establishing which formula was used, but this has so far not proved possible.

Simulating measures of spatial autocorrelation

Cliff and Ord (1969) do not conduct simulation experiments, although their sequels do, notably Cliff and Ord (1973), among many others. Simulation studies are necessarily more demanding computationally, especially if spatially autocorrelated variables are to be created, as in Cliff and Ord (1973, 146–53). In the same book, they also report the use of permutation tests, also known as Monte Carlo or Hope hypothesis testing procedures (Cliff and Ord 1973, 50–52). These procedures provided a way to examine the distribution of the statistic of interest by exchanging at random the observed values between observations, and then comparing the simulated distribution under the null hypothesis of no spatial patterning with the observed value of the statistic in question.

MoranI.boot <- function(var, i, ...) {
  var <- var[i]
  return(moran(x=var, ...)$I)
}
Nsim <- function(d, mle) {
  n <- length(d)
  rnorm(n, mle$mean, mle$sd)
}
f_bperm <- function(x, nsim, listw) {
  boot(x, statistic=MoranI.boot, R=nsim, sim="permutation", listw=listw,
    n=length(x), S0=Szero(listw))
}
f_bpara <- function(x, nsim, listw) {
  boot(x, statistic=MoranI.boot, R=nsim, sim="parametric", ran.gen=Nsim,
    mle=list(mean=mean(x), sd=sd(x)), listw=listw, n=length(x),
    S0=Szero(listw))
}
nsim <- 4999
set.seed(1234)

First let us define a function MoranI.boot just to return the value of Moran’s I for variable var and permutation index i, and a function Nsim to generate random samples from the variable of interest assuming Normality. To make it easier to process the variables in turn, we encapsulate calls to boot in wrapper functions f_bperm and f_bpara. Running 4999 simulations for each of 16 for three different weights specifications and both parametric and permutation bootstrap takes quite a lot of time.

system.time({
MoranNb <- lapply(vars, function(x) f_bpara(x=eire_ge1[[x]], nsim=nsim, listw=nb_B))
MoranRb <- lapply(vars, function(x) f_bperm(x=eire_ge1[[x]], nsim=nsim, listw=nb_B))
Prop_unstdNb  <- lapply(vars, function(x) f_bpara(x=eire_ge1[[x]], nsim=nsim, listw=lw_unstand))
Prop_unstdRb  <- lapply(vars, function(x) f_bperm(x=eire_ge1[[x]], nsim=nsim, listw=lw_unstand))
Prop_stdNb  <- lapply(vars, function(x) f_bpara(x=eire_ge1[[x]], nsim=nsim, listw=lw_std))
Prop_stdRb  <- lapply(vars, function(x) f_bperm(x=eire_ge1[[x]], nsim=nsim, listw=lw_std))
})

res <- lapply(c("MoranNb", "MoranRb", "Prop_unstdNb", "Prop_unstdRb", "Prop_stdNb", "Prop_stdRb"), function(x) sapply(get(x), function(y) (y$t0 - mean(y$t))/sd(y$t)))
res <- t(do.call("rbind", res))
colnames(res) <- c("MoranNb", "MoranRb", "Prop_unstdNb", "Prop_unstdRb", "Prop_stdNb", "Prop_stdRb")
rownames(res) <- vars

We collate the results to compare with the analytical standard deviates under Normality and randomisation, and see that in fact the differences are not at all large, as expressed by the median absolute difference between the tables. We can also see that inferences based on a one-sided α = 0.05 cut-off are the same for the analytical and bootstrap approaches. This indicates that we can, in general, rely on the analytical standard deviates, and that bootstrap methods will not help if assumptions underlying the measures are not met.

print(formatC(res, format="f", digits=4), quote=FALSE)

##    pagval2_10 ln_pagval2_10   pagval10_50     pagval50p     cowspacre  ln_cowspacre 
##        0.8964        0.9776        0.1928        1.2660        1.2649        1.2766 
##  ocattlepacre     pigspacre  ln_pigspacre    sheeppacre ln_sheeppacre     townvillp 
##        0.0536        0.5559        0.5091        0.1743        0.2458        0.5024 
##      carspcap     radiopcap    retailpcap psinglem30_34 
##        1.0811       -0.0093        0.8215        0.6222

oores <- ores - res
apply(oores, 2, mad)

##      MoranN      MoranR Prop_unstdN Prop_unstdR   Prop_stdN   Prop_stdR 
##   1.6470091   1.6241614   0.7681895   0.7170636   1.2615083   1.2723149

alpha_0.05 <- qnorm(0.05, lower.tail=FALSE)
all((res >= alpha_0.05) == (ores >= alpha_0.05))

## [1] FALSE

These assumptions affect the shape of the distribution of the measure in its tails; one possibility is to use a Saddlepoint approximation to find an equivalent to the analytical or bootstrap-based standard deviate for inference (Tiefelsdorf 2002). The Saddlepoint approximation requires the eigenvalues of the weights matrix and iterative root-finding for global Moran’s I, while for local Moran’s I_i, analytical forms are known. Even with this computational burden, the Saddlepoint approximation for global Moran’s I runs quite quickly. First we need to fit null linear models (only including an intercept) to the variables, then apply lm.morantest.sad to the fitted model objects:

lm_objs <- lapply(vars, function(x) lm(formula(paste(x, "~1")), data=eire_ge1))
system.time({
MoranSad <- lapply(lm_objs, function(x) lm.morantest.sad(x, listw=nb_B))
Prop_unstdSad  <- lapply(lm_objs, function(x) lm.morantest.sad(x, listw=lw_unstand))
Prop_stdSad  <- lapply(lm_objs, function(x) lm.morantest.sad(x, listw=lw_std))
})

##    user  system elapsed 
##   0.065   0.000   0.065

res <- sapply(c("MoranSad", "Prop_unstdSad", "Prop_stdSad"), function(x) sapply(get(x), "[[", "statistic"))
rownames(res) <- vars

Although the analytical standard deviates (under Normality) are larger than those reached using the Saddlepoint approximation when measured by median absolute deviation, the differences do not lead to different inferences at this chosen cut-off. This reflects the fact that the shape of the distribution is very sensitive to small n, but for moderate n and global Moran’s I, the effects are seen only further out in the tails. The consequences for local Moran’s I_i are much stronger, because the clique of neighbours of each observation is typically very small. It is perhaps of interest that the differences are much smaller for the case of general weights than for unstandardised binary weights.

print(formatC(res, format="f", digits=4), quote=FALSE)

##               MoranSad Prop_unstdSad Prop_stdSad
## pagval2_10    3.2903   3.1711        3.8283     
## ln_pagval2_10 3.5346   3.3982        4.0441     
## pagval10_50   1.0883   1.3219        1.4791     
## pagval50p     4.4402   4.4258        4.9377     
## cowspacre     4.4366   3.9164        4.7406     
## ln_cowspacre  4.4758   3.9760        4.6493     
## ocattlepacre  0.6030   2.0748        1.8642     
## pigspacre     2.2611   2.7154        3.0775     
## ln_pigspacre  2.1158   2.4271        2.7417     
## sheeppacre    1.0250   1.7040        1.4476     
## ln_sheeppacre 1.2669   2.2921        1.9731     
## townvillp     2.0949   1.2079        1.5872     
## carspcap      3.8500   3.6840        4.1044     
## radiopcap     0.3750   0.6094        0.7906     
## retailpcap    3.0663   2.8049        2.9245     
## psinglem30_34 2.4649   2.6449        2.6089

oores <- res - ores[,c(1,3,5)]
apply(oores, 2, mad)

##      MoranSad Prop_unstdSad   Prop_stdSad 
##    0.37142684    0.10779060    0.05650183

all((res >= alpha_0.05) == (ores[,c(1,3,5)] >= alpha_0.05))

## [1] TRUE

In addition we could choose to use the exact distribution of Moran’s I, as described by Tiefelsdorf (2000); its implementation is covered in R. S. Bivand, Müller, and Reder (2009). The global case also needs the eigenvalues of the weights matrix, and the solution of a numerical integration function, but for these cases, the timings are quite acceptable.

system.time({ 
MoranEx <- lapply(lm_objs, function(x) lm.morantest.exact(x, listw=nb_B))
Prop_unstdEx  <- lapply(lm_objs, function(x) lm.morantest.exact(x, listw=lw_unstand))
Prop_stdEx  <- lapply(lm_objs, function(x) lm.morantest.exact(x, listw=lw_std))
})

##    user  system elapsed 
##   0.079   0.003   0.083

res <- sapply(c("MoranEx", "Prop_unstdEx", "Prop_stdEx"), function(x) sapply(get(x), "[[", "statistic"))
rownames(res) <- vars

The output is comparable with that of the Saddlepoint approximation, and the inferences drawn here are the same for the chosen cut-off as for the analytical standard deviates calculated under Normality.

print(formatC(res, format="f", digits=4), quote=FALSE)

##               MoranEx Prop_unstdEx Prop_stdEx
## pagval2_10    3.2895  3.1660       3.8261    
## ln_pagval2_10 3.5384  3.3979       4.0430    
## pagval10_50   1.0798  1.3131       1.4745    
## pagval50p     4.4568  4.4430       4.9446    
## cowspacre     4.4532  3.9268       4.7453    
## ln_cowspacre  4.4928  3.9875       4.6531    
## ocattlepacre  0.5967  2.0611       1.8596    
## pigspacre     2.2486  2.7033       3.0740    
## ln_pigspacre  2.1031  2.4131       2.7380    
## sheeppacre    1.0168  1.6924       1.4430    
## ln_sheeppacre 1.2575  2.2779       1.9685    
## townvillp     2.0822  1.1999       1.5826    
## carspcap      3.8593  3.6899       4.1037    
## radiopcap     0.3696  0.6054       0.7873    
## retailpcap    3.0616  2.7938       2.9210    
## psinglem30_34 2.4533  2.6321       2.6050

oores <- res - ores[,c(1,3,5)]
apply(oores, 2, mad)

##      MoranEx Prop_unstdEx   Prop_stdEx 
##   0.37187539   0.09751723   0.05419300

all((res >= alpha_0.05) == (ores[,c(1,3,5)] >= alpha_0.05))

## [1] TRUE

Li, Calder, and Cressie (2007) take up the challenge in Cliff and Ord (1969, 31), to try to give the statistic a bounded fixed range. Their APLE measure is intended to approximate the spatial dependence parameter of a simultaneous autoregressive model better than Moran’s I, and re-scales the measure by a function of the eigenvalues of the spatial weights matrix. APLE requires the use of row standardised weights.

vars_scaled <- lapply(vars, function(x) scale(eire_ge1[[x]], scale=FALSE))
nb_W <- nb2listw(lw_unstand$neighbours, style="W")
pre <- spatialreg:::preAple(0, listw=nb_W)
MoranAPLE <- sapply(vars_scaled, function(x) spatialreg:::inAple(x, pre))
pre <- spatialreg:::preAple(0, listw=lw_std, override_similarity_check=TRUE)
Prop_stdAPLE <- sapply(vars_scaled, function(x) spatialreg:::inAple(x, pre))
res <- cbind(MoranAPLE, Prop_stdAPLE)
colnames(res) <- c("APLE W", "APLE Gstd")
rownames(res) <- vars

In order to save time, we use the two internal functions spatialreg:::preAple and spatialreg:::inAple, since for each definition of spatial weights, the same eigenvalue calculations need to be made. The notation using the ::: operator says that the function with named after the operator is to be found in the namespace of the package named before the operator. The APLE values repeat the pattern that we have already seen — for some variables, the measured autocorrelation is very similar irrespective of spatial weights definition, while for others, the change in the definition from binary to general does make a difference.

print(formatC(res, format="f", digits=4), quote=FALSE)

##               APLE W APLE Gstd
## pagval2_10    0.7702 0.6628   
## ln_pagval2_10 0.7615 0.6655   
## pagval10_50   0.3954 0.3446   
## pagval50p     0.7519 0.7313   
## cowspacre     0.8329 0.7408   
## ln_cowspacre  0.8148 0.7487   
## ocattlepacre  0.1468 0.4092   
## pigspacre     0.6227 0.6205   
## ln_pigspacre  0.5582 0.5887   
## sheeppacre    0.1594 0.2841   
## ln_sheeppacre 0.2046 0.4550   
## townvillp     0.3442 0.2644   
## carspcap      0.7140 0.6166   
## radiopcap     0.0083 0.1376   
## retailpcap    0.6376 0.5307   
## psinglem30_34 0.4094 0.4889

Odds and ends …

The differences found in the case of a few variables in inference using the original binary weights, and the general weights proposed by Cliff and Ord (1969) are necessarily related to the the weights thenselves. Figures plot_wts and plot_map show the values of the weights in sparse matrix form, and the sums of weights by county where these sums are not identical by design. In the case of binary weights, the matrix entries are equal, but the sums up-weight counties with many neighbours.

General weights up-weight counties that are close to each other, have more neighbours, and share larger boundary proportions (an asymmetric relationship). There is a further impact of using boundary proportions, in that the boundary between the county and the exterior is subtracted, thus boosting the weights between edge counties and their neighbours, even if there are few of them. Standardised general weights up-weight further up-weight counties with few neighbours, chiefly those on the edges of the study area.

With a small data set, here with n = 25, it is very possible that edge and other configuration effects are relatively strong, and may impact inference in different ways. The issue of egde effects has not really been satisfactorily resolved, and should be kept in mind in analyses of data sets of this size and shape.