# Slope of regression line in Moran scatterplot

In moran scatter plot, I checked

1. Slope of lm((spatial lagged variable) ~ (variable)): moran coefficient,

2. Slope of lm((spatial lagged standardized variable) ~ (standardized variable)): moran coefficient, and

3. Slope of lm((standardized spatial lagged variable) ~ (standardized variable)): correlation coefficient.

However, I think Anselin (1999) remark case [3] is moran coefficient. Is it true?

####################
library(spdep)
data(baltimore)
# head(baltimore)
# plot(baltimore$X, baltimore$Y)
nb <- knn2nb(knearneigh(cbind(baltimore$X, baltimore$Y), k=4))
listW <- nb2listw(nb, style="W")
W <- listw2mat(listW)

y <- baltimore\$PRICE    # as variable
Wy <- W %*% y       # as spatially lagged variable
Z_y <- scale(y)     # as standardized　variable
Z_Wy <- scale(Wy)       # as standardized spatially lagged variable
WZ_y <- W %*% Z_y       # as spatially lagged standardized variable

## Correration coefficient
cor(y, Wy)
## (Global) Moran's I
moran.test(y, listW, randomisation=F, alternative="two.sided")

par(mfrow=c(1,3))
## Moran scatterplot
# Case [1]:
# lm((spatially lagged variable) ~ (variable))
# equal to moran.plot(y, listW)
plot(y, Wy, xlab="variable", ylab="spatially lagged variable",
main=paste("[1] slope of line:", round(coef(lm(Wy~y))[2], 3)))
abline(lm(Wy~y))
abline(v=mean(y), lty=2); abline(h=mean(Wy), lty=2)

# Case [2]:
# lm((spatially lagged standardized variable) ~ (standardized variable))
plot(Z_y, WZ_y, xlab="standardized　variable",
ylab="spatially lagged standardized variable",
main=paste("[2] slope of line:", round(coef(lm(WZ_y~Z_y))[2], 3)))
abline(lm(WZ_y~Z_y))
abline(v=mean(Z_y), lty=2); abline(h=mean(WZ_y), lty=2)

# Case [3]:
# lm((standardized spatially lagged variable) ~ (standardized variable))
plot(Z_y, Z_Wy, xlab="standardized　variable",
ylab="standardized spatially lagged variable",
main=paste("[3] slope of line:", round(coef(lm(Z_Wy~Z_y))[2], 3)))
abline(lm(Z_Wy~Z_y))
abline(v=mean(Z_y), lty=2); abline(h=mean(Z_Wy), lty=2)

par(mfrow=c(1,1))
####################


Anselin, L. (1999) Interactive techniques and exploratory spatial data analysis. In: Longley P.A.

Goodchild M.F., Maguire D.J., Rhind D.W. (eds) Geographic information system: Principles, techniques, management and applications, 253–266, Wiley, New York.

• Can you clarify your question? "Is it true?" what is it? Commented Feb 17, 2016 at 13:34
• Anselin (1999) remarks a slope of line of case 3 is moran coefficient, not correlation coefficient. However, as pointed at R codes, I think the slope is equal to correlation coefficient. Is this correct?
– Tky
Commented Feb 17, 2016 at 14:16