Spatial Autocorrelation and Areal Data

HES 505 Fall 2023: Session 21

Matt Williamson

Objectives

By the end of today you should be able to:

  • Use the spdep package to identify the neighbors of a given polygon based on proximity, distance, and minimum number

  • Understand the underlying mechanics of Moran’s I and calculate it for various neighbors

  • Distinguish between global and local measures of spatial autocorrelation

  • Visualize neighbors and clusters

Revisiting Spatial Autocorrelation

Spatial Autocorrelation

  • Attributes (features) are often non-randomly distributed

  • Especially true with aggregated data

  • Interest is in the relationship between proximity and the feature

  • Difference from kriging and semivariance

From Manuel Gimond

Moran’s I

  • Moran’s I

Finding Neighbors

  • How do we define \(I(d)\) for areal data?

  • What about \(w_{ij}\)?

  • We can use spdep for that!!

::: :::

Using spdep

cdc <- read_sf("data/opt/data/2023/vectorexample/cdc_nw.shp") %>% 
  select(stateabbr, countyname, countyfips, casthma_cr)

::: :::

Finding Neighbors

  • Queen, rook, (and bishop) cases impose neighbors by contiguity

  • Weights calculated as a \(1/ num. of neighbors\)

nb.qn <- poly2nb(cdc, queen=TRUE)
nb.rk <- poly2nb(cdc, queen=FALSE)

Finding Neighbors

Getting Weights

lw.qn <- nb2listw(nb.qn, style="W", zero.policy = TRUE)
lw.qn$weights[1:5]
[[1]]
[1] 0.5 0.5

[[2]]
[1] 0.25 0.25 0.25 0.25

[[3]]
[1] 0.2 0.2 0.2 0.2 0.2

[[4]]
[1] 0.3333333 0.3333333 0.3333333

[[5]]
[1] 1
asthma.lag <- lag.listw(lw.qn, cdc$casthma_cr)
                         asthma.lag        
[1,] "Camas"      "9.9"  "10.3"            
[2,] "Kootenai"   "10.4" "9.575"           
[3,] "Kootenai"   "10"   "9.88"            
[4,] "Kootenai"   "9.5"  "10.2666666666667"
[5,] "Twin Falls" "10.2" "9.5"             
[6,] "Twin Falls" "10.4" "9.9"

Fit a model

  • Moran’s I coefficient is the slope of the regression of the lagged asthma percentage vs. the asthma percentage in the tract

  • More generally it is the slope of the lagged average to the measurement

M <- lm(asthma.lag ~ cdc$casthma_cr)
cdc$casthma_cr 
     0.6467989

Comparing observed to expected

  • We can generate the expected distribution of Moran’s I coefficients under a Null hypothesis of no spatial autocorrelation

  • Using permutation and a loop to generate simulations of Moran’s I

n <- 400L   # Define the number of simulations
I.r <- vector(length=n)  # Create an empty vector

for (i in 1:n){
  # Randomly shuffle income values
  x <- sample(cdc$casthma_cr, replace=FALSE)
  # Compute new set of lagged values
  x.lag <- lag.listw(lw.qn, x)
  # Compute the regression slope and store its value
  M.r    <- lm(x.lag ~ x)
  I.r[i] <- coef(M.r)[2]
}