We often examine data with the aim of making predictions.  Spatial data
analysis is no exception.  Given measurements of a variable at a set of points
in a region, we might like to extrapolate to points in the region where the
variable was not measured or, possibly, to points outside the region that we
believe will behave similarly.  We can base these predictions on our measured
values alone by kriging or we can incorporate covariates and make
predictions using a regression model.

In R, the lme linear mixed-effects regression command in the nlme
R package allows the user to fit a regression model in which the
outcome and the expected errors are spatially autocorrelated.  There are
several different forms that the spatial autocorrelation can take and the most
appropriate form for a given dataset can be assessed by looking at the shape of
the variogram of the data and choosing from the options available.

We will again be using the thick dataset provided in the SAS
documentation for proc variogram, which includes the measured thickness
of coal seams at different coordinates (we have converted this to a .csv file
for easy use in R). To this dataset, we have added a covariate called soil
measuring the soil quality.  We wish to predict thickness (thick)
with soil quality (soil) in a regression model that incorporates the
spatial autocorrelation of our data.

The code below installs and loads the nlme package and reads in the
data we will use.

install.packages("nlme")
library(nlme)
spdata <- read.table("https://stats.idre.ucla.edu/stat/r/faq/thick.csv", header = T, sep = ",")

The lme command requires a grouping variable.  Since we do not
have a grouping variable in our data, we can create a dummy variable that
has the same value for all 75 observations.

dummy <- rep(1, 75) 
spdata <- cbind(spdata, dummy) 
soil.model <- lme(fixed = thick ~ soil, data = spdata, random = ~ 1 | dummy, method = "ML") 
summary(soil.model)

Linear mixed-effects model fit by maximum likelihood
 Data: spdata 
       AIC      BIC    logLik
  342.3182 351.5881 -167.1591

Random effects:
 Formula: ~1 | dummy
         (Intercept) Residual
StdDev: 4.826056e-05 2.247569

Fixed effects: thick ~ soil 
               Value Std.Error DF   t-value p-value
(Intercept) 31.94203 3.1569891 73 10.117878  0.0000
soil         2.25521 0.8655887 73  2.605407  0.0111
 Correlation: 
     (Intr)
soil -0.997

Standardized Within-Group Residuals:
        Min          Q1         Med          Q3         Max 
-2.68798974 -0.53279498  0.03896491  0.66007203  2.20612991 

Number of Observations: 75
Number of Groups: 1

Next, we can run the same model with spatial correlation structures. Let’s
assume that, based on following the steps shown in R FAQ:
How do I fit a variogram model to my spatial data in R using regression commands?, we determined that our outcome
thick appears to have a Guassian spatial correlation form. We can
specify such a structure with the correlation and corGaus options
for lme.

soil.gau <- update(soil.model, correlation = corGaus(1, form = ~ east + north), method = "ML")
summary(soil.gau)

Linear mixed-effects model fit by maximum likelihood
 Data: spdata 
       AIC      BIC    logLik
  91.50733 103.0948 -40.75366

Random effects:
 Formula: ~1 | dummy
         (Intercept) Residual
StdDev: 8.810794e-05 2.088383

Correlation Structure: Gaussian spatial correlation
 Formula: ~east + north | dummy 
 Parameter estimate(s):
   range 
20.43725 
Fixed effects: thick ~ soil 
               Value Std.Error DF  t-value p-value
(Intercept) 40.32797 0.5877681 73 68.61204  0.0000
soil         0.00348 0.0160363 73  0.21693  0.8289
 Correlation: 
     (Intr)
soil -0.102

Standardized Within-Group Residuals:
       Min         Q1        Med         Q3        Max 
-2.9882532 -0.7133776 -0.1146245  0.6745696  2.0877393 

Number of Observations: 75
Number of Groups: 1

In this example, incorporating the Gaussian correlation structure both
improved the model fit and changed the nature of the regression model.
Without the spatial structure, soil is a statistically significant
predictor of thick.  With the spatial structure, this relationship
becomes not significant.  This suggests that after controlling for location
and the known correlation structure, soil does not add much new information.

See also

References