1. Dealing with Spatial Autocorrelation
Spatial Analysis Seminar
Spring 2009

2. Spatial Autocorrelation Defined
“…the property of random variables taking values, at pairs of locations a certain distance apart, that are more similar (positive autocorrelation) or less similar (negative autocorrelation) than expected for randomly associated pairs of observations.”
Legendre (1993)

3. Types of Spatial Autocorrelation
Inherent autocorrelation: caused by “contagious biotic processes”
vs.
Induced spatial dependence: biological variables of interest are functionally dependent on one or more autocorrelated exogenous variable(s)

4. Why Should We Care?
“natural systems almost always have autocorrelation in the form of patchiness or gradients…over a wide range of spatial and temporal scales.”
Fortin & Dale (2005)
→ Autocorrelation is a “fact of life” for ecologists!

5. 2 Views of Spatial Autocorrelation:
It’s a nuisance that complicates statistical hypothesis testing
It’s functionally important in many ecosystems, so we must revise our theories and models to incorporate spatial structure
Either way, the first step involves describing the autocorrelation (i.e., the “spatial structure”)

6. Describing Spatial Autocorrelation
Compute Moran’s I or Geary’s c coefficients over multiple distances
Correlogram: plot distance on X-axis against correlation coefficient on Y-axis
Mantel correlogram: multivariate response
Semi-variogram/variogram

7. Example Data Wetland hardwood forest (5 x 5 m cells)
Response variable: log of non-ground lidar points in 0-1 m vertical height bin
n1 = 217, n2 = 68
Welch’s t-test (unequal variance, unequal sample sizes) results: t = 2.33, df = 181, p-value ≈ 0.021

8. Moran’s I correlograms

9. Now what do I do??? Adjusting the effective sample size
Spatial statistical modeling methods
Restricted randomization
Other methods: canonical ordination, partial Mantel tests, etc.

10. Adjusting the Effective Sample Size
Estimate of effective sample size (Fortin & Dale 2005, p. 223, Equation 5.15):
For first-order autocorrelation ρ and large n:

11. Adjusting the Effective Sample Size
For the “Recently Burned” example data:
For the “Long Unburned” example data:
Welch’s t-test results: t = 1.76, df = 123,
p ≈ 0.080
BUT, this is a very simplistic model!

12. Detour: Autocorrelation Models
Model 1 (“spatial independence”):
Model 2 (“first-order autoregressive”):
Model 3 (“induced autoregressive”):
Model 4 (“doubly autoregressive”):
SOURCE: Fortin & Dale (2005), pp

13. Detour: Autocorrelation Models
The models on the previous slide were one-dimensional, but most spatial data is two-dimensional (Lat-Long, XY-coordinates, etc.)
The two-dimensional spatial autocorrelation model incorporates W, a “proximity matrix” of neighbor weights, which in turn affects the variance-covariance matrix (C):

14. Generalized Least Squares (GLS)
Relatively easy way to introduce spatial autocorrelation structure to linear models
Fits a parametric correlation function (exponential, Gaussian, spherical, etc.) directly to the variance-covariance matrix
Assumes normally distributed errors, but errors are allowed to be correlated and/or have unequal variances
Built-in R package: nlme

15. GLS Model – No Spatial Structure
library(nlme) …
## Model A: spatial independence
ModelA <- gls(LN_COUNT~BURNED,data=SAC_data)
plot(Variogram(ModelA, form=~x+y))

16. GLS Models with Spatial Structure
> ModelB <- gls(LN_COUNT~BURNED,data=SAC_data,corr=corAR1())
> ModelC <- gls(LN_COUNT~BURNED,data=SAC_data,corr=corExp(form=~x+y))
> ModelD <- gls(LN_COUNT~BURNED,data=SAC_data,corr=corGaus(form=~x+y))
> ModelE <- gls(LN_COUNT~BURNED,data=SAC_data,corr=corSpher(form=~x+y))
> AIC(ModelA,ModelB,ModelC,ModelD,ModelE)
df AIC
ModelA
ModelB
ModelC
ModelD
ModelE
> anova(ModelA,ModelC)
Model df AIC BIC logLik Test L.Ratio p-value
ModelA
ModelC vs <.0001
→ Exponential GLS model seems to fit best

17. Other Autocorrelation Models
Conditional autoregressive (CAR), simultaneous autoregressive (SAR), and moving average (MA) models
See pp of Fortin & Dale (2005)
Implemented in R package spdep, as well as SAM (Spatial Analysis for Macroecology) software
Generalized linear mixed models (GLMMs): R built-in packages MASS, nlme
But wait, there’s more: see Dormann et al. (2007) review paper in Ecography (30)

18. Models and Reality
“Much of the treatment of spatial autocorrelation in the statistical literature is predicated on the simplest AR model, which produces an exponential decline in autocorrelation as a function of distance (Figure 5.16).”
Fortin & Dale (2005, pp )
BUT, simple corrections based on first-order AR don’t account for effects of potentially negative autocorrelation at greater distances

19. Restricted Randomization
PROBLEM: randomization tests based on complete spatial randomness will destroy autocorrelation structure
POTENTIAL SOLUTIONS:
“Toroidal shift” randomization (Figure 5.12)
Contiguity-constrained permutations (see Legendre et al for algorithms)

20. Conclusion
Incorporating spatial structure into ecological models was identified by Legendre as a “new paradigm” in 1993, BUT…
…ecologists are still refining their methods for dealing with spatial autocorrelation
OUR LAST HOPE?: Dale, M.R.T. and M.-J. Fortin. (in press). Spatial Autocorrelation and Statistical Tests: Some Solutions. Journal of Agricultural, Biological, and Environmental Statistics.

21. Spatial autocorrelation, don’t make me open this…