Considering Spatial Dependence in Lattice Data: Two Views 

 Drew Thomas 

 Last year during Prof. Rima Izem's Spatial Statistics course, I started to wonder about different analytical techniques for comparing lattice data (say voting results, epidemiological information, or the prevalence of basketball courts) on a map with distinct spatial units such as counties. 

 A set of techniques had been demonstrated to determine spatial autocorrelation through the use of a fixed-value neighbour matrix, with one parameter determining the strength of the autocorrelation. The use of the fixed neighbour matrix perturbed me somewhat, since the practice of geostatistics uses a tool called the empirical variogram - a functional estimate of variance between sample sites through a regression, based on taking each possible pair of points and computing the difference between squared values - which might give a more reasonable estimate of autocorrelation than a simpler model.  

 As it turned out, this same question was asked by  Prof. Melanie Wall  from the Biostatistics Department at the University of Minnesota about a year before I got around to it. In her paper "A close look at the spatial structure implied by the CAR and SAR models" (J. Stat. Planning and Inference, v121, no.2), Prof. Wall tests the idea of using a variogram approach to model spatial structure on SAT data against more common lattice models. And what do you know - the variogram approach holds up to scrutiny. In some cases it outperforms the lattice model, such as in the extreme case of Tennessee and Missouri, which have a bizarrely low correlation due to the fact that each state has eight neighbours. 

 As well as feeling relief that this difficulty with the model wasn't just in my imagination, I'm glad to see that this type of inference crosses so many borders.