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Spatial Statistics

Spatial statistics is really three subjects wearing one name, and the distinction is not decoration — it decides what the data even are. Areal data attach a value to a region, so there are no coordinates at all, only which regions touch which; the model lives on a graph. Geostatistical data are measured at points in continuous space, where distance is meaningful and the object of interest is the surface between the observations. Point-process data have no attached value: the locations themselves are the outcome, and the question is whether they cluster. Fit the wrong family to your data and the answer is not merely imprecise, it is about a different quantity.

What unifies them is the reason to bother. Nearby things resemble each other, so treating observations as independent throws away information and, for rare events in small areas, leaves rate estimates dominated by sample size rather than risk. The Bayesian response is the one used throughout this collection: put the dependence in a prior and let the model borrow strength — from neighbours on a graph, from a covariance function across distance, or from an intensity surface. A recurring theme is being careful about what is genuinely identified: a smoothed map that looks decisive can rest on an assumption the data cannot check, so several of these examples end in an exceedance probability or a sensitivity analysis rather than a single number.

The arc runs foundations first, then the two methodological branches — the epidemiologist's conditional autoregression and the economist's simultaneous one — then geostatistics and point processes, then time, and finally a real application. That last step is the point of the rest: the same machinery, at national scale, used to ask whether the American “disability belt” is an artefact of age, of small-county noise, or of testing three thousand hypotheses at once — and whether it is spreading. It is none of the three, and it is not.

The prior that does the work on a graph. For areal data the whole model rests on one construction. A CAR prior says each region’s value, given all the others, is normal with a mean equal to the average of its neighbours — dependence specified locally, one region at a time, which is what makes it computable on a graph of three thousand counties. The version used almost everywhere here is the intrinsic one, ICAR, which is improper: it constrains only the differences between neighbours and says nothing about the overall level, so the whole field can slide up or down without changing its probability. That is harmless as a prior and fatal if left alone, which is why an ICAR field is always accompanied by a sum-to-zero constraint — the level belongs to the intercept, and without the constraint the two are unidentified and the sampler crawls along the ridge between them. The BYM model then adds an ordinary independent random effect alongside the spatial one, so the data can decide how much of the variation is genuinely spatial and how much is just region-level noise.

How to read a variogram. Geostatistics replaces the graph with distance, and the object that summarises it is the variogram: average squared difference between pairs of observations, plotted against how far apart they are. It normally rises and then flattens, and its three features are the vocabulary of the whole field. The nugget is the intercept — the variance still present at zero distance, which is measurement error plus whatever varies at a finer scale than the sampling. The sill is the plateau, the total variance once points are far enough apart to be unrelated. The range is the distance at which it gets there, so it is the honest answer to “how far does the correlation reach?”. Fitting a covariance function — usually a Matérn, whose extra smoothness parameter controls how rough the surface is allowed to be — is fitting those three numbers, and kriging is what you do with them afterwards.

Why a spatial coefficient is not a marginal effect. This is the trap in the econometric branch and it is easy to walk into. In a spatial-autoregressive model the outcome in one region depends on the outcomes of its neighbours, which in turn depend on theirs — so changing one region’s covariate changes that region, then its neighbours, then their neighbours, and some of the effect returns to where it started. The coefficient β is not the answer to “what happens if this county’s income rises?”. The LeSage–Pace decomposition gives the quantities that are: a direct effect (the average impact on the region itself, including the feedback that comes back to it), an indirect effect or spillover (the total impact on everywhere else), and their sum, the total effect. Reporting β alone systematically understates the answer, which is the single most common error in applied spatial econometrics.

Two caveats that apply to every map here. The first is the modifiable areal unit problem: for areal data the regions are an administrative accident, and both the strength of a correlation and sometimes its sign can change if the same underlying data are aggregated to different boundaries. Nothing in the model can detect this, because the model never sees anything but the boundaries it was given. The second is edge effects. Regions and points at the boundary of the study area have neighbours that were never observed, so they are systematically less informed than interior ones — which for point processes means a naive clustering statistic will find spurious sparseness at the edges unless it is corrected. Neither is a flaw to be fixed so much as a fact to be stated, and both are reasons to prefer an exceedance probability over a point estimate when the conclusion is going to be drawn on a map.