Boundaries, links and clusters: a new paradigm in spatial analysis?

This paper develops and applies new techniques for the simultaneous detection of boundaries and clusters within a probabilistic framework. The new statistic "little b" (written b(ij)) evaluates boundaries between adjacent areas with different values, as well as links between adjacent areas with similar values. Clusters of high values (hotspots) and low values (coldspots) are then constructed by joining areas abutting locations that are significantly high (e.g., an unusually high disease rate) and that are connected through a "link" such that the values in the adjoining areas are not significantly different. Two techniques are proposed and evaluated for accomplishing cluster construction: "big B" and the "ladder" approach. We compare the statistical power and empirical Type I and Type II error of these approaches to those of wombling and the local Moran test. Significance may be evaluated using distribution theory based on the product of two continuous (e.g., non-discrete) variables. We also provide a "distribution free" algorithm based on resampling of the observed values. The methods are applied to simulated data for which the locations of boundaries and clusters is known, and compared and contrasted with clusters found using the local Moran statistic and with polygon Womble boundaries. The little b approach to boundary detection is comparable to polygon wombling in terms of Type I error, Type II error and empirical statistical power. For cluster detection, both the big B and ladder approaches have lower Type I and Type II error and are more powerful than the local Moran statistic. The new methods are not constrained to find clusters of a pre-specified shape, such as circles, ellipses and donuts, and yield a more accurate description of geographic variation than alternative cluster tests that presuppose a specific cluster shape. We recommend these techniques over existing cluster and boundary detection methods that do not provide such a comprehensive description of spatial pattern.