Optimal geographic scales for local spatial statistics

Local spatial statistics are used to test for spatial association in some variable of interest, and to test for clustering around predefined locations. Such statistics require that a neighbourhood be defined around the location of interest. This is done by specifying weights for surrounding regions, and this is tantamount to specification of the scale at which the local dependence or clustering is tested. In practice, weights are usually assigned exogenously, with little thought given to their definition. Most common is the definition of binary adjacency - weights are set equal to one if the region is adjacent to the focal region and to zero otherwise. But this implies a spatial scale that may or may not be the best one to evaluate the variable under study - the actual scale of dependence or clustering is one that is smaller or larger. An alternative strategy is to try different sets of weights corresponding to different spatial scales. The purpose of this article is to provide statistical tests that allow for examination of several local statistics across multiple spatial scales, and yet avoid the need for simulation. Application of these tests leads to a choice of spatial scale through the weights, as well as an assessment of statistical significance. The approach is illustrated using data on leukemia from central New York State.