A Studentized Permutation Test for the Comparison of Spatial Point Patterns

In this study, a new test is proposed for the hypothesis that two (or more) observed point patterns are realizations of the same spatial point process model. To this end, the point patterns are divided into disjoint quadrats, on each of which an estimate of Ripley’s K-function is calculated. The two groups of empirical K-functions are compared by a permutation test using a Studentized test statistic. The proposed test performs convincingly in terms of empirical level and power in a simulation study, even for point patterns where the K-function estimates on neighboring subsamples are not strictly exchangeable. It also shows improved behavior compared with a test suggested by Diggle et al. for the comparison of groups of independently replicated point patterns. In an application to two point patterns from pathology that represent capillary positions in sections of healthy and cancerous tissue, our Studentized permutation test indicates statistical significance, although the patterns cannot be clearly distinguished by the eye.