Nonparametric one-way analysis of variance of replicated bivariate spatial point patterns

A common problem in neuropathological studies is to assess the spatial patterning of cells on tissue sections and to compare spatial patterning between disorder groups. For a single cell type, the cell positions constitute a univariate point process and interest focuses on the degree of spatial aggregation. For two different cell types, the cell positions constitute a bivariate point process and the degree of spatial interaction between the cell types is of interest. We discuss the problem of analysing univariate and bivariate spatial point patterns in the one-way design where cell patterns have been obtained for groups of subjects. A bootstrapping procedure to perform a nonparametric one-way analysis of variance of the spatial aggregation of a univariate point process has been suggested by Diggle, Lange and Benes (1991). We extend their replication-based approach to allow the comparison of the spatial interaction of two cell types between groups, to include planned comparisons (contrasts) and to assess whole groups against complete spatial randomness and spatial independence. We also accommodate several replicate tissue sections per subject. An advantage of our approach is that it can be applied when processes are not stationary, a common problem in brain tissue sections since neurons are arranged in cortical layers. We illustrate our methods by applying them to a neuropathological study to investigate abnormalities in the functional relationship between neurons and astrocytes in HIV associated dementia.