Restricted random labeling: testing for between-group interaction after controlling for joint population and within-group spatial structure

Statistical measures of spatial interaction between multiple types of entities are commonly assessed against a null model of either toroidal shift (TS), which controls for spatial structure of individual subpopulations, or random labeling (RL), which controls for spatial structure of the joint population. Neither null model controls for both types of spatial structure simultaneously, although this may sometimes be desirable when more than two subpopulations are present. To address this, we propose a flexible framework for specifying null models that we refer to as restricted random labeling (rRL). Under rRL, a specified subset of individuals is restricted and other individuals are randomly relabeled. Within this framework, two specific null models are proposed for pairwise analysis within populations consisting of three or more subpopulations, to simultaneously control for spatial structure in the joint population and one or the other of the two subpopulations being analyzed. Formulas are presented for calculating expected nearest neighbor counts and co-location quotients within the proposed framework. Differences between TS, RL and rRL are illustrated by application to six types of generating processes in a simulation study, and to empirical datasets of tree species in a forest and crime locations in an urban setting. These examples show that rRL null models are typically stricter than either TS or RL, which often detect “interactions” that are an expected consequence either of the joint population pattern or of individual subpopulation patterns. Copyright Springer-Verlag Berlin Heidelberg 2015