Directional Clustering Tests Based on Nearest Neighbor Contingency Tables

Spatial interaction between two or more classes or species has important implications in various fields and causes multivariate patterns such as segregation or association. Segregation occurs when members of a class or species are more likely to be found near members of the same class or conspecifics; while association occurs when members of a class or species are more likely to be found near members of another class or species. The null patterns considered are random labeling (RL) and complete spatial randomness (CSR) of points from two or more classes, which is called \emph{CSR independence}, henceforth. The clustering tests based on nearest neighbor contingency tables (NNCTs) that are in use in literature are two-sided tests. In this article, we consider the directional (i.e., one-sided) versions of the cell-specific NNCT-tests and introduce new directional NNCT-tests for the two-class case. We analyze the distributional properties; compare the empirical significant levels and empirical power estimates of the tests using extensive Monte Carlo simulations. We demonstrate that the new directional tests have comparable performance with the currently available NNCT-tests in terms of empirical size and power. We use four example data sets for illustrative purposes and provide guidelines for using these NNCT-tests.

💡 Research Summary

This paper addresses the problem of testing spatial interaction between two or more classes of points using nearest‑neighbor contingency tables (NNCTs). While most existing NNCT‑based methods, such as Dixon’s cell‑specific tests and Ceyhan’s extensions, are two‑sided—testing for any deviation from randomness—the authors argue that many scientific questions are directional: one may be interested specifically in segregation (same‑type points clustering) or association (different‑type points clustering). To fill this gap, the authors develop one‑sided (directional) versions of the cell‑specific tests and introduce two new directional test statistics for the two‑class case.

The paper begins by defining the null hypotheses: random labeling (RL) of fixed locations and complete spatial randomness (CSR) of points from each class, collectively referred to as “CSR independence.” Under either null, the NNCT contains four cell counts N₁₁, N₁₂, N₂₁, N₂₂, with row sums fixed (class sizes) and column sums random. Expected cell counts under RL/CSR are derived analytically, and exact formulas for variances and covariances are provided (extending Dixon’s 1994 results).

The directional tests are constructed as standardized Z‑statistics:
Z₁₁ = (N₁₁ – E