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

The paper addresses a gap in spatial point‑pattern analysis where nearest‑neighbor contingency tables (NNCTs) have traditionally been used only for two‑sided tests of segregation or association. Recognizing that many scientific questions are inherently directional—researchers often wish to test specifically for segregation (excess same‑type neighbors) or for association (excess cross‑type neighbors)—the authors develop one‑sided (directional) versions of NNCT tests for the two‑class case.

First, the authors formalize the two null models commonly employed: random labeling (RL), where class labels are randomly assigned to a fixed set of locations, and complete spatial randomness (CSR) independence, where points of each class arise independently from homogeneous Poisson processes. For each model they derive the expected cell counts, variances, and covariances of the NNCT, providing the theoretical basis for test statistics.

Building on the classic cell‑specific Z‑statistics, the paper introduces two directional statistics, Z⁺ and Z⁻, which respectively capture “more than expected” and “less than expected” deviations in a given cell. In addition, a global directional statistic Q is defined to assess the overall tendency of the pattern toward segregation or association. The authors discuss the asymptotic normality of these statistics under the null hypotheses, but also acknowledge that small sample sizes or highly unbalanced class proportions can invalidate the normal approximation. Consequently, they recommend Monte‑Carlo randomization (10 000 or more replicates) to obtain accurate empirical p‑values when needed.

A comprehensive Monte‑Carlo simulation study evaluates empirical size and power. The design varies (i) total sample size (30–500+ points), (ii) class proportion (balanced vs. highly skewed), and (iii) interaction strength (strong segregation, weak segregation, CSR, weak association, strong association). Results show that the new directional tests maintain nominal significance levels (≈0.05) across all scenarios, and that they achieve substantially higher power than traditional two‑sided NNCT tests when the true alternative is directional, especially for weak association where power gains of 12–18 % are observed. The tests also exhibit robustness to class‑size imbalance because Z⁺ and Z⁻ treat excess and deficit separately.

Four real‑world data sets illustrate practical application: (1) a forest plot of two tree species, (2) a disease‑incidence map in a Korean agricultural region, (3) spatial distribution of soil particle size classes, and (4) urban crime type locations. In each case, the conventional two‑sided NNCT test yields borderline p‑values, whereas the directional tests provide clear evidence for either segregation or association, thereby offering more actionable insights (e.g., targeted disease control measures).

The discussion emphasizes that the choice between RL and CSR independence should be guided by the scientific context, and that researchers should select the appropriate directional statistic (cell‑specific Z⁺/Z⁻ or global Q) based on whether they are interested in a particular class pair or the overall pattern. The authors also outline limitations, such as the need for Monte‑Carlo calibration in small samples and the increased complexity of interpreting Q in multi‑class settings. Future work is suggested on extending directional NNCT methods to heterogeneous environments, multi‑scale interactions, and Bayesian frameworks.

In summary, this study enriches the NNCT methodology by introducing well‑grounded one‑sided tests that retain correct size, improve power for directional alternatives, and provide clearer interpretability for applied spatial analyses. The paper supplies detailed theoretical derivations, extensive simulation evidence, and concrete guidelines, making the new directional NNCT tests ready for immediate use by ecologists, epidemiologists, geographers, and other researchers dealing with multivariate spatial point patterns.