Confidence Set for the Cluster of a Spatial Scan Statistic.

Spatial cluster detection is widely used in fields such as epidemiology and environmental science to identify spatial units that exhibit patterns differing from background variation. While traditional methods like spatial scan statistics are effective in detecting clusters, they often lack tools to quantify and visualize the uncertainty surrounding detected clusters. In this work, we propose a novel framework for constructing confidence sets for spatial clusters, applicable to both Poisson and Normal models. Building on likelihood-based methods, we leverage the p value of the likelihood ratio test statistic to construct confidence sets with empirical coverage close to nominal levels. To address computational complexity, we implement a binary search algorithm to efficiently determine significance thresholds. We also introduce two visualization tools: a confidence heatmap showing the chance of inclusion of each spatial unit in the true cluster, and confidence envelopes delineating the bounds of uncertainty. Simulation studies and an application to real cancer mortality data demonstrate the method's interpretability, robustness to non-circular or multiple clusters, and practical utility. Our approach offers a principled alternative to traditional significance-based methods and opens avenues for extensions to multiple cluster detection, spatio-temporal analysis, and spatially varying regression models.