Abstract
Spatial cluster detection, which is the identification of spatial units adjacent in space associated with distinctive patterns of data of interest relative to background variation, is useful for discerning spatial heterogeneity in regression coefficients. Some real studies with regression-based models on air quality data show that there exists not only spatial heterogeneity but also heteroscedasticity between air pollution and its predictors. Since the low air quality is a well-known risk factor for mortality, various cardiopulmonary diseases, and preterm birth, the analysis at the tail would be of more interest than the center of air pollution distribution. In this article, we develop a spatial cluster detection approach using a threshold quantile regression model to capture the spatial heterogeneity and heteroscedasticity. We introduce two threshold variables in the quantile regression model to define a spatial cluster. The proposed test statistic for identifying the spatial cluster is the supremum of the Wald process over the space of threshold parameters. We establish the limiting distribution of the test statistic under the null hypothesis that the quantile regression coefficient is the same over the entire spatial domain at the given quantile level. The performance of our proposed method is assessed by simulation studies. The proposed method is also applied to analyze the particulate matter (PM2.5) concentration and aerosol optical depth (AOD) data in the Northeastern United States in order to study geographical heterogeneity in the association between AOD and PM2.5 at different quantile levels.