Distinguishing background heterogeneity from excess risk is a central challenge in case-control event data when both covariates and residual spatial or spatio-temporal structure matter. We develop a covariate-adjusted kernel regression framework that embeds an orthogonalized residual risk surface within a semiparametric binary model, and extend the approach from purely spatial to explicit spatio-temporal analysis. We apply the method to 959 gun violence incidents at public schools in the contiguous United States from 2000 to 2024, using incidents from the K-12 School Shooting Database linked to official school records for the corresponding year. The fitted models identify stable school-level associations, including markedly higher risk for larger schools and for middle and high schools, while also revealing substantial residual structure beyond the background distribution of schools. After adjustment for covariates, excess risk is found to remain concentrated in a persistent central-eastern corridor of the United States, with the strongest evidence appearing in recent years. More broadly, the analysis shows how residual risk surfaces can sharpen inference by separating background heterogeneity from anomalous structure in case-control event processes evolving over space and time.

翻译：区分背景异质性与超额风险是病例对照事件数据中的核心挑战，尤其当协变量与空间或时空残余结构共同发挥作用时。本文构建了一个协变量调整的核回归框架，将正交化残余风险曲面嵌入半参数二元模型，并将该方法从纯空间分析扩展至显式时空分析。我们将该方法应用于2000年至2024年间美国本土公立学校发生的959起枪击暴力事件，数据源自K-12校园枪击数据库，并与相应年份的官方学校记录进行关联。拟合模型揭示了稳定的学校层面关联特征，包括大型学校、初中及高中的风险显著升高，同时揭示了超出学校背景分布的显著残余结构。在调整协变量后，超额风险仍集中在美国持续存在的中央东部走廊区域，且近年来呈现最强证据。更广泛而言，该分析展示了残余风险曲面如何通过分离病例对照事件过程在空间与时间演化中的背景异质性与异常结构，从而锐化推断结论。