New Insight of Spatial Scan Statistics via Regression Model

Abstract

The spatial scan statistic is widely used to detect disease clusters in epidemiological surveillance. Since the seminal work by~\cite{kulldorff1997}, numerous extensions have emerged, including methods for defining scan regions, detecting multiple clusters, and expanding statistical models. Notably,~\cite{jung2009} and~\cite{ZHANG20092851} introduced a regression-based approach accounting for covariates, encompassing classical methods such as those of~\cite{kulldorff1997}. Another key extension is the expectation-based approach~\citep{neill2005anomalous,neillphdthesis}, which differs from the population-based approach represented by~\cite{kulldorff1997} in terms of hypothesis testing. In this paper, we bridge the regression-based approach with both expectation-based and population-based approaches. We reveal that the two approaches are separated by a simple difference: the presence or absence of an intercept term in the regression model. Exploiting the above simple difference, we propose new spatial scan statistics under the Gaussian and Bernoulli models. We further extend the regression-based approach by incorporating the well-known sparse L0 penalty and show that the derivation of spatial scan statistics can be expressed as an equivalent optimization problem. Our extended framework accommodates extensions such as space-time scan statistics and detecting multiple clusters while naturally connecting with existing spatial regression-based cluster detection. Considering the relation to case-specific models~\citep{she2011,10.1214/11-STS377}, clusters detected by spatial scan statistics can be viewed as outliers in terms of robust statistics. Numerical experiments with real data illustrate the behavior of our proposed statistics under specified settings.