Family of power divergence spatial scan statistics

The classical spatial scan test, which derived by maximizing the likelihood ratio statistic over a collection of cluster candidates, is widely used in spatial cluster detection. As the likelihood ratio statistic is only a special case in the family of power divergence (PD) goodness-of-fit statistics, the classical spatial scan test is extended to the family of PD spatial scan tests. Therefore, the family of PD spatial scan tests includes not only the classical case but also many other cases. The test statistics, the asymptotic null distribution, and the methods to incorporate overdispersion in the cluster detection are derived. It is found that in the absence of independent variables, the asymptotic null distribution of the PD spatial scan statistics only depends on the ratio of at risks populations and the collection of cluster candidates. Particularly, the focus is on three special cases in the family. They are the deviance, the Pearson, and the Freeman–Tukey spatial scan tests, where the deviance spatial scan test is equivalent to the classical spatial scan test. In simulation studies, it is found that the three test statistics are almost equally powerful for cluster detection. Spatial epidemiologists and spatial statisticians have long used georeferenced data for spatial pattern analysis which includes spatial cluster detection. Recent health care reform legislation in the United States mandates the use of electronic health records (EHRs) and requires that spatial data, such as patient location in latitude and longitude, be explicitly included in an EHR for meaningful use. As EHRs are implemented nationwide, more spatially referenced data will be available. A great opportunity for spatial disease investigations is to include disease surveillance, cluster detection, and etiology discovery. Kulldorff (1997)'s spatial scan statistic is widely accepted by public health agencies for disease surveillance and cluster detection. There have been many extensions of Kulldorff's spatial scan statistic that incorporate different shapes and explanatory variables, and overdispersion (Assuncao et al. Most extensions are based on different statistical frameworks, resulting in parallel or even incompatible developments. In this paper, we propose an extension based on the power divergence (PD) family under the framework of generalized linear models (GLMs), which includes Kulldorff's spatial scan statistic as a special case. Spatial data pose three difficult issues for spatial statisticians: (1) lack of repeated observations at any spatial points or spatial units, (2) high dimensionally varied and arbitrary shapes, and (3) disconnection between a variable of interest and ecological …