Abstract
A common problem in spatial statistics is whether a set of points are randomly distributed or if they show signs of clusters or clustering. When the locations of clusters are of interest, it is natural to use a spatial scan statistic.
Different spatial scan statistics have been proposed. These are discussed and presented in a general framework that incorporates two-dimensional scan statistics on the plane or on a sphere, as well as three-dimensional scan statistics in space or in space—time. Computational issues are then looked at, presenting efficient algorithms that can be used for different scan statistics in connection with Monte Carlo-based hypothesis testing. It is shown that the computational requirements are reasonable even for very large data sets. Which scan statistic to use will depend on the application at hand, which is discussed in terms of past as well as possible future practical applications in areas such as epidemiology, medical imaging, astronomy, archaeology, urban and regional planning, and reconnaissance.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adler, R. J. (1984). The supremum of a particular Gaussian fieldAnnals of Probability 12436–444.
Alm, S. E. (1997) On the Distrinbution of the scan statistic of a two dimensional Poisson processAduances in Applied Probability 291–16
Alm, S. E. (1998). On the distribution of scan statistics for Poisson processes in two and three dimensionsExtremes(to appear).
Alt, K. W. and Vach, W. (1991). The reconstruction of `genetic kinship’ in prehistoric burial complexes-problems and statistics, InClassification Data Analysis and Knowledge Organization (Eds.H. H. Bock and P. Ihm), Berlin: Springer-Verlag.
Anderson, N. H. and Titterington, D. M. (1997). Some methods for investigating spatial clustering with epidemiological applicationsJournal of the Royal Statistical Society Series A 16087–105.
Besag, J. and Newell, J. (1991). The detection of clusters in rare diseasesJournal of the Royal Statistical Society Series A 154143–155.
Chen, J. and Glaz, J. (1996). Two dimensional discrete scan statisticsStatistics EE Probability Letters 3159–68.
Choynowski, M. (1959). Maps based on probabilitiesJournal of the American Statistical Association 54385–388.
Dwass, M. (1957). Modified randomization tests for nonparametric hypothesesAnnals of Mathematical Statistics 28181–187.
Eggleton, P. and Kermack, W. O. (1944). A problem in the random distribution of particlesProceedings of the Royal Society Edinburgh Section 62103–115.
Glaz, J. and Naus, J. (1991). Tight bounds and approximations for scan statistic probabilities for discrete dataAnnals of Applied Probability 1306–318.
Hjalmars, U., Kulldorff, M., Gustafsson, G. and Nagarwalla, N. (1996). Childhood leukemia in Sweden: Using GIS and a spatial scan statistic for cluster detectionStatistics in Medicine 15707–715.
Kulldorff, M. (1997). A spatial scan statisticCommunications in Statistics-Theory and Methods 261481–1496.
Kulldorff, M., Athas, W. F., Feuer, E. J., Miller, B. A. and Key, C. R. (1998). Evaluating cluster alarms: A space-time scan statistic and brain cancer in Los AlamosAmerican Journal of Public Health(submitted).
Kulldorff, M., Feuer, E. J., Miller, B. A. and Freedman, L. S. (1997). Breast cancer clusters in Northeast United States: A geographic analysisAmerican Journal of Epidemiology 146161–170.
Kulldorff, M. and Nagarwalla, N. (1995). Spatial disease clusters: Detection and inferenceStatistics in Medicine 14799–810.
Kulldorff, M. and Williams, G. (1997).SaTScan v 1.0 Software for the Space and Space-Time Scan StatisticsBethesda, MD: National Cancer Institute.
Lawson, A. (1997). Cluster modeling of disease incidence via MCMC methodsJournal of Statistical Planning and Inference(submitted).
Loader, C. R. (1991). Large-deviation approximations to the distribution of scan statisticsAdvances in Applied Probability 23751–771.
Mánsson, M. (1996). On Clustering of Random Points in the Plain and in SpacePh.D. ThesisDepartment of Mathematics, Chalmers University of Technology and Gothenburg University, Gothenburg.
Mantel, N. (1967). The detection of disease clustering and a generalized regression approachCancer Research 27209–220.
Miller, B. A., Gloeckler Ries, L. Y., Hankey, B. F., Kosary, C. L., Harras, A., Devesa, S. S. and Edwards, B. K. (1993).SEER Cancer Statistics Review 1973–1990Bethesda, MD: National Cancer Institute.
Naus, J. (1965a). The distribution of the size of maximum cluster of points on the lineJournal of the American Statistical Association 60532–538.
Naus, J. (1965b). Clustering of random points in two dimensionsBiometrika 52263–267.
Naus, J. (1974). Probabilities for a generalized birthday problemJournal of the American Statistical Association 69810–815.
Openshaw, S., Charlton, M., Wymer, C. and Craft, A. (1987). A Mark 1 Geographical Analysis Machine for the automated analysis of point data setsInternational Journal of Geographical Information Systems 1335–358.
Priebe, C. (1998). A spatial scan statistic for stochastic scan partitionsJournal of the American Statistical Association(to appear).
Sahu, S. K., Bendel, R. B. and Sison, C. P. (1993). Effect of relative risk and cluster configuration on the power of the one-dimensional scan statisticStatistics in Medicine 121853–1865.
Saperstein, B. (1972). The generalized birthday problemJournal of the American Statistical Association 67425–428.
Turnbull, B., Iwano, E. J., Burnett, W. S., Howe, H. L. and Clark, L. C. (1990). Monitoring for clusters of disease: Application to leukemia incidence in Upstate New YorkAmerican Journal of Epidemiology 132S136—S143.
Wallenstein, S., Gould, M. S. and Kleinman, M. (1989a). Use of the scan statistic to detect time-space clusteringAmerican Journal of Epidemiology 1301057–1064.
Wallenstein, S., Weinberg, C. R. and Gould, M. (1989b). Testing for a pulse in seasonal event dataBiometrics 45817–830.
Wallenstein, S., Naus, J. and Glaz, J. (1993). Power of the scan statistic for detection of clusteringStatistics in Medicine 121819–1843.
Wallenstein, S., Naus, J. and Glaz, J. (1994a). Power of the scan statistic in detecting a changed segment in a Bernoulli sequenceBiometrika 81595–601.
Wallenstein, S., Naus, J. and Glaz, J. (1994b). Power of the scan statisticsASA Proceedings of the Section of Epidemiology 8170–75.
Walsh, S. J. and Fenster, J. R. (1997). Geographical clustering of mortality from systemic sclerosis in the Southeastern United States, 1981–90Journal of Rheumatology(to appear).
Weinstock, M. A. (1981). A generalized scan statistic test for the detection of clustersInternational Journal of Epidemiology 10289–293.
Worsley, K. J., Evans, A. C., Marrett, S. and Neelin, P. (1992). A three-dimensional statistical analysis for CBF activation studies in human brainJournal of Cerebral Blood Flow and Metabolism 12900–918.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media New York
About this chapter
Cite this chapter
Kulldorff, M. (1999). Spatial Scan Statistics: Models, Calculations, and Applications. In: Glaz, J., Balakrishnan, N. (eds) Scan Statistics and Applications. Statistics for Industry and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1578-3_14
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1578-3_14
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7201-4
Online ISBN: 978-1-4612-1578-3
eBook Packages: Springer Book Archive