Spatial Scan Statistics: Models, Calculations, and Applications

  • Martin Kulldorff

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.

Keywords and phrases

Spatial statistics geography spatial clusters space—time clusters maximum likelihood likelihood ratio test 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler, R. J. (1984). The supremum of a particular Gaussian fieldAnnals of Probability 12436–444.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Alm, S. E. (1997) On the Distrinbution of the scan statistic of a two dimensional Poisson processAduances in Applied Probability 291–16MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alm, S. E. (1998). On the distribution of scan statistics for Poisson processes in two and three dimensionsExtremes(to appear).Google Scholar
  4. 4.
    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.Google Scholar
  5. 5.
    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.Google Scholar
  6. 6.
    Besag, J. and Newell, J. (1991). The detection of clusters in rare diseasesJournal of the Royal Statistical Society Series A 154143–155.Google Scholar
  7. 7.
    Chen, J. and Glaz, J. (1996). Two dimensional discrete scan statisticsStatistics EE Probability Letters 3159–68.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Choynowski, M. (1959). Maps based on probabilitiesJournal of the American Statistical Association 54385–388.CrossRefGoogle Scholar
  9. 9.
    Dwass, M. (1957). Modified randomization tests for nonparametric hypothesesAnnals of Mathematical Statistics 28181–187.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Eggleton, P. and Kermack, W. O. (1944). A problem in the random distribution of particlesProceedings of the Royal Society Edinburgh Section 62103–115.MathSciNetMATHGoogle Scholar
  11. 11.
    Glaz, J. and Naus, J. (1991). Tight bounds and approximations for scan statistic probabilities for discrete dataAnnals of Applied Probability 1306–318.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    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.CrossRefGoogle Scholar
  13. 13.
    Kulldorff, M. (1997). A spatial scan statisticCommunications in Statistics-Theory and Methods 261481–1496.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    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).Google Scholar
  15. 15.
    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.CrossRefGoogle Scholar
  16. 16.
    Kulldorff, M. and Nagarwalla, N. (1995). Spatial disease clusters: Detection and inferenceStatistics in Medicine 14799–810.CrossRefGoogle Scholar
  17. 17.
    Kulldorff, M. and Williams, G. (1997).SaTScan v 1.0 Software for the Space and Space-Time Scan StatisticsBethesda, MD: National Cancer Institute.Google Scholar
  18. 18.
    Lawson, A. (1997). Cluster modeling of disease incidence via MCMC methodsJournal of Statistical Planning and Inference(submitted).Google Scholar
  19. 19.
    Loader, C. R. (1991). Large-deviation approximations to the distribution of scan statisticsAdvances in Applied Probability 23751–771.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    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.Google Scholar
  21. 21.
    Mantel, N. (1967). The detection of disease clustering and a generalized regression approachCancer Research 27209–220.Google Scholar
  22. 22.
    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.Google Scholar
  23. 23.
    Naus, J. (1965a). The distribution of the size of maximum cluster of points on the lineJournal of the American Statistical Association 60532–538.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Naus, J. (1965b). Clustering of random points in two dimensionsBiometrika 52263–267.MathSciNetMATHGoogle Scholar
  25. 25.
    Naus, J. (1974). Probabilities for a generalized birthday problemJournal of the American Statistical Association 69810–815.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    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.CrossRefGoogle Scholar
  27. 27.
    Priebe, C. (1998). A spatial scan statistic for stochastic scan partitionsJournal of the American Statistical Association(to appear).Google Scholar
  28. 28.
    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.CrossRefGoogle Scholar
  29. 29.
    Saperstein, B. (1972). The generalized birthday problemJournal of the American Statistical Association 67425–428.MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    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.Google Scholar
  31. 31.
    Wallenstein, S., Gould, M. S. and Kleinman, M. (1989a). Use of the scan statistic to detect time-space clusteringAmerican Journal of Epidemiology 1301057–1064.Google Scholar
  32. 32.
    Wallenstein, S., Weinberg, C. R. and Gould, M. (1989b). Testing for a pulse in seasonal event dataBiometrics 45817–830.MATHCrossRefGoogle Scholar
  33. 33.
    Wallenstein, S., Naus, J. and Glaz, J. (1993). Power of the scan statistic for detection of clusteringStatistics in Medicine 121819–1843.CrossRefGoogle Scholar
  34. 34.
    Wallenstein, S., Naus, J. and Glaz, J. (1994a). Power of the scan statistic in detecting a changed segment in a Bernoulli sequenceBiometrika 81595–601.MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Wallenstein, S., Naus, J. and Glaz, J. (1994b). Power of the scan statisticsASA Proceedings of the Section of Epidemiology 8170–75.MathSciNetGoogle Scholar
  36. 36.
    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).Google Scholar
  37. 37.
    Weinstock, M. A. (1981). A generalized scan statistic test for the detection of clustersInternational Journal of Epidemiology 10289–293.CrossRefGoogle Scholar
  38. 38.
    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.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Martin Kulldorff
    • 1
  1. 1.National Cancer InstituteBethesdaUSA

Personalised recommendations