1. Global Clustering Tests

2. Tests for Spatial Randomness H 0 : The risk of disease is the same everywhere after adjustment for age, gender and/or other covariates.

3. Tests for Global Clustering Evaluates whether clustering exist as a global phenomena throughout the map, without pinpointing the location of specific clusters.

4. Tests for Global Clustering More than 100 different tests for global clustering proposed by different scientists in different fields. For example: Whittemore’s Test, Biometrika 1987 Cuzick-Edwards k-NN, JRSS 1990 Besag-Newell’s R, JRSS 1991 Tango’s Excess Events Test, StatMed 1995 Swartz Entropy Test, Health and Place 1998 Tango’s Max Excess Events Test, StatMed 2000

5. Cuzick-Edward’s k-NN Test  i  c i  j  c j I(d ij <d ik(i) ) where c i = number of deaths in county i d ij = distance from county i to county j k(i) = the county with the ‘k-nearest neighbor’ to an individual in county i, defined in terms of expected cases rather than individuals.

6. Cuzick-Edward’s k-NN Test Special case of the Weighted Moran’s I Test, proposed by Cliff and Ord, 1981

7. Tango’s Excess Events Test  i  j  c j -E(c j )]  c j -E(c j )] e -4d 2 ij / 2 where c i = number of deaths in county i E(c j ) = expected cases in county i | H 0 d ij = distance from county i to county j = clustering scale parameter

8. Whittemore's Test Whittemore et al. proposed the statistic

9. Besag- Newell’s R For each case, find the collection of nearest counties so that there are a total of at least k cases in the area of the original and neighboring counties. Using the Poisson distribution, check if this area is statistically significant (not adjusting for multiple testing) R is the the number of cases for which this procedure creates a significant area

10. Besag-Newell's R Let um(i)=min{j:(D j(i) +1) k}. Under null hypothesis, the case number s will have Poisson distribution with probability where p=C/N. For each county R is defined as

11. Swartz ’ s Entropy Test The test statistic is defined as where n i is the population in county I, and N is the total population

12. Global Clustering Tests Power Evaluation Joint work with Toshiro Tango, Peter Park and Changhong Song

13. Power Evaluation, Setup 245 counties and county equivalents in Northeastern United States Female population 600 randomly distributed cases, according to different probability models

14. Note Besag-Newell’s R and Cuzick-Edwards k-NN tests depend on a clustering scale parameter. For each test we evaluate three different parameters.

15. Global Chain Clustering Each county has the same expected number of cases under the null and alternative hypotheses 300 cases are distributed according to complete spatial randomness Each of these have a twin case, located at the same or a nearby location.

16. Power Zero Distance Besag-Newell 0.480.490.42 Cuzick-Edwards1.000.920.73 Tango’s MEET0.99 Swartz Entropy1.00 Whittemore’s Test0.13 Spatial Scan0.79

17. Power Fixed Distance, 1% Besag-Newell0.060.080.23 Cuzick-Edwards0.160.320.38 Tango’s MEET0.41 Swartz Entropy0.14 Whittemore’s Test0.12 Spatial Scan0.28

18. Power Fixed Distance, 4% Besag-Newell 0.060.060.12 Cuzick-Edwards0.060.060.07 Tango’s MEET0.17 Swartz Entropy0.06 Whittemore’s Test0.10 Spatial Scan0.12

19. Power Random Distance, 1% Besag-Newell 0.140.210.27 Cuzick-Edwards0.530.520.47 Tango’s MEET0.56 Swartz Entropy0.39 Whittemore’s Test0.12 Spatial Scan0.35

20. Power Random Distance, 4% Besag-Newell 0.080.100.12 Cuzick-Edwards0.140.170.18 Tango’s MEET0.25 Swartz Entropy0.13 Whittemore’s Test0.10 Spatial Scan0.18

21. Hot Spot Clusters One or more neighboring counties have higher risk that outside. Constant risks among counties in the cluster, as well as among those outside the cluster

22. Power Grand Isle, Vermont (RR=193) Besag-Newell 0.710.390.09 Cuzick-Edwards0.750.170.04 Tango’s MEET0.20 Swartz Entropy0.94 Whittemore’s Test0.02 Spatial Scan1.00

23. Power Grand Isle +15 neigbors (RR=3.9) Besag-Newell 0.820.880.50 Cuzick-Edwards0.760.620.25 Tango’s MEET0.23 Swartz Entropy0.71 Whittemore’s Test0.01 Spatial Scan0.97

24. Power Pittsburgh, PA (RR=2.85) Besag-Newell 0.040.020.98 Cuzick-Edwards0.650.920.90 Tango’s MEET0.92 Swartz Entropy0.27 Whittemore’s Test0.00 Spatial Scan0.94

25. Power Pittsburgh + 15 neighbors (RR=2.1) Besag-Newell 0.290.280.91 Cuzick-Edwards0.600.720.84 Tango’s MEET0.83 Swartz Entropy0.35 Whittemore’s Test0.00 Spatial Scan0.95

26. Power Manhattan (RR=2.73) Besag-Newell 0.040.030.95 Cuzick-Edwards0.630.860.89 Tango’s MEET0.94 Swartz Entropy0.26 Whittemore’s Test0.27 Spatial Scan0.92

27. Power Manhattan + 15 neighbors (RR=1.53) Besag-Newell 0.010.060.37 Cuzick-Edwards0.260.650.80 Tango’s MEET0.99 Swartz Entropy0.05 Whittemore’s Test0.87 Spatial Scan0.93

28. Power, Three Clusters Grand Isle (RR=193), Pittsburgh (RR=2.85), Manhattan (RR=2.73 Besag-Newell 0.540.181.00 Cuzick-Edwards0.991.000.99 Tango’s MEET1.00 Swartz Entropy0.99 Whittemore’s Test0.01 Spatial Scan1.00

29. Power, Three Clusters Grand Isle +15, Pittsburgh +15, Manhattan +15 Besag-Newell 0.640.770.84 Cuzick-Edwards0.910.960.96 Tango’s MEET0.98 Swartz Entropy0.74 Whittemore’s Test0.12 Spatial Scan0.98

30. Conclusions Besag-Newell’s R and Cuzick-Edward’s k-NN often perform very well, but are highly dependent on the chosen parameter Moran’s I and Whittemore’s Test have problems with many types of clustering Tango’s MEET perform well for global clustering The spatial scan statistic perform well for hot-spot clusters

31. Limitations Only a few alternative models evaluated, on one particular geographical data set. Results may be different for other types of alternative models and data sets.