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Global Clustering Tests. Tests for Spatial Randomness. H 0 : The risk of disease is the same everywhere after adjustment for age, gender and/or other covariates. Tests for Global Clustering. Evaluates whether clustering exist as a global phenomena throughout the map, without

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tests for spatial randomness
Tests for Spatial Randomness

H0: The risk of disease is the same everywhere after adjustment for

age, gender and/or other covariates.

tests for global clustering
Tests for Global Clustering

Evaluates whether clustering exist as a global

phenomena throughout the map, without

pinpointing the location of specific clusters.

tests for global clustering4
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
cuzick edward s k nn test
Cuzick-Edward’s k-NN Test

åici åjcj I(dij<dik(i))

where

ci = number of deaths in county i

dij = 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.

cuzick edward s k nn test6
Cuzick-Edward’s k-NN Test

Special case of the Weighted Moran’s I Test,

proposed by Cliff and Ord, 1981

tango s excess events test
Tango’s Excess Events Test

åi åj [cj-E(cj)] [cj-E(cj)] e-4d2ij/l2

where

ci = number of deaths in county i

E(cj) = expected cases in county i | H0

dij = distance from county i to county j

l = clustering scale parameter

whittemore s test
Whittemore\'s Test

Whittemore et al. proposed the statistic

besag newell s r
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
besag newell s r10
Besag-Newell\'s R

Let um(i)=min{j:(Dj(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

swartz s entropy test
Swartz’s Entropy Test

The test statistic is defined as

where ni is the population in county I, and N is the total population

global clustering tests power evaluation
Global Clustering TestsPower Evaluation

Joint work with Toshiro Tango, Peter Park and Changhong Song

power evaluation setup
Power Evaluation, Setup
  • 245 counties and county equivalents in Northeastern United States
  • Female population
  • 600 randomly distributed cases, according to different probability models
slide14
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.

global chain clustering
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.
power zero distance
PowerZero Distance

Besag-Newell 0.48 0.49 0.42

Cuzick-Edwards 1.00 0.92 0.73

Tango’s MEET 0.99

Swartz Entropy 1.00

Whittemore’s Test 0.13

Spatial Scan 0.79

power fixed distance 1
PowerFixed Distance, 1%

Besag-Newell 0.06 0.08 0.23

Cuzick-Edwards 0.16 0.32 0.38

Tango’s MEET 0.41

Swartz Entropy 0.14

Whittemore’s Test 0.12

Spatial Scan 0.28

power fixed distance 4
PowerFixed Distance, 4%

Besag-Newell 0.06 0.06 0.12

Cuzick-Edwards 0.06 0.06 0.07

Tango’s MEET 0.17

Swartz Entropy 0.06

Whittemore’s Test 0.10

Spatial Scan 0.12

power random distance 1
PowerRandom Distance, 1%

Besag-Newell 0.14 0.21 0.27

Cuzick-Edwards 0.53 0.52 0.47

Tango’s MEET 0.56

Swartz Entropy 0.39

Whittemore’s Test 0.12

Spatial Scan 0.35

power random distance 4
PowerRandom Distance, 4%

Besag-Newell 0.08 0.10 0.12

Cuzick-Edwards 0.14 0.17 0.18

Tango’s MEET 0.25

Swartz Entropy 0.13

Whittemore’s Test 0.10

Spatial Scan 0.18

hot spot clusters
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
power grand isle vermont rr 193
PowerGrand Isle, Vermont (RR=193)

Besag-Newell 0.71 0.39 0.09

Cuzick-Edwards 0.75 0.17 0.04

Tango’s MEET 0.20

Swartz Entropy 0.94

Whittemore’s Test 0.02

Spatial Scan 1.00

power grand isle 15 neigbors rr 3 9
PowerGrand Isle +15 neigbors (RR=3.9)

Besag-Newell 0.82 0.88 0.50

Cuzick-Edwards 0.76 0.62 0.25

Tango’s MEET 0.23

Swartz Entropy 0.71

Whittemore’s Test 0.01

Spatial Scan 0.97

power pittsburgh pa rr 2 85
PowerPittsburgh, PA (RR=2.85)

Besag-Newell 0.04 0.02 0.98

Cuzick-Edwards 0.65 0.92 0.90

Tango’s MEET 0.92

Swartz Entropy 0.27

Whittemore’s Test 0.00

Spatial Scan 0.94

power pittsburgh 15 neighbors rr 2 1
PowerPittsburgh + 15 neighbors (RR=2.1)

Besag-Newell 0.29 0.28 0.91

Cuzick-Edwards 0.60 0.72 0.84

Tango’s MEET 0.83

Swartz Entropy 0.35

Whittemore’s Test 0.00

Spatial Scan 0.95

power manhattan rr 2 73
PowerManhattan (RR=2.73)

Besag-Newell 0.04 0.03 0.95

Cuzick-Edwards 0.63 0.86 0.89

Tango’s MEET 0.94

Swartz Entropy 0.26

Whittemore’s Test 0.27

Spatial Scan 0.92

power manhattan 15 neighbors rr 1 53
PowerManhattan + 15 neighbors (RR=1.53)

Besag-Newell 0.01 0.06 0.37

Cuzick-Edwards 0.26 0.65 0.80

Tango’s MEET 0.99

Swartz Entropy 0.05

Whittemore’s Test 0.87

Spatial Scan 0.93

power three clusters grand isle rr 193 pittsburgh rr 2 85 manhattan rr 2 73
Power, Three ClustersGrand Isle (RR=193), Pittsburgh (RR=2.85), Manhattan (RR=2.73

Besag-Newell 0.54 0.18 1.00

Cuzick-Edwards 0.99 1.00 0.99

Tango’s MEET 1.00

Swartz Entropy 0.99

Whittemore’s Test 0.01

Spatial Scan 1.00

power three clusters grand isle 15 pittsburgh 15 manhattan 15
Power, Three ClustersGrand Isle +15, Pittsburgh +15, Manhattan +15

Besag-Newell 0.64 0.77 0.84

Cuzick-Edwards 0.91 0.96 0.96

Tango’s MEET 0.98

Swartz Entropy 0.74

Whittemore’s Test 0.12

Spatial Scan 0.98

conclusions
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
limitations
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.
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