Global Clustering Tests

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Global Clustering Tests
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Tests for Spatial Randomness
H0 The risk of disease is the same everywhere
after adjustment for age, gender and/or other
covariates.
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Tests for Global Clustering
Evaluates whether clustering exist as a global
phenomena throughout the map, without
pinpointing the location of specific clusters.
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Tests for Global Clustering

• More than 100 different tests for global
  clustering proposed by different scientists in
  different fields. For example
• Whittemores Test, Biometrika 1987
• Cuzick-Edwards k-NN, JRSS 1990
• Besag-Newells R, JRSS 1991
• Tangos Excess Events Test, StatMed 1995
• Swartz Entropy Test, Health and Place 1998
• Tangos Max Excess Events Test, StatMed 2000

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Cuzick-Edwards k-NN Test
åi ci åj cj I(dijltdik(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.
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Cuzick-Edwards k-NN Test
Special case of the Weighted Morans I
Test, proposed by Cliff and Ord, 1981
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Tangos 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
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Whittemore's Test

• Whittemore et al. proposed the statistic

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Besag- Newells 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

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Besag-Newell's R

• Let um(i)minj(Dj(i)1) k. Under null
  hypothesis, the case number s will have Poisson
  distribution with probability
• where pC/N. For each county
• R is defined as

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Swartzs Entropy Test

• The test statistic is defined as

where ni is the population in county I, and N is
the total population
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Global Clustering TestsPower Evaluation
Joint work with Toshiro Tango, Peter Park and
Changhong Song
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Power Evaluation, Setup

• 245 counties and county equivalents in
  Northeastern United States
• Female population
• 600 randomly distributed cases, according to
  different probability models

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

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PowerZero Distance

• Besag-Newell 0.48 0.49 0.42
• Cuzick-Edwards 1.00 0.92 0.73
• Tangos MEET 0.99
• Swartz Entropy 1.00
• Whittemores Test 0.13
• Spatial Scan 0.79

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PowerFixed Distance, 1

• Besag-Newell 0.06 0.08 0.23
• Cuzick-Edwards 0.16 0.32 0.38
• Tangos MEET 0.41
• Swartz Entropy 0.14
• Whittemores Test 0.12
• Spatial Scan 0.28

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PowerFixed Distance, 4

• Besag-Newell 0.06 0.06 0.12
• Cuzick-Edwards 0.06 0.06 0.07
• Tangos MEET 0.17
• Swartz Entropy 0.06
• Whittemores Test 0.10
• Spatial Scan 0.12

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PowerRandom Distance, 1

• Besag-Newell 0.14 0.21 0.27
• Cuzick-Edwards 0.53 0.52 0.47
• Tangos MEET 0.56
• Swartz Entropy 0.39
• Whittemores Test 0.12
• Spatial Scan 0.35

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PowerRandom Distance, 4

• Besag-Newell 0.08 0.10 0.12
• Cuzick-Edwards 0.14 0.17 0.18
• Tangos MEET 0.25
• Swartz Entropy 0.13
• Whittemores Test 0.10
• Spatial Scan 0.18

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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

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PowerGrand Isle, Vermont (RR193)

• Besag-Newell 0.71 0.39 0.09
• Cuzick-Edwards 0.75 0.17 0.04
• Tangos MEET 0.20
• Swartz Entropy 0.94
• Whittemores Test 0.02
• Spatial Scan 1.00

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PowerGrand Isle 15 neigbors (RR3.9)

• Besag-Newell 0.82 0.88 0.50
• Cuzick-Edwards 0.76 0.62 0.25
• Tangos MEET 0.23
• Swartz Entropy 0.71
• Whittemores Test 0.01
• Spatial Scan 0.97

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PowerPittsburgh, PA (RR2.85)

• Besag-Newell 0.04 0.02 0.98
• Cuzick-Edwards 0.65 0.92 0.90
• Tangos MEET 0.92
• Swartz Entropy 0.27
• Whittemores Test 0.00
• Spatial Scan 0.94

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PowerPittsburgh 15 neighbors (RR2.1)

• Besag-Newell 0.29 0.28 0.91
• Cuzick-Edwards 0.60 0.72 0.84
• Tangos MEET 0.83
• Swartz Entropy 0.35
• Whittemores Test 0.00
• Spatial Scan 0.95

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PowerManhattan (RR2.73)

• Besag-Newell 0.04 0.03 0.95
• Cuzick-Edwards 0.63 0.86 0.89
• Tangos MEET 0.94
• Swartz Entropy 0.26
• Whittemores Test 0.27
• Spatial Scan 0.92

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PowerManhattan 15 neighbors (RR1.53)

• Besag-Newell 0.01 0.06 0.37
• Cuzick-Edwards 0.26 0.65 0.80
• Tangos MEET 0.99
• Swartz Entropy 0.05
• Whittemores Test 0.87
• Spatial Scan 0.93

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Power, Three ClustersGrand Isle (RR193),
Pittsburgh (RR2.85), Manhattan (RR2.73

• Besag-Newell 0.54 0.18 1.00
• Cuzick-Edwards 0.99 1.00 0.99
• Tangos MEET 1.00
• Swartz Entropy 0.99
• Whittemores Test 0.01
• Spatial Scan 1.00

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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
• Tangos MEET 0.98
• Swartz Entropy 0.74
• Whittemores Test 0.12
• Spatial Scan 0.98

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Conclusions

• Besag-Newells R and Cuzick-Edwards k-NN often
  perform very well, but are highly dependent on
  the chosen parameter
• Morans I and Whittemores Test have problems
  with many types of clustering
• Tangos MEET perform well for global clustering
• The spatial scan statistic perform well for
  hot-spot clusters

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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.