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Empirical/Asymptotic P-values for Monte Carlo-Based Hypothesis Testing:
an Application to Cluster Detection Using the Scan Statistic
Allyson Abrams, Martin Kulldorff, Ken Kleinman
Department of Ambulatory Care and Prevention,
Harvard Medical School and Harvard Pilgrim Health Care
Presented at EVA, August 15, 2005
This work was funded by the United States National Cancer Institute, grant number RO1-CA95979.
Background: Scan Statistics
• Spatial scan statistic – used to identify geographic clusters
• Use moving circular window on map– Any point on map can be the center of a cluster– Each circle includes a different set of points– If the centroid of a region is included in the
circle, the whole region is included
Background: Scan Statistics
For each distinct window, calculate the likelihood, proportional to:
n = number of cases inside circle
N = total number of cases
= expected number of cases inside circle
n N nn N n
N
Background: Scan Statistics
• The scan statistic is the maximum likelihood over all possible circles– Identifies the most unusual cluster
• To find p-value, use Monte Carlo hypothesis testing– Redistribute cases randomly and recalculate the scan
statistic many times– Proportion of scan statistics from the Monte Carlo
replicates which are greater than or equal to the scan statistic for the true cluster is the p-value
Background: Scan Statistics
Background: Scan Statistics
• That discussion only considered spatial clustering• To extend to clustering in space and time, use
cylinders instead of circles– The height of the cylinder represents time
• The rest of the process is unchanged• SaTScan is a freely available software that uses the
scan statistic to detect clusters in space, time, or space-time (www.satscan.org)
Background: SaTScan
• Main drawback to Monte Carlo hypothesis testing: increased precision for p-values can only be obtained through greatly increasing the number of Monte Carlo replicates– A big problem for small p-values
• SaTScan can take anywhere from seconds to hours to run, depending on the data, the type of analysis, and the number of Monte Carlo replicates
Background
• We use SaTScan for 2 main reasons1. Daily surveillance for disease outbreaks
2. Evaluating systems that use SaTScan for surveillance
• In both cases, we need to limit the amount of time it takes to generate each p-value while still retaining enough precision in the p-value to determine how unusual a cluster is
Goal
• Estimate distribution of the scan statistic using fewer Monte Carlo replicates– See how the p-values obtained from the
distributional parameters compares with the true p-value
Methods
• Sample map – 245 counties in the northeast United States with 600 cases
• Ran SaTScan on the sample map using 100,000,000 Monte Carlo replicates to find the 'true' log-likelihood needed to obtain p-values of 0.01, 0.001, 0.0001, 0.00001– Corresponds to the following order statistics
from the 100,000,000 Monte Carlo replicates: 1,000,000; 100,000; 10,000; 1,000
Methods
• Ran SaTScan 1000 times on the same map, each time generating 999 Monte Carlo replicates
• For each of the 1000 SaTScan runs:– Found maximum likelihood estimates of the
parameters for each distribution based on the 999 Monte Carlo replicates
• Distributions used: Normal, Lognormal, Gamma, Gumbel
Methods
• The empirical/asymptotic p-value for each distribution is the area to the right of the observed log-likelihood for a given distribution
• For each distribution, we generated:1. empirical/asymptotic p-values based on the 'true'
log-likelihood value2. the log-likelihoods that would have been required
to generate p-values of 0.01, 0.001, 0.0001, 0.000013. The usual Monte Carlo-based p-value reported in
SaTScan
Methods
• Repeated the entire process using 60 and 6000 cases– Results were almost identical
• Using 600 cases, repeated entire process with 99 and 9999 Monte Carlo replicates in each of the 1000 simulations– Again, very similar results
Results
0
2
4
6
8
10Percent
Gamma
0
2
4
6
8
10Percent
Gumbel
0
2
4
6
8
10Percent
Lognormal
0. 0002 0. 00146 0. 00272 0. 00398 0. 00524 0. 0065 0. 00776 0. 00902 0. 01028 0. 01154 0. 0128 0. 01406
0
2
4
6
8
10Percent
Normal
p
True p-value = 0.01
Results
0
20
40
60
80
100Percent
Gamma
0
20
40
60
80
100Percent
Gumbel
0
20
40
60
80
100Percent
Lognormal
0 0. 000196 0. 000392 0. 000588 0. 000784 0. 00098 0. 001176 0. 001372 0. 001568 0. 001764
0
20
40
60
80
100Percent
Normal
p
True p-value = 0.001
Results
0
20
40
60
80
100Percent
Gamma
0
20
40
60
80
100Percent
Gumbel
0
20
40
60
80
100Percent
Lognormal
0 0. 000024 0. 000048 0. 000072 0. 000096 0. 00012 0. 000144 0. 000168 0. 000192 0. 000216 0. 00024
0
20
40
60
80
100Percent
Normal
p
True p-value = 0.0001
Results
0
20
40
60
80
100Percent
Gamma
0
20
40
60
80
100Percent
Gumbel
0
20
40
60
80
100Percent
Lognormal
0 3. 6E- 06 7. 2E- 06 0. 0000108 0. 0000144 0. 000018 0. 0000216 0. 0000252 0. 0000288
0
20
40
60
80
100Percent
Normal
p
True p-value = 0.00001
Results
0
2
4
6
8
10
12
14
Percent
Gumbel
0. 004 0. 00526 0. 00652 0. 00778 0. 00904 0. 0103 0. 01156 0. 01282 0. 01408 0. 01534 0. 0166 0. 01786 0. 01912
0
2
4
6
8
10
12
14
Percent
SaTScan
p
True p-value = 0.01
Results
0
10
20
30
40
50
Percent
Gumbel
0. 0004 0. 0008 0. 0012 0. 0016 0. 002 0. 0024 0. 0028 0. 0032 0. 0036 0. 004 0. 0044 0. 0048
0
10
20
30
40
50
Percent
SaTScan
p
True p-value = 0.001
Results
0
20
40
60
80
100
Percent
Gumbel
0 0. 0002 0. 0004 0. 0006 0. 0008 0. 001 0. 0012 0. 0014 0. 0016 0. 0018 0. 002
0
20
40
60
80
100
Percent
SaTScan
p
True p-value = 0.0001
Results
0
20
40
60
80
100
Percent
Gumbel
0 0. 000098 0. 000196 0. 000294 0. 000392 0. 00049 0. 000588 0. 000686 0. 000784 0. 000882 0. 00098
0
20
40
60
80
100
Percent
SaTScan
p
True p-value = 0.00001
Results
• The empirical/asymptotic p-values from the Gumbel distribution appear only slightly conservatively biased
• Other tested distributions all resulted in anti-conservatively biased p-values
• The ordinary Monte Carlo p-values reported from SaTScan had greater variance than the Gumbel-based p-values
Conclusions
• Empirical/asymptotic p-values based on the Gumbel distribution can be preferable to true Monte Carlo p-values
• Empirical/asymptotic p-values can accurately generate p-values smaller than is possible with Monte Carlo p-values with a given number of replicates
• We suggest empirical/asymptotic p-values as a hybrid method to accurately obtain small p-values with a relatively small number of Monte Carlo replicates
Future work
• Results shown today are based on purely spatial analyses – we will also look at space-time analyses
• An option will be added in SaTScan to allow the user to request the Gumbel-based p-value
