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Tests for Spatial Clustering
global statistic aggregate / points
k-function Grimson’s method Cuzick & Edward’s method Join Count
aggregate data Geary’s C Moran’s I
local statistic spatial scan statistic LISA statistic geographical analysis machine (GAM)
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K - Function
summary of local dependence of spatial process -> second order process
expresses number of expected events within given distance of randomly chosen event
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Example: k – Function for Newcastle Disease Outbreak
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TB Case-Control Study in Central North Island of NZ
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Cuzick and Edward’s Test applied to TB Case-Control Study
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Legend0 - 0.690.69 - 2.032.03 - 3.583.58 - 5.85.8 - 10.18
Local Spatial Autocorrelation
Local MoranLocal Geary
Legend-0.59 - -0.21-0.21 - 0.120.12 - 0.60.6 - 2.12.1 - 6.75
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Spatial Scan Statistic
no pre-specified cluster sizecan take confounding into accountalso does time - space clusteringmethod
increasing circles (cylinders if including time) compare risk within with outside circle most likely cluster -> circle with maximum
likelihood (more than expected number of cases)
SaTScan software (public domain)
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Example - SaTScan
locations of den sites of tuberculous and non-tuberculous possums
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Example - SaTScan cont.
MOST LIKELY CLUSTER1. Coordinates / radius..: (348630,708744) / 126.65 Population............: 56 Number of cases.......: 34 (16.44 expected) Overall relative risk.: 2.07 Log likelihood ratio..: 15.86 P-value...............: 0.001SECONDARY CLUSTERS2. Coordinates / radius..: (348491,708496) / 33.35 Population............: 5 Number of cases.......: 5 (1.47 expected) Overall relative risk.: 3.41 Log likelihood ratio..: 6.25 P-value...............: 0.3373. Coordinates / radius..: (348369,708453) / 80.55 Population............: 8 Number of cases.......: 7 (2.35 expected) Overall relative risk.: 2.98 Log likelihood ratio..: 6.13 P-value...............: 0.365
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Example - SaTScan cont.
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Space-Time Scan Statistic
MOST LIKELY CLUSTER
1.Census areas included.: 75, 26, 77, 76, 29, 32
Coordinates / radius..: (389631,216560) / 59840.47
Time frame............: 1997/1/1 - 1999/12/31
Population............: 4847
Number of cases.......: 1507 (632.85 expected)
Overall relative risk.: 2.38
Log likelihood ratio..: 509.4
Monte Carlo rank......: 1/1000
P-value...............: 0.001
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Framework for Spatial Data Analysis
Visualization
Exploration
Modelling
Attribute data
Feature data
Databases
Maps
Describe patterns
Test hypothese
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GISDBMS
StatisticalSoftware
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Modelling
explain and predict spatial structure hypothesis testing
methods data mining statistical and simulation modelling multi-criteria/multi-objective decision
modellingproblem -> spatial dependence
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3D Risk Map for FMD Outbreak Occurrence in Thailand(based on random effects logistic regression analysis)
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Recent Developments in Spatial Regression Modelling
generalised linear mixed models (GLMM) use random effect term to reflect spatial
structureimpose spatial covariance structuresBayesian estimation, Markov chain Monte
Carlo (MCMC), Gibbs sampling
autologistic regression include spatial covariate MCMC estimation
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Bayesian Regression Modelling
Bayesian inference combines
information from data (likelihood)prior distributions for unknown parameters
to generateposterior distribution of dependent variable
allows modelling of data heterogeneity, addresses multiplicity issues
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TB Reactor Risk Modelling
dependent variable -> observed TB reactors per county in 1999 in GB
Poisson regression model MCMC estimation expected no. TB reactors two random effects (convolution prior)
spatial – conditionally autoregressive (CAR) prior
non-spatial – exchangeable normal prior
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Raw Standardised Morbidity Ratio
BUGS softwarewith GeoBUGS extension
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Example – Kernel Density Plots
RR[25] sample: 5000
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Raw SMR and Posterior Relative Risk Maps
raw SMR Bayes’ RRestimates
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Medians and 95% CI of Posterior Relative Risks
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Model Residuals and RR Significance
RR[31] sample: 10000
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Relative Importance of Structured versus Unstructured Random Effect
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Multi-Criteria Decision Making using GIS
decision -> choice between alternatives vaccinate wildlife or not
criterion -> evidence used to decide on decision factors and constraints
presence of wildlife reservoircattle stocking densityaccess to wildlife for vaccine delivery
decision rule -> procedure for selection and combination of criteria
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Multi-Criteria Decision Making in GIS cont.
evaluation -> application of decision rules multi-criteria evaluations
boolean overlaysweighted linear combinations
uncertainty database uncertainty decision rule uncertainty -> fuzzy versus crisp
setsdecision risk -> likelihood of decision being
wrong -> Bayesian probability theory, Dempster-Shafer Theory
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Dempster - Shafer Theory
extension of Bayesian probability theorydata uncertainty included in calculation ->
belief in hypothesis not complement of belief in negation (sensitivity of diagnosis)
collect different sources of evidence for presence/absence (data, expert knowledge) re-express as probability
combine evidence as mass of support for particular hypothesis
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More about Dempster-Shafer Theory
belief total support for hypothesis degree of hard evidence supporting hypothesis
plausibility degree to which hypothesis cannot be
disbelieved degree to which conditions appear to be right
for hypothesis, even though hard evidence is lacking
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Even more about Dempster-Shafer Theory
belief interval range between belief and plausibility degree of uncertainty in establishing
presence/absence of hypothesis areas with high belief interval suitable for
collection of new data
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Example – East Coast Fever Occurrence in Zimbabwe
Belief in T.parva Presence
Belief interval for T.parva Presence(Degree of uncertainty)
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Landscape Structure
quantify landscape structure/composition
habitat features as a whole
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TB Infected Herds around Hauhungaroa Ranges in NZ
Vegetation classespasturepasture/scrubscrublandforest
Farm boundaries
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Framework for Spatial Data Analysis
Visualization
Exploration
Modelling
Attribute data
Feature data
Databases
Maps
Describe patterns
Test hypothese
s
GISDBMS
StatisticalSoftware
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Conclusion
spatial analysis essential component of epidemiological analysis
key ideas visualization -> extremely effective for
analysis and presentation exploration -> cluster detection
methods (beware of type I error) modelling -> Bayesian modelling and
decision analysis techniques