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1 Tests for Spatial Clustering zglobal statistic yaggregate / points xk-function xGrimsons method xCuzick & Edwards method xJoin Count yaggregate data xGearys C xMorans I zlocal statistic yspatial scan statistic yLISA statistic ygeographical analysis machine (GAM)

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2 K - Function zsummary of local dependence of spatial process -> second order process zexpresses number of expected events within given distance of randomly chosen event

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3 Example: k – Function for Newcastle Disease Outbreak

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4 TB Case-Control Study in Central North Island of NZ cases = red controls = blue

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5 Cuzick and Edwards Test applied to TB Case-Control Study

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6 Local Spatial Autocorrelation Local Moran Local Geary

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7 Spatial Scan Statistic zno pre-specified cluster size zcan take confounding into account zalso does time - space clustering zmethod yincreasing circles (cylinders if including time) ycompare risk within with outside circle ymost likely cluster -> circle with maximum likelihood (more than expected number of cases) zSaTScan software (public domain)

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8 Example - SaTScan zlocations of den sites of tuberculous and non- tuberculous possums

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9 Example - SaTScan cont. MOST LIKELY CLUSTER 1. Coordinates / radius..: (348630,708744) / Population : 56 Number of cases : 34 (16.44 expected) Overall relative risk.: 2.07 Log likelihood ratio..: P-value : SECONDARY CLUSTERS 2. Coordinates / radius..: (348491,708496) / Population : 5 Number of cases : 5 (1.47 expected) Overall relative risk.: 3.41 Log likelihood ratio..: 6.25 P-value : Coordinates / radius..: (348369,708453) / 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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10 Example - SaTScan cont.

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11 Space-Time Scan Statistic MOST LIKELY CLUSTER 1.Census areas included.: 75, 26, 77, 76, 29, 32 Coordinates / radius..: (389631,216560) / Time frame : 1997/1/ /12/31 Population : 4847 Number of cases : 1507 ( expected) Overall relative risk.: 2.38 Log likelihood ratio..: Monte Carlo rank......: 1/1000 P-value : 0.001

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12 Framework for Spatial Data Analysis Visualization Exploration Modelling Attribute data Feature data Databases Maps Describe patterns Test hypothese s GIS DBMS Statistical Software

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13 Modelling zexplain and predict spatial structure yhypothesis testing zmethods ydata mining ystatistical and simulation modelling ymulti-criteria/multi-objective decision modelling zproblem -> spatial dependence

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14 3D Risk Map for FMD Outbreak Occurrence in Thailand (based on random effects logistic regression analysis)

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15 Recent Developments in Spatial Regression Modelling zgeneralised linear mixed models (GLMM) yuse random effect term to reflect spatial structure ximpose spatial covariance structures xBayesian estimation, Markov chain Monte Carlo (MCMC), Gibbs sampling zautologistic regression yinclude spatial covariate yMCMC estimation

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16 Bayesian Regression Modelling zBayesian inference ycombines xinformation from data (likelihood) xprior distributions for unknown parameters yto generate xposterior distribution of dependent variable yallows modelling of data heterogeneity, addresses multiplicity issues

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17 TB Reactor Risk Modelling zdependent variable -> observed TB reactors per county in 1999 in GB zPoisson regression model yMCMC estimation yexpected no. TB reactors ytwo random effects (convolution prior) xspatial – conditionally autoregressive (CAR) prior xnon-spatial – exchangeable normal prior

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18 Raw Standardised Morbidity Ratio BUGS software with GeoBUGS extension

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19 Example – Kernel Density Plots

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20 Raw SMR and Posterior Relative Risk Maps raw SMR Bayes RR estimates

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21 Medians and 95% CI of Posterior Relative Risks

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22 Model Residuals and RR Significance

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23 Relative Importance of Structured versus Unstructured Random Effect

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24 Multi-Criteria Decision Making using GIS zdecision -> choice between alternatives yvaccinate wildlife or not zcriterion -> evidence used to decide on decision yfactors and constraints xpresence of wildlife reservoir xcattle stocking density xaccess to wildlife for vaccine delivery zdecision rule -> procedure for selection and combination of criteria

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25 Multi-Criteria Decision Making in GIS cont. zevaluation -> application of decision rules ymulti-criteria evaluations xboolean overlays xweighted linear combinations zuncertainty ydatabase uncertainty ydecision rule uncertainty -> fuzzy versus crisp sets zdecision risk -> likelihood of decision being wrong -> Bayesian probability theory, Dempster-Shafer Theory

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26 Dempster - Shafer Theory zextension of Bayesian probability theory zdata uncertainty included in calculation -> belief in hypothesis not complement of belief in negation (sensitivity of diagnosis) zcollect different sources of evidence for presence/absence (data, expert knowledge) yre-express as probability zcombine evidence as mass of support for particular hypothesis

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27 More about Dempster- Shafer Theory zbelief ytotal support for hypothesis ydegree of hard evidence supporting hypothesis zplausibility ydegree to which hypothesis cannot be disbelieved ydegree to which conditions appear to be right for hypothesis, even though hard evidence is lacking

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28 Even more about Dempster-Shafer Theory zbelief interval yrange between belief and plausibility ydegree of uncertainty in establishing presence/absence of hypothesis yareas with high belief interval suitable for collection of new data

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29 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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30 Landscape Structure zquantify landscape structure/composition zhabitat features as a whole

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31 TB Infected Herds around Hauhungaroa Ranges in NZ

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32 Framework for Spatial Data Analysis Visualization Exploration Modelling Attribute data Feature data Databases Maps Describe patterns Test hypothese s GIS DBMS Statistical Software

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33 Conclusion zspatial analysis essential component of epidemiological analysis zkey ideas yvisualization -> extremely effective for analysis and presentation yexploration -> cluster detection methods (beware of type I error) ymodelling -> Bayesian modelling and decision analysis techniques

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