Tests for spatial clustering
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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). K - Function.

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Tests for Spatial Clustering

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Tests for spatial clustering

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)

K function

K - Function

  • summary of local dependence of spatial process -> second order process

  • expresses number of expected events within given distance of randomly chosen event

Example k function for newcastle disease outbreak

Example: k – Function for Newcastle Disease Outbreak

Tb case control study in central north island of nz

TB Case-Control Study in Central North Island of NZ

cases = redcontrols = blue

Cuzick and edward s test applied to tb case control study

Cuzick and Edward’s Test applied to TB Case-Control Study

Local spatial autocorrelation

Local Spatial Autocorrelation

Local Moran

Local Geary

Spatial scan statistic

Spatial Scan Statistic

  • no pre-specified cluster size

  • can take confounding into account

  • also does time - space clustering

  • method

    • 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)

Example satscan

Example - SaTScan

  • locations of den sites of tuberculous and non-tuberculous possums

Example satscan cont

Example - SaTScan cont.


1. 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...............: 0.001


2. 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...............: 0.337

3. 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

Example satscan cont1

Example - SaTScan cont.

Space time scan statistic

Space-Time Scan Statistic


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

Framework for spatial data analysis

Framework for Spatial Data Analysis

Attribute data

Feature data





Describe patterns



Test hypotheses




  • explain and predict spatial structure

    • hypothesis testing

  • methods

    • data mining

    • statistical and simulation modelling

    • multi-criteria/multi-objective decision modelling

  • problem -> spatial dependence

Tests for spatial clustering

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

Recent developments in spatial regression modelling

Recent Developments in Spatial Regression Modelling

  • generalised linear mixed models (GLMM)

    • use random effect term to reflect spatial structure

      • impose spatial covariance structures

      • Bayesian estimation, Markov chain Monte Carlo (MCMC), Gibbs sampling

  • autologistic regression

    • include spatial covariate

    • MCMC estimation

Bayesian regression modelling

Bayesian Regression Modelling

  • Bayesian inference

    • combines

      • information from data (likelihood)

      • prior distributions for unknown parameters

    • to generate

      • posterior distribution of dependent variable

    • allows modelling of data heterogeneity, addresses multiplicity issues

Tb reactor risk modelling

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

Raw standardised morbidity ratio

Raw Standardised Morbidity Ratio

BUGS softwarewith GeoBUGS extension

Example kernel density plots

Example – Kernel Density Plots

Raw smr and posterior relative risk maps

Raw SMR and Posterior Relative Risk Maps

Bayes’ RRestimates

raw SMR

Medians and 95 ci of posterior relative risks

Medians and 95% CI of Posterior Relative Risks

Model residuals and rr significance

Model Residuals and RR Significance

Relative importance of structured versus unstructured random effect

Relative Importance of Structured versus Unstructured Random Effect

Multi criteria decision making using gis

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 reservoir

      • cattle stocking density

      • access to wildlife for vaccine delivery

  • decision rule -> procedure for selection and combination of criteria

Multi criteria decision making in gis cont

Multi-Criteria Decision Making in GIS cont.

  • evaluation -> application of decision rules

    • multi-criteria evaluations

      • boolean overlays

      • weighted linear combinations

  • uncertainty

    • database uncertainty

    • decision rule uncertainty -> fuzzy versus crisp sets

  • decision risk -> likelihood of decision being wrong -> Bayesian probability theory, Dempster-Shafer Theory

Dempster shafer theory

Dempster - Shafer Theory

  • extension of Bayesian probability theory

  • data 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

More about dempster shafer theory

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

Even more about dempster shafer theory

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

Example east coast fever occurrence in zimbabwe

Example – East Coast Fever Occurrence in Zimbabwe

Belief interval for T.parva Presence(Degree of uncertainty)

Belief in T.parva Presence

Landscape structure

Landscape Structure

  • quantify landscape structure/composition

  • habitat features as a whole

Tb infected herds around hauhungaroa ranges in nz

TB Infected Herds around Hauhungaroa Ranges in NZ

Framework for spatial data analysis1

Framework for Spatial Data Analysis

Attribute data

Feature data





Describe patterns



Test hypotheses




  • 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

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