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

  • 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

  • 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


TB Case-Control Study in Central North Island of NZ

cases = redcontrols = blue


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


Local Spatial Autocorrelation

Local Moran

Local Geary


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

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


Example - SaTScan cont.

MOST LIKELY CLUSTER

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

SECONDARY CLUSTERS

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 cont.


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


Framework for Spatial Data Analysis

Attribute data

Feature data

Databases

GISDBMS

Visualization

Maps

Describe patterns

Exploration

StatisticalSoftware

Test hypotheses

Modelling


Modelling

  • explain and predict spatial structure

    • hypothesis testing

  • methods

    • data mining

    • statistical and simulation modelling

    • multi-criteria/multi-objective decision modelling

  • problem -> spatial dependence


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


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 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

  • 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

BUGS softwarewith GeoBUGS extension


Example – Kernel Density Plots


Raw SMR and Posterior Relative Risk Maps

Bayes’ RRestimates

raw SMR


Medians and 95% CI of Posterior Relative Risks


Model Residuals and RR Significance


Relative Importance of Structured versus Unstructured Random Effect


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.

  • 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

  • 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

  • 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

  • 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

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

Belief in T.parva Presence


Landscape Structure

  • quantify landscape structure/composition

  • habitat features as a whole


TB Infected Herds around Hauhungaroa Ranges in NZ


Framework for Spatial Data Analysis

Attribute data

Feature data

Databases

GISDBMS

Visualization

Maps

Describe patterns

Exploration

StatisticalSoftware

Test hypotheses

Modelling


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


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