Model Based Geostatistics

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Model Based Geostatistics

• Archie Clements
• University of Queensland
• School of Population Health

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Overview

• Introduction to geostatistics
• Assumptions
• Variogram components
• Variogram models
• Kriging
• Assumptions
• Model-based geostatistics
• Principles
• Building the model
• Prediction
• Validation
• Applications parasitic disease control in Africa

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Spatial variation
Z
Y
X
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First and second order variation

• First-order variation
• Trend
• Large-scale variation
• Can be due to large-scale environmental drivers
  (e.g. temperature for vector-borne diseases)
• Second-order variation
• Localised variation clustering
• Modelled using geostatistics

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Spatial dependence

• Observations close in space are more similar than
  observations far apart
• The variance of pairs of observations that are
  close together (small h) tends to be smaller than
  the variance of pairs far apart (large h)
• Basis of the semivariogram
• Spatial decomposition of the sample variance

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Semivariance statistical notation
Semivariance is half the average squared
difference of values observed at locations
separated by a given distance (and direction)
Function of distance (and direction) distance in
bins, direction in sectors of compass azimuth
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Modelling spatial correlation semivariogram
Partial Sill
Semivariance
Sill
Nugget
Lag (h)
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Nugget

• Random variation (white noise) non-spatial
  measurement error
• Microvariation (spatial variation at a scale
  smaller than the smallest bin)
• If no spatial correlation
• Nugget sill (flat semivariogram)

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Semivariogram decisions to be made

• How many/what sized bins?
• Depends on density of data points
• For regular-spaced (grid-sampled) data bin size
  size of cells in the grid
• For irregular sampling modify according to
  range of spatial correlation (big range, big
  bins small range, small bins)
• What maximum lag(h) to use?
• Should be estimated up to half the length of the
  shortest side of study area
• Which parametric model to use?
• Visual fit
• Statistical fit

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Variogram models
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Schistosoma mansoni, Uganda
Omnidirectional semivariograms
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Anisotropy

• Spatial dependence is different in different
  directions
• Semivariogram calculated in one direction is
  different from semivariogram calculated in
  another direction
• Should check for anisotropy and, if present,
  accommodate it in interpolation
• Range or sill (or both) can differ

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Schistosoma mansoni, Uganda directional
semivariograms
Direction Range (km) Sill Nugget
Omni- directional 43.4 7E-2 4E-2
0 39.4 1E-1 -3E-3
45 43.6 7E-2 2E-2
90 35.8 8E-2 3E-2
135 39.5 1E-1 2E-2
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Schistosoma haematobium, Northwestern Tanzania
Direction Range (km) Sill Nugget
Omni- directional 36.0 5E-2 0
0 260.1 2E-2 3E-2
45 163.9 6E-3 3E-2
90 56.2 5E-2 0
135 97.7 3E-2 7E-3
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Schistosoma haematobium, Northwestern Tanzania
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Trended and skewed data

• Data should be de-trended
• Polynomials (regression on XY coordinates)
• Generalised linear models (regression on
  covariates)
• Generalised additive models (can over-fit)
• If directional variograms are calculated range
  in one direction is gt3 X range in perpendicular,
  sign of trend
• If skewed, consider transformation (e.g. log
  transformation, normal score transformation)
• Otherwise, extreme values overly influence
  interpolated map
• Have to back-transform interpolated values
• Called disjunctive Kriging

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Non-stationarity

• Spatial correlation structure cannot be
  generalised to the whole study area
• Why does it occur?
• Different factors may operate in different parts
  of the study area
• Different ecological zones with different disease
  epidemiology
• Need to estimate the spatial correlation
  structure separately in each homogeneous zone

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Kriging

• Z(si) is the measured value at the ith location
• ?i is the weight attributed to the measured value
  at the ith location (calculated using
  semivariogram)
• So is the prediction location

For formulae on how the weights are estimated
using the variogram http//en.wikipedia.org/wiki/
Kriging
Prediction standard error/variance gives an
indication of precision of the prediction
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Geostatistics summary

• Geostatistics involves 3 steps
• Exploratory data analysis
• Definition of a variogram
• Using the variogram for interpolation (Kriging)
• Technique applicable for
• Point-referenced data
• Spatially continuous processes
• Disease risk
• Rainfall, elevation, temperature, other climate
  variables
• Wildlife, vegetation, geology (mineral deposits)

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Bayesian model-based geostatistics

• Seminal paper
• Diggle, Tawn and Moyeed (1998). Model-based
  geostatistics. Appl. Stat. 473299-350
• Observed a need for addressing non-Gaussian
  observational error
• Idea is to embed linear Kriging methodology
  within a more general distributional framework
• Generalised linear models with an unobserved
  Gaussian process in the linear predictor
• Implemented in a Bayesian framework

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Advantages of the Bayesian approach

• Natural framework for incorporation of parameter
  uncertainty into spatial prediction
• Can build uncertainty into parameters using
  priors
• Non-informative
• Informative (based on exploratory analysis,
  additional sources of information)
• Convenient for modelling hierarchical data
  structures

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Bayesian model-based geostatistics
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Predictions

• Can predict at specified validation locations
  (with observed outcomes for comparison)
• Can predict at non-sampled locations, e.g. a
  prediction grid
• Might be interested in
• outcome
• spatial random effect
• Standard error of predicted outcome

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Validation

• Jack-knifing sampling with replacement
• Remove one observation, do prediction at that
  location and store predicted value
• Repeat for all observations
• Compare predicted to observed using statistical
  measures of fit (RMSE) and discriminatory
  performance (AUC)
• Not feasible with MBG other than with v. small
  datasets
• Cross-validation sampling without replacement
• Set aside a subset for validation (ideally 50)
• Use remaining data to train model
• Compare predicted and observed for the validation
  subset using statistical measures
• Can then recombine the validation and training
  subsets for final model build
• External validation using other prospective or
  retrospective dataset

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Model-based geostatistics summary

• Model-based geostatistics involves
• Visual and exploratory data analysis
• Variography (to determine if there is
  second-order spatial variation)
• Variable selection (for deterministic component)
• Building model (e.g. in WinBUGS)
• Model selection (e.g. using DIC)
• Prediction and validation

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Application Schistosomiasis in Sub-Saharan
Africa
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• Schistosomiasis
• 779 million people at risk
• 207 million infected
• Most in Africa
• Significant illness and mortality

• Two main forms in Africa
• Urinary schistosomiasis caused by Schistosoma
  haematobium
• Intestinal schistosomiasis caused by S. mansoni

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Life cycle of Schistosoma haematobium

Cercariae released
Adult worm in human bladder wall
Sporocysts in snail
Eggs in urine
Miracidia
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Diagnosis of infection

• S. haematobium
• Microscopic examination of urine slides Presence
  of eggs and egg counts
• Macrohaematuria (visible blood)
• Microhaematuria (invisible blood) tested using
  chemical reagent strips
• Blood in urine questionnaire
• S. mansoni and soil-transmitted helminths
• microscopic examination of stool samples

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School-based control programmes

• School-aged children have highest prevalence
  (proportion infected) and intensity (severity) of
  infection
• Education system is convenient for control
  central location to access target population

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How do we determine which schools should be
targeted?

• World Health Organisation guidelines treat
  communities biannually where prevalence in
  school-age children is gt10 and annually where
  prevalence gt50

• No surveillance
• Need to do surveys

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Field survey northwest Tanzania
Lake Victoria

• 153 schools surveyed
• 60 children per school
• What about non-sampled locations? Need to predict
  (interpolate) values

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MBG model for S. haematobium prevalence
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S. haematobium model results
Variable Coefficient Odds Ratio
Intercept 1.9 (-2.3 - 10.3)
LST gt35-39C 0.4 (-0.3 - 1.1) 1.5 (0.8 - 2.9)
LST gt39C 0.3 (-1.5 - 2.2) 1.4 (0.2 - 8.6)
Rainfall gt1050mm -1.1 (-3.4 - 1.1) 0.3 (3.3 x 10-2 - 3.1)
? 0.9 (0.6 - 1.3)
f 0.2 (0.1 - 1.0)
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Clements et al. TMIH 2006
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Uncertainty
Lower bound 95 PI
Upper bound 95 PI
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• Co-ordinated surveys in 3 contiguous countries
• 418 schools
• gt26,000 children

Variable
Variable Mean (95 CI) SD
Sex Female 0.70 (0.65, 0.76) 0.03
Age 910 years 1.16 (1.00, 1.33) 0.08
Age 1112 years 1.51 (1.31, 1.73) 0.10
Age 1316 years 1.79 (1.53, 2.06) 0.14
Distance to perennial water body 0.34 (0.21, 0.54) 0.08
Land surface temperature 0.80 (0.51, 1.21) 0.18
Land surface temperature2 1.10 (0.85, 1.40) 0.14
Rate of decay of spatial correlation 2.03 (1.48, 2.74) 0.32
Variance of the spatial random effect (sill) 7.03 (5.36, 9.31) 1.01
Probability that prevalence is gt50 Clements et
al. EID 2008
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Other outcomes co-infection
East Africa Brooker and Clements, Int. J.
Parasitol., in press S. mansoni mono-infection
7.9 Hookworm mono-infection 40.5 Co-infection
8.1
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Model for co-infection
,
YijkMultinomial(pijk,nijk),
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Variable S. mansoni mono-infection posterior mean (95 posterior CI) Hookworm mono-infection posterior mean (95 posterior CI) S. mansoni/hookworm co-infection posterior mean (95 posterior CI)
Intercept -3.8 (-4.7 - -2.9) -0.6 (-1.1 - -0.3) -4.4 (-5.0 - -3.7)
OR Elevation 0.35 (0.22 - 0.58) 0.77 (0.65 - 0.89) 0.30 (0.20 - 0.47)
OR DPWB 0.23 (0.10 - 0.45) 0.94 (0.76 - 1.15) 0.30 (0.18 - 0.58)
OR Rural vs urban 0.43 (0.21 - 0.79) 0.98 (0.68 - 1.37) 0.61 (0.36 - 1.02)
OR Ext. rural vs urban 0.62 (0.23 - 1.44) 1.16 (0.82 - 1.81) 0.75 (0.31 - 1.62)
OR LST 0.88 (0.62 - 1.25) 0.60 (0.50 - 0.72) 0.57 (0.31 - 0.87)
OR Female 0.86 (0.76 - 0.96) 0.91 (0.86 - 0.97) 0.70 (0.63 - 0.77)
OR Age (9-10 years) 1.67 (1.37 - 2.06) 1.17 (1.04 - 1.30) 1.82 (1.52 - 2.21)
OR Age (11-13 years) 2.44 (2.06 - 2.89) 1.55 (1.39 - 1.71) 2.99 (2.55 - 3.52)
OR Age (14 years) 2.87 (2.19 - 3.71) 1.88 (1.63 - 2.14) 3.83 (3.01 - 4.86)
Phi (rate of decay) 3.52 (1.73 - 7.21) 4.98 (3.38 - 7.33) 3.76 (2.10 - 7.36)
Sill 6.39 (3.52 - 11.78) 1.31 (0.98 - 1.76) 6.34 (3.98 - 9.95)
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Co-infection
Hookworm monoinfection
S. mansoni - Hookworm coinfection
S. mansoni monoinfection
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Other outcomes Intensity of infection

• Prevalence is used (currently) for disease
  control planning
• Intensity of infection (eggs/ml urine or /g
  faeces) is more indicative of
• Morbidity (anaemia, urine tract, hepatic
  pathology)
• Transmission

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Model for intensity of infection
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Intensity of S. mansoni infection, East Africa
Clements et al. Parasitol 2006
Variable Posterior Mean (95 CI)
Intercept 10.06 (5.77 - 13.22)
Female -0.41 (-0.72 - -0.11)
Elevation (m) -0.007 (-0.01 - -0.004)
DPWB (dec deg) -5.36 (-7.51 - -3.30)
Sill 23.96 (19.06 - 32.07)
Range 0.134 (0.09 - 0.20)
Overdispersion 0.06 (0.058 - 0.062)
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Conclusions

• In disease control we need evidence-based
  framework for deciding on where to allocate
  limited control resources
• Maps are useful tools for highlighting
  sub-national variation targeting interventions
  advocacy (national and local) integrated control
  programmes estimating heterogeneities in disease
  burden
• Model-based geostatistics enables rich inference
  from spatial data uncertainty