Statistical modelling and inference for spatio-temporal ...mhazelto/talks/IDReC-2012.pdf · Statistical modelling and inference for spatio-temporal disease processes Martin Hazelton1

Statistical modelling and inference forspatio-temporal disease processes

Martin Hazelton1

Statistics and Bioinformatics GroupInstitute of Fundamental Sciences, Massey University

24 October 2012

1Presenter: [email protected]

IDReC Symposium October 2012, Palmerston North 1 / 31

Disease, Prediction and Chance

Spread of infectious disease is a complexprocess typically involving a plethora offactors.Some are explicable/predictable.Others are essentially random.Plenty of work for statisticians decidingwhich is which.


Spatial Patterns ... of Disease?

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C

Complete spatial randomnessLocations of gastroenteritis casesMystery spatial point pattern


Spatial Patterns ... of Disease?

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Gastroenteritis cases

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Complete spatial randomness

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Sea anemone locations

Complete spatial randomness (B)Locations of gastroenteritis cases (A)Mystery spatial point pattern (C)


Measuring Spatial Risk

To understand spatial patterns of disease we must adjust forunderlying population density.To this end, Bithell (1990) defined the relative risk function:

r(z) =f (z)g(z)

z ∈ R

where f is density of cases and g density of controls.Usual to work on log-scale:

ρ(z) = log{r(z)} = log{f (z)} − log{g(z)}

Bithell J.F. (1990). An application of density estimation to geographical epidemiology. Statistics in

Medicine 9:691–701.


Kernel Estimation

Straightforward approach to estimation of r is to replacingunknown densities by kernel estimates thereof:

r(z) =f (z)g(z)

z ∈ R

Kernel density estimate constructed from bivariate datax1, . . . ,xn:

f (z) = n−1n∑

i=1

Kh(z − x i)

I Kh(z) = h−2K (z/h), with kernel K being spherically symmetric;I h is the bandwidth, controlling degree of smoothing.


Constructing Kernel Density EstimatesAn example using the gastroenteritis data

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Smoothing of Kernel Density EstimatesSmall bandwidth (undersmoothing)








Smoothing of Kernel Density EstimatesLarge bandwidth (oversmoothing)








Example: Cancer of the Larynx in South Lancashire

Data are geographicalcoordinates for 58 casesof cancer of the larynx,and 978 controls (casesof lung cancer).Data collected between1974 and 1983 byChorley and SouthRibble Health Authorityin Lancashire, England.

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The Effect of Bandwidth SelectionFor the log-relative risk function

−15

−10

−5

0

h=1

−15

−10

−5

0

h=10


Research on Bandwidth Selection

Bandwidth selection is a challenging problem for both theoreticaland practical reasons.

I Typical inhomogeneity of populations.I Want stable estimates of risk even where data are sparse.I Asymptotic theory very delicate.

Recent progress on spatially adaptive smoothing regimens Davies& Hazelton (2010).

Davies, T.M. and Hazelton, M.L. (2010). Adaptive kernel estimation of spatial relative risk.

Statistics in Medicine 29, 2423–2437.


Estimation with Spatially Adaptive BandwidthsEstimation of case density

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Estimation with Spatially Adaptive BandwidthsEstimation of log-relative risk surface

−15

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−5

0


Boundary Bias

Disease data collectedover finite region.For points near edge,some of kernel weightspills over boundary andis lost.Has significant impact onkernel estimates.Theoretical analysis isintricate.

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Foundations of Asymptotic Analysis of Boundary Bias


Asymptotics with Adaptive Boundary KernelOr ... Why Postdocs are Great

bias(f (z))

h20

=[D10f (z)]2

4f (z)

[ b0

q

(a(20)

40 + 2a(11)31 + a(02)

22 + 7a(10)30 + 7a(01)

21 + 8a20

)+

b1

q

(a(20)

50 + 2a(11)41 + a(02)

32 + 9a(10)40 + 9a(01)

31 + 15a30

)+

b2

q

(a(20)

41 + 2a(12)32 + a(02)

23 + 9a(10)31 + 9a(01)

22 + 15a21

)]

+D10f (z)D01f (z)

4f (z)

[ b0

q

(2a(20)

31 + 4a(11)22 + 2a(02)

13 + 14a(10)21 + 14a(01)

12 + 16a11

)+

b1

q

(2a(20)

41 + 4a(11)22 + 2a(02)

23 + 18a(10)31 + 18a(01)

22 + 30a21

)+

b2

q

(2a(20)

32 + 4a(11)23 + 2a(02)

14 + 18a(10)22 + 18a(01)

13 + 30a12

)]

+[D01f (z)]2

4f (z)

[ b0

q

(a(20)

22 + 2a(11)13 + a(02)

04 + 7a(10)12 + 7a(01)

03 + 8a02

)+

b1

q

(a(20)

32 + 2a(11)23 + a(02)

14 + 9a(10)22 + 9a(01)

13 + 15a12

)+

b2

q

(a(20)

23 + 2a(11)14 + a(02)

05 + 9a(10)13 + 9a(01)

04 + 15a03

)]

+D20f (z)

4

[ b0

q

(2a(10)

30 + 2a(01)21 + 8a20

)+

b1

q

(2a(10)

40 + 2a(01)31 + 10a30

)+

b2

q

(2a(10)

31 + 2a(01)22 + 10a21

)]

+D11f (z)

4

[ b0

q

(4a(10)

21 + 4a(01)12 + 16a11

)+

b1

q

(4a(10)

31 + 4a(01)22 + 20a21)

)+

b2

q

(4a(10)

22 + 4a(01)13 + 20a12

)]

+D02f (z)

4

[ b0

q

(2a(10)

12 + 2a(01)03 + 8a02

)+

b1

q

(2a(10)

22 + 2a(01)13 + 10a12

)+

b2

q

(2a(10)

13 + 2a(01)04 + 10a03

)],


Computer Implementation

Important practical problems in spatial epidemiology can quicklylead to fascinating theoretical work.Need to make this stuff available to users.Released R package sparr. See Davies, Hazelton & Marshall(2011).

Davies, T.M., Hazelton, M.L. and Marshall, J.C. (2011). sparr: Analyzing spatial relative risk

using fixed and adaptive kernel density estimation in R. Journal of Statistical Software 39, 1–14.


Screenshot of R Package Sparr


Statistics and Bioinformatics Group Research

Methodological research with applications to infectious diseaseI Spatial statisticsI Markov chain Monte Carlo methodsI Simulation based inferenceI Smoothing methodsI Semiparametric modellingI Statistical software development

Variety of applications


Some Members of the Research Group

Postgraduate students

Sarojinie FernandoPhD (recently submitted)

Brigid Betz-StableinPhD (year 3)

Kate RichardsPhD (year 2)

Khair JonesPhD (year 1)

Lyndal HendenHonours

Staff with interest in infectious disease applications

Prof Martin Hazelton Dr Chris JewellA/Prof Geoff Jones Dr Jonathan Marshall












Improved Spatio-Temporal Risk Estimation

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Local linear versus density ratio estimation of relative risk.Application to myrtle wilt in Tasmanian myrtle beech (Nothofaguscunninghamii).












Modelling Progression of Eye Disease

−4

−3

−2

−1

0

(a) SPROG method

−4

−3

−2

−1

0

(b) PLR method

Using ideas from geographical disease mapping to examinespatial patterns of damage over retina.Interpretation of visual field data must account for physiology andoptics.












Modelling Foot and Mouth Disease in Vietnam

Data quality issues for FMD inVietnam.Appropriate level of dataaggregation for modelling?Developing methods foridentifying anomalies

I High risk provinces.I Under-reporting.












Bayesian Constrained Smoothing Problems

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1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3

0.0

0.2

0.4

0.6

0.8

1.0

Rainfall (m)

Toxo

plas

mos

is p

ositi

ve

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MonotonicUnconstrained

In some problems want flexible smooth fit under shape constraints.Application to toxoplasmosis data from 34 cities in El Salvador.












Methods of Simulation Based Inference

Modern complex stochasticmodels often have intractabledistribution theory.Standard methods of inferenceinfeasible.We are working on improvingsimulation based approaches(e.g. ABC).Application is estimation ofeffective population sizes in Baliusing coalescent model.


Statistical modelling and inference for spatio-temporal ...mhazelto/talks/IDReC-2012.pdf · Statistical modelling and inference for spatio-temporal disease processes Martin Hazelton1

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