Statistical Analysis of Spatio-temporal Point Process Data. Peter J Diggle

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1 Statistical Analysis of Spatio-temporal Point Process Data Peter J Diggle Department of Medicine, Lancaster University and Department of Biostatistics, Johns Hopkins University School of Public Health

2 Gastroenteric disease in Hampshire, UK

3 Gastroenteric disease in Hampshire, UK 3374 incident cases, 1 August 2000 to 26 August largely sporadic incidence pattern concentration in population centres occasional clusters of cases?

4 Questions establish normal spatio-temporal pattern of reported cases (NHS Direct) identify spatially and temporally localised anomalies in incidence pattern (real-time surveillance)

5 The 2001 UK FMD epidemic First confirmed case 20 February 2001 Approximately 140,000 at-risk farms in the UK (cattle and/or sheep) Outbreaks in 44 counties, epidemic particularly severe in Cumbria and Devon Last confirmed case 30 September 2001 Consequences included: more than 6 million animals slaughtered (4 million for disease control, 2 million for welfare reasons ) estimated direct cost 8 billion

6 y y March February x 30 April 31 May y y x x x y y June x x

7 Progress of the epidemic in Cumbria predominant pattern is of transmission between nearneighbouring farms but also some apparently spontaneous outbreaks qualitatively similar pattern in other English counties

8 Questions What factors affected the spread of the epidemic? How effective were control strategies in limiting the spread?

9 Analysis strategies for continuous-time processes 1. Empirical: log-gaussian Cox process models Poisson process with space-time intensity Λ(x, t) = exp{s(x, t)} 2. Mechanistic: work with conditional intensity function H t = λ(x, t H t ) = complete history (locations and times of events) conditional intensity (hazard) for new event at location x, time t, given history H t

10 Analysis strategies for continuous-time processes (1) log-gaussian Cox process model relatively tractable (eg closed-form expressions for second-moment structure) also able to generate a wide range of aggregated patterns scientifically natural if major determinant of pattern is environmental variation otherwise, often still a sensible empirical model

11 Model for gastroenteric disease data Notation λ 0 (x, t) = λ(x, t) = R(x, t) = normal intensity of incident cases actual intensity of incident cases spatio-temporal variation from normal pattern λ(x, t) = λ 0 (x, t)r(x, t) Scientific objective Use incident data up to time t to construct predictive distribution for current risk surface, R(x, t), hence identify anomalies, for further investigation.

12 Spatio-temporal model formulation λ(x, t) = λ 0 (x, t)r(x, t) λ 0 (x, t) = λ 0 (x)µ 0 (t) R(x, t) = exp{s(x, t)} S(x, t) = spatio-temporal Gaussian process: E[S(x, t)] = 0.5σ 2 Var{S(x, t)} = σ 2 Corr{S(x, t), S(x u, t v)} = ρ(u, v) conditional on R(x, t), incident cases form an inhomogeneous Poisson process with intensity λ(x, t)

13 Parameter estimation λ 0 (x) : locally adaptive kernel smootihng µ 0 (t) : Poisson log-linear regression σ 2, ρ(u, v) : matching empirical and theoretical second moments (but could also use Monte Carlo MLE)

14 Spatial prediction plug-in for estimated model parameters MCMC to generate samples from conditional distribution of S(x, t) given data up to time t choose critical threshold value c > 1 map empirical exceedance probabilities, p t (x) = P (exp{s(x, t)} > c data) web-reporting with daily updates Do we need to take account of parameter uncertainty?

15 Spatial prediction : results for 6 March 2003 c = 2

16 Analysis strategies for continuous-time processes (2) Analysis via conditional intensity function H t = λ(x, t H t ) = complete history (locations and times of events) conditional intensity (hazard) for new event at location x, time t, given history H t

17 Likelihood analysis Log-likelihood for data (x i, t i ) A [0, T ] : i = 1,..., n, with t 1 < t 2 <... < t n, is L(θ) = n log λ(x i, t i H ti ) i=1 T 0 A λ(x, t H t )dxdt Rarely tractable, but Monte Carlo methods are becoming available in special cases (eg log-gaussian Cox processes)

18 Partial likelihood analysis Data (x i, t i ) A [0, T ] : i = 1,..., n, with t 1 < t 2 <... < t n Condition on locations x i and times t i, derive log-likelihood for observed ordering 1, 2,..., n can allow for right-censored event-times if relevant R i = risk-set at time t i p i = λ(x i, t i H ti )/ j R i λ(x j, t i H ti ) (discrete R i ) p i = λ(x i, t i H ti )/ R i λ(x j, t i H ti )dx (continuous R i ) partial log-likelihood: L p (θ) = n i=1 log p i

19 A model for the FMD epidemic (after Keeling et al, 2001) Notation H t = history of process up to t λ(x, t H t ) = conditional intensity λ jk (t) = rate of transmission from farm j to farm k Farm-specific covariates for farm i n 1i = number of cows n 2i = number of sheep

20 Transmission kernel f(u) = exp{ (u/φ) 0.5 } + ρ At-risk indicator for transmission of infection I jk (t) = 1 if farm k not infected and not slaughtered by time t, and farm j infected and not slaughtered by time t Reporting delay Simplest assumption is that reporting date is infection date plus τ (latent period of disease plus reporting delay if any)

21 Resulting statistical model λ jk (t) = λ 0 (t)a j B k f( x j x k )I jk (t) λ 0 (t) = arbitrary A j = (αn 1j + n 2j ) B k = (βn 1k + n 2k )

22 Fitting the model rate of infection for farm k at time t is λ k (t) = j λ jk (t) partial likelihood contribution from ith case is p i = λ i (t i )/ k λ k (t i )

23 FMD results Common parameter values in Cumbria and Devon? Likelihood ratio test: χ 2 4 = 2.98 Parameter estimates (ˆα, ˆβ, ˆφ, ˆρ) = (4.92, 30.68, 0.39, ) But note that likelihood ratio test rejects ρ = 0.

24 Model extensions sub-linear dependence of infectivity/susceptibility on stock size A j = (αn γ 1j + nγ 2j ) B k = (βn γ 1k + nγ 2k ) Likelihood ratio test: χ 2 1 = other farm-specific covariates, eg z j = area of farm j and similarly for B k. A j = (αn γ 1j + nγ 2j ) exp(z j δ) Likelihood ratio test: χ 2 1 = 3.26

25 Baseline intensity: Nelson-Aalen estimator Write λ ij (t) as λ ij (t) = λ 0 (t)ρ ij (t) Nelson-Aalen estimator is ˆΛ 0 (t) = t 0 ˆρ(u) 1 dn(u) = i:t i t ˆρ(t i ) 1 where ˆρ(t) is plug-in from fitted model.

26 Nelson-Aalen estimates for Cumbria (solid line) and Devon (dotted line) cumulative hazard time (days since 1 Feb)

27 An ecological application data record locations x i and arrival times t i of nesting birds on several small off-shore islands birds known to prefer higher ground for nesting physical limit on distance between any two nests 25cm does spatio-temporal pattern of nesting sites show any evidence of spatial interaction beyond minimum separation distance?

28 Model for the pattern of nesting sites Interaction function h(u) = Conditional intensity is 0 : u δ 0 θ : δ < u δ 1 : u > δ λ(x, t H t ) = λ 0 (t) exp{z(x)β} g(x, t H t ) z(x) = elevation u (t) = min j:tj <t x x j g(x, t i H ti ) = h{u (t i )}

29 Final pattern on four islands Island 84 Island 74 y y x x Island 61 Island 56 y y x x

30 Confidence envelope for h(u) exp(theta) Distance (meter)

31 Conclusions spatio-temporal point process data-sets becoming widely available different problems require different modelling strategies temporal should often take precedence over spatial routine implementation is an important consideration