These spatial survival models can be implemented in a Bayesian framework. One way to do so is via MCMC methods, where the posterior of interest has the following form:
π(ψ|data)∝π(data|ψ)π(ψ).
We can use MCMC to draw samples from the target posterior and find the posterior expectation
Eπ(ψ|data)[g(ψ)]. For the purpose of this thesis, we will briefly introduce an advanced adaptive MCMC
method for spatial survival modelling which will be used later in this thesis (see Chapter 5). This method can be applied to spatial survival datasets and the computational cost is shown in Taylor (2015) to reduce from O(n3) toO(n) with the cost of increasing grid size beingO(mlogm).
Suppose that for a vector of parameters Φ = (ω, β, η), the hazard function follows
h(ti; Φ, Zi) = exp{Xiβ+Zi}h0(ti, ω),
whereZis some spatially continuous stationary latent Gaussian field andZiis the value at location of observationi,ω is a vector of parameters inh0, andη denotes parameters of the covariance function
CHAPTER 3. SPATIAL SURVIVAL ANALYSIS
ofZ. The Exponential model is a suitable proposal for Cov(Z); ie. σ2exp{−d/φ}withσ2 being the
marginal variance of fields andφthe ‘spatial decay’ parameter.
The MCMC inference of this particular type follows Metropolis-Hastings scheme with the following adjustments. As the parameters are mostly defined on positive numbers, we will be working with log-transformed versions. In the MCMC scheme, Z’s are not worked with directly, rather with a vector of transformed variables,γ= (γ1, . . . , γm), such thatZ=−σ2/2 + Σ
(1/2)
σ,φ γ. Here, Σ
(1/2)
σ,φ is the
Cholesky decomposition of the covariance matrix. Apriori,γ0(s)∼ N(0,1) andγ’s can be generated by the simulation of multivariate Gaussian variables with mean−σ2/2 and variance Σ
η. Appropriate Z’s are then constructed via simulatedγ’s.
Samples of MCMC are drawn from the posterior π(Φ, γ|data)∝π(data|Φ, γ)π(Φ, γ), using MCMC Gamerman and Lopes (2006); Gilks et al. (1995) where the parameters are transformed; eg. ˜ω= logω.
π(data|Φ, γ) =π(data|β,ω, γ˜ ) due to the conditional independence. The MCMC scheme has Langevin kernels forβ, ˜ωf,γ and a random walk kernel for ˜η. The algorithm follows:
• Initialise the chain at{β(0),ω˜(0),η˜(0), γ(0)};
• Proposal density forζ= (β,ω,˜ η, γ˜ ) isq(ζ(i∗)|ζ(i−1)) =N(ζ(i∗);µ
ζ(i−1), h2Σ),where µζ(i−1)= (β,ω˜)(i−1)+h2h2β,ω˜ 2 Σβ,ω˜ ∂log{π(ζ(i−1)|Y)} ∂(β,ω˜) ˜ η(i−1) γ(i−1)+ h2h2 γ 2 Σγ ∂log{π(ζ(i−1)|Y)} ∂γ and Σ = h2 β,ω˜Σβ,ω˜ 0 0 0 ch2 ˜ ηΣη˜ 0 0 0 h2 γΣγ .
• Here the constantsh2−are optimal scalings in MALA proposals (Roberts and Rosenthal, 2001); More details can be seen in Taylor and Rowlingson (2014) and Taylor (2015) . The optimal value ofhshould give asymptotic acceptance rate 0.574.
Other inferential methods for spatial survival data can be seen in theRpackagespBayesSurv where spatial copula linear dependent Dirichlet process mixture models, anova Dirichlet process mixtures and marginal proportional hazard models can be implemented (Zhou et al., 2018). PackageBayesX
(Umlauf et al., 2015; Belitz et al., 2015) fits different types of spatial survival models based on MCMC simulation techniques. Examples of model classes supported byBayesX include generalised additive mixed models, dynamic models, geoadditive models and models for space-time regression.
CHAPTER 3. SPATIAL SURVIVAL ANALYSIS
The package provides different smoothness priors including Markov random field priors for spatial effects and allows flexible parametric baseline hazards using penalised splines. Also,BayesX allows handling of time-varying coefficients. INLApackage also allows the fit of survival models but instead of using MCMC techniques, a Laplace approximation is used. Although this is computationally faster, it does not allow one to make exact inference over parameters and problems occur when models are of higher hierarchies.
This chapter aims to provide essential background knowledge of (spatial) survival analyses and their relevant inferential methods.The information contained in this Chapter is relevant to Chapter 5. The survival data considered in Chapter 5 are censored data. Non-parametric approach (Kaplan- meier curve) is used to provide exploratory analysis. Semi-parametric approach (Cox PH model) is the root model which the hazard model introduced in Chapter 5. Frailty models provide the basics of how spatial random effects can be involved using frailty terms and spatial survival model is what the extension in Chapter 5 is based on. The main inference method for this thesis is via Bayesian framework. More particularly, based on MH scheme, an advanced adaptive MCMC method is employed to proceed inference for spatio-temporal survival model inR. More details can be found in Chapter 5 orspatsurv package vignette.
Chapter 4
Spatiotemporal Modelling of
Tuberculosis Incidence in Urban
Portugal from 2000–2013
This Chapter delivers an analysis of space-time tuberculosis data in Urban Portugal area between 2000 and 2013 at a lower administrative level than other studies. This Chapter delivers an analysis of space-time tuberculosis data in Urban Portugal area between 2000 and 2013 at a lower administrative level than other studies. The novelty shows in application of INLA approximation over space-time tuberculosis data at a finer geographical scale. This is a submitted paper to the International Jour- nal of Tuberculosis and Lung Disease. Background and previously done research on similar area are shown in Section 4.2. Section 4.1 presents the analysis using INLA method on urban tuberculosis incidence rates discovering spatial clusters, identifying high risk areas and understanding the rela- tionship between risk factors. Section 8.1 shows a few other approaches pursued on the same data as exploratory analyses.
CHAPTER 4. TB IN PORTUAL
4.1
Spatiotemporal Modelling of Tuberculosis Incidence in
Urban Portugal from 2000–2013
4.1.1
Abstract
Despite being one of the medium-to-low endemic countries, Portugal still shows one of the highest tu- berculosis incidences in the European Union. Although it has seen progressively decreasing incidences, the regional differences suggest that better understanding at sub-regional epidemiology would assist control over tuberculosis. In this chapter, the analysis looked at tuberculosis incidences at freguesia levels (lower administrative levels than municipalities in Portugal) in Lisbon and Oporto from 2000- 2013. It used Poisson mixed effect models with both spatial and temporal correlated random effects, interaction terms and unstructured random effects. Models are adjusted for socioeconomic covariate effects with inferential method based on INLA. Both regions have identified areas of higher incidence; freguesias near the central Lisbon city area and a few areas in the north. Clear spatial clusters are detected in both Lisbon and Oporto Metropolitan areas. Areas showing high relative risks are not necessarily associated with high population densities but are surrounding areas of poverty zones.