Model-Based Geostatistics for Prevalence Mapping in Low-Resource Settings

In low-resource settings, prevalence mapping relies on empirical prevalence data from a finite, often spatially sparse, set of surveys of communities within the region of interest, possibly supplemented by remotely sensed images that can act as proxies for environmental risk factors. A standard geostatistical model for data of this kind is a generalized linear mixed model with binomial error distribution, logistic link and a combination of explanatory variables and a Gaussian spatial stochastic process in the linear predictor. In this paper, we first review statistical methods and software associated with this standard model, then consider several methodological extensions whose development has been motivated by the requirements of specific applications. These include: methods for combining randomised survey data with data from non-randomised, and therefore potentially biased, surveys; spatio-temporal extensions; spatially structured zero-inflation. Throughout, we illustrate the methods with disease mapping applications that have arisen through our involvement with a range of African public health programmes.Comment: Submitte

On The Inverse Geostatistical Problem of Inference on Missing Locations

The standard geostatistical problem is to predict the values of a spatially continuous phenomenon, $S(x)$ say, at locations $x$ using data $(y_i,x_i):i=1,..,n$ where $y_i$ is the realization at location $x_i$ of $S(x_i)$, or of a random variable $Y_i$ that is stochastically related to $S(x_i)$. In this paper we address the inverse problem of predicting the locations of observed measurements $y$. We discuss how knowledge of the sampling mechanism can and should inform a prior specification, $\pi(x)$ say, for the joint distribution of the measurement locations $X = \{x_i: i=1,...,n\}$, and propose an efficient Metropolis-Hastings algorithm for drawing samples from the resulting predictive distribution of the missing elements of $X$. An important feature in many applied settings is that this predictive distribution is multi-modal, which severely limits the usefulness of simple summary measures such as the mean or median. We present two simulated examples to demonstrate the importance of the specification for $\pi(x)$, and analyze rainfall data from Paran\'a State, Brazil to show how, under additional assumptions, an empirical of estimate of $\pi(x)$ can be used when no prior information on the sampling design is available.Comment: Under revie

Julian Ernst Besag, 26 March 1945 -- 6 August 2010, a biographical memoir

Julian Besag was an outstanding statistical scientist, distinguished for his pioneering work on the statistical theory and analysis of spatial processes, especially conditional lattice systems. His work has been seminal in statistical developments over the last several decades ranging from image analysis to Markov chain Monte Carlo methods. He clarified the role of auto-logistic and auto-normal models as instances of Markov random fields and paved the way for their use in diverse applications. Later work included investigations into the efficacy of nearest neighbour models to accommodate spatial dependence in the analysis of data from agricultural field trials, image restoration from noisy data, and texture generation using lattice models.Comment: 26 pages, 14 figures; minor revisions, omission of full bibliograph

INLA or MCMC? A Tutorial and Comparative Evaluation for Spatial Prediction in log-Gaussian Cox Processes

We investigate two options for performing Bayesian inference on spatial log-Gaussian Cox processes assuming a spatially continuous latent field: Markov chain Monte Carlo (MCMC) and the integrated nested Laplace approximation (INLA). We first describe the device of approximating a spatially continuous Gaussian field by a Gaussian Markov random field on a discrete lattice, and present a simulation study showing that, with careful choice of parameter values, small neighbourhood sizes can give excellent approximations. We then introduce the spatial log-Gaussian Cox process and describe MCMC and INLA methods for spatial prediction within this model class. We report the results of a simulation study in which we compare MALA and the technique of approximating the continuous latent field by a discrete one, followed by approximate Bayesian inference via INLA over a selection of 18 simulated scenarios. The results question the notion that the latter technique is both significantly faster and more robust than MCMC in this setting; 100,000 iterations of the MALA algorithm running in 20 minutes on a desktop PC delivered greater predictive accuracy than the default \verb=INLA= strategy, which ran in 4 minutes and gave comparative performance to the full Laplace approximation which ran in 39 minutes.Comment: This replaces the previous version of the report. The new version includes results from an additional simulation study, and corrects an error in the implementation of the INLA-based method

A Partial Likelihood for Spatio-temporal Point Processes

Spatio-temporal point process data arise in many fields of application. An intuitively natural way to specify a model for a spatio-temporal point process is through its conditional intensity at location x and time t, given the history of the process up to time t. Typically, this results in an analytically intractable likelihood. Likelihood-based inference therefore relies on Monte Carlo methods which are computationally intensive and require careful tuning to each application. We propose a partial likelihood alternative which is computationally straightforward and can be applied routinely. We apply the method to data from the 2001 foot-and-mouth epidemic in the UK, using a previously published model for the spatio-temporal spread of the disease

Bayesian Geostatistical Design

This paper describes the use of model-based geostatistics for choosing the optimal set of sampling locations, collectively called the design, for a geostatistical analysis. Two types of design situations are considered. These are retrospective design, which concerns the addition of sampling locations to, or deletion of locations from, an existing design, and prospective design, which consists of choosing optimal positions for a new set of sampling locations. We propose a Bayesian design criterion which focuses on the goal of efficient spatial prediction whilst allowing for the fact that model parameter values are unknown. The results show that in this situation a wide range of inter-point distances should be included in the design, and the widely used regular design is therefore not the optimal choice