7 Continuous Spatial Variation 7.4 Kriging 7.6 Continuous vs Discrete Spatial Variation Models?

7.5 Generalized linear geostatistical models

A different way to think about the geostatistical model is in its conditional form:

Y_{i}|S(x),\mu(x)\sim\mathrm{N}\{\mu(x_{i})+S(x_{i}),\tau^{2}\}

This led Diggle et al. (1998) to embed the model within the class of generalized linear mixed models (Breslow and Clayton, 1993b), to allow for example:

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Poisson log-linear geostatistical models

$\displaystyle S(x)$ $\displaystyle\sim$ $\displaystyle{\rm SGP}\{m,\sigma^{2},\rho(u)\},\qquad m=0$

$\displaystyle\alpha_{i}$ $\displaystyle=$ $\displaystyle\sum_{k=1}^{p}z_{k}(x_{i})\beta_{k}+S(x_{i})$

$\displaystyle Y_{i}|S(x_{i}),\beta_{1},\ldots,\beta_{p}$ $\displaystyle\sim$ $\displaystyle\text{Poisson}\{\exp(\alpha_{i})\}$
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Binary logistic-linear geostatistical models

$\displaystyle S(x)$ $\displaystyle\sim$ $\displaystyle{\rm SGP}\{m,\sigma^{2},\rho(u)\},\qquad m=0$

$\displaystyle\alpha_{i}$ $\displaystyle=$ $\displaystyle\sum_{k=1}^{p}z_{k}(x_{i})\beta_{k}+S(x_{i})$

$\displaystyle Y_{i}|S(x_{i}),\beta_{1},\ldots,\beta_{p}$ $\displaystyle\sim$ $\displaystyle\text{Bernoulli}\left(\frac{\exp(\alpha_{i})}{1+\exp(\alpha_{i})}\right)$

Example 7.7.

Childhood malaria in Gambia

A survey was conducted in village communities throughout Gambia (Diggle et al., 2002), see Figure 7.6.

Figure 7.6: Link, Caption: The Gambia is a long, narrow country on the west coast of Africa. This plot shows the border of the country and also the locations (as black hollow circles) of village communities in Gambia from which malaria samples, and other local information were collected.

For each child in the survey, the following information was recorded:

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$Y_{ij}=$ presence/absence of malarial parasites in blood-sample, for $j$ th child in $i$ th village
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age, sex, bednet use
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satellite-derived vegetation green-ness index

A logistic regression model, allowing for residual spatial variation and residual non-spatial variation between villages, is:

\mathrm{logit}\{\mathbb{P}(Y_{ij}=1)\}=\sum_{k=1}^{p}z_{ijk}\beta_{k}+U_{i}+S(% x_{i})

where $U_{i}\sim{\rm N}(0,\nu^{2})$ and $S(x)$ is a zero-mean stationary Gaussian process on $\mathbb{R}^{2}$ .

Results:

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inclusion of either $U_{i}$ or $S(x_{i})$ term materially affects inferences about $\beta_{k}$
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map of $\hat{U}_{i}$ against village locations suggests strong spatial structure
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map of $\hat{S}(x)$ shows spatial smoothing of unexplained spatial variation in risk

Figure 7.7: First Link, Second Link, Caption: Top: a map of the Gambia, green areas here have a higher density of vegetation, something that mosquitos require as a source of food. Bottom (pair of plots): Showing spatial predictions of the probability that a malaria blood sample will be positive; the country has been split in half in order to be able to zoom in more effectively. The more westerly area, of the country shown in the topmost of these two plots, is the coastal region of the Gambia, the large black area is the Atlantic Ocean. In these two plots, blue areas are areas of low risk and red areas are areas of high risk, with the intensity of the colour being in some sense proportional to the amount of risk.

Further extensions of spatial geostatistical models are often required for all but the simplest problems. The assumption of stationarity is often inappropriate, and may datasets suitable for geostatistical analysis also have a temporal dimension, for example in environmental and healthcare monitoring. Much recent research in the subject has focused on developing suitable methods to extend geostatistical methods into the spatio-temporal domain.

$\displaystyle S(x)$	$\displaystyle\sim$	$\displaystyle{\rm SGP}\{m,\sigma^{2},\rho(u)\},\qquad m=0$
$\displaystyle\alpha_{i}$	$\displaystyle=$	$\displaystyle\sum_{k=1}^{p}z_{k}(x_{i})\beta_{k}+S(x_{i})$
$\displaystyle Y_{i}\|S(x_{i}),\beta_{1},\ldots,\beta_{p}$	$\displaystyle\sim$	$\displaystyle\text{Poisson}\{\exp(\alpha_{i})\}$