5 Discrete Spatial Variation 5 Discrete Spatial Variation 5.2 Including both spatial and non-spatial variation

5.1 Discrete spatial variation

In models of discrete spatial variation, the geographical space under study is regarded as a fixed set of spatial sampling units, typically defined by a partitioning of a continuous region into politically defined sub-regions.

In the following example, the region is the north of England and the sub-regions are counties.

Figure 5.1: Link, Caption: Showing the county boundaries in the north of England.

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Models for discrete spatial variation are usually defined in terms of their so-called full conditional distributions, incorporating notions of “local” dependence between spatial units.
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More formally, if $Y_{i}:i=1,...,n$ denote a set of outcome variables associated with each of $n$ spatial units, the model is specified by the $n$ univariate distributions

$[Y_{i}|Y_{1},...,Y_{i-1},Y_{i+1},...,Y_{n}],$

where $(Y_{1},\ldots,Y_{n})\sim$ a multivariate distribution
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Note that a mutually consistent specification of the full conditionals involves non-obvious constraints on the allowable forms of distribution, which are set out in the celebrated Hammersley-Clifford Theorem (Besag (1974)).
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Typically, simplifying assumptions are made so that only a few of the $n-1$ terms in the conditioning set play any part. A neighbourhood structure needs to be defined. For example, we may assume that there is correlation between a county and its adjacent counties only, ignoring the rest.
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The following shows one example of how this might be done.

Figure 5.2: Link, Caption: Showing counties in the north of England and illustrating one potential definition of a neighbourhood for the county of Yorkshire (the largest pictured county above the centre of the plot): namely all counties that share a boundry with Yorkshire. The black dots are located at the centroid of each county (anticlockwise from the top surrounding Yorkshire are: Durham, Cumbria, Lancashire, West Yorkshire, East Riding of Yorkshire and Cleveland) the solid lines joining Yorkshire to these counties have been added to make clear which of the polygons shares a border. Note that in practice, shapefiles like this can be unreliable: it is not necessarily the case that boundaries exactly touch, though they may appear to do so by eye - it is only by zooming substantially that these more subtle features are revealed.

Example 5.1.

Lip cancer in Scotland; this example was originally analysed in Clayton and Kaldor (1987), with further comment and analysis in Clayton and Bernardinelli (1992) and in Breslow and Clayton (1993a).

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spatial units are the counties of Scotland;
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the response from each county $i$ is $Y_{i}$ , the total number of cases during the years 1975-1980 inclusive;
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let $R_{i}$ denote the risk for county $i$ , and $N_{i}$ the size of the population at risk
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a natural model to fit to the data is that

$Y_{i}|R_{i}\sim\text{Poisson}(N_{i}R_{i})$
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an available covariate is $x_{i}$ , the percentage of the population of county $i$ who are engaged in agriculture, fishing or forestry
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stronger predictors of lip cancer would be tobacco and alcohol consumption, but these are not available

Figure 5.3: Link, Caption: Showing the counties of Scotland - we will analyse the Scottish lip cancer data in the labs.

To model residual spatial variation in risk, after adjusting for the available covariate, we assume that

\log R_{i}=\alpha+\beta x_{i}+S_{i}

where the $S_{i}$ are spatially correlated random effects that follow a discrete spatial variation model in which:

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two counties are neighbours if they share a common boundary
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the full conditionals of county $i$ depend only on the neighbours of county $i$
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$S_{i}|$ neighbours $\sim{\rm N}(m_{i},v_{i})$ where

$m_{i}=$ mean of $S_{j}$ from counties $j$ which are neighbours of county $i$ ;

$v_{i}=\sigma^{2}/n_{i}$ , where $n_{i}=$ number of neighbours of county $i$

Note that this specification (an example of a conditional autoregressive (CAR) model) corresponds to an improper joint distribution for $(S_{1},...,S_{n})$ , with joint pdf

f(s_{1},...,s_{n})\propto\exp\left\{-\sum_{i\sim j}\frac{(s_{i}-s_{j})^{2}}{2% \sigma^{2}}\right\}

where $i\sim j$ indicates that counties $i$ and $j$ are neighbours.