5 Discrete Spatial Variation 5.1 Discrete spatial variation 5.3 Continuous vs discrete spatial variation models?

5.2 Including both spatial and non-spatial variation

When there is a ‘natural’ statistical model for the data, such as the Poisson model for count data or the binomial for binary data, it is often the case that:

•

after adjusting for all known covariates, the residual variation is bigger than can be explained by the assumed distributional model
•

but this extra-variation may or may not be spatially structured

A cautious strategy is to contemplate a model with both spatial and non-spatial sources of extra-variation.

5.2.1 Monte Carlo test for residual spatial variation

A simple test for the presence of residual spatial variation for area-level data is as follows. Given residuals $T_{i}$ :

1.

For each $i\neq j$ , calculate $T^{*}_{ij}=T_{i}-T_{j}$ and $u^{*}_{ij}=\|u_{i}-u_{j}\|$ , where $u_{i}$ is the centroid of region $i$ .
2.

Calculate the sample correlation $c^{*}=\textrm{Corr}(u^{*},T^{*})$ .
3.

Randomly reallocate the $T_{i}$ values to the sampled areas, and repeat steps 1 and 2 to find the sample correlation $c^{\dagger}$ .
4.

Carry out step 3 a total of $m-1$ times (a large number).
5.

Calculate a p-value for a test of residual spatial correlation as $(n^{*}+1)/m$ , where $n^{*}$ is the number of $c^{\dagger}$ values from the sample that are greater than or equal to $c^{*}$ .

Note that there are considerable drawbacks to this method, including the lack of power to detect residual spatial correlation in small samples and the inability to detect directional effects. However, a small p-value is usually a clear indication that there is residual spatial correlation present, which may necessitate further modelling.

Example 5.2.

Lip cancer in Scotland (continued)

Replace the previous model for risk by

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

where

•

$S_{i}$ again follow a discrete spatial variation model, and
•

$U_{i}$ are mutually independent ${\rm N}(0,\nu^{2})$ without any spatial structure.

The following table of results is adapted from Breslow and Clayton (1993a). Three different models are considered:

1.

Poisson regression (include neither $U_{i}$ nor $S_{i}$ terms)
2.

non-spatial extra-Poisson variation (include $U_{i}$ )
3.

spatial extra-Poisson variation (include $S_{i}$ )

Model	Estimates $\pm$ standard errors
	$\alpha$	$\beta\times 10$	$\sigma$	$\nu$
1	$-0.54\pm 0.07$	$0.74\pm 0.06$	—	—
2	$-0.44\pm 0.16$	$0.68\pm 0.14$	—	$0.60\pm 0.08$
3	$-0.18\pm 0.12$	$0.35\pm 0.12$	$0.73\pm 0.13$	—

1.

Poisson regression model seriously underestimates standard errors of $\alpha$ and $\beta$ ;
2.

non-spatial extra-Poisson variation model fixes up standard errors, point estimates only change slightly;
3.

spatial extra-Poisson variation model gives comparable standard errors to model 2 but materially changes point estimates.

To assess the goodness of fit of each model and compare between different models standard diagnostic tests for Poisson regression models can be used. See for example Section 4.3.