In Example 8.6, assess the impact on the profile of neighbourhood psychosis prevalence rates per \(1000, R_{k}=1000 \theta_{k, n e i}\) (e.g. levels of skewness in such prevalence rates) and exceedance probabilities \(\operatorname{Pr}\left(R_{k}>7.75\right)\) (where 7.75 per 1000 is the regional prevalence rate), of using a standard Student \(t\) kernel with 5 degrees of freedom, namely

\[K(d)=\frac{\Gamma(3)}{\Gamma(2.5)(5 \pi)^{0.5}}\left[1+0.2 d^{2}\right]^{-3}\]

Data from Example 8.6

Health data (e.g. on prevalence patterns) may be collected by practitioner agencies (e.g. general practitioner or GP practices responsible for primary care) for the populations they are

responsible for. Unless catchment areas are defined, these populations may be spatially dispersed over several neighbourhoods. However, spatial contrasts between residential neighbourhoods in prevalence may be of interest in assessing health equity. In the UK, population prevalence totals for a range of chronic diseases are maintained by general practices for their populations under a scheme called the Quality and Outcomes Framework. Only total disease counts for GP practices are publicly released, with no additional disaggregation of prevalence by patient demographic category, and no totals provided for small areas. In England there are around 8200 general practices with populations averaging 6600, and around 32000 neighbourhoods with populations typically of 1500-2000 (census units known as Lower level super output areas, or LSOAs).

The analysis here concerns prevalence registers for psychotic illness (schizophrenia, bipolar disorder) for 190 GP practice populations in a sector of London, known as Outer North East London (ONEL), consisting of four London boroughs (Barking and Dagenham, Havering, Redbridge, Waltham Forest). These practice populations are generally dispersed over several neighbourhoods, with population centroids \(s_{i}=\left(s_{1 i}, s_{2 i}\right)\) based on weighted averages over the subpopulations in each LSOA. The same sector has 562 LSOA neighbourhoods, with population centroids \(s_{n e i, k}=\left(s_{n e i, 1 k}, s_{n e i, 2 k}\right)\). A discrete process convolution model is applied to counts \(y_{i}\) of prevalent psychosis, assumed Poisson with
\[y_{i} \sim \operatorname{Po}\left(P_{i} \theta_{i}\right)\]

for \(i=1, \ldots, 190\) GP practice populations of size \(P_{i}\). The aim is to interpolate prevalence counts \(y_{n e i, k}\) for the \(k=1, \ldots, 562\) residential neighbourhoods with populations \(P_{n e i, k}\).

The DPC model uses a grid of \(m=90\) points encompassing the outer NE London sector. The discrete grid effects \(\left\{x_{j}, j=1, \ldots, m\right\}\) are assigned an intrinsic CAR prior based on rook contiguity of the grid points. A baseline model for the mean prevalence in each GP population includes a spatial effect \(z\left(s_{i}\right)\) generated by the DPC prior, and an additional iid effect to represent overdispersion, and measurement variation (e.g. variable effectiveness in disease case finding between GP practices). The model for GP practice prevalence rates then has the form
\[\begin{aligned}
\log \left(\theta_{i}\right) & =\beta_{0}+z\left(s_{i}\right)+u_{i}, \\
z\left(s_{i}\right) & =\sum_{j=1}^{m} K\left(d_{i j}\right) x_{j}, \\
x_{1: n} & \sim \operatorname{ICAR}\left(1,1 / \tau_{x}\right), \\
u_{i} & \sim N\left(0,1 / \tau_{u}\right) .
\end{aligned}\]

The random effects have precisions \(\tau_{x}\) and \(\tau_{u}\) that are assigned gamma \(G a(1,0.001)\) priors. A normal kernel of known form, namely
\[K(d)=\frac{1}{2 \pi} \exp \left(-0.5 d^{2}\right)\]

is assumed. An alternative would be to take the \(x_{j}\) to have known variance and the kernel function to have an unknown scaling parameter. Interpolated prevalence totals for neighbourhoods are generated according to
\[\begin{aligned}
y_{n e i, k} & \sim \operatorname{Po}\left(P_{n e i, k} \theta_{n e i, k}\right) \\
\log \left(\theta_{n e i, k}\right) & =\beta_{0}+z\left(s_{n e i, k}\right)
\end{aligned}\]

\[z\left(s_{n e i, k}\right)=\sum_{j=1}^{m} K\left(d_{k j}\right) x_{j}\]

where \(d_{k j}\) are distances between LSOA population centroids \(s_{n e i, k}\) and the grid point locations \(w_{j}=\left(w_{1 j}, w_{2 j}\right)\).

A two chain run of 5000 iterations shows a posterior mean \(\sigma_{x}\) of 0.063 , with modelled GP practice psychosis prevalences rates per 1000 (i.e. \(1000 \times \theta_{i}\) ) having 2.5 th and 97.5 th percentiles of 4 and 13.6 per 1000 . Some variation between GP practices may reflect effectiveness in case finding or clinical variation, and the variation between interpolated neighbourhood rates (based on the smooth spatial signal underlying practice variations) is smaller, with \(95 \%\) of neighbourhoods between 5.1 and 8.9 per 1000. The average deviance and DIC are respectively 214 and 365, with the LPML at -764 .

A second model uses a socioeconomic deprivation score available both for GP practice populations and for neighbourhoods. Let \(D_{i}\) denote the GP practice score, and \(D_{n e i, k}\) the neighbourhood score. Then a second model has
\[\begin{aligned}
\log \left(\theta_{i}\right) & =\beta_{0}+\beta_{1} D_{i}+z\left(s_{i}\right)+u_{i} \\
\log \left(\theta_{n e i, k}\right) & =\beta_{0}+\beta_{1} D_{n e i, k}+z\left(s_{n e i, k}\right)
\end{aligned}\]

This model shows a significant deprivation effect on prevalence with \(95 \%\) interval for \(\beta_{1}\) of \((0.07,0.19)\). Variation in the discrete grid effects is reduced, with mean for \(\sigma_{x}\) now at 0.043 , but with a wider \(95 \%\) interval for neighbourhood rates, from 4.9 to 10.4 per 1000 . The average deviance and DIC are now respectively 210 and 355, with the LPML at -736 .