The overall model used for this analysis is a binary logistic model at the individual level for each of the k outcomes. A Bayesian formulation is assumed and so all parameters in the model have prior distributions, as follows:

This can model spatial structure (see, e.g., Besag [

13]; Lawson [

10], chapter 5) and essentially assumes that neighboring areas are positively correlated. It is assumed that each outcome has a different variance of the spatial field and hence, the precisions have a

k subscript. Note that while this ICAR formulation is dependent on neighborhood definition, it is an adaptive prior specification as the variance depends on the number of neighbors of any given region. This allows for edge regions to have larger variance and smaller precision. To estimate the probability of the diseases for each province, a binomial model on aggregated province data was used as in Aregay, Lawson, Faes, Kirby, Carroll and Watjou [

8], Aregay, Lawson, Faes and Kirby [

9]. The binomial model assumed here is an approximation and we assume that the random effect structure employed makes allowance for this misspecification. The outcome was defined as cases of each disease per province out of the number of individuals sampled per province. The mean sampling weight per province (

$s{w}_{j}^{p}$), the mean age per province (

$Ag{e}_{j}^{p}$), the percentage of male individuals per province (

$Se{x}_{j}^{p}$) and their interaction were defined as adjusting variables. The same uncorrelated and correlated random spatial effects from the individual model were used in the aggregated model. This was done to use the spatial information gained from the individual analysis for the aggregated model. Hence, for the

k th outcome, with age x sex adjustment,

where

${n}_{j}^{p}$ is the sample size in the

${j}^{th}$ province and

${N}_{j}^{p}$ is the population of the province.

${p}_{jk}^{p}$ is the probability of the outcome and

${y}_{jk}^{p}={\displaystyle \sum _{i\in j}{y}_{ik}}$ where the sum is over all the individuals within the

j th province. A crude unadjusted estimate of prevalence could be computed as

${y}_{jk}^{p}/{n}_{j}^{p}$.