To run the code, start R and source('analyze.R') in this directory. The code depends on these packages available from CRAN:

• parallel
• rstan
• maps
• viridis

To account for the variable sampling between counties and potential similarities between neighboring counties, we estimated the underlying proportion of deer testing positive within each county of the $m$ counties using a Bayesian conditional autoregressive model. The number of positive tests, $y_{i}$, out of $n_{i}$ total tests within each county $i$ was modeled as:

$y_{i}\sim \text{Binomial}(p_{i},n_{i})$

where $p_{i}$ is the proportion of deer expected to test positive in that county and:

$p_{i} = \text{logit}^{- 1}( \alpha + \beta_{i} )$

Here, $\alpha$ represents the average proportion positive for a county and the vector of differences from this average for each county, $\beta$, is distributed as a multivariate normal:

$\beta\sim \text{Normal}_{\text{prec}}(0,\frac{1}{\sigma}(D-\theta A))$

where $D$ is a $m \times m$ matrix with 0s on the off diagonal and the diagonal element on each row $i$, $D_{i,i}$, equal to the number of counties that are adjacent to county $i$, $A$ is a $m \times m$ adjacency matrix with element $A_{i,j}$ is 1 if county $i$ neighbors county $j$ and 0 otherwise and the diagonal set to 0 and $\text{Normal}_{\text{prec}}(x,y)$ is a multivariate normal distribution with means $x$ and precision matrix $y$. For priors, $\theta$ was given a uniform prior between 0 and 1, $\sigma\sim \text{Gamma}(1,1)$ and $\alpha\sim \text{Normal}(-2,10)$.

Posterior probability distributions were estimated using Markov chain Monte Carlo sampling using Stan v2.21.0.

County adjacency data was obtained from the US Census Bureau.