9.13 Further considerations

  1. Prior Distributions - Prior distributions (e.g., non-informative N (0, 100,000)) for each of the parameters, and over the entire real line for μ (e.g., an improper (flat) prior).  Prior distributions for the random effect describing region-wide heterogeneity and to describe the spatial structure (e.g., intrinsic Gaussian conditional autoregressive (ICAR) prior with a sum to zero constraint) should also be determined. Because of the marginal specification for region-wide heterogeniety and spatial structure, prior distributions for the precisions using simulations in WinBUGS should be determined using the psi metric (Eberly and Carlin 2000).

  2. Model Selection - candidate models can consist of different structures with strictly additive effects, environmental predictors can be grouped, such that they can all be entered or removed from the models together. Also, to account for the spatial structure of the data, random effects parameters can be included in some models to represent region-wide heterogeneity and local clustering. Therefore, all models can consist of all possible combinations of the grouped variables, other covariates, and random effects. Deviance information criterion (DIC) can then be used to evaluate this candidate set of models with DIC weights allowing for an intuitive comparison of the evidence in the data for each candidate model. The weights are considered a measure of the strength of evidence in the data for ith model being the "best" model of those within the candidate set, and therefore provide a measure of model selection uncertainty (Burnham and Anderson 2002, Spiegelhalter et al. 2002).

  3. Goodness-of-Fit - to examine the goodness-of-fit of the top model from candidate sets, a numerical posterior predictive check can be conducted (Gelman et al. 2004). We can use the total number of positive subjects (farm’s in our case) conditioned on the observed covariate values in our sample as our test statistic. Generating the posterior distribution of this statistic using parameter estimates from the marginal posterior distribution contained in MCMC chains. Thus given each farm’s covariate values, we generated estimates of individual infection probabilities for every location for each of the 250,000 iterations of our MCMC chains, and created a Bernoulli random variable using this probability of M. bovis infection. We then summed these random variables across all farms to create our test statistic. The posterior distribution of the test statistic was created from the values of these test statistics across all iterations of our MCMC chains.  Finally, we can calculate the posterior predictive P-value as the probability of having fewer M. bovis-positive farms then the total number of infected farms observed in the sample based on this posterior distribution of the test statistic.

  4. Convergence and prior sensitivity - examination of correlation and trace plots, as well as estimates of the corrected scale reduction factor for each parameter and multivariate potential scale reduction factors can provide evidence that chains for each model had converged. Additionally, the posterior distributions can be assessed to determine if they are overly sensitive to prior specification.