Efficient estimation and prediction for the Bayesian binary spatial model with flexible link functions

Spatial generalized linear mixed models (SGLMMs) are popular models for
spatial data with a non-Gaussian response. Binomial SGLMMs with logit or
probit link functions are often used to model spatially dependent
binomial random variables. It is known that for independent Binomial
data, the robit regression model provides a more robust (against extreme
observations) alternative to the more popular logistic and probit models.
In this article, we introduce a Bayesian spatial robit model for
spatially dependent binomial data. Since constructing a meaningful prior
on the link function parameter as well as the spatial correlation
parameters in SGLMMs is difficult, we propose an empirical Bayes (EB)
approach for the estimation of these parameters as well as for the
prediction of the random effects. The EB methodology is implemented by
efficient importance sampling methods based on Markov chain Monte Carlo
(MCMC) algorithms. Our simulation study shows that the robit model is
robust against model misspecification, and our EB method results in
estimates with less bias than full Bayesian (FB) analysis.
The methodology is applied to a Celastrus Orbiculatus data, and a
Rhizoctonia root data. For the former, the robit model is shown to do better
for predicting the spatial distribution of an invasive species which is
known to contain outlying observations. For the latter, our approach is
doing as well as the classical models for predicting the disease severity
for a root disease, as the probit link is shown to be appropriate.

Though this paper is written for Binomial SGLMMs for brevity, the EB
methodology is more general and can be applied to other types of SGLMMs.
In the accompanying R package geoBayes, implementations for other SGLMMs
such as Poisson and Gamma SGLMMs are provided.