  Spatial generalized linear mixed models (SGLMMs) are popular for analyzing
non-Gaussian spatial data. These models assume a prescribed link function that
relates the underlying spatial field with the mean response. There are
circumstances, such as when the data contain outlying observations, where the
use of a prescribed link function can result in poor fit, which can be improved
by using a parametric link function. Some popular link functions, such as the
Box-Cox, are unsuitable because they are inconsistent with the Gaussian
assumption of the spatial field. We present sensible choices of parametric link
functions which possess desirable properties. It is important to estimate the
parameters of the link function, rather than assume a known value. To that end,
we present a generalized importance sampling (GIS) estimator based on multiple
Markov chains for empirical Bayes analysis of SGLMMs. The GIS estimator,
although more efficient than the simple importance sampling, can be highly
variable when used to estimate the parameters of certain link functions. Using
suitable reparameterizations of the Monte Carlo samples, we propose modified
GIS estimators that do not suffer from high variability. We use Laplace
approximation for choosing the multiple importance densities in the GIS
estimator. Finally, we develop a methodology for selecting models with
appropriate link function family, which extends to choosing a spatial
correlation function as well. We present an ensemble prediction of the mean
response by appropriately weighting the estimates from different models. The
proposed methodology is illustrated using simulated and real data examples.
