Optimal Interpolation and the Appropriateness of Cross-Validating Variogram in Spatial Generalized Linear Mixed Models

In this work, we consider some computational issues related to the minimum mean-squared error (MMSE) prediction of non-Gaussian variables under a spatial generalized linear mixed model (GLMM). This model has been used to model spatial non-Gaussian variables by Diggle et al. (1998) and Zhang (2002), under which MMSE prediction of non-Gaussian variables can be computed. Since the MMSE prediction is non-linear and cannot be computed in closed form, Markov chain Monte Carlo techniques are employed to approximate the predictor. We first establish some analytical results and show through three examples how these results can be utilized to make the MMSE predictions computationally more efficient. We then examine the appropriateness of cross-validating variogram in a spatial GLMM through a simulation study. Cross-validation is closely related to prediction since it uses partial data to predict the remaining. Our results show that cross-validating variogram fails to indicate the lack of fit even when the lack is apparent, and is not a useful diagnosing tool for a spatial GLMM despite its usefulness for other spatial models.