Spatial Simulation and Estimation of Generalized Linear Mixed Models with Non-Normal Data

It is often of interest to predict spatially correlated discrete data, such as counts arising from disease incidence or mortality rates, or indicator variables arising from population thresholds or measuring presence or absence of a given phenomenon. A generalized linear mixed models (GLMM) approach to prediction using Poisson and Bernoulli response variables conditional on the spatial lo- cation is simulated using G-side models. We simulated data from a Poisson and Bernoulli distribution with spherical correlation structure, and separately simulated covariates correlated with the origi- nal variable from Gaussian, Binomial, and Beta distributions. This was accomplished using NORTA (Normal to Anything) after simulating a spatial Gaussian structure. We then compared prediction of unobserved spatial locations under various conditions: with the entire response variable (Poisson or Bernoulli) available or various fractions of it missing, and with the entire covariate variable (Gaussian, Binomial, Beta) or some of it missing. We also fit a multivariate GLMM with both the response variable and the covariate as outcome variables to compare its prediction with the other scenarios as described. We found, as expected, the addition of a covariate improved prediction in the GLMM models. However, the comparison of interest is looking at the effect of the various covariate distributions. Moreover, spatial modeling of non-normal data with GLMM presents some unique challenges, and should not be pursued without prior understanding.