Two-scale spatial models for binary data

A spatial lattice model for binary data is constructed from two spatial scales linked through conditional probabilities. A coarse grid of lattice locations is specified, and all remaining locations (which we call the background) capture fine-scale spatial dependence. Binary data on the coarse grid are modelled with an autologistic distribution, conditional on the binary process on the background. The background behaviour is captured through a hidden Gaussian process after a logit transformation on its Bernoulli success probabilities. The likelihood is then the product of the (conditional) autologistic probability distribution and the hidden Gaussian-Bernoulli process. The parameters of the new model come from both spatial scales. A series of simulations illustrates the spatial-dependence properties of the model and likelihood-based methods are used to estimate its parameters. Presence-absence data of corn borers in the roots of corn plants are used to illustrate how the model is fitted.