A Centered Bivariate Spatial Regression Model for Binary Data with an Application to Presettlement Vegetation Data in the Midwestern United States

Spatially structured discrete data arise in diverse areas of application, such as forestry, epidemiology, or soil sciences. Data from several binary variables are often collected at each location. Variation in distributional properties across the spatial domain is of interest. The specific application that motivates our work involves characterizing historical distributions of two species of Oak in the Driftless Area in the Midwestern United States. Scientists are interested in understanding the patterns of interaction between species, as well as their relationships to spatial covariates. Accounting for spatial dependence is not only of inherent interest but also reduces prediction mean squared error, and is necessary for obtaining appropriate measures of uncertainty (i.e., standard errors and confidence intervals). To address the needs of the application, we introduce a centered bivariate autologistic model, which accounts for the statistical dependence in two response variables simultaneously, for the association between them and for the effect of spatial covariates. The model proposed here offers a relatively stable large-scale model structure, with model parameters which can be interpreted in the usual sense across levels of dependence. Since the model allows for separate dependence parameters for each variable, it offers, in essence, the equivalent of a model with a non-separable covariance function. The flexible model framework permits straightforward generalizations to structures with more than two variables, a temporal component, or an irregular lattice domain. Supplementary materials accompanying this paper appear on-line.