Maximum Likelihood Estimation of Regression Parameters With Spatially Dependent Discrete Data

Generalized estimating equations (GEEs) have been successfully used to estimate regression parameters from discrete longitudinal data. GEEs have been adapted for spatially correlated count data with less success. It is convenient to model correlated counts as lognormal-Poisson, where a latent lognormal random process carries all correlation. This model limits correlation and can lead to negative bias of standard errors. Moreover, correlation is not the best dependence measure for highly nonnormal data. This article proposes a model which yields maximum likelihood (ML) estimates of regression parameters when the response is discrete and spatially dependent. This model employs a spatial Gaussian copula, bringing the discrete distribution into the Gaussian geostatistical framework, where correlation completely describes dependence. The model yields a log-likelihood for regression parameters that can be maximized using established numerical methods. The proposed procedure is used to estimate the relationship between Japanese beetle grub counts and soil organic matter. These data exhibit residual correlation well above the lognormal-Poisson correlation limit, so that model is not appropriate. The data and MATLAB code are available online. Simulations demonstrate that negative bias in GEE standard errors leads to nominal 95% confidence coverage less than 62% for moderate or strong spatial dependence, whereas ML coverage remains above 82%.