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Poisson cokriging as a Generalized Linear Mixed Model
It is often of interest to predict spatially correlated count outcomes/responses that follow a Poisson distribution. In a public health setting this would include disease incidence or mortality rates. One such disease that we would like to predict the incidence of is West Nile virus. West Nile virus is an arthropod-borne virus most commonly spread by infected mosquitoes, with most infections occurring from June to September. Count data for the number of West Nile virus cases in humans are available at the county level for each state in the U.S, from 2000-2014. Also collected are counts of infected birds as well as environmental predictors. ^ To predict a Poisson outcome variable in the presence of an auxiliary variable, Poisson cokriging as a Generalized Linear Mixed Model (GLMM) is proposed. This model has a bivariate structure with a Poisson outcome variable and an auxiliary variable. A covariance matrix similar to that used in cokriging is assumed. In a simulation study, prediction results from the Poisson cokriging model are compared to a GLMM model with a Poisson outcome treating the auxiliary variable as a fixed effect covariate. Various distributions for the auxiliary variable are also considered. Poisson cokriging was examined in the context of a real data example using the human West Nile virus data from Nebraska with sentinel bird and percent irrigated land as auxiliary variables. Leave one out cross-validation was used to assess prediction in Nebraska counties. ^ The simulation study showed that Poisson cokriging methodology can be applied successfully in practice and will allow for a variety of distributions for the auxiliary variable, including normal, gamma, and Poisson. The back-transformation method of prediction showed smaller average prediction errors compared to the direct method, but showed slightly larger root prediction mean square error and coverage. When comparing the Poisson cokriging to the Poisson fixed model, the results were similar between the two models. When predicting human West Nile virus incidence the Poisson cokriging model showed excellent results with small average errors and coverage closed to 95%. The Poisson cokriging model works well and is an excellent tool.^
Smith, Lynette M, "Poisson cokriging as a Generalized Linear Mixed Model" (2015). ETD collection for University of Nebraska - Lincoln. AAI3718106.