The autologistic model with covariates for sample data and robust sampling designs using predicted probability of presence

Abstract

We extend the autologistic model for spatially correlated binary response lattice data by including covariates related to presence and using sample data from a subset of sites instead of complete data. We assume the binary response is presence or absence of a species. The model output is the predicted probability of presence for sites over the lattice. We present a Bayesian framework and develop a Gibbs sampling estimation procedure. We demonstrate using three examples that the autologistic model with covariates for sample data reproduces the truth more accurately than either the autologistic model, which uses only spatial relation information, or the logistic regression model, which uses only covariate information. We further extend our model to account for possibly imperfect detection in the sample observations. We assume there are covariates which are related to detectability of the species and modify the likelihood function to a logistic regression form. We expand the Bayesian set-up and Gibbs sampling estimation procedure to include the modified likelihood function. With two examples we demonstrate that also using sighting-related covariates achieves good reproduction of the truth. We investigate the use of the predicted probability in development of robust sampling designs whose goal is maximization of detection. The model output can be used to determine inclusion probabilities for unequal probability samples, define homogeneous partitions for stratified designs and be included in a ratio estimator of number of occupied sites. We present and compare four unequal probability sample design estimators, two strata construction methods, two within strata sampling methods, seven allocation strategies, and a fixed top stratum design. These comparisons are made with respect to the expected number of observed presence sites and the variance of the estimator of occupied sites. The fixed top stratum design, which samples all sites in the strata with the highest predicted probability, achieves the greatest detection and also attains a small variance for the estimator. This work serves to advance statistical methods used to map and detect rare species. These methods are applicable as well to mapping, or image analysis, and sample designs for lattice data.