Dependence Between Effort Offsets and Process Intensity in Generalized Models for Count Data: Spatial Models of Animal Abundance

Researchers often use offsets to control for variation in exposure or effort in GLMM‐style models for count data. In this paper, we study the case where effort is dependent on the underlying process of interest (e.g., Poisson intensity) and where inference focuses on prediction. We first examine a simple Poisson GLM, where we use simulation to show that dependence can result in biased GLM predictions, presumably because observations with greater offsets receive more weight within the fitting process. Possible solutions in this case include (i) including inverse offset values as weights within the fitting process, or (ii) dividing counts by the observed level of effort prior to analysis (we term these “Horvitz‐Thompson‐like responses”). We then consider an ecological application involving the estimation of animal abundance using transect sampling. In this case, a GLMM fitted to animal counts is used to predict abundance over a gridded study area, where environmental covariates account for variation in animal density, and the offset is a product of area surveyed and detection probability. We used simulation to assess the performance of several approaches for estimating abundance and variance when there is possible dependence between detection probability and abundance, showing that models with Horvitz‐Thompson–like responses can potentially outperform other alternatives when such dependence exists. However, when applied to a beluga whale data set where turbidity was related to both detection and abundance, there seemed to be little evidence of bias. Nevertheless, we suggest that analysts first test for dependence between offsets and the underlying count process, and consider remedies such as Horvitz–Thompson responses if such dependence exists. We note connections between our research and the topic of preferential sampling in spatial statistics literature.