Abundance estimation with sightability data: A Bayesian data augmentation approach

Steinhorst & Samuel (Biometrics 1989; 45, 415-425) showed how logistic regression models, fit to detection data collected from radiocollared animals, can be used to estimate and adjust for visibility bias in wildlife population surveys. Population abundance is estimated using a modified Horvitz-Thompson (mHT) estimator in which counts of observed animal groups are divided by their estimated inclusion probabilities (determined by plot-level sampling probabilities and detection probabilities estimated from radiocollared individuals). The sampling distribution of the mHT estimator is typically right-skewed, and statistical inference relies on asymptotic theory that may not be appropriate with small samples. We develop an alternative, Bayesian model-based approach which we apply to data collected from moose (Alces alces) in Minnesota. We model detection probabilities as a function of visual obstruction, informed by data from 124 sightability trials involving radiocollared moose. These sightability data, along with counts of moose from a stratified random sample of aerial plots, are used to estimate moose abundance in 2006 and 2007 and the log rate of change between the 2 years. Unlike traditional design-based estimators, model-based estimators require assumptions regarding stratum-specific distributions of the detection covariates, the number of animal groups per plot and the number of animals per animal group. We demonstrate numerical and graphical methods for assessing the validity of these assumptions and compare two different models for the distribution of the number of animal groups per plot, a beta-binomial model and a logistic-t model. Estimates of the log rate of change (95% CI) between 2006 and 2007 were -0·21 (-0·53, 0·12), -0·24 (-0·64, 0·16), and -0·25 (-0·64, 0·15) for the beta-binomial model, logistic-t model and mHT estimator, respectively. Plots of posterior-predictive distributions and goodness-of-fit measures both suggest the beta-binomial model provides a better fit to the data. The Bayesian framework offers many inferential advantages, including the ability to incorporate prior information and perform exact inference with small samples. More importantly, the model-based approach provides additional flexibility when designing and analysing multi-year surveys (e.g. rotational sampling designs could be used to focus sampling effort in important areas, and random effects could be used to share information across years).