Estimating northern fur seal pup production: A case study of batch mark abundance estimation

We describe a hierarchical N-mixture model for estimating northern fur seal pup production from batch mark-resight data. Our goal was to improve upon a traditional design-based estimation method used for over 50 years. To this end, we propose a hierarchical N-mixture model to account for differences in animal availability for resighting and observer detection probabilities. A Bayesian approach is used for inference with three separate methods proposed for necessary computations. First a straightforward posterior sample is drawn using MCMC. This was considered the gold standard for this analysis. However, we also consider an approximate model-based on Gaussian approximation of the Poisson and binomial distributions used in the exact hierarchical model. By using the Gaussian approximations, analytic integration can be used to marginalize over latent components. Inference can then be made by maximizing the posterior to find the mode. Following this we investigate both delta-method and parametric bootstrap approaches for calculating abundance and the associated standard errors with batch sampling data collected on northern fur seals in 2016. Each of the three newly proposed methods produced virtually identical estimates and standard errors. In addition, the parametric bootstrap approximation to a posterior sample was also virtually the same as the MCMC derived sample. There were a handful of small differences in the point estimates of abundance but these were not large enough to be especially notable. There were some noticeable differences in uncertainty estimates. The production estimate coefficients of variation ranged from 2.5-9.5% for the model-based estimates, but 0.0-12.5% for the design-based estimates. The design-based estimates of uncertainty were more volatile using the design-based method with some standard error estimates unreasonably small. Through a case study we have provided support for using Gaussian approximation in N-mixture type models when abundance is relatively large. These situations are typically challenging for fitting models using any sort of marginalization because there is no analytic solution for the exact models. However, when abundances are large using a Gaussian approximated model allows for analytic integration over latent abundance states which can greatly facilitate methodology such as maximum likelihood or posterior inference.