02-Methods.Rmd

# Analysis Methods


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Long-term trends in migration counts were estimated independently for each species, site and season using a Bayesian hierarchical GAM framework with Integrated Nested Laplace Approximation (R-INLA, Rue et al. 2017) in R (version 3.1.3; R Core Team 2014). We estimated trends using log-linear regression, which included a smooth effect for year (i) and day of year (j) to model temporal patterns in population trajectories and the seasonal distributions of counts, respectively. We fit a cubic regression spline using the package mgcv (Wood 2011) with approximately 1 knot for every 4 years of data (following Smith and Edwards 2020), and 3 knots for days of data as a seasonal smooth for migration patterns. The number of knots controls for the upper limit on the complexity of the smoother. We also include a fixed effect for survey duration (hours) to control for differences in daily effort, a random annual fluctuations around the smoothed year effects (YE) to model population trajectories (similar to Smith and Edwards 2020), with an independent and identically distributed (IID) hierarchical year term (i.e., GAMYE).

Indices of Abundance: The population trajectory for each species, site and season were defined be the collection of estimated annual indices of abundance (ni). These indices are calculated as the exponential mean daily count from the posterior distribution of the GAMYE model. 

Trends: Multiple metrics can be used to estimate a 'trend' from the estimated population trajectories. We provide 'slope' trends calculated post-hoc from the population trajectories. These linear slope trends provide an estimate of the medium and longer-term patterns of population change, removing the effect of the random annual fluctuations.

Trend estimates and 95% credible intervals were back-transformed to annual rates of population change using 100\*exp(estimate)-1. Trend were calculated using the full dataset, as well as for the most recent 10-years, and three generations (Bird et al. 2020), the latter of which are informative to status assessments in Canada (e.g., see COSEWIC's Quantitative criteria and guideline for the status assessment of wildlife species). If the three-generation length was less than 10 years, the output reflects the 10-year trend.

Literature Cited

Bird, J. P., Martin, R., Akçakaya, H. R., Gilroy, J., Burfield, I. J., Garnett, S. T., Symes, A., Taylor, J., ekerciolu, Ç. H., & Butchart, S. H. M. (2020). Generation lengths of the world's birds and their implications for extinction risk. Conservation Biology, 34(5), 1252-1261. ttps://doi.org/10.1111/cobi.13486

Edwards, B., Smith, A. (2021). "bbsBayes: An R Package for Hierarchical Bayesian Analysis of North American Breeding Bird Survey Data." Journal of Open Research Software, 9. <doi:10.5334/jors.329>.

R Core Team. (2014). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. [online] URL: <http://www.r-project.org>.

Rue, H., S. Martino, F. Lindgren, D. Simpson, and A. Riebler. (2014). INLA: Functions which allow to perform full Bayesian analysis of latent Gaussian models using Integrated Nested Laplace Approximation. [online] URL: <http://www.r-inla.org>.

Smith, A., Edwards, B. (2020). North American Breeding Bird Survey status and trend estimates to inform a wide-range of conservation needs, using a flexible Bayesian hierarchical generalized additive model. bioRxiv. doi: <https://doi.org/10.1101/2020.03.26.010215>

Wood, S.N. (2011) Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. Vol. 73, Journal of the Royal Statistical Society (B). p. 3--36.