60 Let Ni,j be an observed number of nests with a standard deviation of Si,j. The distribution of Ni,j may be close to a Gaussian distribution when the number of nests and monitoring coverage are high, but it can also be a positive skew when the number of nests or monitoring coverage is low. For this reason, a gamma distribution was used to model the data; the gamma distribution is always positive and can show a positive skew when the standard deviation is high compared to the mean. The fit of the parameters was then completed using maximum likelihood with a gamma distribution. For the final estimate, the expected number of nests Ei,j was only used when no observation was available; in other situations, the number of nests fitted using the phenology model was preferred. The log likelihood of the observations is the sum of the log likelihood for each observation Ni,j within the gamma model. This model was implemented using the R package ‘Phenology’ (Girondot 2023b). 3.3.4 Strategy for parameter fitting The parameters used in the models to implement the nesting phenology and to estimate the annual number of nests for islets that were not monitored over a given season were fitted using the same statistical approach. First, the parameters are fitted using maximum likelihood, and then the models are selected using the Akaike Information Criterion (AIC) (Akaike 1974) and Akaike weight (Burnham and Anderson 1998). In short, AIC evaluates the quality of the fit that penalizes for overfitting too many parameters, while the Akaike weight gives the relative support of the different models, i.e. the probability for each model being the best one. Finally, the distribution of parameters was assessed using the Metropolis-Hastings algorithm, which is a Markov chain Monte Carlo method for obtaining a sequence of random samples from a probability distribution (Metropolis et al. 1953; Hastings 1970). The adaptive proposal distribution (Rosenthal 2011) as implemented in R package ‘HelpersMG’ (Girondot 2023a) ensures that the acceptance rate is close to 0.234, which is the optimal acceptance rate (Roberts and Rosenthal 2001). A total of 20,000 iterations was run. Priors were all uniform with a range of proposals large enough to ensure that it did not constrain the limits of the parameters. From the 20,000 sets of parameters, we calculated the posterior predictive mean and standard deviation of the statistics of interest. The adjustments were performed using the R package ‘Phenology’ (Girondot 2023b). Comparisons between the observed and modeled values were based on the adjusted coefficient of determination (Nagelkerke 1991).
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