Data analyses
Generalized Mixed-Effects Linear Models were chosen assuming binomial family and logit link function to investigate how the selection of a nest-site (1, presence of nest vs 0, absence of nest) was influenced by four of the variables measured. These were: water depth, vegetation height, offshore distance of the nest, and grazing intensity. The identity of the pair of nests was included as a random intercept in the models to control for potential non-independence of paired nests. Test for multi-collinearity (using the Variance Inflation Factor, VIF) and over-dispersion (computing the parameter Φ) were performed in all models, and predictor variables were log-transformed for normalization.
An information-theoretic approach was used to examine the importance of each predictor variable (Burnham & Anderson, 2002). The full model including the four variables (i.e., water depth, vegetation height, offshore distance of the nest and grazing intensity) was compared against simpler models with all possible combination of predictor variables (random factor and intercept always included) using the Akaike Information Criterion corrected for small sample sizes (AICc; Burnham & Anderson, 2002). Interaction and quadratic terms were not included in the analyses because no prior hypotheses justify including these terms, and the small sample size limits the capacity of inferring complex interactions between predictor variables. The significance values of the predictor variables in the best candidate model (i.e., the one with the lowest AICc) were then computed using Type III Wald chi-square test. Wald test helps to find out if explanatory variables in a model are significant (i.e. if they add something to the model) (Glen, 2020). Mixed-effect models were fitted using the R package blme (Chung et al., 2013) in R v3.5.3 (R Core Team, 2019).
Google Earth Pro was used to approximate distances between any two neighbouring territorial pairs within a group of nests in each of the three sections of the lake.