Fig. 2: Global test of Allen’s rule on avian beak length across 9962 (99.7%) bird species.

The null allometric model (a) used to scale the absolute (log-transformed) beak length with log body size, the residuals from which were used as the relative beak length. Distribution of relative beak length across species geographic ranges (b). Model selection procedure for predicting log beak length (c), involving models with log body mass and either of six temperature measures within species geographic ranges included as fixed and interaction terms; AIC—Akaike Information Criterion, r²—coefficient of determination. An exemplar Allen’s model (d) showing increasing beak length with max temperature of all months, while controlling for body size as fixed term. An exemplar model with interaction of body size and max temperature of all months (e) illustrating how Allen’s rule operates across steeping quantiles of body size (left) and how allometry varies across quantiles of temperature (right). See Supplementary Fig. 2 for surrogate models based on the other temperature measures (evaluated in c). The p values refer to the significance of model’s fixed (d) or interaction terms (e) derived from two-tailed tests. The shaded area around the trend line is simple shading to facilitate reading. The results were obtained with phylogenetic linear regression by phylolm models on a single maximum clade credibility phylogenetic tree.