Methods

To conduct these analyses, we initially constructed a phylogeny of all extant hummingbird species in R using the phylogenetic data provided by BirdTree.org \cite{Jetz_2012,Jetz_2014,team2013r}. We created a species, and genus level tree to use for both rule analyses. The first rule we tested was Bergmann's rule, first described in 1847, it posits that species that live at higher absolute latitudes are larger than species that live closer to the equator.  To test Bergmann's hypothesis, we created a simple General Linear Model (GLM) testing the relationship between the overall body mass of the birds, with the mean absolute latitude of their ecographic range. Body mass data was extracted from \cite{Wilman_2014}, and the mean latitude of each species was extracted from Bird Life \cite{world2017} Two of these models were tested, one that examined Bergmann's rule over the entirety of the American continents, and one which only tested 10° around the equator, 5° in each direction. This second analysis was chosen because if the Latitudinal Diversity Gradient (LDG) holds to be true, we expected there to be a much larger concentration of hummingbirds within this range than outside of it, and this second analysis will allow the opportunity to exclude potential outliers from the far reaches of the continents. Many previous studies have shown that such an effect can be examined over as small as a 5° latitudinal difference in each orientation, so this analysis should still provide enough power to show an effect\cite{Salewski_2016}. These two GLMs were modelled using  R, and simple plots were created showing the relationship between body mass and latitude.  Finally for Bergmann's hypothesis, we tested the relationship between body size and latitude within the largest hummingbird genus, Phaethornis, which comprises of 25 species, by creating a subset of the data within R and creating a simple GLM to test the relationship.
The test whether a Trochilidae diversity gradient exists over the equator, we created a number of generalized linear models. Firstly, we created a simple GLM that examined the relationship between the mean absolute latitude of the genus, using our previously constructed genus level phylogeny, and examining whether there was a relationship between the number of species within a genus, and the average absolute latitude of that genus. We also created a second model, sorted the species into 'bins' between each 5°  latitude gradient, where mirror latitudes are sorted into the same bin. From this we created a plot to show how species richness varies as you move further away from the equator, we a gradient, where mirror latitudes are sorted into the same bin. From this we created a plot and GLM to show how species richness varies as you move further from the equator.  Finally,  using separate data  acquired from the Bird Life conservation project, we plotted the density of hummingbird species across the Americas, using the raster package in R, to better represent the diversity gradients across the Americas in a more visually pleasing way.

Results

Bergmann's Rule

The results of the Bergmann's rule analysis shows a complete lack of correlation between the midpoint latitude of the species and overall body mass. The GLM showed a negligibly small negative correlation between the two over both continents (R = -0.00538, intercept = 1.698).