Mist nets and baited traps were used to capture the birds. With few exceptions (see below), each bird was measured for beak length (anterior edge of nostril to tip of upper mandible), beak depth (at the nares) and beak width (base of lower mandible). Most birds were then banded, which minimized remeasurement of the same individuals within years. After measurement, all birds were released at the site of capture. Within each time- and site-specific ‘sample’, G. fortis
could be reliably distinguished from related species based on morphological discontinuities and concordance with previous work (Ford et al. 1973
; Abbott et al. 1977
; Grant 1986
). At Academy Bay in 2004, for example, discriminant functions based on beak dimensions upheld our original species designations for 40 out of 41 G. fuliginosa
, 169 out of 173 G. fortis
and 11 out of 11 G. magnirostris
Data for Academy Bay were collected by David Snow (February 1963–May 1964, N=110; beak length measured differently), Hugh Ford and colleagues (August–September 1968, N=327; beak width not measured), Ian Abbott (February–March 1973, N=35; beak depth not measured), Peter Grant (November–December 1973, N=58), Peter Boag (January–February 1988, N=53) and Jeffrey Podos, Andrew Hendry and colleagues (January–March 1999–2002, N=119; February–March 2003, N=125; January–March 2004, N=173). Data for Borrero Bay were collected by Ian Abbott and Peter Grant (April–May 1973, N=122; culmen depth rather than beak depth), Peter Grant (December 1973, N=72; July 1975, N=84) and Andrew Hendry and colleagues (March 2004, N=128). Data for El Garrapatero were collected by Jeffrey Podos, Andrew Hendry and colleagues (February–March 2003, N=55; January–March 2004, N=114; January–April 2005, N=180).
Different investigators occasionally measured beak dimensions in different ways, as noted above. We therefore avoided direct beak size comparisons among samples collected by different investigators. Our main inferences were instead based on the degree of bimodality within each sample. For each beak size dimension in each sample, we first calculated coefficients of variation, which should be robust to differences in mean values. We then used principal components analysis to combine all beak dimensions into a single composite measure of ‘beak size’ (PC1) for each bird within a given sample. Because beak width was not measured in 1968, a critical sample for our inferences, we also recalculated PC1 for each bird after excluding beak width measurements from each sample.
We used three complementary methods to infer bimodality. First, we examined frequency histograms of PC1 for samples of more than 100 birds. Second, we plotted observed cumulative proportions for PC1 against cumulative proportions expected under normality. These plots take the form of a straight line for a single normal distribution but have a characteristically curved shape for a bimodal distribution (see §3
). Third, we tested statistically whether each sample was better represented by a single normal distribution or by a mixture of two normal distributions. To do this, we fitted a two-component normal mixture model, having data
are two normal density functions having means
and a common variance
, and p
represents the probability that observation
lies in component 1 (described by
). Mixtures were fitted to the data via discretization of parameters and an efficient summation method (Brewer 2003
). This approach enabled direct calculation of Bayesian estimates of the means, variance and proportions (p
) for each normal distribution, which amounts to a form of numerical integration of the posterior densities (Brewer 2003
For each sample, we next compared the fit of a single normal distribution to that of a mixture of two normal distributions based on Akaike's Information Criterion corrected for sample size (AICc
; Burnham & Anderson 2002
). For each sample, we calculated ΔAIC as AICc
for the single normal distribution minus AICc
for the fitted mixture of two normal distributions. Our interpretation of these ΔAIC values was similar to established guidelines (Burnham & Anderson 2002
). Specifically, we interpreted ΔAIC<−8 as strong support for a single normal distribution, −8≤ΔAIC<−5 as moderate support for a single normal distribution, −5≤ΔAIC≤5 as roughly equivalent support for a single normal distribution or a mixture of two normal distributions, 5<ΔAIC≤8 as moderate support for a mixture of two normal distributions, and ΔAIC>8 as strong support for a mixture of two normal distributions. We also calculated AICw
(Burnham & Anderson 2002
) for each sample, here representing the likelihood that a mixture of two normal distributions fits the data better than a single normal distribution.
Inferring bimodality in finite samples is a notoriously difficult statistical endeavour (Brewer 2003
). One problem is that overlap between the tails of two distributions can fill the gap between them, making the two modes difficult to distinguish. Another problem is that a distribution with fewer individuals can be obscured by the tail of a distribution with more individuals. Both of these properties characterized our finch data to the extent that frequency histograms were limited in their ability to discriminate between unimodal and bimodal distributions. Fortunately, the normal probability plots and Bayesian mixture models proved very effective at doing so (simulation results not shown). We therefore use frequency histograms as visual representations when sample sizes are large, but specifically base our primary inferences on the probability plots and mixture models.