2.5 Comparison of factors influencing relative specialization
We analyzed the relative influence of the factors Sex, Month, Location,
Year, and Sample Size on \(\text{PS}_{i}\) using generalized linear
mixed models (GLMMs). We chose mixed models because they allowed us to
include Sample Size, Location, and Year as random variables. Restricted
maximum likelihood estimation was used because it considers the loss of
degrees of freedom when estimating fixed effects and thus offers a more
unbiased estimate than maximum likelihood methods (West, Welch, &
Galecki, 2015). Before modeling the data, we performed a logit
transformation (\(log(\frac{\text{PS}_{i}}{1-\text{PS}_{i}}\))) on the\(\text{PS}_{i}\) values to normalize them. This transformation was
necessary because \(\text{PS}_{i}\) is bounded by a theoretical minimum
and one(Bolnick et al., 2002). When numbers are bounded, the variance
distribution is shifted towards the mean (Sokal & Rohlf, 2012). A logit
transformation is an excellent choice for addressing this because it
extends the tails of the distribution more than other alternatives
(Warton & Hui, 2011).
All models were tested in the R 3.3.1 package lme4 (Bates, Mächler,
Bolker, & Walker, 2015). This package provides basic measurements of
goodness-of-fit including AIC and coefficients. The R 3.3.1 package
MuMIn was used to determine the r² values for mixed models. Subsequent
calculations of \(AIC\), and \(w_{i}\) (positive Akaike weights or
likelihood of being the best model (Anderson, 2008)) were completed
using Excel. \(AIC\) was calculated as the difference between two AIC
scores; \(w_{i}\) was calculated following Burnham and Anderson (2010).
To more clearly understand the relationship between sex ratio of the
population and\(\ \text{PS}_{i}\), sex ratios were produced for every
paired group (groups of males and females from the same location, month,
and year) by calculating the percent of scat identified as female. The
average \(\text{PS}_{i}\) for each paired group was then compared with
this female percentage using a Spearman’s rank correlation. Spearman’s
rank correlation was used to account for the heteroscedasticity of the
dataset, and was completed using R 3.3.1. Additionally, the average
proportion of female scat for each month and location are visualized in
the supplemental material (Figure S1).