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).