Dominance and plumage traits: meta-analysis and metaregression analysis
Procedures of Meta-analysis and Metaregression
All meta-analyses were conducted in the statistical software S-Plus (TIBCO; http://www.tibco.com/) and R (version 2.11.1; R Development Core Team 2010), using LMMs to perform a random-effect meta-analysis (Nakagawa et al. 2007). In all analyses, we accounted for the hierarchical structure in the data (e.g. with multiple effect sizes from the same population) by including study population and species as nested random effects (Nakagawa et al. 2007). This statistical procedure allowed us to use multiple effect sizes from a study or population in the same analysis without violating the assumption of independence (Nakagawa & Hauber 2011). Our meta-analytical LMM was calculated as an intercept-only model with the restricted maximum likelihood (REML) method (nlme package; Pinheiro & Bates 2000). Reported P values are for the main effects (intercepts) only.To determine whether a set of effect sizes was homogeneous, we calculated the residual heterogeneity QREML, as random-effects models were used (Nakagawa et al. 2007). When residual heterogeneity was significant, the variance among effect sizes was greater than expected from sampling error, suggesting the existence of important moderator variables. It is worth emphasizing that even though our metaregressions accounted for some moderator variables (Table 1), it is still possible that other (unaccounted) moderators could have introduced heterogeneity in the data. Furthermore, we decided not to include interaction terms among predictor variables in our metaregressions, as the models would be overparameterized (i.e. models would have too many parameters and not enough data points in each category to be robust; Ginzburg & Jensen 2004). We conducted contrast analyses (LMMs) for all metaregression models to check for the effect of different levels of an explanatory variable on the relationship between dominance and plumage. We show the results of contrast analyses only if the difference between levels of a variable (contrasts) was statistically significant (all other results are in Table A2 in Appendix 2).
We also conducted a randomization test to evaluate the importance of the variance components of our random factors (i.e. study and species). If a variance component was significantly different from zero, it indicated the existence of either study or species effects, the latter of which may, but does not necessarily, imply phylogenetic signal in the data (cf. Hadfield & Nakagawa 2010). We tested the null hypothesis that the variance component = 0, against the alternative hypothesis that the variance component > 0 (Nakagawa & Schielzeth 2010). We randomized the original effect size vector 100 000 times, each followed by fitting the meta-analytical LMM to estimate randomized random factor variance components. The P value was determined as the proportion of randomizations that yielded a variance component larger than or equal to the variance component of the original data. Furthermore, we conducted a phylogenetic meta-analysis described in Hadfield & Nakagawa (2010) to account for the lack of independence across species caused by their evolutionary relationships, using the MCMCglmm package in R (Hadfield 2010). As our conclusions did not change with the use of the phylogenetic meta-analysis, we only present results based on the general meta-analysis (the results of the phylogenetic meta-analysis can be found in Appendix 3).
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