Supplemental Appendix Meta-Analysis Model Form

Supplemental Appendix – Meta-analysis model form

General form of the meta-analytical models (MCMCglmm package using R syntax; Hadfield 2010; R Development Core Team 2011; R version 2.13.0):

Intercept-only model form considering the magnitude and direction of the selection coefficient. prior = list(R = list(V = 1, n = 0.002), G = list(G1 = list(V = 1, n = 0.002), G2 = list(V = 1, n = 0.002))) measurement_error_variance = selection_coefficient_standard_error^2 model = MCMCglmm(variance_standardized_selection_coefficient ~ 1, mev = measurement_error_variance, data = data, prior = prior, random = ~ study + species)

The magnitude (absolute value) of selection coefficients follows a folded normal distribution (see Hereford et al. 2004; Morrissey and Hadfield 2012). In cases where we were interested in the magnitude of selection, we extracted the posterior distribution of solutions, and the posterior distribution of variance matrices; we took the absolute values of the solutions, computed standard deviations from the variance matrices, and applied these to the folded normal distribution. From this posterior distribution, we obtained the posterior mode and 95% credible interval.

Moderator model form considering the magnitude and direction of the selection coefficient. Here, we model heterogeneous variances, allowing study, species and residual variances to differ among levels of the moderator.

For example, examining the influence of the moderator “trait class”: prior = list(R = list(V = diag(4), n = 0.002), G = list(G1 = list(V = diag(4), n = 0.002), G2 = list(V = diag(4), n = 0.002))) measurement_error_variance = selection_coefficient_standard_error^2 model = MCMCglmm(variance_standardized_selection_coefficient ~ trait_class, mev = measurement_error_variance, data = data, prior = prior, random=~idh(trait_class):study + idh(trait_class):species, rcov=~idh(trait_class):units)

For the moderators, fitness component and taxonomic group, where we were interested in the magnitude of linear selection, we used the methods described above involving the folded normal distribution.

In each model form, “variance_standardized_selection_coefficient” refers to variance standardized linear and quadratic selection gradients and differentials (β, γ, g, s) which serve as our metric of effect size in the meta-analysis.

References

Hadfield J (2010) MCMC methods for multi-response generalized linear mixed models: The MCMCglmm R package. J Stat Softw 33:1-22 Hereford J, Hansen TF, Houle D (2004) Comparing strengths of directional selection: How strong is strong? Evolution 58:2133-2143 Morrissey MB, Hadfield JD (2012) Directional selection in temporally replicated studies is remarkably consistent. Evolution 66:435–442 R Development Core Team (2011) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria ISBN 3-900051-07-0:URL http://www.R- project.org/ Supplemental material

Table S1. Summary of results from models of linear selection gradients: overall (intercept-only) and as functions of moderator variables, including trait class, fitness component and taxonomic group. All models account for sampling error and study- and species-level autocorrelation. Note that estimates of the magnitude of selection are applied to the folded normal distribution.

Model form Moderator level Posterior mode 95% Credible interval β ~ 1 0.058 0.018 0.089 ~ trait class Size 0.118 0.051 0.170 Other MO 0.072 0.037 0.128 Phenology -0.064 -0.115 0.011 Other LH 0.032 -0.053 0.125 |β| ~ 1 0.139 0.127 0.162 ~ taxon Invertebrates 0.117 0.083 0.192 Plants 0.186 0.156 0.235 Vertebrates 0.133 0.112 0.158 ~ fitness component Survival 0.090 0.076 0.116 Mating success 0.198 0.144 0.287 Fecundity 0.188 0.166 0.218 Total 0.229 0.143 0.611

Table S2. Posterior modes with 95% credible intervals for “corrected” estimates (linear and quadratic selection gradients and differentials) from models which consider the effects of sampling error and study- and species-level autocorrelation (note that estimates of the magnitude of selection are applied to the folded normal distribution), and modes of the kernel density with bootstrapped (n=10,000) 95% confidence intervals for “uncorrected” estimates which do not consider these effects.

β (95% CI) s γ g Corrected 0.05 0.018 0.08 0.11 0.011 0.24 0.002 -0.039 0.04 -0.013 -0.084 0.059 0 8 9 6 1 Uncorrected 0.00 -0.004 0.02 0.03 0.016 0.05 0.008 -0.001 0.01 -0.025 -0.044 0.021 9 1 1 0 8 |β| (95% CI) |s| |γ| |g| Corrected 0.13 0.127 0.16 0.22 0.166 0.31 0.192 0.175 0.20 0.111 0.085 0.194 9 9 2 3 0 Uncorrected 0.04 0.041 0.05 0.05 0.044 0.06 0.029 0.019 0.03 0.057 0.044 0.065 6 8 4 2 4