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.Rproject.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
β
Moderator level
~ 1
~ trait class
0.018 0.089
Size
0.118
0.051 0.170
Other MO
0.072
0.037 0.128
Phenology
-0.064
-0.115 0.011
0.032
-0.053 0.125
0.139
0.127 0.162
Invertebrates
0.117
0.083 0.192
Plants
0.186
0.156 0.235
Vertebrates
0.133
0.112 0.158
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
~ 1
~ taxon
~ fitness component
95% Credible interval
0.058
Other LH
|β|
Posterior mode
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
Corrected
(95% CI)
0.058 0.018
Uncorrected 0.009 -0.004
|β|
Corrected
(95% CI)
s
γ
g
0.089 0.116 0.011 0.241 0.002
-0.039
0.040 -0.013
-0.084
0.059
0.021 0.031 0.016 0.050 0.008
-0.001
0.018 -0.025
-0.044
0.021
|s|
| γ|
|g|
0.139 0.127
0.162 0.223 0.166 0.310 0.192
0.175
0.209 0.111
0.085
0.194
Uncorrected 0.048 0.041
0.054 0.052 0.044 0.064 0.029
0.019
0.036 0.057
0.044
0.065