Strategic Female Reproductive Investment in...

Supplementary material

Strategic Female Reproductive Investment in Response to Male

Attractiveness in Birds

Terezia Horvathova1,2

, Shinichi Nakagawa3, & Tobias Uller

1,4

1. Edward Grey Institute, Department of Zoology, University of Oxford, OX1

3PS

2. Department of Zoology, Faculty of Natural Sciences, Comenius University,

Mlynska dolina, 842 15 Bratislava, Slovakia

3. Department of Zoology, University of Otago, PO Box 56, Dunedin, 9054,

New Zealand

4. Corresponding author, Email: [email protected], Phone: +44(0)1865-

281194, Fax: +44(0)1865-271168

mailto:[email protected]

Statistical procedures: more details

Bayesian mixed-effects meta-analysis, BMM, and Bayesian phylogenetic mixed-

effects meta-analysis, BPMM (Models 1-10 in Table 1 and Tables S1-S4; Fig. S1)

along with Egger’s regression test (see below) were all implemented using the

function MCMCglmm in the R package MCMCglmm [1]. MCMCglmm uses a Markov

chain Monte Carlo (MCMC) algorithm to fit generalized linear mixed models with an

inverse Wishart prior. The inverse Wishart prior has two parameters, V (which

specifies variance) and nu(which specifies the degree of belief in V). For the random

effects of all the models, we used V = 1 and nu = 0.002, which are equivalent to an

inverse gamma prior, widely used in the statistical literature [2]. For each statistical

model, we ran three MCMC chains (i.e. three independent runs of MCMCglmm

models) to be able to test the convergence of model parameters among the chains. For

every chain, we used the same settings for sampling: 1) the number of iterations of

2,000,000, 2) the thinning interval of 500 and 3) the number of burn-in of 1,500,000.

This setting resulted in 1000 samples, which constituted our posterior distributions for

all parameter estimates (fixed and random effects along with deviance information

criterion, DIC [3]). We checked convergence of model parameters (or posterior

distributions) by using the Gelman-Rubin statistic (the potential scale reduction, PSR,

factor should be less than 1.1 among chains; [4]). All sets of three chains showed the

PSR factor less than 1.02, meaning model convergence were appropriate. We only

used posterior distributions from the first of three chains, to report our results. Also, it

is noted that we reported posterior means rather than posterior modes for fixed factors

in the main text (except the results related to variance components; all detailed results

are presented in Tables S1-S4). If 95% credible (or credibility) intervals (also known

as highest posterior densities) of fixed effects did not span across zero, we deemed

these effects were statistically significant.

We reported the proportion of the total variance in variance components for a

particular variance (Table 1) as an index of heterogeneity. This is because more

traditional measures of heterogeneity such as Q and I2, both of which were developed

for medical/social sciences [5], were not easily calculable for our meta-analytic

models (BMM and BPMM, which included species and/or phylogeny as random

effects). Also, conventional tests for publication bias were difficult to implement due

to the nature of our dataset (including non-independent data points and the existence

of heterogeneity). Therefore, we applied Egger’s regression to the results of Model 6

(Table 1) according to Roberts & Stanley [6] (Fig. 3). By this way, we did not violate

any statistical assumptions such as independence of data points. If the intercept of

Egger’s regression is not significantly different from zero, one can conclude there is

little evidence for publication bias [7] (in our case, b0 = 0.291, 95% CI -0.340 to

0.699).

REFERENCES

1. Hadfield, J. D. 2010 MCMC methods for multi-response Generalised Linear

Mixed Models: the MCMCglmm R package. J. Stat. Softw. 33, 1-22

2. Gelman, A. & Hill, J. 2007 Data analysis using regression and

multilevel/hierarchical models. Cambridge: Cambridge University Press.

3. Lunn, D. J., Thomas, A., Best, N. & Spiegelhalter, D. 2000 WinBUGS - A

Bayesian modelling framework: Concepts, structure, and extensibility.

Statistics and Computing 10, 325-337

4. Gelman, A. & Rubin, D. B. 1992 Inference from iterative simulation using

multiple sequences (with discusiion). Statistical Science 7, 457-511.

5. Higgins, J. P. T., Thompson, S. G., Deeks, J. J. & Altman, D. G. 2003

Measuring inconsistency in meta-analyses. Brit. Med. J. 327, 557-560.

6. Robert, C. J. & Stanley, T. D. Meta-regression analysis: issues of publication

bias in economics. Oxford: Blackwell Publishing

7. Egger, M., Smith, G. D., Schneider, M. & Minder, C. 1997 Bias in meta-

analysis detected by a simple, graphical test. Brit. Med. J. 315, 629-634.

Table S1. Estimates from Bayesian mixed-effects meta-analysis (Model 1) and phylogenetic mixed-effects meta-analysis (Model 2) for the

relationship between male attractiveness and (SD: standard deviation of estimate, CI: credible interval).

Fixed effects Posterior mode Posterior mean (SD) Lower CI Upper CI

Model 1

Intercept (meta-analytic mean) 0.128 0.146 (0.034) 0.078 0.213

Model 2

Intercept (meta-analytic mean) 0.157 0.171 (0.070) 0.045 0.310

Random effects Posterior mode Posterior mean (SD) Lower CI Upper CI

Model 1

Study variance 0.001 0.008 (0.008) <0.001 0.023

Species variance 0.002 0.007 (0.009) <0.001 0.022

Residual variance 0.001 0.007 (0.006) <0.001 0.018

Model 2

Study variance 0.001 0.008 (0.007) <0.001 0.022


Phylogenetic variance 0.002 0.12 (0.017) <0.001 0.046


Sample sizes at different levels are: 17 species, 45 studies and 113 effect sizes (these are true for all the models we ran).

Table S2. Estimates from Bayesian mixed-effects meta-regression analysis (Model 3) and phylogenetic mixed-effects meta-regression analysis

(Model 4) for the relationship between male attractiveness and female investment with the moderator, female traits (SD: standard deviation of a

parameter estimate, CI: credible interval).


Model 3

Clutch size 0.113 0.118 (0.048) 0.025 0.208

Egg size 0.169 0.179 (0.054) 0.068 0.283

Feeding rate 0.259 0.278 (0.087) 0.105 0.444

Laying onsets 0.101 0.092 (0.075) -0.050 0.241

Yolk androgen 0.171 0.148 (0.063) 0.029 0.271

Immunity stimulants 0.055 0.070 (0.089) -0.108 0.238

Model 4

Clutch size 0.137 0.150 (0.082) <0.001 0.339

Egg size 0.182 0.211 (0.085) 0.048 0.381

Feeding rate 0.300 0.314 (0.113) 0.131 0.559

Laying onsets 0.131 0.122 (0.099) -0.081 0.307

Yolk androgen 0.224 0.176 (0.089) -0.023 0.332

Immunity stimulants 0.102 0.094 (0.108) -0.136 0.295


Model 3

Study variance 0.001 0.008 (0.008) <0.001 0.023


Residual variance 0.002 0.008 (0.006) 0.001 0.020

Model 4

Study variance 0.001 0.007 (0.006) <0.001 0.019




Note that the intercepts were removed from Models 3 &4 for better interpretability.


(Model 6) for the relationship between male attractiveness and female investment with the moderator, female traits, parental care types and their

interactions (SD: standard deviation of a parameter estimate, CI: credible interval).


Model 5

Intercept (clutch size & both care) 0.108 0.133 (0.522) 0.030 0.230

Egg size (Δ between egg size & clutch size) 0.005 -0.002 (0.062) -0.120 0.121

Feeding rate (Δ between rate & clutch size) 0.198 0.140 (0.093) -0.049 0.311

Laying onset (Δ between onset & clutch size) -0.003 -0.044 (0.080) -0.195 0.111

Yolk androgen (Δ between androgen & clutch size) -0.054 -0.071 (0.078) -0.219 0.085

Immunity stimulants (Δ between immu. & clutch size) -0.141 -0.157 (0.119) -0.410 0.051

Female care (Δ between female care & both care) -0.058 -0.068 (0.111) -0.307 0.132

Egg size : Female care 0.226 0.258 (0.134) 0.018 0.541

Laying onset : Female care 0.116 -0.027 (0.311) -0.672 0.549

Yolk androgen : Female care 0.479 0.442 (0.171) 0.120 0.765

Immunity stimulants : Female care 0.254 0.279 (0.175) -0.056 0.620

Model 6

Intercept (clutch size & both care) 0.127 0.168 (0.096) -0.026 0.347

Egg size (Δ between egg size & clutch size) 0.006 -0.002 (0.062) -0.120 0.118

Feeding rate (Δ between rate & clutch size) 0.156 0.132 (0.092) -0.056 0.304

Laying onset (Δ between onset & clutch size) -0.034 -0.050 (0.077) -0.189 0.107

Yolk androgen (Δ between androgen & clutch size) -0.106 -0.072 (0.077) -0.223 0.079

Immunity stimulants (Δ between immu. & clutch size) -0.143 -0.162 (0.120) -0.374 0.088

Female care (Δ between female care & both care) -0.159 -0.109 (0.142) -0.400 0.148

Egg size : Female care 0.265 0.249 (0.135) 0.001 0.522

Laying onset : Female care -0.119 -0.044 (0.323) -0.644 0.606

Yolk androgen : Female care 0.464 0.436 (0.166) 0.141 0.794

Immunity stimulants : Female care 0.255 0.272 (0.174) -0.081 0.581


Model 5

Study variance 0.002 0.009 (0.008) <0.001 0.025



Model 6

Study variance 0.001 0.008 (0.008) <0.001 0.025




The character Δ stands for ‘difference’.


(Model 8) for the relationship between male attractiveness and female investment with the moderator, female traits, parental care types and their

interactions and experimental designs (both how females were exposed to males and whether or not male traits were manipulated) (SD: standard

deviation of a parameter estimate, CI: credible interval).


Model 7

Intercept (manipulation & after-paring) 0.144 0.168 (0.068) 0.048 0.299

Male trait (Δ between manip. & non-manip.) 0.040 0.018 (0.075) -0.123 0.159

Female 1 (Δ between after-paring & random-access) -0.083 -0.115 (0.100) -0.300 0.075

Female 2 (Δ between after-paring & within-female) -0.004 -0.055 (0.099) -0.251 0.127

Model 8

Intercept (manipulation & after-paring) 0.142 0.191 (0.103) 0.001 0.399

Male trait (Δ between manip. & non-manip.) 0.017 0.017 (0.080) -0.139 0.171

Female 1 (Δ between after-paring & random-access) -0.065 -0.085 (0.113) -0.313 0.156

Female 2 (Δ between after-paring & within-female) -0.045 -0.0224 (0.113) -0.248 0.202


Model 7

Study variance 0.002 0.011 (0.010) <0.001 0.032



Model 8

Study variance 0.002 0.011 (0.010) <0.001 0.032




Note that fixed effects estimates included in Models 5 & 6 were omitted as qualitative results remain the exactly the same in the corresponding

models. The character Δ stands for ‘difference’.

Figure. S1. The topology of the phylogenetic tree used for Bayesian phylogenetic

mixed-effects meta-analysis (BPMM; Models 2, 4, 6, & 8 in Table 1).

Strategic Female Reproductive Investment in...

Documents

Transcript of Strategic Female Reproductive Investment in...