The evolution of cerebellum structure correlates with nest complexity Zachary J. Hall, Sally E. Street, and Susan D. Healy Supplementary Material 2. Supplementary analyses. All PGLS parameters are model-averaged estimates across the tree-block. Main effects were calculated by comparing models with and without nest structure as a predictor using ANOVA. F and p values for main effects could not be model-averaged across the tree-block because they are comparisons between models rather than PGLS parameters with associated model likelihoods. Rather we report maximum or minimum values, whichever is more conservative for the specific analysis. Main regression model parameters in full: Model: CFI predicted by nest structure and log cerebellum volume Variable Cerebellum volume Nest structure (main effect) Cup vs. platform nests Platform vs. no nests Cup vs. no nests Test statistic T: 10.056 F > 3.875 T: 2.019 T: 2.047 T: 3.165 Significance p < 0.001 p < 0.026 p = 0.049 p = 0.047 p = 0.003 R2 = 0.615 = 0.852 Specificity of Nest Structure predicting CFI Although both cerebellar foliation and log cerebellum volume are associated with increases in log whole brain volume (model-averaged slopes: 0.441 and 0.898 respectively, both p < 0.001), nest type did not predict log cerebellum volume (F1,60 < 1.686, p > 0.194; (fixed) = 0.95, using log whole brain volume and log body size as co-variates) nor log whole brain volume (F1,60 < 2.144 p > 0.126; (fixed) = 0.95, using log body size as a co-variate). Therefore, the effect of nest structure we report is specific to cerebellar foliation, using cerebellum volume as a co-variate. Furthermore, the effect of nest structure on cerebellar foliation is not associated with an increase in whole brain volume. Results are not confounded by flightlessness Our main results were not affected by removing two flightless species (Rhea americana and Struthio camelus), in terms of either the main effect of nest structure on cerebellar foliation: (F1,58 > 4.589, p < 0.028, across 3000 trees, λ (fixed) = 0.85, using log cerebellum volume as a co-variate), or in any of the planned contrasts (all model-averaged p < 0.05). Re-analysis using Bayesian Markov-Chain Monte Carlo (MCMC) Using model averaging to summarise PGLS parameters across a treeblock accounts for phylogenetic uncertainty, but does not account for potential uncertainty in the statistical model. In order to account for both phylogenetic and model uncertainty, we re-ran our main PGLS analyses using Bayesian Markov-Chain Monte Carlo (MCMC) methods in BayesTraits (Version 1, [1,2]). We estimated posterior probability distributions for parameters including regression coefficients (β), model R2 and phylogenetic signal (λ). We report average values for parameters and the percentage of posterior estimates above zero and in the predicted direction (% β +ve). Prior to analysis, we determined that >95% of posterior estimates for regression coefficients above zero and in the predicted direction would be interpreted as ‘strong evidence’ for a statistical relationship between variables, as, for example, in [3]. As for our main analyses, we used CFI as the outcome variable, predicted by log transformed cerebellar volume and nest type. We ran MCMC chains for 5,000,000 iterations, sampling every 100 generations. We used uniform prior distributions for regression coefficients (-100, +100). Mean acceptance rates were between 20-40%, as recommended by Pagel & Meade [2], and all effective sample sizes were >5,000. We found strong evidence for greater CFI relative to cerebellar foliation in cup nesters relative to platform nesters (average β: 0.24, 100% β +ve), platform nesters relative to species with no nests (average β: 0.22, 96% β +ve) and cup nesters relative to species with no nests (average β: 0.46, 100% β +ve). We also found strong evidence that cerebellar volume predicted cerebellar foliation (average β: 0.51, 100% β +ve). The model R2 was 0.62. 1 Pagel, M. & Meade, A. 2006 Bayesian analysis of correlated evolution of discrete characters by reversible-jump Markov Chain Monte Carlo. Amer. Nat. 167, 808-825. 2 Pagel, M., & Meade, A. 2007 BayesTraits (Version 1.0). Reading, UK. www.evolution.rdg.ac.uk 3 Ross, C.F., Iriarte-Diaz, J. & Nunn, C.L. 2012 Innovative approaches to the relationship between diet and mandibular morphology in primates, Int. J. Primatol. 33, 632-660.

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# Supplementary Material 2