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New phylogenetic methods for studying the phenotypic axis of adaptive radiation Liam J. Revell University of Massachusetts Boston

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Outline 1.The ‘phytools’ package. 2.New approaches for the analysis of quantitative trait data: a)Phylogenetic analysis of the evolutionary correlation. b)Bayesian method for locating rate shifts in the tree. c)Incorporating intraspecific variability in phylogenetic analyses. 3.Luke!

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Outline 1.The ‘phytools’ package. 2.New approaches for the analysis of quantitative trait data: a)Phylogenetic analysis of the evolutionary correlation. b)Bayesian method for locating rate shifts in the tree. c)Incorporating intraspecific variability in phylogenetic analyses. 3.Luke!

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Phylogenetics in R

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Simulation Visualization Tree input/output/manipulation Inference Comparative biology -> Major functions of ‘phytools’

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Outline 1.The ‘phytools’ package. 2.New approaches for the analysis of quantitative trait data: a)Phylogenetic analysis of the evolutionary correlation. b)Bayesian method for locating rate shifts in the tree. c)Incorporating intraspecific variability in phylogenetic analyses. 3.Luke!

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Outline 1.The ‘phytools’ package. 2.New approaches for the analysis of quantitative trait data: a)Phylogenetic analysis of the evolutionary correlation. b)Bayesian method for locating rate shifts in the tree. c)Incorporating intraspecific variability in phylogenetic analyses. 3.Luke!

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The evolutionary correlation Revell & Collar 2009, Evolution

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non-piscivorous piscivorous

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Likelihood For 2-Correlation Model Revell & Collar 2009, Evolution

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Table. Model selection for the one and two rate matrix models. Modelrlog(L)AICc One matrix model R =0.41572.19-131.7 Two matrix model R 1 (Non-piscivory) =-0.05882.16-140.7 R 2 (Piscivory) =0.779 Likelihood ratio test -2·log(L 1 /L 2 ) = 19.94P(χ 2,df=3) < 0.001 Revell & Collar 2009, Evolution

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Outline 1.R phylogenetics and the ‘phytools’ package. 2.New approaches for the analysis of quantitative trait data: a)Phylogenetic analysis of the evolutionary correlation. b)Bayesian method for locating rate shifts in the tree. c)Incorporating intraspecific variability in phylogenetic analyses. 3.Luke!

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Bayesian MCMC method for rate variation x 20 Revell, & al. 2012

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Bayesian MCMC method for rate variation 3.52 3.45 3.55 3.86 4.06 3.40 3.67 5.15 2.74 2.75 7.95 4.63 5.99 7.70 8.77 8.62 1.04 1.00 2.27 2.85 1.16 2.34 2.12 4.03 4.65 4.53 4.33 4.15 4.10 3.98 Revell, & al. 2012

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Bayesian MCMC method for rate variation 3.52 3.45 3.55 3.86 4.06 3.40 3.67 5.15 2.74 2.75 7.95 4.63 5.99 7.70 8.77 8.62 1.04 1.00 2.27 2.85 1.16 2.34 2.12 4.03 4.65 4.53 4.33 4.15 4.10 3.98 ? ? ? Revell, & al. 2012

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Bayesian MCMC chain: evol.rate.mcmc() posterior sample Starting values σ12σ12 σ22σ22 Evolutionary rates & rate- shift Proposal σ12σ12 σ22σ22 Propose new rate-shift (or rates) XLP XLP | |,1min Compute posterior odds ratio Retain proposal with probability α Repeat Reject proposal with probability 1-α

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Bayesian MCMC chain

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MCMC proposal 3.52 3.45 3.55 3.86 4.06 3.40 3.67 5.15 2.74 2.75 7.95 4.63 5.99 7.70 8.77 8.62 1.04 1.00 2.27 2.85 1.16 2.34 2.12 4.03 4.65 4.53 4.33 4.15 4.10 3.98 σ22σ22 σ12σ12 Rate shift point with two evolutionary rates: the rate tipward (σ 2 2, in this case) and rootward of the rate- shift.

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MCMC proposal 1. Propose shift from exponential distribution. 2. Go right or left with equal probability; reflect back down the tree from the tips.

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MCMC proposal 3.52 3.45 3.55 3.86 4.06 3.40 3.67 5.15 2.74 2.75 7.95 4.63 5.99 7.70 8.77 8.62 1.04 1.00 2.27 2.85 1.16 2.34 2.12 4.03 4.65 4.53 4.33 4.15 4.10 3.98 σ12σ12 σ22σ22 1. Propose shift from exponential distribution. 2. Go right or left with equal probability; reflect back down the tree from the tips.

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Averaging the posterior sample: min.split() To find the median shift-point in our sample, we first computed the patristic distance between all the shifts in our posterior sample. We then picked the split with the lowest summed distance to all the other sample. (We might have instead found the shift with the lowest sum of squared distances, or found a point on a tree that minimized the sum of squared distances.)

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Averaging the posterior sample: min.split() We can also compute the posterior probabilities of the shift being on any edge. We just calculate these as the frequency of the edge in the posterior sample. 0.97 0.01 0.02

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Averaging the posterior: posterior.evolrate() Averaging the posterior sample of rates is also non-trivial. This is because our posterior sample is a mixture of rates comprising different parts of the tree and different tips. How can we average the rates from these different samples?

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Averaging the posterior: posterior.evolrate() σ 2 2 σ 1 2

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σ 2 2 σ 1 2 XX

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X X x σ 1 2 x σ 2 2

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Averaging the posterior: posterior.evolrate()

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X X x σ 2 2 x σ 1 2 In this case, σ 2 2 = σ 2 2 ; while σ 1 2 > σ 1 2

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Identification of the “correct” edge Somewhat surprisingly, identification of the “correct” edge was effectively independent of the number of tips in the tree for a given rate shift. However, relative patristic distance from the true shift point does decline with increasing N.

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Estimating the evolutionary rates We do get better at estimating the evolutionary rates unbiasedly (and their ratio) for increased N. The evolutionary rates tend to be biased towards each other for small N, which we think is a natural consequence of integrating over uncertainty in the location of the rate shift.

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Outline 1.The ‘phytools’ package. 2.New approaches for the analysis of quantitative trait data: a)Phylogenetic analysis of the evolutionary correlation. b)Bayesian method for locating rate shifts in the tree. c)Incorporating intraspecific variability in phylogenetic analyses. 3.Luke!

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Phylogenetic comparative analyses are conducted with species means. But data in empirical studies are uncertain estimates obtained by measuring one or a few individuals. **Ignoring intraspecific variability can cause bias in various types of comparative analysis. Our solution: sample both species means & variances, and the parameters of the evolutionary model, from their joint posterior distribution using Bayesian MCMC. Introduction

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First, we need an equation for the likelihood:

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How does it work?

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MCMC phytools:: fitBayes Posterior sample (Revell & Reynolds, 2012) How does it work?

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> results gen sig2 a 6.000000e+04 1.574812e+00 3.401690e-01 t1 t2... -2.088050e-01 7.424047e-01... How does it work?

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0123 -2 0 1 2 3 xbar estimated means How does it work?

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0123 -2 0 1 2 3 xbar estimated means > results gen sig2 a 6.000000e+04 1.574812e+00 3.401690e-01 t1 t2... -2.088050e-01 7.424047e-01... > SS.bayes [1] 2.176141 > SS.arith [1] 2.980928 How does it work?

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Is this result general.... YES! Generating σ 2 0.2 0.0 0.2 0.3 Mean square error -- MSE Bayesian means -- MSE arithmetic means 0.4 0.1 0.40.60.81.0

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Outline 1.The ‘phytools’ package. 2.New approaches for the analysis of quantitative trait data: a)Phylogenetic analysis of the evolutionary correlation. b)Bayesian method for locating rate shifts in the tree. c)Incorporating intraspecific variability in phylogenetic analyses. 3.Luke!

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