New phylogenetic methods for studying the phenotypic axis of adaptive radiation

Liam J. RevellUniversity of Massachusetts Boston

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!

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!

Phylogenetics in R

Simulation

Visualization

Tree input/output/manipulation

Inference

Comparative biology

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Major functions of ‘phytools’

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!

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!

The evolutionary correlation

Revell & Collar 2009, Evolution

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Likelihood For 2-Correlation Model

Revell & Collar 2009, Evolution

41091.273.1

73.197.5

Table. Model selection for the one and two rate matrix models.

41084.412.0

12.062.8

41084.412.3

12.330.3

Model r log(L) AICc

One matrix model

R = 0.415 72.19 -131.7

Two matrix model

R1(Non-piscivory) = -0.058 82.16 -140.7

R2(Piscivory) = 0.779

Likelihood ratio test

-2·log(L1/L2) = 19.94 P(χ2,df=3) < 0.001

Revell & Collar 2009, Evolution

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!

Bayesian MCMC method for rate variation

x 20

Revell, & al. 2012

Bayesian MCMC method for rate variation

3.523.453.553.864.063.403.675.152.742.757.954.635.997.708.778.621.041.002.272.851.162.342.124.034.654.534.334.154.103.98

Revell, & al. 2012

Bayesian MCMC method for rate variation

3.523.453.553.864.063.403.675.152.742.757.954.635.997.708.778.621.041.002.272.851.162.342.124.034.654.534.334.154.103.98

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Revell, & al. 2012

Bayesian MCMC chain: evol.rate.mcmc()

posterior sample

Startingvalues

σ12 σ2

2

Evolutionary rates & rate-

shift

Proposalσ1

2 σ22

Propose new rate-shift (or

rates)

σ12 σ2

2σ12σ12 σ2

2σ22 σ1

2σ12 σ2

2σ22

σ12σ12 σ2

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σ12 σ2

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2σ22

σ12 σ2

2σ12σ12 σ2

2σ22

Compute posterior odds ratio

σ12 σ2

2σ12σ12 σ2

2σ22

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Retain proposal with probability α

Repeat

σ12σ12 σ2

2σ22

)1,0(~Urif

Reject proposal with probability 1-α

posterior sample

Startingvalues

σ12 σ2

2

Evolutionary rates & rate-

shift

Startingvalues

σ12σ12 σ2

2σ22

Evolutionary rates & rate-

shift

Proposalσ1

2 σ22

Propose new rate-shift (or

rates)

Proposalσ1

2σ12 σ2

2σ22

Propose new rate-shift (or

rates)

σ12 σ2

2σ12σ12 σ2

2σ22 σ1

2σ12 σ2

2σ22σ1

2σ12 σ2

2σ22

σ12σ12 σ2

2σ22

XLP

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|

|

,1min

σ12 σ2

2σ12σ12 σ2

2σ22

σ12σ12 σ2

2σ22

σ12 σ2

2σ12σ12 σ2

2σ22

Compute posterior odds ratio

σ12σ12 σ2

2σ22

XLP

XLP

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,1min

σ12 σ2

2σ12σ12 σ2

2σ22

σ12σ12 σ2

2σ22

σ12 σ2

2σ12σ12 σ2

2σ22

Compute posterior odds ratio

σ12 σ2

2σ12σ12 σ2

2σ22

)1,0(~Urif

Retain proposal with probability α

σ12 σ2

2σ12σ12 σ2

2σ22

)1,0(~Urif

Retain proposal with probability α

Repeat

σ12σ12 σ2

2σ22

)1,0(~Urif

Reject proposal with probability 1-α

σ12σ12 σ2

2σ22

)1,0(~Urif

Reject proposal with probability 1-α

Bayesian MCMC chain

MCMC proposal

3.523.453.553.864.063.403.675.152.742.757.954.635.997.708.778.621.041.002.272.851.162.342.124.034.654.534.334.154.103.98

σ22

σ12

Rate shift point with two evolutionary rates: the rate tipward (σ2

2, in this case) and rootward of the rate-shift.

3.523.453.553.864.063.403.675.152.742.757.954.635.997.708.778.621.041.002.272.851.162.342.124.034.654.534.334.154.103.98

MCMC proposal

1. Propose shift from exponential distribution.

2. Go right or left with equal probability; reflect back down the tree from the tips.

MCMC proposal

3.523.453.553.864.063.403.675.152.742.757.954.635.997.708.778.621.041.002.272.851.162.342.124.034.654.534.334.154.103.98

σ12

σ22

1. Propose shift from exponential distribution.

2. Go right or left with equal probability; reflect back down the tree from the tips.

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.)

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.

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0.01

0.02

Averaging the posterior: posterior.evolrate()

σ22

σ12

σ12

σ22

σ12

σ22

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?

Averaging the posterior: posterior.evolrate()

σ22

σ12

Averaging the posterior: posterior.evolrate()

σ22

σ12

X X

Averaging the posterior: posterior.evolrate()

X X

x σ12

x σ22

Averaging the posterior: posterior.evolrate()

Averaging the posterior: posterior.evolrate()

X

X

x σ22

x σ12

In this case, σ22 = σ2

2;

while σ12 > σ1

2

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.

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.

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!

• 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

First, we need an equation for the likelihood:

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model & tree

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means species | dataP

model evol.

variances,

means, species

likelihood

-0.3530.906

0.552,0.371,1.155,0.483-0.329,0.521,-0.013,-1.08

-2.322,-1.809

-0.707

1.31

-1.444,-0.889-1.277,-0.545

-1.246,-0.335,-0.086-0.583

0.218,0.721,1.144,0.4982.547,2.071

-0.8,-1.111

-0.346

-1.292-1.381,-1.547

0.621,0.795,0.065,0.697

0.172,-0.303,0.137-0.264,-0.005

-0.783,0.101,-0.7390.376,-0.278,-0.094,0.667

2.6582.683,2.656

0.679,-0.288,-0.404-0.456

3.249,2.157,2.845,2.2992.596,3.375,3.843

-0.275,0.179,-0.605,-0.206-0.269,0.373

How does it work?

-0.3530.906

0.552,0.371,1.155,0.483-0.329,0.521,-0.013,-1.08

-2.322,-1.809

-0.707

1.31

-1.444,-0.889-1.277,-0.545

-1.246,-0.335,-0.086-0.583

0.218,0.721,1.144,0.4982.547,2.071

-0.8,-1.111

-0.346

-1.292-1.381,-1.547

0.621,0.795,0.065,0.697

0.172,-0.303,0.137-0.264,-0.005

-0.783,0.101,-0.7390.376,-0.278,-0.094,0.667

2.6582.683,2.656

0.679,-0.288,-0.404-0.456

3.249,2.157,2.845,2.2992.596,3.375,3.843

-0.275,0.179,-0.605,-0.206-0.269,0.373

MCMC

phytools::fitBayes

Posterior sample

(Revell & Reynolds, 2012)

How does it work?

> results gen sig2 a 6.000000e+04 1.574812e+00 3.401690e-01 t1 t2 ... -2.088050e-01 7.424047e-01 ...-0.353

0.906

0.552,0.371,1.155,0.483-0.329,0.521,-0.013,-1.08

-2.322,-1.809

-0.707

1.31

-1.444,-0.889-1.277,-0.545

-1.246,-0.335,-0.086-0.583

0.218,0.721,1.144,0.4982.547,2.071

-0.8,-1.111

-0.346

-1.292-1.381,-1.547

0.621,0.795,0.065,0.697

0.172,-0.303,0.137-0.264,-0.005

-0.783,0.101,-0.7390.376,-0.278,-0.094,0.667

2.6582.683,2.656

0.679,-0.288,-0.404-0.456

3.249,2.157,2.845,2.2992.596,3.375,3.843

-0.275,0.179,-0.605,-0.206-0.269,0.373

How does it work?

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me

an

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How does it work?

-1 0 1 2 3

-2-1

01

23

xbar

est

ima

ted

me

an

s

> 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?

Is this result general . . . . YES!

Generating σ2

0.2

0.0

0.2

0.3

Mea

n sq

uare

err

or-- MSE Bayesian means-- MSE arithmetic means

0.4

0.1

0.4 0.6 0.8 1.0

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!