Brian O’Mearahttp://www.brianomeara.info

http://www.youtube.com/watch?v=9R8hpPY_9kYGet out laptop,

fire up R

install.packages("ctv")library(ctv)

install.views("Phylogenetics")install.packages("corHMM")

Model selection

Likelihood ratio test

test statistic = 2(ln L1 - ln L0)

Likelihood ratio test

Posada and Crandal 1998

Likelihood ratio test

Akaike information criterion

AICi = -2 ln Li + ki

“Truth drops out as a constant” -- Burnham and Anderson 2004

AIC is estimator as distance between truth and approximating model

Bayes Factors

Reversible jump MCMC

Model 1

Model 2 Model 2

Model 1

Organize by:Question Method

• Correlation of herbivory with group living

• Crepuscular foraging being intermediate between nocturnal and diurnal

• Biogeography • Causes of

diversification• Rate of trait evolution• What limits the number

of species

• Continuous time Markov Chain

• Birth death process• Multivariate normal• BiSSE and friends• Tree stretching

Flour, sugar, egg, butter, leavening, liquid

Continuous time Markov chain finite state space

A, T, G, Cwoody, herbaceous

susceptible, infected, recovered

herbivorous, omnivorous, carnivorous

0, 2, 4, 6, 8, ..., 100 legs

Per day:

What is probability of it leaving the store that day?If it leaves, what is the probability it was paid for?What is the probability it stays in the store ≥

two days?

Action Bought by adult

Bought by child Stolen

Probability 0.20 0.10 0.05

Per ∆t:

Action Bought by adult

Bought by child Stolen

Probability

0.20 /scaling

0.10 /scaling

0.05 /scaling

Per ∆t:

Action Bought by adult

Bought by child Stolen

Rate radult rchild rstolen

Per ∆t:From \

To Store Adult Child Thief

Store - rstore-adult rstore-child rstore-thief

Per ∆t:From \

To Store Adult Child Thief

Store - rstore-adult rstore-child rstore-thief

Adult

Child

Thief

Per ∆t:From \

To Store Adult Child Thief

Store - rstore-adult rstore-child rstore-thief

Adult radult-store - radult-child radult-thief

Child rchild-store rchild-adult - rchild-thief

Thief rthief-store rthief-adult rthief-child -

From \ To Store Adult Child Thief

Store - rstore-adult rstore-child rstore-thief

Adult radult-store - radult-child radult-thief

Child rchild-store rchild-adult - rchild-thief

Thief rthief-store rthief-adult rthief-child -

From \ To Store Adult Child Thief

Store - rstore-adult rstore-child rstore-thief

Adult radult-store - radult-child radult-thief

Child rchild-store rchild-adult - rchild-thief

Thief rthief-store rthief-adult rthief-child -

Does the store ever get Twinkies back? [Do people return Twinkies for a refund?]

H0: r*-store = 0H1: r*-store > 0

From \ To Store Adult Child Thief

Store - rstore-adult rstore-child rstore-thief

Adult radult-store - radult-child radult-thief

Child rchild-store rchild-adult - rchild-thief

Thief rthief-store rthief-adult rthief-child -

Do adults give to kids at the same rate kids give to adults?H0: rchild-adult = radult-child H1: rchild-adult ≠ radult-child

A B C DA - rAB rAC rAD

B rBA - rBC rBD

C rCA rCB - rCD

D rDA rDB rDC -

•Hypotheses about rates for a single character (are some equal? are some zero?)•Hypotheses about correlation between characters•Tree inference•Ancestral state inference

From this basic model:

Continuous time Markov chain finite state space

Currie et al. 2010 Nature

Currie et al. 2010 Nature

Currie et al. 2010 Nature

From \ To Store Adult Child Thief

Store - rstore-adult rstore-child rstore-thief

Adult radult-store - radult-child radult-thief

Child rchild-store rchild-adult - rchild-thief

Thief rthief-store rthief-adult rthief-child -

Currie et al. 2010 Nature

From \ To

A(acephalous)

sC(simple

chiefdom)

cC(complex chiefdom)

S(state)

A - rA-sC rA-cC rA-S

sC rsC-A - rsC-cC rsC-S

cC rcC-A rcC-sC - rcC-S

S rS-A rS-sC rS-cC -

Currie et al. 2010 Nature

From \ To

A(acephalous)

sC(simple

chiefdom)

cC(complex chiefdom)

S(state)

A - rA-sC rA-cC rA-S

sC rsC-A - rsC-cC rsC-S

cC rcC-A rcC-sC - rcC-S

S rS-A rS-sC rS-cC -

0

0

0

0

0 0

0 0

0

Currie et al. 2010 Nature

From \ To

A(acephalous)

sC(simple

chiefdom)

cC(complex chiefdom)

S(state)

A - rA-sC rA-cC rA-S

sC rsC-A - rsC-cC rsC-S

cC rcC-A rcC-sC - rcC-S

S rS-A rS-sC rS-cC -

0 0

0

0

0 0

Currie et al. 2010 Nature

From \ To

A(acephalous)

sC(simple

chiefdom)

cC(complex chiefdom)

S(state)

A - rA-sC rA-cC rA-S

sC rsC-A - rsC-cC rsC-S

cC rcC-A rcC-sC - rcC-S

S rS-A rS-sC rS-cC -

Currie et al. 2010 Nature

Currie et al. 2010 Nature

Currie et al. 2010 Nature

From \ To

A(acephalous)

sC(simple

chiefdom)

cC(complex chiefdom)

S(state)

A - med small small

sC med - large med

cC med large - med

S med med med -

0

1

A

01 A

B

B

No sex play

No sex play

Yes sex play

Yes sex play

No social play

No social play

Yes social play

Yes social play

From \ To 0A 0B 1B 1A

0A - r0A-0B r0A-1B r0A-1A

0B r0B-0A - r0B-1B r0B-1A

1B r1B-0A r1B-0B - r1B-1A

1A r1A-0A r1A-0B r1A-1B -

From \ To 0A 0B 1B 1A

0A - r0A-0B r0A-1B r0A-1A

0B r0B-0A - r0B-1B r0B-1A

1B r1B-0A r1B-0B - r1B-1A

1A r1A-0A r1A-0B r1A-1B -

0

0

0

0

0

1

A

01 A

B

B

No sex play

No sex play

Yes sex play

Yes sex play

No social play

No social play

Yes social play

Yes social play

0

1

A

01 A

B

B

No sex play

No sex play

Yes sex play

Yes sex play

No social play

No social play

Yes social play

Yes social play

Pagel 1994

From \ To 0A 0B 1B 1A

0A - r0A-0B 0 r0A-1A

0B r0B-0A - r0B-1B 0

1B 0 r1B-0B - r1B-1A

1A r1A-0A 0 r1A-1B -

Barker & Pagel 2005

0

1

1

01 1

0

0

Barker & Pagel 2005

Barker & Pagel 2005

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 a

2 a b

3 b c

4 c d

5 d e

6 e f

7 f g

8 g h

9 h i

10 i j

11 j k

12 k l

13 l m

14 m n

15 n o

16 o

... up to maximum number of genes

...

......

where a, b, etc. are just f(i, j, birthdeath rate)

Ree & Smith 2008

Ree & Smith 2008

Courtesy Nicolas Salamin

Joint: Choose values for x, y, z, w that together maximize likelihood

Joint: Choose values for x, y, z, w that together maximize likelihood

Marginal: Choose value for x (and repeat for others) that maximizes

likelihood while integrating over all values for y, z, w

Joint: Choose values for x, y, z, w that together

maximize likelihood

Marginal: Choose value for x (and

repeat for others) that maximizes likelihood while

integrating over all values for y, z, w

Henry Hargreaves

Finnigan, G. C., V. Hanson-Smith, T. H. Stevens, and J. W. Thornton. 2012. Evolution of increased complexity in a molecular machine. Nature 481:360-U143.

Courtesy Nicolas Salamin

Equal: all states equally likely

Empirical: count the proportion of each state in the tip taxa

Fixed: make them up (ideally, based on knowledge)

Equilibrium: what they'd be if the process ran forever

This assumption can have a major effect on results

Schluter et al. 1997

Beaulieu, O'Meara, and Donoghue, 2013

Beaulieu, O'Meara, and Donoghue, 2013

Beaulieu, O'Meara, and Donoghue, 2013

Beaulieu, O'Meara, and Donoghue, 2013

Tree stretching

Continuous

- rAB × t

rAC × t

rAD × t

rBA × t - rBC ×

trBD ×

t

rCA × t

rCB × t - rCD ×

t

rDA × t

rDB × t

rDC × t -

e( )

Discrete

Lambda = multiply internal branch lengths1

0.5

0

Delta = speed up or slow down

1

1.5

0.5

Kappa = raise each branch to kappa.

Punctuational models.

0.5

0

1

Eldredge and Gould 1971

Eldredge and Gould 1971

¿=?≠

Schematic illustration of evolution of one phenotype on a phylogeny leading to three extant species. Cladogenetic change appears as vertical lines as it is modeled here as an instantaneous event on a geological time scale. Anagenetic change appears as Brownian motion of the phenotype on a logarithmic scale. Sh indi- cates the speciation events that do not appear on a reconstructed phylogeny but did contribute to phenotypic evolution of the extant species, and S◦ indicates a speciation event that can be “ob- served” in a reconstructed phylogeny. In the resulting “branching Brownian motion,” species E and F are separated by three speciation events of which two contributed to the phenotypic difference between E and F. F and G are separated by four events that all four contributed to the present phenotypic difference between F and G.

Bokma 2008

Bokma 2008Eldredge and Gould 1971

=

Two rate: apply different rate before and after some point

(in this case, midpoint)

1

2after

0.2after

What questions can we answer with tree stretching?

Heterogeneity

Smith and Donoghue. Rates of Molecular Evolution Are Linked to Life History in Flowering Plants. Science (2008)

Trees/shrubs

Herbs

Subs

titut

ions

/MY

Meredith et al. 2011, Science

O’Meara 2012

O’Meara 2012

O’Meara 2012

O’Meara 2012

Yang 1994

O’Meara 2012

Pagel and Meade 2004

Familiar

Mixture model

Note: likely used a window, not

mentioned, though

O’Meara 2012

O’Meara 2012

Tree stretching

Heterogeneity

Tree stretching + Heterogenei

ty

Continuous methods Y some some

Discrete methods Y nope nope

O’Meara 2012

library(geiger)?fitContinuous?fitDiscrete

#Look at some Geospiza examples. Is the rate of beak depth evolution dropping in Darwin’s finches?

http://www.youtube.com/watch?v=BZG14R5p6Kg http://www.youtube.com/watch?v=h6fqrDShlMM

MADDISON. Confounding asymmetries in evolutionary diversification and character change. Evolution (2006) vol. 60 (8) pp. 1743-1746

MADDISON. Confounding asymmetries in evolutionary diversification and character change. Evolution (2006) vol. 60 (8) pp. 1743-1746

MADDISON. Confounding asymmetries in evolutionary diversification and character change. Evolution (2006) vol. 60 (8) pp. 1743-1746

MADDISON. Confounding asymmetries in evolutionary diversification and character change. Evolution (2006) vol. 60 (8) pp. 1743-1746

0 1q01

q10

speciation0 speciation1

extinction0extinction1

MADDISON. Confounding asymmetries in evolutionary diversification and character change. Evolution (2006) vol. 60 (8) pp. 1743-1746

MADDISON. Confounding asymmetries in evolutionary diversification and character change. Evolution (2006) vol. 60 (8) pp. 1743-1746

MADDISON. Confounding asymmetries in evolutionary diversification and character change. Evolution (2006) vol. 60 (8) pp. 1743-1746

MADDISON. Confounding asymmetries in evolutionary diversification and character change. Evolution (2006) vol. 60 (8) pp. 1743-1746

Goldberg et al. 2010