a new phylogenetic comparative method: detecting niches and transitions with continuous characters

A new phylogenetic comparative method:detecting niches and transitions with

continuous characters.

Carl Boettiger

UC Davis

June 26, 2010

Carl Boettiger, UC Davis Niches & Transitions 1/35

Size in Lesser Antilles Anoles

Log Body Size

Fre

quen

cy

2.6 2.8 3.0 3.2 3.4

01

23

45

67

Carl Boettiger, UC Davis Niches & Transitions 2/35

3.073.062.992.942.933.353.12.982.932.982.913.053.363.353.333.33.162.72.672.652.662.662.6

Carl Boettiger, UC Davis Niches & Transitions 3/35

Goals

1 Demonstrate selecting models byinformation criteria is inadequate

2 I’ll propose a more robust framework

Carl Boettiger, UC Davis Niches & Transitions 4/35

Goals

1 Demonstrate selecting models byinformation criteria is inadequate

2 I’ll propose a more robust framework

Carl Boettiger, UC Davis Niches & Transitions 4/35

Goals

1 Demonstrate selecting models byinformation criteria is inadequate

2 I’ll propose a more robust framework

Carl Boettiger, UC Davis Niches & Transitions 4/35

Types of models

Carl Boettiger, UC Davis Niches & Transitions 5/35

Comparing Models

t2

t1

md

mgli

fe

oc

gm

gagb

sa

nula

lb

bc

bn

be

wawb

sn

sc

se

po

t2

t1

md

mgli

fe

oc

gm

gagb

sa

nula

lb

bc

bn

be

wawb

sn

sc

se

po

t2

t1

md

mgli

fe

oc

gm

gagb

sa

nula

lb

bc

bn

be

wawb

sn

sc

se

po

Carl Boettiger, UC Davis Niches & Transitions 6/35

Comparing Models

−70 −60 −50 −40 −30 −20

−2 Log Likelihood

BM OU.1OU.2OU.3

( Better Scores)Carl Boettiger, UC Davis Niches & Transitions 7/35

Comparing Models: AIC

−70 −60 −50 −40 −30 −20

AIC

BM OU.1OU.2OU.3

( Better Scores)Carl Boettiger, UC Davis Niches & Transitions 8/35

Comparing Models: Estimating Uncertainty

−70 −60 −50 −40 −30 −20

AIC

BM OU.1OU.2OU.3

( Better Scores)Carl Boettiger, UC Davis Niches & Transitions 9/35

Comparing Models: Uncertainty dominates

−70 −60 −50 −40 −30 −20

AIC

BM OU.1OU.2OU.3

( Better Scores)Carl Boettiger, UC Davis Niches & Transitions 10/35

Sources of this uncertainty

Small datasetsUninformative topologyModel details (i.e. high rates)

−70 −60 −50 −40 −30 −20

AIC

BM OU.1OU.2OU.3

Carl Boettiger, UC Davis Niches & Transitions 11/35

Information criteria alone may be misleading

Carl Boettiger, UC Davis Niches & Transitions 12/35

A Better Way: Comparing Models Directly

Likelihood Ratio

Fre

quen

cy

−10 0 10 20 30 40 50

050

100

150

200

250

300

−2 logL(BM)

L(OU.2)

−70 −60 −50 −40 −30 −20

−2 Log Likelihood

BM OU.2

Carl Boettiger, UC Davis Niches & Transitions 13/35

Method

Likelihood Ratio

Fre

quen

cy

−10 0 10 20 30 40 50

050

100

150

200

250

300

−70 −60 −50 −40 −30 −20

−2 Log Likelihood

BM OU.2

1 Simulate many datasetsunder model A

2 Re-fit both A & B to eachsimulated dataset

3 Write log Likelihood(A) -log Likelihood(B)

Carl Boettiger, UC Davis Niches & Transitions 14/35

Method

Likelihood Ratio

Fre

quen

cy

−10 0 10 20 30 40 50

050

100

150

200

250

300

−70 −60 −50 −40 −30 −20

−2 Log Likelihood

BM OU.2

1 Simulate many datasetsunder model A

2 Re-fit both A & B to eachsimulated dataset

3 Write log Likelihood(A) -log Likelihood(B)

Carl Boettiger, UC Davis Niches & Transitions 14/35

Method

Likelihood Ratio

Fre

quen

cy

−10 0 10 20 30 40 50

050

100

150

200

250

300

−70 −60 −50 −40 −30 −20

−2 Log Likelihood

BM OU.2

1 Simulate many datasetsunder model A

2 Re-fit both A & B to eachsimulated dataset

3 Write log Likelihood(A) -log Likelihood(B)

Carl Boettiger, UC Davis Niches & Transitions 14/35

BM vs OU.2

Likelihood Ratio

Fre

quen

cy

−10 0 10 20 30 40 50

050

100

150

200

250

300

p = 0.002

−70 −60 −50 −40 −30 −20

−2 Log Likelihood

BM OU.2

Carl Boettiger, UC Davis Niches & Transitions 15/35

OU.2 vs BM: Simulating under OU.2

Likelihood Ratio

Fre

quen

cy

−80 −60 −40 −20

050

100

150

p = 0.237

−70 −60 −50 −40 −30 −20

−2 Log Likelihood

BM OU.2

Carl Boettiger, UC Davis Niches & Transitions 16/35

Model A rejects Model B.Model B doesn’t reject Model A.

Carl Boettiger, UC Davis Niches & Transitions 17/35

How about two very similar models? BM vs OU.1

Likelihood Ratio

Fre

quen

cy

0 10 20 30

010

020

030

040

0

p = 0.778

−70 −60 −50 −40 −30 −20

−2 Log Likelihood

BM OU.1

Carl Boettiger, UC Davis Niches & Transitions 18/35

How would AIC rule compare?

Likelihood Ratio

Fre

quen

cy

0 10 20 30

010

020

030

040

0AIC

p = 0.779−70 −60 −50 −40 −30 −20

−2 Log Likelihood

BM OU.1

Carl Boettiger, UC Davis Niches & Transitions 19/35

When data is insufficient to distinguish,method can say “I don’t know”

Carl Boettiger, UC Davis Niches & Transitions 20/35

Can we distinguish between OU.2 and OU.3?

Likelihood Ratio

Fre

quen

cy

−80 −60 −40 −20 0 20

050

100

150

200

250

−70 −60 −50 −40 −30 −20

−2 Log Likelihood

OU.2OU.3

Carl Boettiger, UC Davis Niches & Transitions 21/35

Simulate under OU.2 and compare to OU.3 . . .

Likelihood Ratio

Fre

quen

cy

−80 −60 −40 −20 0 20

050

100

150

200

250

p = 0.051

−70 −60 −50 −40 −30 −20

−2 Log Likelihood

OU.2OU.3

Carl Boettiger, UC Davis Niches & Transitions 22/35

OU.3 vs OU.2: Now we’re preferring OU.2!

Likelihood Ratio

Fre

quen

cy

−60 −40 −20 0 20

050

100

150

200

250

300

350

p = 0.003

−70 −60 −50 −40 −30 −20

−2 Log Likelihood

OU.2OU.3

Carl Boettiger, UC Davis Niches & Transitions 23/35

Model A rejects Model B.Model B rejects Model A.

Carl Boettiger, UC Davis Niches & Transitions 24/35

Non-nested models

t2t1mdmglifeocgmgagbsanulalbbcbnbewawbsnscsepo

t2t1mdmglifeocgmgagbsanulalbbcbnbewawbsnscsepo

Carl Boettiger, UC Davis Niches & Transitions 25/35

Replacing paintings with a transition model

Carl Boettiger, UC Davis Niches & Transitions 26/35

All models are nestedEstimate number of nichesAlso estimate rates of transitions

Carl Boettiger, UC Davis Niches & Transitions 27/35

A hard problem in two easy pieces

P( | ) = P( | )P( | )

Carl Boettiger, UC Davis Niches & Transitions 28/35

Summary

1 Quantifiable, robust model choiceLikelihood Ratio

Fre

quen

cy

−10 0 10 20 30 40 50

050

100

150

200

250

300

p = 0.002

2 Identify when data is insufficientLikelihood Ratio

Fre

quen

cy

0 10 20 30

010

020

030

040

0

p = 0.778

3 New framework avoids painting &non-nested comparison

Carl Boettiger, UC Davis Niches & Transitions 29/35

Thanks!

Chris MartinPeter WainwrightSamantha PriceRoi Holzman

Graham CoopPeter RalphAlan HastingsDoE CSGF

Carl Boettiger, UC Davis Niches & Transitions 30/35

Time

Sta

te

Carl Boettiger, UC Davis Niches & Transitions 31/35

Time

Sta

te

Carl Boettiger, UC Davis Niches & Transitions 32/35

Time

Sta

te

Carl Boettiger, UC Davis Niches & Transitions 33/35

Comparing Models

t2

t1

md

mgli

fe

oc

gm

gagb

sa

nula

lb

bc

bn

be

wawb

sn

sc

se

po

t2

t1

md

mgli

fe

oc

gm

gagb

sa

nula

lb

bc

bn

be

wawb

sn

sc

se

po

t2

t1

md

mgli

fe

oc

gm

gagb

sa

nula

lb

bc

bn

be

wawb

sn

sc

se

po

Carl Boettiger, UC Davis Niches & Transitions 34/35

OU.3 vs OU.4: Why you should mistrust painting trees

Likelihood Ratio

Fre

quen

cy

−50 0 50 100

050

100

150

200

250

300

350

p = 0

−120 −100 −80 −60 −40 −20

−2 Log Likelihood

OU.3OU.4

Carl Boettiger, UC Davis Niches & Transitions 35/35