Talk given at the annual Evolution meeting in Portland, Oregon, June 26th

1. 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

2. Size in Lesser Antilles Anoles
7
6
5
Frequency
4
3
2
1
0
2.6 2.8 3.0 3.2 3.4
Log Body Size
Carl Boettiger, UC Davis Niches & Transitions 2/35

3. 2.6
2.66
2.66
2.65
2.67
2.7
3.16
3.3
3.33
3.35
3.36
3.05
2.91
2.98
2.93
2.98
3.1
3.35
2.93
2.94
2.99
3.06
3.07
Carl Boettiger, UC Davis Niches & Transitions 3/35

4. Goals
Carl Boettiger, UC Davis Niches & Transitions 4/35

5. Goals
1
Demonstrate selecting models by
information criteria is inadequate
Carl Boettiger, UC Davis Niches & Transitions 4/35

6. Goals
1
Demonstrate selecting models by
information criteria is inadequate
2
I’ll propose a more robust framework
Carl Boettiger, UC Davis Niches & Transitions 4/35

7. Types of models
Carl Boettiger, UC Davis Niches & Transitions 5/35

8. Comparing Models
po po po
se se se
sc sc sc
sn sn sn
wb wb wb
wa wa wa
be be be
bn bn bn
bc bc bc
lb lb lb
la la la
nu nu nu
sa sa sa
gb gb gb
ga ga ga
gm gm gm
oc oc oc
fe fe fe
li li li
mg mg mg
md md md
t1 t1 t1
t2 t2 t2
Carl Boettiger, UC Davis Niches & Transitions 6/35

9. Comparing Models
BM
OU.1
OU.2
OU.3
−70 −60 −50 −40 −30 −20
−2 Log Likelihood
Carl Boettiger, UC Davis Niches & Transitions
( Better Scores) 7/35

10. Comparing Models: AIC
BM
OU.1
OU.2
OU.3
−70 −60 −50 −40 −30 −20
AIC
Carl Boettiger, UC Davis Niches & Transitions
( Better Scores) 8/35

11. Comparing Models: Estimating Uncertainty
BM
OU.1
OU.2
OU.3
−70 −60 −50 −40 −30 −20
AIC
Carl Boettiger, UC Davis Niches & Transitions
( Better Scores) 9/35

12. Comparing Models: Uncertainty dominates
BM
OU.1
OU.2
OU.3
−70 −60 −50 −40 −30 −20
AIC
Carl Boettiger, UC Davis Niches & Transitions
( Better Scores) 10/35

13. Sources of this uncertainty
BM
OU.1
OU.2
OU.3
Small datasets
−70 −60 −50 −40 −30 −20
Uninformative topology AIC
Model details (i.e. high rates)
Carl Boettiger, UC Davis Niches & Transitions 11/35

14. Information criteria alone may be misleading
Carl Boettiger, UC Davis Niches & Transitions 12/35

15. A Better Way: Comparing Models Directly
300
BM
OU.2
250
200
−70 −60 −50 −40 −30 −20
Frequency
−2 Log Likelihood
L(BM)
150
−2 log
L(OU.2)
100
50
0
−10 0 10 20 30 40 50
Likelihood Ratio
Carl Boettiger, UC Davis Niches & Transitions 13/35

16. Method
300
BM
OU.2
250
1 Simulate many datasets
200
under model A −70 −60 −50 −40 −30 −20
Frequency
−2 Log Likelihood
150
100
50
0
−10 0 10 20 30 40 50
Likelihood Ratio
Carl Boettiger, UC Davis Niches & Transitions 14/35

17. Method
300
BM
OU.2
250
1 Simulate many datasets
200
under model A −70 −60 −50 −40 −30 −20
Frequency
−2 Log Likelihood
2 Re-fit both A & B to each
150
simulated dataset
100
50
0
−10 0 10 20 30 40 50
Likelihood Ratio
Carl Boettiger, UC Davis Niches & Transitions 14/35

18. Method
300
BM
OU.2
250
1 Simulate many datasets
200
under model A −70 −60 −50 −40 −30 −20
Frequency
−2 Log Likelihood
2 Re-fit both A & B to each
150
simulated dataset
100
3 Write log Likelihood(A) -
log Likelihood(B)
50
0
−10 0 10 20 30 40 50
Likelihood Ratio
Carl Boettiger, UC Davis Niches & Transitions 14/35

19. BM vs OU.2
300
BM
OU.2
250
200
−70 −60 −50 −40 −30 −20
Frequency
−2 Log Likelihood
150
p = 0.002
100
50
0
−10 0 10 20 30 40 50
Likelihood Ratio
Carl Boettiger, UC Davis Niches & Transitions 15/35

20. OU.2 vs BM: Simulating under OU.2
BM
OU.2
150
−70 −60 −50 −40 −30 −20
Frequency
100
−2 Log Likelihood
p = 0.237
50
0
−80 −60 −40 −20
Likelihood Ratio
Carl Boettiger, UC Davis Niches & Transitions 16/35

21. Model A rejects Model B.
Model B doesn’t reject Model A.
Carl Boettiger, UC Davis Niches & Transitions 17/35

22. How about two very similar models? BM vs OU.1
BM
OU.1
400
300
−70 −60 −50 −40 −30 −20
Frequency
−2 Log Likelihood
200
p = 0.778
100
0
0 10 20 30
Likelihood Ratio
Carl Boettiger, UC Davis Niches & Transitions 18/35

23. How would AIC rule compare?
BM
OU.1
400
300 AIC
p = 0.779
−70 −60 −50 −40 −30 −20
Frequency
−2 Log Likelihood
200
100
0
0 10 20 30
Likelihood Ratio
Carl Boettiger, UC Davis Niches & Transitions 19/35

24. When data is insufficient to distinguish,
method can say “I don’t know”
Carl Boettiger, UC Davis Niches & Transitions 20/35

25. Can we distinguish between OU.2 and OU.3?
OU.2
250
OU.3
200
−70 −60 −50 −40 −30 −20
150
Frequency
−2 Log Likelihood
100
50
0
−80 −60 −40 −20 0 20
Likelihood Ratio
Carl Boettiger, UC Davis Niches & Transitions 21/35

26. Simulate under OU.2 and compare to OU.3 . . .
OU.2
250
OU.3
200
−70 −60 −50 −40 −30 −20
150
Frequency
−2 Log Likelihood
p = 0.051
100
50
0
−80 −60 −40 −20 0 20
Likelihood Ratio
Carl Boettiger, UC Davis Niches & Transitions 22/35

27. OU.3 vs OU.2: Now we’re preferring OU.2!
350
OU.2
OU.3
300
250
−70 −60 −50 −40 −30 −20
200
Frequency
−2 Log Likelihood
p = 0.003
150
100
50
0
−60 −40 −20 0 20
Likelihood Ratio
Carl Boettiger, UC Davis Niches & Transitions 23/35

28. Model A rejects Model B.
Model B rejects Model A.
Carl Boettiger, UC Davis Niches & Transitions 24/35

29. Non-nested models
po po
se se
sc sc
sn sn
wb wb
wa wa
be be
bn bn
bc bc
lb lb
la la
nu nu
sa sa
gb gb
ga ga
gm gm
oc oc
fe fe
li li
mg mg
md md
t1 t1
t2 t2
Carl Boettiger, UC Davis Niches & Transitions 25/35

30. Replacing paintings with a transition model
Carl Boettiger, UC Davis Niches & Transitions 26/35

31. All models are nested
Estimate number of niches
Also estimate rates of transitions
Carl Boettiger, UC Davis Niches & Transitions 27/35

32. A hard problem in two easy pieces
P( | ) =
P( | )P( | )
Carl Boettiger, UC Davis Niches & Transitions 28/35

33. Summary
300
250
200
Frequency
150
p = 0.002
100
Quantifiable, robust model choice
50
1
0
−10 0 10 20 30 40 50
Likelihood Ratio
400
300
Frequency
Identify when data is insufficient
200
2 p = 0.778
100
0
0 10 20 30
Likelihood Ratio
3
New framework avoids painting &
non-nested comparison
Carl Boettiger, UC Davis Niches & Transitions 29/35

34. Thanks!
Chris Martin Graham Coop
Peter Wainwright Peter Ralph
Samantha Price Alan Hastings
Roi Holzman DoE CSGF
Carl Boettiger, UC Davis Niches & Transitions 30/35

35. State
Time
Carl Boettiger, UC Davis Niches & Transitions 31/35

36. State
Time
Carl Boettiger, UC Davis Niches & Transitions 32/35

37. State
Time
Carl Boettiger, UC Davis Niches & Transitions 33/35

38. Comparing Models
po po po
se se se
sc sc sc
sn sn sn
wb wb wb
wa wa wa
be be be
bn bn bn
bc bc bc
lb lb lb
la la la
nu nu nu
sa sa sa
gb gb gb
ga ga ga
gm gm gm
oc oc oc
fe fe fe
li li li
mg mg mg
md md md
t1 t1 t1
t2 t2 t2
Carl Boettiger, UC Davis Niches & Transitions 34/35

39. OU.3 vs OU.4: Why you should mistrust painting trees
100 150 200 250 300 350
OU.3
OU.4
−120 −100 −80 −60 −40 −20
Frequency
−2 Log Likelihood
p= 0
50
0
−50 0 50 100
Likelihood Ratio
Carl Boettiger, UC Davis Niches & Transitions 35/35