Why do models matter?
• Model-based methods including ML and Bayesian
inference (typically) make a consistent estimate of
the phylogeny (estimate converges to true tree as
number of sites increases toward infinity)
... even when you’re in the “Felsenstein Zone”
A
C
(Felsenstein, 1978)
B
D
In the Felsenstein Zone
Proportion Correct
1
0.8
parsimony
0.6
ML-GTR
0.4
0.2
0
0
5000
Sequence Length
Sequence Length
Simulation model = GTR
10000
Why do models matter (continued)?
• Parsimony is inconsistent in the Felsenstein zone
(and other scenarios)
• Likelihood is consistent in any “zone” (when certain
requirements are met)
But this guarantee requires that the
model be specified correctly!
Likelihood can also be inconsistent if the
model is oversimplified
• Real data always evolve according to processes
more complex than any computationally feasible
model would permit, so we have to choose “good”
rather than “correct” models
What is a “good” model?
• A model that appropriately balances fit of the data
with simplicity (parsimony, in a different sense)
i.e., if a simpler model fits the data almost as well as
a more complex model, prefer the simpler one
100
120
B
80
80
60
B
B
y
40
y
B
40
B
25
B
B
-40
B
B
0
B
B
B
B
0 B
B
20
0
B
50
x
75
100
-80
0
25
50
x
75
100
y = - 330 +134x - 15.5x 2 +0.816x 3
y =1.30 + 0.965x
- 0.0225x 4 + 0.000335x 5
(r 2 = 0.963)
- 0.00000255x 6 +0.00000000777x 7
(r 2 =1.000)
“The Principle of Parsimony” in the
world of statistics
•
Burnham and Anderson (1998): Model Selection and Inference
– Parsimony lies between the evils of underfitting and overfitting. The
concept of parsimony has a long history in in the sciences. Often this has
been expressed as “Occam’s razor”—shave away all that is not necessary.
Parsimony in statistics represents a tradeoff between bias and variance as
a function of the dimension of the model. A good model is a balance
between under- and over-fitting.
Why models don’t have to be perfect
Assertion: In most situations, phylogenetic inference is relatively
robust to model misspecification, as long as critical factors
influencing sequence evolution are accommodated
Caveat: There are some kinds of model misspecification that
are very difficult to overcome (e.g., “heterotachy”)
E.g.:
A
C
B
D
Half of sites
A
C
D
B
Other half
Likelihood can be consistent in Felsenstein zone, but will be
inconsistent if a single set of branch lengths are assumed when
there are actually two sets of branch lengths (Chang 1996)
GTR Family of Reversible DNA Substitution Models
(general time-reversible)
GTR
3 substitution types
(transversions, 2 transition classes)
Equal base frequencies
TrN
SYM
(Tamura-Nei)
3 substitution types
(transitions,
2 transversion classes)
2 substitution types
(transitions vs.
transversions)
(Hasegawa-Kishino-Yano)
(Felsenstein)
HKY85
F84
Equal base
frequencies
Single substitution type
(Felsenstein)
K3ST
2 substitution types
(transitions vs. transversions)
K2P
F81
Equal base frequencies
(Kimura 3-subst. type)
(Kimura 2-parameter)
Single substitution type
JC
Jukes-Cantor
Among site rate heterogeneity
equal rates?
Lemur
Homo
Pan
Goril
Pongo
Hylo
Maca
•
…TTACATCATCCA
…TTACATCCTCAT
…TTACATCCTCAT
…CCCACGGACTTA
…GCAACCACCCTC
…TGCAACCGTCCT
…CGCAACCATCCT
Some sites extremely unlikely to change due to strong functional or structural
constraint (Hasegawa et al., 1985)
Gamma-distributed rates
–
•
TTGCATCATCCA
TTGCATCATCCA
TTACGCCATCCA
TTACGCCATCCA
TTACGCCATCCT
TTACATTATCCG
TTACATTATCCG
Proportion of invariable sites
–
•
AAGCTTCATAG
AAGCTTCACCG
AAGCTTCACCG
AAGCTTCACCG
AAGCTTCACCG
AAGCTTTACAG
AAGCTTTTCCG
Rate variation assumed to follow a gamma distribution with shape parameter α
Site-specific rates (another way to model ASRV)
–
Different relative rates assumed for pre-assigned subsets of sites
Modeling ASRV with gamma distribution
0.08
α=200
Frequency
0.06
α=0.5
α=2
0.04
α=50
0.02
0
0
1
2
Rate
…can also include a proportion of “invariable” sites (pinv)
Performance of ML when its model is violated
Tree
α = 0.5, pinv=0.5
α = 1.0, pinv=0.5
1
1
1
0.9
0.9
0.9
0.8
0.8
0.7
GTRig
0.6
HKYig
0.4
HKYg
GTRi
0.3
HKYi
0.6
0.1
0.1
0.4
0.3
HKYer
0.2
parsimony
0.1
0
0
100
1000
10000
100000
100
1000
10000
100
100000
1
1
1
0.9
0.9
0.9
0.8
0.8
0.7
GTRig
GTRig
0.6
HKYig
GTRg
0.5
GTRg
0.5
0.2
parsimony
0
100
0.4
HKYg
GTRi
0.3
HKYi
GTRer
0.2
HKYer
0.1
0.8
0.7
0.5
GTRer
HKYer
0.1
10000
100000
100
0.4
GTRg
HKYg
GTRi
0.3
HKYi
GTRer
0.2
parsimony
HKYer
parsimony
0.1
0
0
1000
1000
10000
100000
100
1
1
1
0.9
0.9
0.9
0.8
0.8
0.7
GTRig
HKYig
0.6
GTRg
HKYg
GTRi
HKYi
GTRer
0.5
0.4
0.3
0.2
0.1
100
GTRig
0.7
0.6
HKYig
0.6
GTRg
0.3
10000
100000
0.2
0.3
0.2
HKYer
0.1
100
100000
0.4
GTRer
parsimony
0
1000
10000
0.5
HKYg
GTRi
HKYi
0.4
0
1000
0.8
0.7
0.5
HKYer
parsimony
100000
HKYig
0.6
0.3
10000
GTRig
HKYig
0.4
1000
0.7
0.6
HKYg
GTRi
HKYi
GTRig
HYYig
GTRg
HKYg
GTRi
HKYi
GRTer
HKYer
parsimony
0.5
GTRer
0.2
0
0.6
GTRi
HKYi
0.3
Parsimony
0.7
HKYig
HKTg
0.4
HKYer
GTRig
GTRg
0.5
GTRer
0.2
0.8
0.7
GTRg
0.5
Proportion Correct
α = 1.0, pinv=0.2
0.1
0
1000
10000
Sequence Length
100000
100
1000
10000
100000
“MODERATE”–Felsenstein zone
α = 1.0, pinv=0.5
1
0.9
0.8
JCer
0.7
JC+G
0.6
JC+I
JC+I+G
GTRer
GTR+G
0.5
0.4
GTR+I
0.3
GTR+I+G
parsimony
0.2
0.1
0
100
1000
10000
100000
“MODERATE”–InverseFelsenstein zone
1
0.9
0.8
JCer
0.7
JC+G
JC+I
JC+I+G
0.6
0.5
GTRer
GTR+G
0.4
GTR+I
0.3
GTR+I+G
parsimony
0.2
0.1
0
100
1000
10000
100000
Model selection criteria
• Likelihood ratio tests
δ = −2(ln L0 − ln L1 )
€
If model L0 is nested within model L1, δ is distributed
as χ2 with degrees-of-freedom equal to difference in
number of free parameters
• Akaike information criterion (AIC)
AICi = −2ln Li + 2K
where K is the number of free parameters estimated
• Bayesian information criterion (BIC)
€
BICi = −2ln Li + K ln n
where K is the number of free parameters estimated
and n is the “sample size” (typically number of sites)
€