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Phylogenetic comparative methods Comparative studies (nuisance) Evolutionary studies (objective) Community ecology (lack of alternatives). Current growth of phylogenetic comparative methods New statistical methods Availability of phylogenies Culture. One of many possible types of problems. - PowerPoint PPT Presentation
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Phylogenetic comparative methods
Comparative studies (nuisance)
Evolutionary studies (objective)
Community ecology (lack of alternatives)
Current growth of phylogenetic comparative methods
New statistical methods
Availability of phylogenies
Culture
One of many possible types of problems
y=b0 +b1x+ε
€
y = b0 + ε
or as a special case
This model structure can be used for a variety of types of problems
y=b0 +b1x+εAssumptions:
y takes continuous values
x can be a random variable or a set of known values (continuous or not)
y is linearly related to x
are random variables with expectation 0 and finite (co)variances that are known
y=b0 +b1x+εStatistical methods
(P)IC = GLS
Phylogenetic independent contrastsGeneralized Least Squares
(these are methods, not models)
Other methods for other statistical models
ML, REML, EGLS, GLM, GLMM, GEE, “Bayesian” methods
y=b0 +b1x+ε
are random variables with expectation 0 and finite (co)variances that are known
Phylogeny provides a hypothesis for these covariances
Close Relatives Tend to Resemble Each Other
A
B
C
D
EF
G
H
A
B
C
D
E
F
G
H
I
0 1 2 3 4
-1
0
1
2
3
4
X
Y
A
B
C
D
EF
G
H
A
B
C
D
E
F
G
H
I
0 1 2 3 4
-1
0
1
2
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4
X
Y
What does this represent?
How is it constructed?
Is it known for certain?
A
B
C
D
EF
G
H
A
B
C
D
E
F
G
H
I
0 1 2 3 4
-1
0
1
2
3
4
X
Y
Assume that this represents time and
is known without error
Translate into the pattern of covariances
in among species
V
Hypothetical trait for a single species under Brownian motion evolution
Tra
it va
lue
Time
possible course of evolution
Tra
it va
lue
Time
another possible course of evolution
Tra
it va
lue
Time
another possible course of evolution
Brownian motion evolution gives the hypothetical variance of a trait
Tra
it va
lue
Time
Variance
Brownian motion evolutionT
rait
valu
e
Time
Variance
Brownian motion evolution of a hypothetical trait during speciation
Variance between species = Time
Total variance = Total time
Variance between species = Time
Covariance = Shared time
Total variance = Total time
Variance between species = Time
€
⇒ VBrownia
n motion
Covariance matrix giving phylogenetic covariances among species
diagonal elements give the total variance for species i
off-diagonal elements give covariances between species i and species j
€
v i i
€
v ij
€
V
A
B
C
D
EF
G
H
A
B
C
D
E
F
G
H
I
0 1 2 3 4
-1
0
1
2
3
4
X
Y
I am confused by the authors use of "branch lengths" on page 3023. I'm not sure if "different types of branch lengths" mean different phylogenetic analyses or something else I'm not aware of.
Digression - non-Brownian models of evolution
Ornstein-Uhlenbeck evolution
Stabilizing selection with strength given
by d
Time
selection
Variance between species < Time
Variance between species < Time
Total variance << Total time
Ornstein-Uhlenbeck evolution
Time
Variance
Stabilizing selection means information is “lost” through time
Phylogenetic correlations between species decrease
Phylogenetic Signal(Blomberg, Garland, and Ives 2003)
€
⇒ V(d)
€
V(d) =
measures the strength of signal
OU process
€
V(d) =
y=b0 +b1x+εAssumptions:
y takes continuous values
x can be a random variable or a set of known numbers
y is linearly related to x
are random variables with expectation 0 and finite (co)variances that are known
If d must be estimated, cannot be analyzed using PIC or GLS
If we are dealing with a recent, rapid radiation, (supported clade but with short branches) will the lack of branch length data render any PIC not very informative biologically, because we would expect non-significant probabilities, based solely on the branch lengths alone? page 3022, second paragraph.
Phylogenetic Signal(Blomberg, Garland, and Ives 2003)
€
⇒ V(d)
€
V(d) =
measures the strength of signal
OU process
y=b0 +b1x+εStatistical methods
(P)IC = GLS
Phylogenetic independent contrastsGeneralized Least Squares
(these are methods, not models)
Other methods for other statistical models
ML, REML, EGLS, GLM, GLMM, GEE, “Bayesian” methods
PIC
y1
y2
y3
y4
1
2
3
4
€
Δy ij = β1Δx ij + ν 'i +ν ' jε ij
€
'i = ν i +ν 'k ν 'l
ν 'k +ν 'l
y1
y2
y3
y4
1
2
3
4
€
y4 =y1 ν 1 + y2 ν 2
1 ν 1 +1 ν 2
=y1
ν 1
+y2
ν 2
⎛
⎝ ⎜
⎞
⎠ ⎟
ν 1ν 2
ν 1 + ν 2
⎛
⎝ ⎜
⎞
⎠ ⎟
€
Δy12 = y1 − y2
€
Δy34 = y3 − y4
€
'4 = ν 4 +ν 1ν 2
ν 1 + ν 2
PIC
€
Δy ij
ν 'i +ν ' j
= β1
Δx ij
ν 'i +ν ' j
+ ε ij
Regression through the origin
€
Δy ij = β1Δx ij + ν 'i +ν ' jε ij
PIC
€
Δy ij
ν 'i +ν ' j
= β1
Δx ij
ν 'i +ν ' j
+ ε ij
€
Δy ij
ν 'i +ν ' j
= β1
Δ˜ x iju'i +u' j
+ ε ij
You could also use different branch lengths for x:
Branch lengths of y
Branch lengths of x
PIC
€
Δy ij
ν 'i +ν ' j
= β1
Δx ij
ν 'i +ν ' j
+ ε ij
When could this be justified?
You could also use different branch lengths for x:
€
Δy ij
ν 'i +ν ' j
= β1
Δ˜ x iju'i +u' j
+ ε ij
When could this be justified?
€
Δy ij = β1Δx ij + ν 'i +ν ' jε ij
Never (?)
€
Δy ij
ν 'i +ν ' j
= β1
Δ˜ x iju'i +u' j
+ ε ij
y=b0 +b1x+εStatistical methods
(P)IC = GLS
Phylogenetic independent contrastsGeneralized Least Squares
(these are methods, not models)
Other methods for other statistical models
ML, REML, EGLS, GLM, GLMM, GEE, “Bayesian” methods
Elements of V are given by shared branch lengths under the assumption of “Brownian motion” evolution
E εε'[ ] =σ 2V≠σ 2I
y=b0 +b1x+ε
y= y1,y2,...,yn[ ]'
X= 1,x[ ]
b= b0,b1[ ]'
ˆ b = X'V−1X( )−1
X'V−1y( )
ˆ σ 2 = y−Xˆ b ( )'V−1 y−Xˆ b ( ) n−2( )
Generalized Least Squares, GLS
Ordinary least squares
€
ˆ b = X'X( )−1
X'y( )
ˆ σ 2 = y − X ˆ b ( )'
y − X ˆ b ( ) n − 2( )
V = I
DVD'=I
z=Dy
U =DX
Related to ordinary least squares
y=Xb+ε
Dy=DXb+Dε
z=Ub+α
€
z = Ub + α
E αα'[ ]=E Dε Dε( )'[ ]
=DE εε'[ ]D'
=Dσ 2VD'=σ 2I
€
z = Ub + α
Values of
€
z = Dy
are linear combinations of yi€
E αα '[ ] = σ 2I
A
B
C
D
EF
G
H
A
B
C
D
E
F
G
H
I
0 1 2 3 4
-1
0
1
2
3
4
X
Y
GLS LS
parameter true value estimate 95% confidence
interval
estimate 95% confidence
interval
b0 0 2.28 [-0.82, 5.38] -1.10 [-3.69, 1.49]
b1 0 -0.43 [-1.45, 0.60] 1.45 [0.28, 2.62]
σ2 2 3.35 1.39
{E Yh} 2.84 [ -0.35 , 6.03] 3.84 [0.35 , 7.33]
If IC and GLS can yield identical results and the authors refer to IC as "a special case of GLS models" (p. 3032), in what situation(s) would GLS be a more appropriate method? In other words, why not just use IC?
Divergence time for desert and montane ringtail populations assumed to be 10,000 years
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Predicting values for ancestral and new species
€
Δy ij = β1Δx ij + ν 'i +ν ' jε ij
A
B
C
D
EF
G
H
A
B
C
D
E
F
G
H
I
0 1 2 3 4
-1
0
1
2
3
4
X
Y
Is the prediction of the estimate of y for species I more or less precise than what you would expect from a standard regression analysis?
When dealing with multiple, incongruent gene trees, we can perform multiple PIC's on each tree, and find a correlation or not. How do we know which is the "right" answer?
The three main phylogenetically based statistical methods described in the reading (IC, GLS, and Monte Carlo simulations) rely on correct information about tree topology and branch lengths. If we are unsure of the correctness of these basic assumptions, what is the best way to analyze our data?
I'm unclear how data can be statistically significant when transformed, but not significant otherwise. This seems like cheating/lying.
The paper discussed researchers' decisions about branch lengths, especially in terms of transformations (OU, ACDC). Do researchers use ultrametric trees for these analyses?