Post on 23-Aug-2014
description
The adequacy of phylogenetic trait modelsMatthew Pennell @mwpennell
In collaboration withRich FitzJohn Will Cornwell Luke Harmon
●
●●
●●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●●
●
●●
●
●●
●
●●
●●
●
●●
●
●
●
●
●
●
R2=0.67; p=0.002 R2=0.67; p=0.002
R2=0.67; p=0.002 R2=0.67; p=0.002Anscombe 1973
Is the model appropriate?
If not, what are we missing?
Is the model appropriate?
And if not, what are we missing?
●
● ● ● ● ● ● ● ●● ●
For simple regression models
●
●
●
●
●
●
●
●
●
●
●
Coo
k’s d
istan
ce
Observation
●
●
●●
●
●
●
●
●
●
●
For simple regression modelsRe
sidua
ls
Fitted values
●
●
●
●
●
●
●
● ●
●
●
Statistical tests of model adequacycompliment visual intuition
For phylogenetic trait models
Plotting the relevant data is challenging
No general methods for assessing model adequacy
Especially for complex models
θ1
θ2
θ3
For phylogenetic trait models
Plotting the relevant data is challenging
No general methods for assessing model adequacy
Our approach
Establishing scope
Quantitative traits
Univariate trait models
Tip states assume to ~ multivariate Gaussian
Fit a model to comparative data
Use "tted parameters to simulate data
Compare observed to simulated data
The general idea
The general idea
Fit a model to comparative data
Use "tted parameters to simulate data
Compare observed to simulated data
The general idea
Fit a model to comparative data
Use "tted parameters to simulate data
Compare observed to simulated data
Old statistical idea
θ
Pr(D
|θ)
θ
Pr(θ
|D)
Parametric bootstrapping
Posterior predictive simulation
If we re-ran evolution, how likely are we to see a dataset like ours?
Simulated data similar to observedModel likely adequate
Simulated data very different from observedModel likely inadequate
Comparing observed to simulated data
No two datasets are exactly alike
Use test statistics to summarize data in meaningfulways
No two datasets are exactly alike
Use test statistics to summarize data in meaningfulways
Comparing observed to simulated data
Species are not independent data points
Calculate test-statistics on contrasts
Comparing observed to simulated data
Species are not independent data points
Calculate test statistics on contrasts
Comparing observed to simulated data
Independent contrasts
A
B
C
Ci
Cj
n-1contrasts for n tips
Under BM modelC ~ Gaussian(0, σ)
When model is not Brownian motion
Contrasts no longer expected to be ~ Gaussian
Rescale branch lengths of phylogeny
When model is not Brownian motion
Contrasts no longer expected to be ~ Gaussian
Rescale branch lengths of phylogeny
For models that predict tip states to be multivariate Gaussian
ln L = -0.5[n ln(2π) + ln|Σ| + (Y - μX)’Σ-1(Y - μX)]
For models that predict tip states to be multivariate Gaussian
ln L = -0.5[n ln(2π) + ln|Σ| + (Y - μX)’Σ-1(Y - μX)]
Y is the observed tip states for the n species
μ is the mean of observed data
X is a column vector of 1
Σ is the expected variance-covariance matrixfor the tip states under the model
For models that predict tip states to be multivariate Gaussian
ln L = -0.5[n ln(2π) + ln|Σ| + (Y - μX)’Σ-1(Y - μX)]
Y is the observed tip states for the n species
μ is the mean of observed data
X is a column vector of 1
Σ is the expected variance-covariance matrixfor the tip states under the model
For models that predict tip states to be multivariate Gaussian
ln L = -0.5[n ln(2π) + ln|Σ| + (Y - μX)’Σ-1(Y - μX)]
Y is the observed tip states for the n species
μ is the mean of observed data
X is a column vector of 1
Σ is the expected variance-covariance matrixfor the tip states under the model
For models that predict tip states to be multivariate Gaussian
ln L = -0.5[n ln(2π) + ln|Σ| + (Y - μX)’Σ-1(Y - μX)]
Y is the observed tip states for the n species
μ is the mean of observed data
X is a column vector of 1
Σ is the expected variance-covariance matrixfor the tip states under the model
The Σ matrix
If we "t a Ornstein-Uhlenbeck model
Σij = σ2/2α(1-e-2αT)e-αCij
The Σ matrix
If we "t a Ornstein-Uhlenbeck model
Σij = σ2/2α(1-e-2αT)e-αCij
σ2 rate of diffusion
α pull towards optimum
T tree height
Cij shared branch lengthbetween tips i and j
The Σ matrix
If we "t a Ornstein-Uhlenbeck model
Σij = σ2/2α(1-e-2αT)e-αCij
σ2 rate of diffusion
α pull towards optimum
T tree height
Cij shared branch lengthbetween tips i and j
The Σ matrix
If we "t a Ornstein-Uhlenbeck model
Σij = σ2/2α(1-e-2αT)e-αCij
σ2 rate of diffusion
α pull towards optimum
T tree height
Cij shared branch lengthbetween tips i and j
The Σ matrix
If we "t a Ornstein-Uhlenbeck model
Σij = σ2/2α(1-e-2αT)e-αCij
σ2 rate of diffusion
α pull towards optimum
T tree height
Cij shared branch lengthbetween tips i and j
Building a unit tree
Rescale branch lengths by the amount of co(variance) we expect to accumulate under the model
A
B
C
vi’ = ΣAB - ΣAC
vi
Unit tree example
Ornstein-Uhlenbeck modelσ2 = 0.5 | α = 1
A
B
C
A
B
C
The nice thing about unit trees
Transformation applies to most* models ofcontinuous trait evolution
If model is adequate, contrasts on unit tree will beI.I.D. ~ Gaussian(0, 1)
Also applies to PGLS-style models
Create unit tree from parameter estimates
Compute contrasts on the residuals
If model is adequate contrasts of residuals will beGaussian(0,1) - same test statistics apply
Can compute test statistics onunit tree contrasts to assess adequacy
Var(contrasts)
|Con
tras
ts|
Ancestral state Node height
Contrasts2
Den
sity
Den
sity
Contrasts XCu
mul
ativ
e Pr
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
}
|Con
tras
ts|
|Con
tras
ts|
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Var(contrasts)
|Con
tras
ts|
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
Ancestral state
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Node height
Contrasts2
Den
sity
Den
sity
Contrasts XCu
mul
ativ
e Pr
}
|Con
tras
ts|
|Con
tras
ts|
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Var(contrasts)
|Con
tras
ts|
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
Ancestral state
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Node height
Contrasts2
Den
sity
Den
sity
Contrasts XCu
mul
ativ
e Pr
}
|Con
tras
ts|
|Con
tras
ts|
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Var(contrasts)
|Con
tras
ts|
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
Ancestral state
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Node height
Contrasts2
Den
sity
Den
sity
Contrasts XCu
mul
ativ
e Pr
}
|Con
tras
ts|
|Con
tras
ts|
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Var(contrasts)
|Con
tras
ts|
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
Ancestral state
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Node height
Contrasts2
Den
sity
Den
sity
Contrasts XCu
mul
ativ
e Pr
}
|Con
tras
ts|
|Con
tras
ts|
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Var(contrasts)
|Con
tras
ts|
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
Ancestral state
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Node height
Contrasts2
Den
sity
Den
sity
Contrasts XCu
mul
ativ
e Pr
}
|Con
tras
ts|
|Con
tras
ts|
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Var(contrasts)
|Con
tras
ts|
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
Ancestral state
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Node height
Contrasts2
Den
sity
Den
sity
Contrasts XCu
mul
ativ
e Pr
}
|Con
tras
ts|
|Con
tras
ts|
Simulating new datasets
Tree has already been transformed
Simulate m new datasets under BM with σ2 = 1
Calculate test statistics on contrasts of simulated data
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
}●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
}
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
}
Compare observed test statistics todistribution of simulated test statistics
Putting it all together
Estimate θ
Estimate θ
Build unit tree
Estimate θ
Build unit tree
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
Test statisticsobs data
Estimate θ
Build unit tree
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
Test statisticsobs data
Simulate BM data
Estimate θ
Build unit tree
Test statisticsobs data
Simulate BM data
Test statisticssim data
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
Estimate θ
Build unit tree
Test statisticsobs data
Simulate BM data
Test statisticssim data
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
Compare sim to obstest statistics
https://github.com/mwpennell/arbutus
arbutus R package
Designed to interact with other R packages
Object-oriented
New models and test statistics can easily be added
arbutus R package
library(diversitree)lik <- make.bm(phy, data)div.fit <- find.mle(lik, x.init=1)
arbutus(div.fit)
library(geiger)g.fit <- fitContinuous(phy, data, model = “BM”)
arbutus(g.fit)
E.g.: seed mass evolution in Fagaceae
Ornstein-Uhlenbeck model
}
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Ornstein-Uhlenbeck model
}
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Are common trait models adequatefor real comparative data?
Analysis of 337 comparative datasets
Brownian motion
ree important plant functional traits
72 datasets (20-2,200 spp.) for speci"c leaf area
226 datasets (20-22,817 spp.) for seed mass
39 datasets (20-936 spp.) for leaf nitrogen
Wright et al. 2004Kleyer et al. 2008
Kew SID 2014
Brownian motionZanne et al. 2014
For each dataset
Fit three simple models of trait evolution (Brownian Motion, Ornstein-Uhlenbeck, Early Burst)
Compared model "t using AIC
Assessed the adequacy of the best-supported model
Model comparison using AIC
Datasets (1-337)
AIC
w
Brownian motion
Brownian motion Ornstein-Uhlenbeck Early Burst
Here’s the dark side
Best model rejected (p>0.05) - ML
72/72 speci"c leaf area datasets
185/226 seed mass datasets
39/39 leaf nitrogen datasets
p-values -- REML est. of σ2
p-value0 0.80
Den
sity
Speci"c leaf area Seed mass Leaf nitrogen
Models get worse as trees get bigger
Log(Tree Size)20 11,000
Dist
(sim
, obs
)
Speci"c leaf area Seed mass Leaf nitrogen
Simple, commonly used modelsare often woefully inadequate
But...we already knew that
We are (often) here
●●
●
●●
●
●●
●
●●
●
●●●
●
●
●
●
●
●
●
This is how we learn about biology!
Learn about issues with the data
●
●
●
●
●
●
●
●
●
●
●
Common issues with data
Phylogenetic error (topology & branch lengths)
Measurement error
Biologically interesting ‘outlier’ species
Learn about evolutionary processes
●
●
●●
●
●
●
●
●
●
●
Many ways to add complexity
Time heterogeneous models
Different models for different parts of the tree
Biologically motivated models
Test statistics can help us make informed decisions
May suggest types of models that have not even beendeveloped yet
Does it matter if a model is inadequate?
It depends on the question...
}
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
What is the rate of seed mass evolution?
Single optimum OU model is very misleading
It depends on the question...
What is the rate of seed mass evolution?
Single optimum OU model is very misleading
}
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
It depends on the question...
Was there an “early burst” in seed mass evolution?
Inadequate OU model likely doesn’t affect inference
}
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
It depends on the question...
Was there an “early burst” in seed mass evolution?
Inadequate OU model likely doesn’t affect inference
}
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
● ●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●●
●
●
● ●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
Model adequacy is not binary
Whether the model is “good enough” depends on what questions you are asking
Some concluding thoughts
Understanding how a model fails can provide interesting biological insights
Pay attention to parameter estimates
Look carefully at the data
Plot the test statistics
Keep the question in mind
Pay attention to parameter estimates
Look carefully at the data
Plot the test statistics
Keep the question in mind
Pay attention to parameter estimates
Look carefully at the data
Plot the test statistics
Keep the question in mind
Pay attention to parameter estimates
Look carefully at the data
Plot the test statistics
Keep the question in mind
Advice and encouragementJosef UyedaDaniel CaetanoPaul JoyceGraham Slater
Amy ZanneRoxana HickeyAnahi EspindolaSimon Uribe-Convers
FundingNSFNSERC
NESCentUniversity of Idaho
NESCent Tempo & mode working group
arbutus R packagehttps://github.com/mwpennell/arbutus
code to reproduce all analyseshttps://github.com/rich"tz/modeladequacy
preprinthttp://biorxiv.org/content/early/2014/04/07/004002