The document discusses assessing the adequacy of phylogenetic trait models by comparing observed trait data to simulated trait data generated under the fitted model. It proposes building a "unit tree" by rescaling branch lengths based on the expected covariance structure implied by the model, so that contrasts on this tree should be independent and normally distributed if the model is adequate. A variety of test statistics can then be computed on the contrasts of the observed data and compared to the distribution of those statistics on simulated contrasts to evaluate the model.

1. The adequacy of phylogenetic
trait models
Matthew Pennell
@mwpennell

2. In collaboration with
Rich FitzJohn Will Cornwell Luke Harmon

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R2=0.67; p=0.002 R2=0.67; p=0.002
R2=0.67; p=0.002 R2=0.67; p=0.002
Anscombe 1973

5. Is the model appropriate?
If not, what are we missing?

6. Is the model appropriate?
And if not, what are we missing?

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For simple regression models
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Cook’sdistance
Observation

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For simple regression models
Residuals
Fitted values
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9. Statistical tests of model adequacy
compliment visual intuition

10. For phylogenetic trait models
Plotting the relevant data is challenging
No general methods for assessing model adequacy

11. Especially for complex models
θ1
θ2
θ3

12. For phylogenetic trait models
Plotting the relevant data is challenging
No general methods for assessing model adequacy

13. Our approach

14. Establishing scope
Quantitative traits
Univariate trait models
Tip states assume to ~ multivariate Gaussian

15. Fit a model to comparative data
Use fitted parameters to simulate data
Compare observed to simulated data
The general idea

16. The general idea
Fit a model to comparative data
Use fitted parameters to simulate data
Compare observed to simulated data

17. The general idea
Fit a model to comparative data
Use fitted parameters to simulate data
Compare observed to simulated data

18. Old statistical idea
θ
Pr(D|θ)
θ
Pr(θ|D)
Parametric bootstrapping
Posterior predictive simulation

19. If we re-ran evolution, how likely
are we to see a dataset like ours?

20. Simulated data similar to observed
Model likely adequate
Simulated data very different from observed
Model likely inadequate

21. Comparing observed to simulated data
No two datasets are exactly alike
Use test statistics to summarize data in meaningful
ways

22. No two datasets are exactly alike
Use test statistics to summarize data in meaningful
ways
Comparing observed to simulated data

23. Species are not independent data points
Calculate test-statistics on contrasts
Comparing observed to simulated data

24. Species are not independent data points
Calculate test statistics on contrasts
Comparing observed to simulated data

25. Independent contrasts
A
B
C
Ci
Cj
n-1contrasts for n tips
Under BM model
C ~ Gaussian(0, σ)

26. When model is not Brownian motion
Contrasts no longer expected to be ~ Gaussian
Rescale branch lengths of phylogeny

27. When model is not Brownian motion
Contrasts no longer expected to be ~ Gaussian
Rescale branch lengths of phylogeny

28. 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)]

29. 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 matrix
for the tip states under the model

30. 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 matrix
for the tip states under the model

31. 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 matrix
for the tip states under the model

32. 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 matrix
for the tip states under the model

33. The Σ matrix
If we fit a Ornstein-Uhlenbeck model
Σij = σ2/2α(1-e-2αT)e-αCij

34. The Σ matrix
If we fit 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 length
between tips i and j

35. The Σ matrix
If we fit 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 length
between tips i and j

36. The Σ matrix
If we fit 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 length
between tips i and j

37. The Σ matrix
If we fit 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 length
between tips i and j

38. 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

39. Unit tree example
Ornstein-Uhlenbeck model
σ2 = 0.5 | α = 1
A
B
C
A
B
C

40. The nice thing about unit trees
Transformation applies to most* models of
continuous trait evolution
If model is adequate, contrasts on unit tree will be
I.I.D. ~ Gaussian(0, 1)

41. 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 be
Gaussian(0,1) - same test statistics apply

42. Can compute test statistics on
unit tree contrasts to assess adequacy

43. Var(contrasts)
|Contrasts|
Ancestral state Node height
Contrasts2
Density
Density
Contrasts X
CumulativePr
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}
|Contrasts|
|Contrasts|

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Var(contrasts)
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Ancestral state
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Node height
Contrasts2
Density
Density
Contrasts X
CumulativePr
}
|Contrasts|
|Contrasts|

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Node height
Contrasts2
Density
Density
Contrasts X
CumulativePr
}
|Contrasts|
|Contrasts|

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Ancestral state
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Node height
Contrasts2
Density
Density
Contrasts X
CumulativePr
}
|Contrasts|
|Contrasts|

47. ●
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Var(contrasts)
|Contrasts|
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Ancestral state
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Node height
Contrasts2
Density
Density
Contrasts X
CumulativePr
}
|Contrasts|
|Contrasts|

48. ●
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Var(contrasts)
|Contrasts|
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Ancestral state
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Node height
Contrasts2
Density
Density
Contrasts X
CumulativePr
}
|Contrasts|
|Contrasts|

49. ●
●
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Var(contrasts)
|Contrasts|
●
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Ancestral state
●
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Node height
Contrasts2
Density
Density
Contrasts X
CumulativePr
}
|Contrasts|
|Contrasts|

50. Simulating new datasets
Tree has already been transformed
Simulate m new datasets under BM with σ2 = 1

51. Calculate test statistics on contrasts
of simulated data
●
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}

52. Compare observed test statistics to
distribution of simulated test statistics

53. Putting it all together

54. Estimate θ

55. Estimate θ
Build unit tree

56. Estimate θ
Build unit tree
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Test statistics
obs data

57. Estimate θ
Build unit tree
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Test statistics
obs data
Simulate BM data

58. Estimate θ
Build unit tree
Test statistics
obs data
Simulate BM data
Test statistics
sim data
●
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●
●
●
●
● ●
●
●

59. Estimate θ
Build unit tree
Test statistics
obs data
Simulate BM data
Test statistics
sim data
●
●
●
●
●
●
●
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●
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●
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Compare sim to obs
test statistics

60. https://github.com/mwpennell/arbutus

61. arbutus R package
Designed to interact with other R packages
Object-oriented
New models and test statistics can easily be added

62. 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)

63. E.g.: seed mass evolution in Fagaceae

64. Ornstein-Uhlenbeck model
}
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65. Ornstein-Uhlenbeck model
}
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66. Are common trait models adequate
for real comparative data?

67. Analysis of 337 comparative datasets
ree important plant functional traits
72 datasets (20-2,200 spp.) for specific leaf area
226 datasets (20-22,817 spp.) for seed mass
39 datasets (20-936 spp.) for leaf nitrogen
Wright et al. 2004
Kleyer et al. 2008
Kew SID 2014

68. Zanne et al. 2014

69. For each dataset
Fit three simple models of trait evolution
(Brownian Motion, Ornstein-Uhlenbeck, Early Burst)
Compared model fit using AIC
Assessed the adequacy of the best-supported model

70. Model comparison using AIC
Datasets (1-337)
AICw
Brownian motion Ornstein-Uhlenbeck Early Burst

71. Here’s the dark side

72. Best model rejected (p>0.05) - ML
72/72 specific leaf area datasets
185/226 seed mass datasets
39/39 leaf nitrogen datasets

73. p-values -- REML est. of σ2
p-value0 0.80
Density
Specific leaf area Seed mass Leaf nitrogen

74. Models get worse as trees get bigger
Log(Tree Size)20 11,000
Dist(sim,obs)
Specific leaf area Seed mass Leaf nitrogen

75. Simple, commonly used models
are often woefully inadequate

76. But...we already knew that

77. We are (often) here
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78. This is how we learn about biology!

79. Learn about issues with the data
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80. Common issues with data
Phylogenetic error (topology & branch lengths)
Measurement error
Biologically interesting ‘outlier’ species

81. Learn about evolutionary processes
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82. Many ways to add complexity
Time heterogeneous models
Different models for different parts of the tree
Biologically motivated models

83. Test statistics can help us make
informed decisions
May suggest types of models that have not even been
developed yet

84. Does it matter if a model is inadequate?

85. It depends on the question...
}
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What is the rate of seed mass evolution?
Single optimum OU model is very misleading

86. It depends on the question...
What is the rate of seed mass evolution?
Single optimum OU model is very misleading
}
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87. It depends on the question...
Was there an “early burst” in seed mass evolution?
Inadequate OU model likely doesn’t affect inference
}
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88. It depends on the question...
Was there an “early burst” in seed mass evolution?
Inadequate OU model likely doesn’t affect inference
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89. Model adequacy is not binary
Whether the model is “good enough”
depends on what questions you are asking

90. Some concluding thoughts

91. Understanding how a model fails can
provide interesting biological insights

92. Pay attention to parameter estimates
Look carefully at the data
Plot the test statistics
Keep the question in mind

93. Pay attention to parameter estimates
Look carefully at the data
Plot the test statistics
Keep the question in mind

94. Pay attention to parameter estimates
Look carefully at the data
Plot the test statistics
Keep the question in mind

95. Pay attention to parameter estimates
Look carefully at the data
Plot the test statistics
Keep the question in mind

96. Advice and encouragement
Josef Uyeda
Daniel Caetano
Paul Joyce
Graham Slater
Amy Zanne
Roxana Hickey
Anahi Espindola
Simon Uribe-Convers
Funding
NSF
NSERC
NESCent
University of Idaho
NESCent Tempo & mode working group

97. arbutus R package
https://github.com/mwpennell/arbutus
code to reproduce all analyses
https://github.com/richfitz/modeladequacy
preprint
http://biorxiv.org/content/early/2014/04/07/004002