BioInfoSummer 2013

Introduction to Phylogenetics
Barbara Holland
What is phylogenetics?
The goal of phylogenetics is to infer
evolutionary relationships between species.
This includes bot...
Ernst Haeckel’s Tree of Life (1866)

Since the publication of Origin
of the Species in 1859 people have been
trying to inf...
Why might we care?
• Understanding human origins

• Understanding biogeography, e.g. what’s the relative importance of
dis...
Understanding the emergence of disease

http://evolution.berkeley.edu/evolibrary/news/081101_hivorigins
Special reasons for Australians to care…

http://archive.peabody.yale.edu/exhibits/treeoflife/predictions.html
How to “read” a tree
Cat

Dog

Rat

Mouse

A tree is a connected acyclic graph
Composed of:
Nodes / vertices
Edges / arcs ...
Tree basics
•
•
•

Trees can be rooted or unrooted
Rooted trees are more interpretable
But, most software returns unrooted...
Tree basics
•

Trees can be weighted or unweighted

Cat

Rat

Dog
Cat

Rat
Mouse

•

Dog

Edge weights are used to represe...
Tree basics
•

Trees can be bifurcating (binary) or multifurcating (non-binary)

polytomy
(multifurcation)

Human Chimp Go...
The same tree can look different

A

B

C

D

E

F

F

E

A

D

C

B
Newick format
A computer readable format for
describing trees that uses brackets
and commas.

((A,(B,(C,D))),(E,F));

A

B...
What data can we use?
• Molecular sequence data
– DNA alignments*
– Amino acid alignments

• Presence absence data
– Gene ...
The molecular phylogeny problem
ACCGCTTA

ACTGCTTA

Time

ACCCCTTA

ACCCCTTA
We see the aligned
modern day sequences

ACCC...
Sometimes the data agrees
ACCGCTTA

ACTGCTTA

Time

ACCCCTTA

ACCCCTTA

ACCCCATA ACTGCTTA

ACCCCTTA
ACCCCATA
ACTGCTTA
ACTG...
Sometimes not
ACCGCTTA

ACTGCTTA

Time

ACCCCTTA

ACCCCTTC

ACCCCATA ACTGCTTC

ACCCCTTC
ACCCCATA
ACTGCTTC
ACTGCTAA

ACTGCT...
How can we choose the best tree?
• Distance data
– Quick methods that take a greedy clustering approach (e.g.
Neighbour-Jo...
Parsimony






To decide which tree is best we can use an optimality criterion. This
means we need a way of assigning ...
S1
S2
S3
S4
(1,2),(3,4)
(1,3),(2,4)

1

2

3

4

ACCCCTTC
ACCCCATA
ACTGCTTC
ACTGCTAA

1

3

2

4
S1
S2
S3
S4
(1,2),(3,4)
(1,3),(2,4)

A

A

A

A

ACCCCTTC
ACCCCATA
ACTGCTTC
ACTGCTAA
0
0

A

A

A

A
S1
S2
S3
S4
(1,2),(3,4)
(1,3),(2,4)

C

T
C

C

ACCCCTTC
ACCCCATA
ACTGCTTC
ACTGCTAA
001
002

C

C

T

T

T

T
S1
S2
S3
S4
(1,2),(3,4)
(1,3),(2,4)

C

G
C

C

ACCCCTTC
ACCCCATA
ACTGCTTC
ACTGCTAA
0011
0022

C

C

G

G

G

G
S1
S2
S3
S4
(1,2),(3,4)
(1,3),(2,4)

T

A

T

T

ACCCCTTC
ACCCCATA
ACTGCTTC
ACTGCTAA
001101
002201

T

T

A

T
S1
S2
S3
S4
(1,2),(3,4)
(1,3),(2,4)

T

T

T

A

ACCCCTTC
ACCCCATA
ACTGCTTC
ACTGCTAA
0011011
0022011

T

T

T

A
S1
S2
S3
S4
(1,2),(3,4)
(1,3),(2,4)

C

C

ACCCCTTC
ACCCCATA
ACTGCTTC
ACTGCTAA
00110112 6
00220111 7

C

A
C

A

A

C

A

...
The “large parsimony problem”




The small parsimony problem – to find the score of a given
tree - can be solved in lin...
How many trees are there?
#species

#unrooted binary
tip-labelled trees

4

3

5

3*5=15

6

3*5*7=105

7

3*5*7*9=945

10...
Counting trees
1

1
2
1

3

1

3

1

2

2
1x3=3

2
1
3

5

4
2

1

4 5

3

3

3

4 4

2 5
43

1

2
4

1
3

2

51
43

4

2
...
Search strategies
• Exact search: possible for small numbers of taxa (~12 or less) only
• Branch and Bound: A smarter way ...
The “Felsenstein Zone”




Parsimony was once a very popular method of phylogenetic inference but it
has fallen from gra...
A

Felsenstein's example

C

• The true tree is ((A,B),C,D)
• The only parsimony informative patterns are of the
form: xxy...
Inconsistency with a clock
Felsenstein’s 4 taxon example appeared in a
1978 paper “Cases in which parsimony or
compatibili...
Enter statistical phylogenetics
•

Over the last 30 years model-based methods of phylogenetic inference
have come to the f...
Sequence evolution is modelled as a
Markov process
A

Consider a single edge in a phylogeny, i.e. evolution of
a single sp...
Continuous time Markov chains
A

G

pAA

pAC

pAG

pCA

pCC

pCG

pCT

G

pGA

pGC

pGG

pGT

T

pTA

pTC

pTG

pTT

C

G
...
Example with a simple model
2-state symmetric model
(R := AG purines, Y := CT pyrimidines)

α
R

Y

Each edge i has probab...
Example with a simple model
Y

R
0.1

0.15
0.2

?

?

0.1

0.1

R

R
R
R

Y
R
Y

…R…
…R…
…Y…
…Y…

a
b
c
d

Y
R
R
R

Y
Y

R...
In theory you can then…
• Multiply the likelihoods over all sites to get the likelihood of a particular
weighted tree (in ...
Computational complexity
• Maximum likelihood is computationally complex.
• Fortunately the “tiny problem”, finding the li...
Models of nucleotide substitution
•

Jukes Cantor (JC)
–
–

•

–

α

Transitions and transversions at different rates
Base...
Models of nucleotide substitution
•

Jukes Cantor (JC)
–
–

•

–

β

Transitions and transversions at different rates
Base...
Models of nucleotide substitution
•

Jukes Cantor (JC)
–
–

•

–

β

Transitions and transversions at different rates
Base...
Models of nucleotide substitution
•

Jukes Cantor (JC)
–
–

•

–

α

Transitions and transversions at different rates
Base...
Extra features that can be modelled
•
•
•

Site to site rate variation (usually modelled by a gamma distribution)
Invarian...
Stochastic versus Systematic error
• Rather than just getting a point estimate of the tree we are usually
interested in ge...
Assessing confidence in trees
• We would like some measure of confidence in the inferred tree.
– Is the tree likely to cha...
The bootstrap
• In 1985 Felsenstein introduced the idea of the bootstrap to phylogenetics.

• For each bootstrap sample
– ...
1234567
ATATAAA
ATTATAA
TAAAATA
TATAAAT

a
b
c
d
1224567
ATTTAAA
ATTATAA
TAAAATA
TAAAAAT

a
b
c
d
a
b

c
d

1334567
AAATAA...
Example where the bootstrap is useful
• Simulate data on the four taxon tree below (JC model)
• Use sequence lengths of 10...
Example where the bootstrap is not so
useful
• Simulate data on the two four-taxon trees below (JC model) in the
proportio...
Mistaking precision for accuracy
106 nuclear genes: Different methods provide conflicting Yeast
topologies, each with 100%...
Phylogenomics
• With increasing quantities of sequence data it is common to get very well
resolved trees, i.e. bootstrap v...
Testing model fit
• In phylogenetic studies we often ask questions of the form “Is model A
better than model B?” (relative...
Absolute GOF tests
• Absolute tests of goodness-of-fit are fiddlier to implement than relative
tests. They typically requi...
Introduction to Phylogenetics - Barbara Holland
Introduction to Phylogenetics - Barbara Holland
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Introduction to Phylogenetics - Barbara Holland

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Molecular sequences provide an excellent source of data for attempting to infer the evolutionary relationships between species. This talk will introduce model-based methods of phylogenetic inference (maximum likelihood and Bayesian inference) and discuss both the strengths and limitations of current approaches.

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Introduction to Phylogenetics - Barbara Holland

  1. 1. BioInfoSummer 2013 Introduction to Phylogenetics Barbara Holland
  2. 2. What is phylogenetics? The goal of phylogenetics is to infer evolutionary relationships between species. This includes both information about order of branching, .e.g., did humans and chimpanzees share a common ancestor more recently than humans, chimps and gorillas? And information about timing of events, e.g., how long ago did humans and chimps share an ancestor? Darwin’s sketch: the first phylogenetic tree?
  3. 3. Ernst Haeckel’s Tree of Life (1866) Since the publication of Origin of the Species in 1859 people have been trying to infer the evolutionary “Tree of Life”.
  4. 4. Why might we care? • Understanding human origins • Understanding biogeography, e.g. what’s the relative importance of dispersal versus vicariance? • Understanding the origin of particular traits • Understanding the processes of molecular evolution • Learning about the tempo of evolution, e.g. was the Cambrian explosion really an explosion? Did mammals and birds wait until dinosaurs went extinct to inherit the earth or were they already started before the asteroid hit? • Origin of disease, e.g. where did humans get AIDs from?
  5. 5. Understanding the emergence of disease http://evolution.berkeley.edu/evolibrary/news/081101_hivorigins
  6. 6. Special reasons for Australians to care… http://archive.peabody.yale.edu/exhibits/treeoflife/predictions.html
  7. 7. How to “read” a tree Cat Dog Rat Mouse A tree is a connected acyclic graph Composed of: Nodes / vertices Edges / arcs / branches Pendent (external) edges versus internal edges In a phylogenetic tree the external (degree 1) nodes are associated with labels
  8. 8. Tree basics • • • Trees can be rooted or unrooted Rooted trees are more interpretable But, most software returns unrooted trees root Cat Dog Cat Dog Rat Mouse Rat Mouse
  9. 9. Tree basics • Trees can be weighted or unweighted Cat Rat Dog Cat Rat Mouse • Dog Edge weights are used to represent – the amount of genetic change along the edge – time Mouse
  10. 10. Tree basics • Trees can be bifurcating (binary) or multifurcating (non-binary) polytomy (multifurcation) Human Chimp Gorilla Monkey Polytomies usually represent uncertainty (a soft polytomy), but sometimes they are used to mean that everything happened at exactly the same time (a hard polytomy).
  11. 11. The same tree can look different A B C D E F F E A D C B
  12. 12. Newick format A computer readable format for describing trees that uses brackets and commas. ((A,(B,(C,D))),(E,F)); A B C D E Treeview / Dendroscope F ( , ); ((A, ),(E,F)); ((A,(B, )),(E,F));
  13. 13. What data can we use? • Molecular sequence data – DNA alignments* – Amino acid alignments • Presence absence data – Gene content – Fragment based methods (AFLP, DArT) – SNP chips • Genetic distances (DNA-DNA hybridisation, immunology) • Rare traits – Gene order – Introns? – SINEs and LINEs (short and long interspersed retro-transposable elements ) • Morphological data * Alignment is an important problem in its own right
  14. 14. The molecular phylogeny problem ACCGCTTA ACTGCTTA Time ACCCCTTA ACCCCTTA We see the aligned modern day sequences ACCCCATA ACTGCTTA …ACCCCTTA… …ACCCCATA… …ACTGCTTA… …ACTGCTAA… ACTGCTAA And want to recover the underlying evolutionary tree.
  15. 15. Sometimes the data agrees ACCGCTTA ACTGCTTA Time ACCCCTTA ACCCCTTA ACCCCATA ACTGCTTA ACCCCTTA ACCCCATA ACTGCTTA ACTGCTAA ACTGCTAA
  16. 16. Sometimes not ACCGCTTA ACTGCTTA Time ACCCCTTA ACCCCTTC ACCCCATA ACTGCTTC ACCCCTTC ACCCCATA ACTGCTTC ACTGCTAA ACTGCTAA
  17. 17. How can we choose the best tree? • Distance data – Quick methods that take a greedy clustering approach (e.g. Neighbour-Joining) – Optimality criteria that assign each tree a score (Minimum evolution, fastME) • Character data (e.g. a sequence alignment) – Non-parametric methods (parsimony) – Model-based methods (maximum likelihood, Bayesian)
  18. 18. Parsimony    To decide which tree is best we can use an optimality criterion. This means we need a way of assigning each possible tree a score. Maximum Parsimony is one such criterion. It chooses the tree which requires the fewest substitutions to explain the data. The Principle of Parsimony is the general scientific principle that accepts the simplest of two explanations as preferable.
  19. 19. S1 S2 S3 S4 (1,2),(3,4) (1,3),(2,4) 1 2 3 4 ACCCCTTC ACCCCATA ACTGCTTC ACTGCTAA 1 3 2 4
  20. 20. S1 S2 S3 S4 (1,2),(3,4) (1,3),(2,4) A A A A ACCCCTTC ACCCCATA ACTGCTTC ACTGCTAA 0 0 A A A A
  21. 21. S1 S2 S3 S4 (1,2),(3,4) (1,3),(2,4) C T C C ACCCCTTC ACCCCATA ACTGCTTC ACTGCTAA 001 002 C C T T T T
  22. 22. S1 S2 S3 S4 (1,2),(3,4) (1,3),(2,4) C G C C ACCCCTTC ACCCCATA ACTGCTTC ACTGCTAA 0011 0022 C C G G G G
  23. 23. S1 S2 S3 S4 (1,2),(3,4) (1,3),(2,4) T A T T ACCCCTTC ACCCCATA ACTGCTTC ACTGCTAA 001101 002201 T T A T
  24. 24. S1 S2 S3 S4 (1,2),(3,4) (1,3),(2,4) T T T A ACCCCTTC ACCCCATA ACTGCTTC ACTGCTAA 0011011 0022011 T T T A
  25. 25. S1 S2 S3 S4 (1,2),(3,4) (1,3),(2,4) C C ACCCCTTC ACCCCATA ACTGCTTC ACTGCTAA 00110112 6 00220111 7 C A C A A C A A According to the parsimony optimality criterion we should prefer the tree (1,2),(3,4) over the tree (1,3),(2,4) as it requires the fewest mutations.
  26. 26. The “large parsimony problem”   The small parsimony problem – to find the score of a given tree - can be solved in linear time in the size of the tree (using the Fitch or Sankoff algorithm). The large parsimony problem, finding the tree with minimum score, is known to be NP-Hard.
  27. 27. How many trees are there? #species #unrooted binary tip-labelled trees 4 3 5 3*5=15 6 3*5*7=105 7 3*5*7*9=945 10 2,027,025 20 2.2*1020 n (2n-5)!! An exact search for the best tree, where each tree is evaluated according to some optimality criterion such as parsimony quickly becomes intractable as the number of species increases
  28. 28. Counting trees 1 1 2 1 3 1 3 1 2 2 1x3=3 2 1 3 5 4 2 1 4 5 3 3 3 4 4 2 5 43 1 2 4 1 3 2 51 43 4 2 5 1 x 3 x 5 = 15
  29. 29. Search strategies • Exact search: possible for small numbers of taxa (~12 or less) only • Branch and Bound: A smarter way of doing exact search, up to ~20 taxa • Greedy search: sequentially build up the tree with no backtracking (e.g. Neighbour-Joining) • Local Search – Heuristics: pick a good starting tree and use moves within a “neighbourhood” to find a better tree. • Meta-heuristics: – Genetic algorithms – Simulated annealing – The ratchet
  30. 30. The “Felsenstein Zone”   Parsimony was once a very popular method of phylogenetic inference but it has fallen from grace in recent years due to the fact that it is statistically inconsistent for many relevant phylogenetic models. Consistency in statistics is the property that as you get more and more data you are more and more likely to get the correct answer. i.e. in the case of phylogenetics this means that as you get longer and longer sequence alignments then you'd expect to be increasingly likely to infer the tree that these sequences evolved on.
  31. 31. A Felsenstein's example C • The true tree is ((A,B),C,D) • The only parsimony informative patterns are of the form: xxyy, xyxy, and xyyx • Let p be the probability of mutation on a short edge and let q be the probability of mutation on a long edge B True tree A D C • Felsenstein showed that if p was small compared to q then the chance of seeing a parsimony informative pattern that “matched” (i.e. cost 1 on) the true tree, xxyy, was smaller than the chance of seeing a pattern, xyxy, that cost 1 on the incorrect tree ((A,C),B,D) • For large datasets Parsimony would be guaranteed to get the incorrect tree B D Long branch attraction tree
  32. 32. Inconsistency with a clock Felsenstein’s 4 taxon example appeared in a 1978 paper “Cases in which parsimony or compatibility methods will be positively misleading.” Syst. Zool. The “misleading zone” was conjectured to depend on the highly non-clocklike nature of Felsenstein’s tree. 1 2 3 4 Outgroup In 1989 Hendy and Penny showed that clocklike trees could also suffer from the same effect. The 4 trees ((1,out),(3,4),2) and ((2,out),(3,4),1) and ((3,out),(1,2),4) and ((4,out),(1,2),3) all have lower expected parsimony score than the true tree ((1,2),(3,4),out)
  33. 33. Enter statistical phylogenetics • Over the last 30 years model-based methods of phylogenetic inference have come to the fore. • These methods assume a model of nucleotide substitution which specifies the rates that nucleotides (A,C,G,T) mutate into other nucleotides. • They aim to find the tree with the highest probability of generating the sequence data observed, i.e. the maximum likelihood tree
  34. 34. Sequence evolution is modelled as a Markov process A Consider a single edge in a phylogeny, i.e. evolution of a single species, and the evolution of a single DNA base amongst the possible states {A, C, G, T}. C A time The probability of mutating from state i to j over a length of time t depends only on the current state i and the potential future state j, not on any of the previous history of the sequence, and can be written pij(t). T time t G
  35. 35. Continuous time Markov chains A G pAA pAC pAG pCA pCC pCG pCT G pGA pGC pGG pGT T pTA pTC pTG pTT C G T A pAT C A -qA* qAC qAG qAT C qCA -qC* qCG qCT G qGA qGC -qG* qGT T qTA qTC qTG -qT* T A M= C Transition matrix Q= Where qi* = Σj qij, j ≠ i i.e. rows sum to zero. Instantaneous rate matrix M = exp(Qt) Typically we restrict to stationary, reversible models, with the stationary distribution denoted by π. So, π Q = 0, and D(π)Q is symmetric.
  36. 36. Example with a simple model 2-state symmetric model (R := AG purines, Y := CT pyrimidines) α R Y Each edge i has probability pi of a mutation determined by its length c a 0.1 0.15 0.2 0.1 b 0.1 d a b c d …R… …R… …Y… …Y…
  37. 37. Example with a simple model Y R 0.1 0.15 0.2 ? ? 0.1 0.1 R R R R Y R Y …R… …R… …Y… …Y… a b c d Y R R R Y Y R Y Y L = 0.9*0.9*0.8*0.15*0.1 + 0.9*0.9*0.2*0.85*0.9 + 0.1*0.1*0.2*0.15*0.1 + 0.1*0.1*0.8*0.85*0.9 R Y R R Y Y Y R Y Y
  38. 38. In theory you can then… • Multiply the likelihoods over all sites to get the likelihood of a particular weighted tree (in practice you sum the log likelihoods). • Optimise the edge lengths to find the maximum likelihood value the tree. • Compare the maximum likelihood score of all trees (in practice you would probably use heuristic search). • Report the weighted tree with the highest overall likelihood.
  39. 39. Computational complexity • Maximum likelihood is computationally complex. • Fortunately the “tiny problem”, finding the likelihood of a site on a tree with fixed branch lengths, is solvable in linear time using Felsenstein’s pruning algorithm (dynamic programming) • However, solving the “small problem”, i.e. computing the likelihood of a particular tree, is time consuming. Unlike parsimony, here the edge lengths matter and must be optimized for each tree. • Finding the best edge weights for a given tree uses local search (hillclimbing), and can get stuck in local optima.
  40. 40. Models of nucleotide substitution • Jukes Cantor (JC) – – • – α Transitions and transversions at different rates Base frequencies different General Time Reversible (GTR) G α Transitions and transversions at different rates Base frequencies equal HKY model – – • A Kimura 2 Parameter (K2P) – • All substitutions equally likely Base frequencies equal α α α C α T
  41. 41. Models of nucleotide substitution • Jukes Cantor (JC) – – • – β Transitions and transversions at different rates Base frequencies different General Time Reversible (GTR) G α Transitions and transversions at different rates Base frequencies equal HKY model – – • A Kimura 2 Parameter (K2P) – • All substitutions equally likely Base frequencies equal α α α C β T
  42. 42. Models of nucleotide substitution • Jukes Cantor (JC) – – • – β Transitions and transversions at different rates Base frequencies different General Time Reversible (GTR) G α Transitions and transversions at different rates Base frequencies equal HKY model – – • A Kimura 2 Parameter (K2P) – • All substitutions equally likely Base frequencies equal α α α C β T
  43. 43. Models of nucleotide substitution • Jukes Cantor (JC) – – • – α Transitions and transversions at different rates Base frequencies different General Time Reversible (GTR) G β Transitions and transversions at different rates Base frequencies equal HKY model – – • A Kimura 2 Parameter (K2P) – • All substitutions equally likely Base frequencies equal γ ε δ C δ T
  44. 44. Extra features that can be modelled • • • Site to site rate variation (usually modelled by a gamma distribution) Invariant sites Different parameters for different genes/codon positons BUT • Some parts of reality are problematic… – Base composition bias (LogDet) – Sites that are free to vary change across the tree – Non independence of sites
  45. 45. Stochastic versus Systematic error • Rather than just getting a point estimate of the tree we are usually interested in getting some measure of confidence • Error could arise in two ways – Stochastic – our sequences are too short – Systematic – our models are wrong • There are good methods for determining whether or not stochastic error is a problem.
  46. 46. Assessing confidence in trees • We would like some measure of confidence in the inferred tree. – Is the tree likely to change if we got more data, or if we had used slightly different data? – Are some parts of the tree more robust than others? • The bootstrap is a useful tool for answering these sorts of questions.
  47. 47. The bootstrap • In 1985 Felsenstein introduced the idea of the bootstrap to phylogenetics. • For each bootstrap sample – Create a new alignment by resampling the columns of the observed alignment – Construct a tree for the ‘bootstrap’ alignment • Can be applied to any method that starts from a sequence alignment, e.g., parsimony, likelihood, clustering methods if the distances are derived from an alignment… • The bootstrap support for each edge is the number of bootstrap trees that edge appears in.
  48. 48. 1234567 ATATAAA ATTATAA TAAAATA TATAAAT a b c d 1224567 ATTTAAA ATTATAA TAAAATA TAAAAAT a b c d a b c d 1334567 AAATAAA ATTATAA TAAAATA TTTAAAT a b c d a 1234567 ATATAAA ATTATAA TAAAATA TATAAAT a b c d a b c d b c a 0.75 b d c d 1244567 ATTTAAA ATAATAA TAAAATA TAAAAAT a b c d a b c d
  49. 49. Example where the bootstrap is useful • Simulate data on the four taxon tree below (JC model) • Use sequence lengths of 100, 1000, and 10000 0.01 ((a,b),(c,d)) 0.2 ((a,c),(b,d)) a b c d ((a,d),(b,c)) 100 5.7% 42.8% 49.8% 1000 97% <5% <5% 10000 100% 0 0
  50. 50. Example where the bootstrap is not so useful • Simulate data on the two four-taxon trees below (JC model) in the proportion 55%, 45% and concatenate the sequences • Use total sequence lengths of 100, 1000, and 10000 0.05 0.1 55% a b c d ((a,b),(c,d)) ((a,c),(b,d)) 0.05 0.1 45% a c b d ((a,d),(b,c)) 100 64% 33% 3% 1000 80% 20% 0% 10000 98% <5 <5
  51. 51. Mistaking precision for accuracy 106 nuclear genes: Different methods provide conflicting Yeast topologies, each with 100% bootstrap support Phillips et al. (MBE, 2004) The results show the importance of systematic error
  52. 52. Phylogenomics • With increasing quantities of sequence data it is common to get very well resolved trees, i.e. bootstrap values (or posterior probabilities) close to 100% (or 1) • HOWEVER, slight changes to the model or inference method used can mean you get 100% support for different trees?! • This suggests that it is very important to know how well our models fit our data.
  53. 53. Testing model fit • In phylogenetic studies we often ask questions of the form “Is model A better than model B?” (relative goodness-of-fit tests, based on likelihood ratio tests or information criteria such as the AIC or BIC) • We less frequently ask questions of the form “Is model C a good fit to the data?” (absolute goodness-of-fit tests) • The alst question is clearly important as if our models fit poorly our results may be subject to systematic errors • So why is the last question less popular?
  54. 54. Absolute GOF tests • Absolute tests of goodness-of-fit are fiddlier to implement than relative tests. They typically require doing many parametric simulations under the best model and then summarizing each simulated dataset by a single test statistic. • With very large datasets you almost always will be able to reject your model. • If the best model fails an absolute GOF test, it’s not clear what you should do next…
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