Likelihoods, Bootstraps and Testing Trees Joe Felsenstein Depts. of - - PowerPoint PPT Presentation

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Likelihoods, Bootstraps and Testing Trees Joe Felsenstein Depts. of - - PowerPoint PPT Presentation

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Likelihoods, Bootstraps and Testing Trees Joe Felsenstein Depts. of Genome Sciences and of Biology, University of Washington Likelihoods, Bootstraps and Testing Trees p.1/60 Odds ratio justification for maximum likelihood D the data H 1

slide-1
SLIDE 1

Likelihoods, Bootstraps and Testing Trees

Joe Felsenstein

  • Depts. of Genome Sciences and of Biology, University of Washington

Likelihoods, Bootstraps and Testing Trees – p.1/60

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SLIDE 2

Odds ratio justification for maximum likelihood

D the data H1 Hypothesis 1 H2 Hypothesis 2 | the symbol for “given”

Prob (H1) Prob (H2)

  • Prior odds ratio

Prob (D | H1) Prob (D | H2)

  • Likelihood ratio

=

Prob (H1 | D) Prob (H2 | D)

  • Posterior odds ratio

Likelihoods, Bootstraps and Testing Trees – p.2/60

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SLIDE 3

If a space probe finds no Little Green Men on Mars

priors

no yes no yes

4 1 1 4

Likelihoods, Bootstraps and Testing Trees – p.3/60

slide-4
SLIDE 4

If a space probe finds no Little Green Men on Mars

priors

no yes no yes

likelihoods

no yes 1

4 1 1 4

Likelihoods, Bootstraps and Testing Trees – p.3/60

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SLIDE 5

If a space probe finds no Little Green Men on Mars

priors

no yes no yes

likelihoods

no yes 1

4 1 × 1/3 1 1 4 × 1/3 1

Likelihoods, Bootstraps and Testing Trees – p.3/60

slide-6
SLIDE 6

If a space probe finds no Little Green Men on Mars

priors

posteriors

no yes no yes no yes no yes

likelihoods

no yes 1

4 1 × 1/3 1

=

4 3 1 4 × 1/3 1

=

1 12

Likelihoods, Bootstraps and Testing Trees – p.3/60

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SLIDE 7

The likelihood ratio term ultimately dominates

If we see one Little Green Man, the likelihood calculation does the right thing: 1 4 × 2/3 = ∞ 1 (put this way, this is OK but not mathematically kosher) If we send n space probes and keep seeing none, the likelihood ratio term is 1 3 n It dominates the calculation, overwhelming the prior. Thus even if we don’t have a prior we can believe in, we may be interested in knowing which hypothesis the likelihood ratio is recommending ...

Likelihoods, Bootstraps and Testing Trees – p.4/60

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SLIDE 8

Likelihood in Simple Coin-Tossing

Tossing a coin n times, with probability p of heads, the probability of

  • utcome HHTHTTTTHTTH is

pp(1 − p)p(1 − p)(1 − p)(1 − p)(1 − p)p(1 − p)(1 − p)p which is L = p5(1 − p)6 Plotting L against p to find its maximum:

0.0 0.2 0.4 0.6 0.8 1.0

Likelihood p

0.454

Likelihoods, Bootstraps and Testing Trees – p.5/60

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SLIDE 9

Differentiating to find the maximum:

Differentiating the expression for L with respect to p and equating the derivative to 0, the value of p that is at the peak is found (not surprisingly) to be p = 5/11: ∂L ∂p = 5 p − 6 1 − p

  • p5(1 − p)6 = 0

5 − 11 p = 0 ˆ p = 5 11

Likelihoods, Bootstraps and Testing Trees – p.6/60

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SLIDE 10

A log-likelihood curve

Ln (Likelihood)

length of a branch in the tree

A Likelihood curve in one parameter

Likelihoods, Bootstraps and Testing Trees – p.7/60

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SLIDE 11

Its maximum likelihood estimate

Ln (Likelihood)

length of a branch in the tree

maximum likelihood estimate (MLE)

A Likelihood curve in one parameter and the maximum likelihood estimate

Likelihoods, Bootstraps and Testing Trees – p.8/60

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SLIDE 12

The (approximate, asymptotic) confidence interval

Ln (Likelihood)

length of a branch in the tree

1/2 the value of

a chi−square with 1 d.f. significant at 95% 95% confidence interval

maximum likelihood estimate (MLE)

A Likelihood curve in one parameter and the maximum likelihood estimate and confidence interval derived from it

Likelihoods, Bootstraps and Testing Trees – p.9/60

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SLIDE 13

Contours of a log-likelihood surface in two dimensions

length of branch 1 length of branch 2

Likelihoods, Bootstraps and Testing Trees – p.10/60

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SLIDE 14

Contours of a log-likelihood surface in two dimensions

length of branch 1 length of branch 2

MLE

Likelihoods, Bootstraps and Testing Trees – p.11/60

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SLIDE 15

Log-likelihood-based confidence set for two variables

length of branch 1 length of branch 2

height of this contour is less than at the peak by an amount equal to 1/2 the chi−square value with two degrees of freedom which is significant at 95% level shaded area is the joint confidence interval

Likelihoods, Bootstraps and Testing Trees – p.12/60

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SLIDE 16

Confidence interval for one variable

length of branch 1

height of this contour is less than at the peak by an amount equal to 1/2 the chi−square value with

length of branch 2

  • ne degree of freedom which is significant at 95% level

Likelihoods, Bootstraps and Testing Trees – p.13/60

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SLIDE 17

Confidence interval for the other variable

length of branch 1

height of this contour is less than at the peak by an amount equal to 1/2 the chi−square value with

length of branch 2

  • ne degree of freedom which is significant at 95% level

Likelihoods, Bootstraps and Testing Trees – p.14/60

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SLIDE 18

Calculating the likelihood of a tree

If we have molecular sequences on a tree, the likelihood is the product

  • ver sites of the data D[i] for each site (if those evolve independently):

L = Prob (D | T) = sites

  • i=1

Prob (D[i] | T) With log-likelihoods, the product becomes a sum: ln L = ln Prob (D | T) = sites

  • i=1

ln Prob (D[i] | T)

Likelihoods, Bootstraps and Testing Trees – p.15/60

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SLIDE 19

Calculating the likelihood for site i on a tree

A C C C G

x y z w

t1 t t t t t t 2 3 4 5 6 ti are

"branch lengths",

t7 8

(rate time) X

Sum over all possible states (bases) at interior nodes: L(i) =

  • x
  • y
  • z
  • w

Prob (w) Prob (x | w, t7) × Prob (A | x, t1) Prob (C | x, t2) Prob (z | w, t8) × Prob (C | z, t3) Prob (y | z, t6) Prob (C | y, t4) Prob (G | y, t5)

Likelihoods, Bootstraps and Testing Trees – p.16/60

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SLIDE 20

Calculating the likelihood for site i on a tree

We use the conditional likelihoods: L(i)

j (s)

These compute the probability of everything at site i at or above node j

  • n the tree, given that node j is in state s. Thus it assumes something

(s) that we don’t know in practice – so we compute these for all states s. At the tips we can define these quantities: if the observed state is (say) C, the vector of L’s is (0, 1, 0, 0) . If we observe an ambiguity, say R (purine), they are (1, 0, 1, 0), not (1/2, 0, 1/2, 0)

Likelihoods, Bootstraps and Testing Trees – p.17/60

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SLIDE 21

The “pruning" algorithm:

vj vk

k j

l

L(i)

ℓ (s)

=

  • sj

Prob (sj | s, vj) L(i)

j (sj)

  • ×
  • sk

Prob (sk | s, vk) L(i)

k (sk)

  • (Felsenstein, 1973; 1981).

Likelihoods, Bootstraps and Testing Trees – p.18/60

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SLIDE 22

and at the bottom of the tree: L(i) =

  • s

πs L(i)

0 (s)

(Felsenstein, 1973, 1981) and having gotten the likelihoods for each site:

L =

sites

  • i=1

L(i)

Likelihoods, Bootstraps and Testing Trees – p.19/60

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SLIDE 23

What does “tree space" (with branch lengths) look like?

t1 t2 t1 t2

an example: three species with a clock

A B C t 1 t 2 t 1 t 2 OK not possible

trifurcation

etc. when we consider all three possible topologies, the space looks like:

Likelihoods, Bootstraps and Testing Trees – p.20/60

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SLIDE 24

For one tree topology

The space of trees varying all 2n − 3 branch lengths, each a nonegative number, defines an “orthant" (open corner) of a (2n − 3)-dimensional real space:

A B C D E F

v1

v v v v v v2

3

v4

5 6 7 8

v9

f l

  • r

v9

wall wall

Likelihoods, Bootstraps and Testing Trees – p.21/60

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SLIDE 25

Through the looking-glass

Shrinking one of the n − 1 interior branches to 0, we arrive at a trifurcation:

A B C D E F

v1

v v v v v v2

3

v4

5 6 7 8

v9

Here, as we pass “through the looking glass" we are also touch the space for two other tree topologies, and we could enter either.

Likelihoods, Bootstraps and Testing Trees – p.22/60

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SLIDE 26

Through the looking-glass

Shrinking one of the n − 1 interior branches to 0, we arrive at a trifurcation:

A B C D E F

v1

v v v v v v2

3

v4

5 6 7 8

v9

A B C E F

v1

v v v v v2

3

D

v v4

5 6 7 8

Here, as we pass “through the looking glass" we are also touch the space for two other tree topologies, and we could enter either.

Likelihoods, Bootstraps and Testing Trees – p.22/60

slide-27
SLIDE 27

Through the looking-glass

Shrinking one of the n − 1 interior branches to 0, we arrive at a trifurcation:

A B C D E F

v1

v v v v v v2

3

v4

5 6 7 8

v9

A B C E F

v1

v v v v v2

3

D

v v4

5 6 7 8

A B C D E F

v1

v v v v v v2

3

v4

5 6 7 8

v9

Here, as we pass “through the looking glass" we are also touch the space for two other tree topologies, and we could enter either.

Likelihoods, Bootstraps and Testing Trees – p.22/60

slide-28
SLIDE 28

Through the looking-glass

Shrinking one of the n − 1 interior branches to 0, we arrive at a trifurcation:

A B C D E F

v1

v v v v v v2

3

v4

5 6 7 8

v9

A B C E F

v1

v v v v v2

3

D

v v4

5 6 7 8

A B C D E F

v1

v v v v v v2

3

v4

5 6 7 8

v9

A B C D E F

v1

v v v v v v2

3

v4

5 6 7 8

v9

Here, as we pass “through the looking glass" we are also touch the space for two other tree topologies, and we could enter either.

Likelihoods, Bootstraps and Testing Trees – p.22/60

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SLIDE 29

The graph of all trees of 5 species

C D B E A D B C E A D B E C A C E D A B D C A E B A C D E B E B C D A B C D E A C B D E A A B D E C A B E C D B C E D A B D C E A E B D C A E C B D A

The Schoenberg graph (all 15 trees of size 5 connected by NNI’s)

Likelihoods, Bootstraps and Testing Trees – p.23/60

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SLIDE 30

A data example: mitochondrial D-loop sequences

Bovine CCAAACCTGT CCCCACCATC TAACACCAAC CCACATATAC AAGCTAAACC AAAAATACCA Mouse CCAAAAAAAC ATCCAAACAC CAACCCCAGC CCTTACGCAA TAGCCATACA AAGAATATTA Gibbon CTATACCCAC CCAACTCGAC CTACACCAAT CCCCACATAG CACACAGACC AACAACCTCC Orang CCCCACCCGT CTACACCAGC CAACACCAAC CCCCACCTAC TATACCAACC AATAACCTCT Gorilla CCCCATTTAT CCATAAAAAC CAACACCAAC CCCCATCTAA CACACAAACT AATGACCCCC Chimp CCCCATCCAC CCATACAAAC CAACATTACC CTCCATCCAA TATACAAACT AACAACCTCC Human CCCCACTCAC CCATACAAAC CAACACCACT CTCCACCTAA TATACAAATT AATAACCTCC TACTACTAAA AACTCAAATT AACTCTTTAA TCTTTATACA ACATTCCACC AACCTATCCA TACAACCATA AATAAGACTA ATCTATTAAA ATAACCCATT ACGATACAAA ATCCCTTTCG CACCTTCCAT ACCAAGCCCC GACTTTACCG CCAACGCACC TCATCAAAAC ATACCTACAA CAACCCCTAA ACCAAACACT ATCCCCAAAA CCAACACACT CTACCAAAAT ACACCCCCAA CACCCTCAAA GCCAAACACC AACCCTATAA TCAATACGCC TTATCAAAAC ACACCCCCAA CACTCTTCAG ACCGAACACC AATCTCACAA CCAACACGCC CCGTCAAAAC ACCCCTTCAG CACCTTCAGA ACTGAACGCC AATCTCATAA CCAACACACC CCATCAAAGC ACCCCTCCAA CACAAAAAAA CTCATATTTA TCTAAATACG AACTTCACAC AACCTTAACA CATAAACATA TCTAGATACA AACCACAACA CACAATTAAT ACACACCACA ATTACAATAC TAAACTCCCA CACAAACAAA TGCCCCCCCA CCCTCCTTCT TCAAGCCCAC TAGACCATCC TACCTTCCTA TTCACATCCG CACACCCCCA CCCCCCCTGC CCACGTCCAT CCCATCACCC TCTCCTCCCA CATAAACCCA CGCACCCCCA CCCCTTCCGC CCATGCTCAC CACATCATCT CTCCCCTTCA CACAAATTCA TACACCCCTA CCTTTCCTAC CCACGTTCAC CACATCATCC CCCCCTCTCA CACAAACCCG CACACCTCCA CCCCCCTCGT CTACGCTTAC CACGTCATCC CTCCCTCTCA CCCCAGCCCA ACACCCTTCC ACAAATCCTT AATATACGCA CCATAAATAA CA TCCCACCAAA TCACCCTCCA TCAAATCCAC AAATTACACA ACCATTAACC CA GCACGCCAAG CTCTCTACCA TCAAACGCAC AACTTACACA TACAGAACCA CA

Likelihoods, Bootstraps and Testing Trees – p.24/60

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SLIDE 31

which gives the ML tree

Maximum likelihood tree for the Hasegawa 232-site mitochondrial D-loop data set, with Ts/Tn set to 2, analyzed with maximum likelihood (DNAML)

Mouse Human Chimp Gorilla Orang Gibbon Bovine

0.792 0.902 0.486 0.336 0.121 0.049 0.304 0.153 0.075 0.172 0.106

ln L = −1405.6083

Likelihoods, Bootstraps and Testing Trees – p.25/60

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SLIDE 32

Models with amino acids

Dayhoff PAM model Jones−Taylor−Thornton model specific models for secondary−structure contexts or membrane proteins Models adapted from Henikoff BLOSUM scoring But ... how to take DNA sequence into account? Constraints of code?

A C D E F G H I K L M N P Q R S T V W Y A C D E F G H I K L M N P Q R S T V W Y etc.

Likelihoods, Bootstraps and Testing Trees – p.26/60

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SLIDE 33

Codon models

phe phe leu leu leu leu leu leu ile ile ile met val val val val ser stop stop U C U C C U U C A G A G A G A G U C A G U C A G UUU UUC UUA UUG CUU CUC CUA CUG AUU AUC AUA AUG GUU GUC GUA GUG UCA UAA UGA

Goldman & Yang, 1994; Muse & Gaut, 1994) 1 ω

Probabilities of change vary depending on whether amino acid is changing, and to what

Likelihoods, Bootstraps and Testing Trees – p.27/60

slide-34
SLIDE 34

Covarion models?

A G T A A G G T T T A A G T C A A G A A G G T T T A A G T C A A G A A G T T T A A G T C A A G A A G G T T T A A G T C A A G A A G T T T A A G T C A A G A A G G T T A A G T C A

(Fitch and Markowitz, 1970)

C A C A A T T T

Which sites are available for substitutions changes as one moves along the tree

Likelihoods, Bootstraps and Testing Trees – p.28/60

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SLIDE 35

How to calculate likelihood with rate variation

Easy! Since branch lengths always come into transition probability formulas as r × t , can just multiply lengths of branches by the appropriate factor to calculate the likelihood for a site. (Branch lengths are usually scaled by assuming a rate of 1.)

Likelihoods, Bootstraps and Testing Trees – p.29/60

slide-36
SLIDE 36

Rate variation among sites

C C C C A A G G A A C T A A G G A G A T A A A A G C C C C C G G G G G A A G G C

10.0 2.0 0.3 Rates

  • f

evolution

Phylogeny

1 2 3 4 5 6 7 8

Sites

...

...

Rates at different sites:

Likelihoods, Bootstraps and Testing Trees – p.30/60

slide-37
SLIDE 37

Hidden Markov Model of rate variation among sites

C C C C A A G G A A C T A A G G A G A T A A A A G C C C C C G G G G G A A G G C

10.0 2.0 0.3 Rates

  • f

evolution

Phylogeny

1 2 3 4 5 6 7 8

Sites

...

...

Hidden Markov chain that assigns rates:

Likelihoods, Bootstraps and Testing Trees – p.31/60

slide-38
SLIDE 38

Hidden Markov Models sum up over all paths The Hidden Markov Chain method sums up likelihoods over all possible paths through the states:

Prob (Data | tree) = paths Prob(Data| tree, path)

  • ne path

another path

Σ

Prob(path)

This is done using a recursive algorithm known as the Forwards

Likelihoods, Bootstraps and Testing Trees – p.32/60

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SLIDE 39

The rate combination contributing the most:

We can leave behind pointers that allow us to backtrack This can be done by a dynamic programming algorithm called the Viterbi Algorithm, well-known in the HMM literature. (Of course, this one might account for only 0.001 of the likelihood)

Likelihoods, Bootstraps and Testing Trees – p.33/60

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SLIDE 40

Forwards-Backwards algorithm (marginal probabilities)

The Forwards−Backwards algorithm at a given site to the overall likelihood can calculate the contribution of one rate (a little different from the Viterbi calculation)

Likelihoods, Bootstraps and Testing Trees – p.34/60

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SLIDE 41

The Gamma distribution, used for rates

α = 1/4 CV = 2 α = 1 CV = 1 α = 4 CV = 1/2

0.5 1.5 2.5

rate

1 2 3

Likelihoods, Bootstraps and Testing Trees – p.35/60

slide-42
SLIDE 42

A numerical example. Cyochrome B

We analyze 31 cytochrome B sequences, aligned by Naoko Takezaki, using the Proml protein maximum likelihood program. Assume a Hidden Markov Model with 3 states, rates: category rate probability 1 0.0 0.2 2 1.0 0.4 3 3.0 0.4 and expected block length 3. We get a reasonable, but not perfect, tree with the best rate combination inferred to be

Likelihoods, Bootstraps and Testing Trees – p.36/60

slide-43
SLIDE 43

The cytochrome B tree from the above run

seaurchin2 seaurchin1 lamprey trout loach carp xenopus chicken

  • possum

wallaroo platypus rat mouse cat hseal gseal bovine whalebp whalebm dhorse horse rhinocer gibbon sorang borang gorilla2 gorilla1 cchimp pchimp african caucasian

wrong dubious

(It’s not perfect).

Likelihoods, Bootstraps and Testing Trees – p.37/60

slide-44
SLIDE 44

Rates inferred from Cytochrome B

1333333311 3222322313 3321113222 2133111111 1331133123 112211111 african M-----TPMRK INPLMKLINH SFIDLPTPSN ISAWWNFGSL LGACLILQIT TGLFLAMH caucasian .......... .........R .......... .......... ..T....... ......... cchimp .......T.. .......... .......... .......... .......... ......... pchimp .......T.. .......... .......... ..T....... .......... ......... gorilla1 .......... T...A..... .......... ..T....... .......... ......... gorilla2 .......... T...A..... .......... ..T....... .......... ......... borang .......... T......... .L........ .......... ......I.TI ......... sorang ......ST.. T......... .L........ .......... ......I... ......... gibbon .......L.. T......... .L....A... ..M....... .........I ......... bovine ......NI.. SH....IV.N A.....A... ..S....... ..I......L ......... whalebm ......NI.. TH....I..D A......... ..S....... ..L...V..L ......... whalebp ......NI.. TH....IV.D A.V....... ..S....... ..L...M..L ......... dhorse ......NI.. SH..I.I... ......A... ..S....... ..I......L ......... horse ......NI.. SH..I.I... .......... ..S....... ..I......L ......... rhinocer ......NI.. SH..V.I... .......... ..S....... ..I......L ......... cat ......NI.. SH..I.I... ......A... .......... ..V..T...L ......... gseal ......NI.. TH....I..N .......... .......... ..I......L ......... hseal ......NI.. TH....I..N .......... .......... ..I......L ......... mouse ......N... TH..F.I... ......A... ..S....... ..V..MV..I ......... rat ......NI.. SH..F.I... ......A... ..S....... ..V..MV..L ......... platypus .....NNL.. TH..I.IV.. .......... ..S....... ..L...I..L ......... wallaroo ......NL.. SH..I.IV.. ......A... .......... ......I..L .........

  • possum

......NI.. TH....I..D .......... .......... ..V...I..L ......... chicken ....APNI.. SH..L.M..N .L....A... .......... .AV..MT..L ...L..... xenopus ....APNI.. SH..I.I..N .......... ..SL...... ..V...A..I ......... carp ....A-SL.. TH..I.IA.D ALV....... .......... ..L...T..L ......... loach ....A-SL.. TH..I.IA.D ALV...A... ..V....... ..L...T..L ......... trout ....A-NL.. TH..L.IA.D ALV...A... ..V....... ..L..AT..L ......... lamprey .SHQPSII.. TH..LS.G.S MLV...S.A. .......... .SL......I ...I..... seaurchin1

  • ...LG.L.. EH.IFRIL.S T.V...L... L.I....... ..L...T..L .........

seaurchin2

  • ...AG.L.. EH.IFRIL.S T.V...L... L.M....... ..L...I.LI ..I......

Likelihoods, Bootstraps and Testing Trees – p.38/60

slide-45
SLIDE 45

Rates inferred from Cytochrome B

2223311112 2222222222 2222232112 2222222223 1222221112 333311112 african PDASTAFSSI AHITRDVNYG WIIRYLHANG ASMFFICLFL HIGRGLYYGS FLYSETWNI caucasian .......... .......... .......... .......... .......... ......... cchimp .......... .......... .......... .......... .......... ...L..... pchimp .......... .......... .......... ...L...... .V........ ...L..... gorilla1 .......... .......... .T........ .......... .......... ..HQ..... gorilla2 .......... .......... .T........ .......... .......... ..HQ..... borang ...T...... .......... .M..H..... ...L...... .......... .THL..... sorang .......... .......... .M..H..... .......... .......... .THL..... gibbon .........V .......... .......... .......... .......... ...L..... bovine S.TT.....V T..C...... .....M.... ........YM .V........ YTFL..... whalebm ..TM.....V T..C...... .V........ ........YA .M........ HAFR..... whalebp ..TT.....V T..C...... .......... ........YA .M........ YAFR..... dhorse S.TT.....V T..C...... .......... .........I .V........ YTFL..... horse S.TT.....V T..C...... .......... .........I .V........ YTFL..... rhinocer ..TT.....V T..C...... .M........ .........I .V........ YTFL..... cat S.TM.....V T..C...... .......... ........YM .V...M.... YTF...... gseal S.TT.....V T..C...... .......... ........YM .V........ YTFT..... hseal S.TT.....V T..C...... .......... ........YM .V........ YTFT..... mouse S.TM.....V T..C...... .L...M.... .......... .V........ YTFM..... rat S.TM.....V T..C...... .L....Q... .......... .V........ YTFL..... platypus S.T......V ...C...... .L...M.... ..L..M.I.. .......... YTQT..... wallaroo S.TL.....V ...C...... .L..N..... .....M.... .V...I.... Y..K.....

  • possum

S.TL.....V ...C...... .L..NI.... .....M.... .V...I.... Y..K..... chicken A.T.L....V ..TC.N.Q.. .L..N..... ..F....I.. .......... Y..K....T xenopus A.T.M....V ...CF..... LL..N..... L.F....IY. .......... ...K..... carp S.I......V T..C...... .L..NV.... ..F....IYM ..A....... Y..K..... loach S.I......V ...C...... .L..NI.... ..F.....Y. ..A....... Y..K..... trout S.I......V C..C...S.. .L..NI.... ..F....IYM ..A....... Y..K..... lamprey ANTEL....V M..C....N. .LM.N..... .......IYA .....I.... Y..K....V seaurchin1 A.I.L....A S..C...... .LL.NV.... ..L....MYC .........G SNKI....V seaurchin2 A.INL....V S..C...... .LL.NV...C ..L....MYC .........L TNKI....V

Likelihoods, Bootstraps and Testing Trees – p.39/60

slide-46
SLIDE 46

Likelihood curve and its confidence interval

−2620 −2625 −2630 −2635 −2640 5 10 20 50 100 200

Transition / transversion ratio ln L

(This is for the 14-species primates data available for download).

Likelihoods, Bootstraps and Testing Trees – p.40/60

slide-47
SLIDE 47

Constraints on a tree for a clock

A B C D E

v2 v1 v3 v4 v5 v6 v7 v8

Constraints for a clock

v2 v1 = v4 v5

=

v3 v7 v4 v8

= + +

v1 v6 v3

= +

Does not constrain the branch length on the unrooted tree

Likelihoods, Bootstraps and Testing Trees – p.41/60

slide-48
SLIDE 48

Likelihood-ratio test of molecular clock

Mouse Human Chimp Gorilla Orang Gibbon Bovine

log−likelihood parameters Without clock With clock Difference 11 6 5 −1407.085 df = 5 −1405.608 1.477

χ 2= 2.954

(non−significant) (This is for this 7-species subset of the primates data).

Likelihoods, Bootstraps and Testing Trees – p.42/60

slide-49
SLIDE 49

Likelihood surface for three clocklike trees

0.10 0.20 0.10 −204 −205 −206

x

ln Likelihood

A C B x A x C x B C B A

bigger bigger

(These are “profile likelihoods" as they show the largest likelihood for that value of x , maximizing over the other branch length in the tree.)

Likelihoods, Bootstraps and Testing Trees – p.43/60

slide-50
SLIDE 50

Two trees to be tested using KHT test

Mouse Bovine Gibbon Orang Gorilla Chimp Human Mouse Bovine Gibbon Orang Gorilla Chimp Human

Tree I Tree II

Likelihoods, Bootstraps and Testing Trees – p.44/60

slide-51
SLIDE 51

Table of differences in log-likelihood

site 1 2 3 4 5 6 ln L Tree

I II

231 232

−1405.61 −1408.80

...

Diff

...

+3.19 −2.971 −4.483 −5.673 −5.883 −2.691

...

−8.003 −2.971 −2.691 −2.983 −4.494 −5.685 −5.898 −2.700 −7.572 −2.987 −2.705 +0.012 +0.013 +0.010 −0.431 +0.015 +0.111 +0.012 +0.010

Likelihoods, Bootstraps and Testing Trees – p.45/60

slide-52
SLIDE 52

Histogram of those differences

−0.50 0.0 0.50 1.0 1.5 2.0

Difference in log likelihood at site

Do sign test, or t-test, or similar nonparametric tests.

Likelihoods, Bootstraps and Testing Trees – p.46/60

slide-53
SLIDE 53

Bootstrap sampling (with mixtures of normals)

θ (unknown) true value of (unknown) true distribution empirical distribution of sample estimate of θ

Likelihoods, Bootstraps and Testing Trees – p.47/60

slide-54
SLIDE 54

Bootstrap sampling (with mixtures of normals)

Bootstrap replicates θ (unknown) true value of (unknown) true distribution empirical distribution of sample estimate of θ Distribution of estimates of parameters

Likelihoods, Bootstraps and Testing Trees – p.47/60

slide-55
SLIDE 55

Bootstrap sampling (with mixtures of normals)

Bootstrap replicates θ (unknown) true value of (unknown) true distribution empirical distribution of sample estimate of θ Distribution of estimates of parameters

Likelihoods, Bootstraps and Testing Trees – p.47/60

slide-56
SLIDE 56

Bootstrap sampling (with mixtures of normals)

Bootstrap replicates θ (unknown) true value of (unknown) true distribution empirical distribution of sample estimate of θ Distribution of estimates of parameters

Likelihoods, Bootstraps and Testing Trees – p.47/60

slide-57
SLIDE 57

Bootstrap sampling

To infer the error in a quantity, θ, estimated from a sample of points x1, x2, . . . , xn we can Do the following R times (R = 1000 or so)

Likelihoods, Bootstraps and Testing Trees – p.48/60

slide-58
SLIDE 58

Bootstrap sampling

To infer the error in a quantity, θ, estimated from a sample of points x1, x2, . . . , xn we can Do the following R times (R = 1000 or so) Draw a “bootstrap sample" by sampling n times with replacement from the sample. Call these x∗

1, x∗ 2, . . . , x∗

  • n. Note that some of the
  • riginal points are represented more than once in the bootstrap

sample, some once, some not at all.

Likelihoods, Bootstraps and Testing Trees – p.48/60

slide-59
SLIDE 59

Bootstrap sampling

To infer the error in a quantity, θ, estimated from a sample of points x1, x2, . . . , xn we can Do the following R times (R = 1000 or so) Draw a “bootstrap sample" by sampling n times with replacement from the sample. Call these x∗

1, x∗ 2, . . . , x∗

  • n. Note that some of the
  • riginal points are represented more than once in the bootstrap

sample, some once, some not at all. Estimate θ from each of the bootstrap samples, call these ˆ θ∗

k

(k = 1, 2, . . . , R)

Likelihoods, Bootstraps and Testing Trees – p.48/60

slide-60
SLIDE 60

Bootstrap sampling

To infer the error in a quantity, θ, estimated from a sample of points x1, x2, . . . , xn we can Do the following R times (R = 1000 or so) Draw a “bootstrap sample" by sampling n times with replacement from the sample. Call these x∗

1, x∗ 2, . . . , x∗

  • n. Note that some of the
  • riginal points are represented more than once in the bootstrap

sample, some once, some not at all. Estimate θ from each of the bootstrap samples, call these ˆ θ∗

k

(k = 1, 2, . . . , R) When all R bootstrap samples have been done, the distribution of ˆ θ∗

i

estimates the distribution one would get if one were able to draw repeated samples of n data points from the unknown true distribution.

Likelihoods, Bootstraps and Testing Trees – p.48/60

slide-61
SLIDE 61

Bootstrap sampling of phylogenies

Original Data

sequences sites

Estimate of the tree Estimate of the tree

A C B D

Likelihoods, Bootstraps and Testing Trees – p.49/60

slide-62
SLIDE 62

Bootstrap sampling of phylogenies

Original Data

sequences sites Bootstrap sample #1

Estimate of the tree Bootstrap estimate of

sample same number

  • f sites, with replacement

sequences sites

Bootstrap estimate of the tree, #1

A C B D A C B D

Likelihoods, Bootstraps and Testing Trees – p.49/60

slide-63
SLIDE 63

Bootstrap sampling of phylogenies

Original Data

sequences sites Bootstrap sample #1 Bootstrap sample #2

sample same number

  • f sites, with replacement

sample same number

  • f sites, with replacement

sequences sequences sites sites

(and so on)

Estimate of the tree Bootstrap estimate of the tree, #1

A C B D A C B D

Bootstrap estimate of the tree, #2

A C B D

Likelihoods, Bootstraps and Testing Trees – p.49/60

slide-64
SLIDE 64

Analyzing bootstraps with phylogenies

The sites are assumed to have evolved independently given the tree. They are the entities that are sampled (the xi). The trees play the role of the parameter. One ends up with a cloud of R sampled trees.

Likelihoods, Bootstraps and Testing Trees – p.50/60

slide-65
SLIDE 65

Analyzing bootstraps with phylogenies

The sites are assumed to have evolved independently given the tree. They are the entities that are sampled (the xi). The trees play the role of the parameter. One ends up with a cloud of R sampled trees. To summarize this cloud, we ask, for each branch in the tree, how frequently it appears among the cloud of trees. We make a tree that summarizes this for all the most frequently occurring branches. This is the majority rule consensus tree of the bootstrap estimates of the tree.

Likelihoods, Bootstraps and Testing Trees – p.50/60

slide-66
SLIDE 66

Partitions from branches in an (unrooted) tree

AE | BCDF

E A C F B D

Likelihoods, Bootstraps and Testing Trees – p.51/60

slide-67
SLIDE 67

Partitions from branches in an (unrooted) tree

AE | BCDF ACE | BDF

E A C F B D E A C F B D

Likelihoods, Bootstraps and Testing Trees – p.51/60

slide-68
SLIDE 68

Partitions from branches in an (unrooted) tree

AE | BCDF ACE | BDF ACEF | BD

E A C F B D E A C F B D E A C F B D

Likelihoods, Bootstraps and Testing Trees – p.51/60

slide-69
SLIDE 69

Partitions from branches in an (unrooted) tree

AE | BCDF ACE | BDF ACEF | BD

E A C F B D E A C F B D E A C F B D E A C F B D

A | CEFBD

Likelihoods, Bootstraps and Testing Trees – p.51/60

slide-70
SLIDE 70

Partitions from branches in an (unrooted) tree

AE | BCDF ACE | BDF ACEF | BD

E A C F B D E A C F B D E A C F B D E A C F B D

A | CEFBD

and so on for all the other external (tip) branches

Likelihoods, Bootstraps and Testing Trees – p.51/60

slide-71
SLIDE 71

The majority-rule consensus tree

Trees: How many times each partition of species is found: AE | BCDF ACE | BDF ACEF | BD 1 AC | BDEF AEF | BCD ADEF | BC ABDF | EC ABCE | DF

B D F E C A B D F E C A B D F E C A B D F E C A B D F E A C

Likelihoods, Bootstraps and Testing Trees – p.52/60

slide-72
SLIDE 72

The majority-rule consensus tree

Trees: How many times each partition of species is found: AE | BCDF ACE | BDF ACEF | BD 1 AC | BDEF AEF | BCD ADEF | BC ABDF | EC ABCE | DF

B D F E C A B D F E C A B D F E C A B D F E C A B D F E A C

1

Likelihoods, Bootstraps and Testing Trees – p.52/60

slide-73
SLIDE 73

The majority-rule consensus tree

Trees: How many times each partition of species is found: AE | BCDF ACE | BDF ACEF | BD 1 AC | BDEF AEF | BCD ADEF | BC ABDF | EC ABCE | DF

B D F E C A B D F E C A B D F E C A B D F E C A B D F E A C

1 1

Likelihoods, Bootstraps and Testing Trees – p.52/60

slide-74
SLIDE 74

The majority-rule consensus tree

Trees: How many times each partition of species is found: AE | BCDF ACE | BDF ACEF | BD 1 AC | BDEF AEF | BCD ADEF | BC ABDF | EC ABCE | DF

B D F E C A B D F E C A B D F E C A B D F E C A B D F E A C

1 2 1 1

Likelihoods, Bootstraps and Testing Trees – p.52/60

slide-75
SLIDE 75

The majority-rule consensus tree

Trees: How many times each partition of species is found: AE | BCDF ACE | BDF ACEF | BD AC | BDEF 1 AEF | BCD 1 ADEF | BC ABDF | EC ABCE | DF

B D F E C A B D F E C A B D F E C A B D F E C A B D F E A C

2 1 2 1 1

Likelihoods, Bootstraps and Testing Trees – p.52/60

slide-76
SLIDE 76

The majority-rule consensus tree

Trees: How many times each partition of species is found: AE | BCDF 3 ACE | BDF ACEF | BD 1 AC | BDEF 1 AEF | BCD 1 ADEF | BC 2 ABDF | EC ABCE | DF

B D F E C A B D F E C A B D F E C A B D F E C A B D F E A C

2 2

Likelihoods, Bootstraps and Testing Trees – p.52/60

slide-77
SLIDE 77

The majority-rule consensus tree

Trees: How many times each partition of species is found: AE | BCDF 3 ACE | BDF 3 ACEF | BD 1 AC | BDEF 1 AEF | BCD 1 ADEF | BC 2 ABDF | EC 1 ABCE | DF 3

B D F E C A B D F E C A B D F E C A B D F E C A B D F E A C

Likelihoods, Bootstraps and Testing Trees – p.52/60

slide-78
SLIDE 78

The majority-rule consensus tree

Trees: How many times each partition of species is found: AE | BCDF 3 ACE | BDF 3 ACEF | BD 1 AC | BDEF 1 AEF | BCD 1 ADEF | BC 2 ABDF | EC 1 ABCE | DF 3

B D F E C A B D F E C A B D F E C A B D F E C A B D F E A C

Likelihoods, Bootstraps and Testing Trees – p.52/60

slide-79
SLIDE 79

The majority-rule consensus tree

Trees: How many times each partition of species is found: AE | BCDF 3 ACE | BDF 3 ACEF | BD 1 AC | BDEF 1 AEF | BCD 1 ADEF | BC 2 ABDF | EC 1 ABCE | DF 3 A E C B D F

60 60 60

B D F E C A B D F E C A B D F E C A B D F E C A B D F E A C

Majority−rule consensus tree of the unrooted trees:

Likelihoods, Bootstraps and Testing Trees – p.52/60

slide-80
SLIDE 80

Bootstrap sampling of a phylogeny

Bovine Mouse Squir Monk Chimp Human Gorilla Orang Gibbon Rhesus Mac Jpn Macaq Crab−E.Mac BarbMacaq Tarsier Lemur

80 72 74 99 99 100 77 42 35 49

84

In this example, parsimony was used to infer the tree.

Likelihoods, Bootstraps and Testing Trees – p.53/60

slide-81
SLIDE 81

Potential problems with the bootstrap

Sites may not evolve independently

Likelihoods, Bootstraps and Testing Trees – p.54/60

slide-82
SLIDE 82

Potential problems with the bootstrap

Sites may not evolve independently Sites may not come from a common distribution (but you can consider them to be sampled from a mixture of possible distributions)

Likelihoods, Bootstraps and Testing Trees – p.54/60

slide-83
SLIDE 83

Potential problems with the bootstrap

Sites may not evolve independently Sites may not come from a common distribution (but you can consider them to be sampled from a mixture of possible distributions) If do not know which branch is of interest at the outset, a “multiple-tests" problem means that the most extreme P values are

  • verstated

Likelihoods, Bootstraps and Testing Trees – p.54/60

slide-84
SLIDE 84

Potential problems with the bootstrap

Sites may not evolve independently Sites may not come from a common distribution (but you can consider them to be sampled from a mixture of possible distributions) If do not know which branch is of interest at the outset, a “multiple-tests" problem means that the most extreme P values are

  • verstated

P values are biased (too conservative)

Likelihoods, Bootstraps and Testing Trees – p.54/60

slide-85
SLIDE 85

Potential problems with the bootstrap

Sites may not evolve independently Sites may not come from a common distribution (but you can consider them to be sampled from a mixture of possible distributions) If do not know which branch is of interest at the outset, a “multiple-tests" problem means that the most extreme P values are

  • verstated

P values are biased (too conservative) Bootstrapping does not correct biases in phylogeny methods

Likelihoods, Bootstraps and Testing Trees – p.54/60

slide-86
SLIDE 86

Delete-half jackknife P values

Bovine Mouse Squir Monk Chimp Human Gorilla Orang Gibbon Rhesus Mac Jpn Macaq Crab−E.Mac BarbMacaq Tarsier Lemur

80 99 100

84

98 69 72 80 50 59 32

In this example, parsimony was used to infer the tree.

Likelihoods, Bootstraps and Testing Trees – p.55/60

slide-87
SLIDE 87

A diagram of the parametric bootstrap

  • riginal

data estimate

  • f tree

data set #1 data data data set #2 set #3 set #100

computer simulation estimation

  • f tree

T1 T T 2 T3 100

Likelihoods, Bootstraps and Testing Trees – p.56/60

slide-88
SLIDE 88

References

Likelihood Edwards, A. W. F . and L. L. Cavalli-Sforza. 1964. Reconstruction of evolutionary trees. pp. 67-76 in Phenetic and Phylogenetic Classification, ed.

  • V. H. Heywood and J. McNeill. Systematics Association Publication No. 6.

Systematics Association, London. [The founding paper for parsimony and likelihood for phylogenies, using gene frequencies] Jukes, T. H. and C. Cantor. 1969. Evolution of protein molecules. pp. 21-132 in Mammalian Protein Metabolism, ed. M. N. Munro. Academic Press, New York. [The Jukes-Cantor model, in one formula and a couple of sentences] Neyman, J. 1971. Molecular studies of evolution: a source of novel statistical problems. In Statistical Decision Theory and Related Topics, ed. S.

  • S. Gupta and J. Yackel, pp. 1-27. New York: Academic Press. [First paper
  • n likelihood for molecular sequences. Neyman was a famous statistician.]

Felsenstein, J. 1973. Maximum-likelihood and minimum-steps methods for estimating evolutionary trees from data on discrete characters. Systematic Zoology 22: 240-249. [The pruning algorithm, parsimony is not same as likelihood] Felsenstein, J. 1981. Evolutionary trees from DNA sequences: a maximum likelihood approach. Journal of Molecular Evolution 17: 368-376. [Making likelihood useable for molecular sequences]

Likelihoods, Bootstraps and Testing Trees – p.57/60

slide-89
SLIDE 89

(more references)

Yang, Z. 1994. Maximum-likelihood estimation of phylogeny from DNA sequences when substitution rates differ over sites. Molecular Biology and Evolution 10: 1396-1401. [Use of gamma distribution of rate variation in ML phylogenies] Yang, Z. 1994. Maximum likelihood phylogenetic estimation from DNA sequences with variable rates over sites: approximate methods. Journal of Molecular Evolution 39: 306-314. [Approximating gamma distribution in ML phylogenies by an HMM] Yang, Z. 1995. A space-time process model for the evolution of DNA

  • sequences. Genetics 139: 993-1005. [Allowing for autocorrelated rates

along the molecule using an HMM for ML phylogenies] Felsenstein, J. and G. A. Churchill. 1996. A Hidden Markov Model approach to variation among sites in rate of evolution Molecular Biology and Evolution 13: 93-104. [HMM approach to evolutionary rate variation] Thorne, J. L., N. Goldman, and D. T. Jones. 1996. Combining protein evolution and secondary structure. Molecular Biology and Evolution 13 666-673. [HMM for secondary structure of proteins, with phylogenies] Bootstraps etc. Efron, B. 1979. Bootstrap methods: another look at the jackknife. Annals of Statistics 7: 1-26. [The original bootstrap paper]

Likelihoods, Bootstraps and Testing Trees – p.58/60

slide-90
SLIDE 90

(more references)

Margush, T. and F . R. McMorris. 1981. Consensus n-trees. Bulletin of Mathematical Biology 43: 239-244i. [Majority-rule consensus trees] Felsenstein, J. 1985. Confidence limits on phylogenies: an approach using the bootstrap. Evolution 39: 783-791. [The bootstrap first applied to phylogenies] Zharkikh, A., and W.-H. Li. 1992. Statistical properties of bootstrap estimation of phylogenetic variability from nucleotide sequences. I. Four taxa with a molecular clock. Molecular Biology and Evolution 9: 1119-1147. [Discovery and explanation of bias in P values] Künsch, H. R. 1989. The jackknife and the bootstrap for general stationary

  • bservations. Annals of Statistics 17: 1217-1241. [The block-bootstrap]

Wu, C. F . J. 1986. Jackknife, bootstrap and other resampling plans in regression analysis. Annals of Statistics 14: 1261-1295. [The delete-half jackknife] Efron, B. 1985. Bootstrap confidence intervals for a class of parametric

  • problems. Biometrika 72: 45-58. [The parametric bootstrap]

Templeton, A. R. 1983. Phylogenetic inference from restriction endonuclease cleavage site maps with particular reference to the evolution of humans and the apes. Evolution 37: 221-224. [The first paper

  • n the KHT test]

Likelihoods, Bootstraps and Testing Trees – p.59/60

slide-91
SLIDE 91

(more references)

Goldman, N. 1993. Statistical tests of models of DNA substitution. Journal

  • f Molecular Evolution 36: 182-98. [Parametric bootstrapping for testing

models] Shimodaira, H. and M. Hasegawa. 1999. Multiple comparisons of log-likelihoods with applications to phylogenetic inference. Molecular Biology and Evolution 16: 1114-1116. [Correction of KHT test for multiple hypothesis] Prager, E. M. and A. C. Wilson. 1988. Ancient origin of lactalbumin from lysozyme: analysis of DNA and amino acid sequences. Journal of Molecular Evolution 27: 326-335. [winning-sites test] Hasegawa, M. and H. Kishino. 1994. Accuracies of the simple methods for estimating the bootstrap probability of a maximum-likelihood tree. Molecular Biology and Evolution 11: 142-145. [RELL probabilities] General reading Felsenstein, J. 2004. Inferring Phylogenies. Sinauer Associates, Sunderland, Massachusetts. [Book you and all your friends must rush out and buy] Yang, Z. 2006. Computational Molecular Evolution. Oxford University Press,

  • Oxford. [Well-thought-out book on molecular phylogenies]

Semple, C. and M. Steel. 2003. Phylogenetics. Oxford University Press,

  • Oxford. [Good for a mathematical audience]

Likelihoods, Bootstraps and Testing Trees – p.60/60