= 1 = 4 3. Odds ratio justification for maximum likelihood. Likelihoods, Bootstraps and Testing Trees. Prob (H 2 D) Prob (H 1 D) Prob (D H 2 )

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1 4 1 1/3 1 = /3 1 = 1 12 Odds ratio justification for maximum likelihood Likelihoods, ootstraps and Testing Trees Joe elsenstein the data H 1 Hypothesis 1 H 2 Hypothesis 2 the symbol for given epts. of enome Sciences and of iology, University of Washington Prob (H 1 ) Prob (H 2 ) }{{} Prior odds ratio Prob ( H 1 ) Prob ( H 2 ) }{{} Likelihood ratio = Prob (H 1 ) Prob (H 2 ) }{{} Posterior odds ratio Likelihoods, ootstraps and Testing Trees p.1/60 Likelihoods, ootstraps and Testing Trees p.2/60 If a space probe finds no Little reen Men on Mars yes no yes no priors likelihoods no 1 yes 0 posteriors no yes no yes The likelihood ratio term ultimately dominates If we see one Little reen Man, the likelihood calculation does the right thing: 1 4 2/3 = 0 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 ( ) n 1 3 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...

2 Likelihood in Simple oin-tossing Tossing a coin n times, with probability p of heads, the probability of outcome HHTHTTTTHTTH is which is L = p 5 (1 p) 6 pp(1 p)p(1 p)(1 p)(1 p)(1 p)p(1 p)(1 p)p Plotting L against p to find its maximum: ifferentiating to find the maximum: ifferentiating 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 ) p 5 (1 p) 6 = 0 1 p 5 11 p = 0 Likelihood ˆp = p Likelihoods, ootstraps and Testing Trees p.5/60 Likelihoods, ootstraps and Testing Trees p.6/60 log-likelihoodcurve Likelihood curve in one parameter Its maximum likelihood estimate Likelihood curve in one parameter and the maximum likelihood estimate Ln (Likelihood) Ln (Likelihood) length of a branch in the tree length of a branch in the tree maximum likelihood estimate (ML)

3 The (approximate, asymptotic) confidence interval ontours of a log-likelihood surface in two dimensions Likelihood curve in one parameter and the maximum likelihood estimate and confidence interval derived from it Ln (Likelihood) 1/2 the value of a chi square with 1 d.f. significant at 95% length of branch 2 95% confidence interval length of a branch in the tree maximum likelihood estimate (ML) Likelihoods, ootstraps and Testing Trees p.9/60 length of branch 1 Likelihoods, ootstraps and Testing Trees p.10/60 ontours of a log-likelihood surface in two dimensions Log-likelihood-based confidence set for two variables shaded area is the joint confidence interval length of branch 2 ML 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 length of branch 1 length of branch 1

4 onfidence interval for one variable onfidence interval for the other variable length of branch 2 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 one degree of freedom which is significant at 95% level height of this contour is less than at the peak by an amount equal to 1/2 the chi square value with one degree of freedom which is significant at 95% level length of branch 1 Likelihoods, ootstraps and Testing Trees p.13/60 length of branch 1 Likelihoods, ootstraps and Testing Trees p.14/60 alculating the likelihood of a tree If we have molecular sequences on a tree, the likelihood is the product over sites of the data [i] for each site (if those evolve independently): L = Prob( T) = sites i=1 With log-likelihoods, the product becomes a sum: ln L = ln Prob( T) = sites i=1 Prob ( [i] T) ln Prob ( [i] T) alculating the likelihood for site i on a tree t 1 x t i are t t 4 t 5 2 t 3 y t t 6 7 "branch lengths", z (rate X time) w t 8 Sum over all possible states (bases) at interior nodes: L (i) = Prob (w) Prob(x w, t 7 ) x y z w Prob ( x, t 1 )Prob( x, t 2 )Prob(z w, t 8 ) Prob ( z, t 3 )Prob(y z, t 6 )Prob( y, t 4 )Prob( y, t 5 )

5 alculating the likelihood for site i on a tree We use the conditional likelihoods: L (i) j (s) The pruning" algorithm: j k These compute the probability of everything at site i at or above node j on 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. v j l v k t the tips we can define these quantities: if the observed state is (say), 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) L (i) l (s) = [ s j Prob (s j s, v j ) L (i) j (s j ) [ ] Prob (s k s, v k ) L (i) k (s k) s k ] (elsenstein, 1973; 1981). Likelihoods, ootstraps and Testing Trees p.17/60 Likelihoods, ootstraps and Testing Trees p.18/60 and at the bottom of the tree: What does tree space" (with branch lengths) look like? L (i) 0 = s π s L (i) 0 (s) an example: three species with a clock trifurcation not possible t 1 t 1 etc. (elsenstein, 1973, 1981) t 2 OK and having gotten the likelihoods for each site: L = sites i=1 L (i) 0 when we consider all three possible topologies, the space looks like: t 2 t 1 t 2

6 or 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: v 1 v 6 v 2 3 v v v 8 7 v 9 v 5 v 4 wall floor wall v 9 Through the looking-glass Shrinking one of the n 1 interior branches to 0, we arrive at a trifurcation: v 1 v6 v7 v 2 v3 v8 v 4 v 9 v5 v 2 v 1 v v v3 8 7 v 9 v 4 v 6 v5 v 1 v6 3 v v 8 7 v 4 v v 2 v5 v 1 v6 v 2 v3 v8 v v 5 v 9 7 v 4 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, ootstraps and Testing Trees p.21/60 Likelihoods, ootstraps and Testing Trees p.22/60 The graph of all trees of 5 species dataexample:mitochondrial-loopsequences The Schoenberg graph (all 15 trees of size 5 connected by NNI s) ovine Mouse ibbon Orang orilla himp Human TT T T TT T T T TT TT TTT TT T TT T T T T T TT TTT TTTT T TT T T T T TT TT TTT T T T T TT TTTT TT TTT TTT TTTT TTTTT TT TT TT TT TTTT TTT T TTTT TTT TTT TT TT T T T T TT T TT TT TTT TTT TT T TT TT T TTT T T TTTTT TTT TT TT TT TTT TTT TTT TT T TTTT T TT TTTT TTT T TT T TTT T TT TT TTT TTT TT TT TTTT TT TT TT T TT TTT TT TTT TT TTT TT TT

7 which gives the ML tree Maximum likelihood tree for the Hasegawa 232-site mitochondrial -loop data set, with Ts/Tn set to 2, analyzed with maximum likelihood (NML) ln L = Orang orilla himp Human Mouse ibbon ovine Likelihoods, ootstraps and Testing Trees p.25/60 Models with amino acids H I K L M N P Q R S T V W Y etc. H I K L M N P Q R S T V W Y ayhoff PM model Jones Taylor Thornton model specific models for secondary structure contexts or membrane proteins Models adapted from Henikoff LOSUM scoring ut... how to take N sequence into account? onstraints of code? Likelihoods, ootstraps and Testing Trees p.26/60 odon models oldman & Yang, 1994; Muse & aut, 1994) ovarion models? (itch and Markowitz, 1970) U T T T T U U phe phe leu UUU UU UU ser U stop U stop U T T T T T leu UU U leu UU T T T T T leu leu U U U leu ile U UU T T T T Which sites are available ile ile met U U U for substitutions changes as one moves along the tree U val val val UU U U T T T T val U 1 ω 0 Probabilities of change vary depending on whether amino acid is changing, and to what T T T T T

8 How to calculate likelihood with rate variation asy! Since branch lengths always come into transition probability formulas as r t,canjustmultiplylengthsofbranchesbythe appropriate factor to calculate the likelihood for a site. (ranch lengths are usually scaled by assuming a rate of 1.) Rate variation among sites Sites Phylogeny T T... Likelihoods, ootstraps and Testing Trees p.29/60 Rates at different sites: Rates of evolution Likelihoods, ootstraps and Testing Trees p.30/60 Hidden Markov Model of rate variation among sites Sites Phylogeny T T... Hidden Markov Models sum up over all paths The Hidden Markov hain method sums up likelihoods over all possible paths through the states: Prob (ata tree) = Σ Prob(ata tree, path) Prob(path) paths Hidden Markov chain that assigns rates: Rates of evolution one path another path

9 The rate combination contributing the most: orwards-ackwards algorithm (marginal probabilities) We can leave behind pointers that allow us to backtrack This can be done by a dynamic programming algorithm called the Viterbi lgorithm, well-known in the HMM literature. (Of course, this one might account for only of the likelihood) The orwards ackwards algorithm can calculate the contribution of one rate at a given site to the overall likelihood (a little different from the Viterbi calculation) Likelihoods, ootstraps and Testing Trees p.33/60 Likelihoods, ootstraps and Testing Trees p.34/60 The amma distribution, used for rates α = 1/4 V = 2 numericalexample.yochrome We analyze 31 cytochrome sequences, aligned by Naoko Takezaki, using the Proml protein maximum likelihood program. ssume ahidden Markov Model with 3 states, rates: category rate probability α = 1 V = 1 α = 4 V = 1/2 and expected block length 3. We get a reasonable, but not perfect, tree with the best rate combination inferred to be rate

10 The cytochrome tree from the above run (It s not perfect). wrong dubious seaurchin2 seaurchin1 lamprey trout loach carp xenopus chicken opossum wallaroo platypus rat mouse cat hseal gseal bovine whalebp whalebm horse dhorse rhinocer gibbon sorang borang gorilla2 gorilla1 pchimp cchimp african caucasian Likelihoods, ootstraps and Testing Trees p.37/60 Rates inferred from ytochrome african M-----TPMRK INPLMKLINH SILPTPSN ISWWNSL LLILQIT TLLMH caucasian......r t cchimp...t pchimp...t t gorilla1... T T gorilla2... T T borang... T....L I.TI... sorang...st.. T....L I gibbon...l.. T....L.....M......I... bovine...ni.. SH...IV.N.....S.....I...L... whalebm...ni.. TH...I S.....L...V..L... whalebp...ni.. TH...IV..V.....S.....L...M..L... dhorse...ni.. SH..I.I S.....I...L... horse...ni.. SH..I.I S.....I...L... rhinocer...ni.. SH..V.I S.....I...L... cat...ni.. SH..I.I V..T...L... gseal...ni.. TH...I..N I...L... hseal...ni.. TH...I..N I...L... mouse...n... TH...I S.....V..MV..I... rat...ni.. SH...I S.....V..MV..L... platypus...nnl.. TH..I.IV S.....L...I..L... wallaroo...nl.. SH..I.IV I..L... opossum...ni.. TH...I V...I..L... chicken...pni.. SH..L.M..N.L V..MT..L...L... xenopus...pni.. SH..I.I..N.....SL.....V...I... carp...-sl.. TH..I.I. LV L...T..L... loach...-sl.. TH..I.I. LV.....V.....L...T..L... trout...-nl.. TH..L.I. LV.....V.....L..T..L... lamprey.shqpsii.. TH..LS..S MLV...S......SL...I...I... seaurchin1 -...L.L.. H.IRIL.S T.V...L... L.I.....L...T..L... seaurchin2 -...L.. H.IRIL.S T.V...L... L.M... Likelihoods,..L...I.LI ootstraps and Testing..I... Trees p.38/60 Rates inferred from ytochrome can PSTSSI HITRVNY WIIRYLHN SMILL HIRLYYS LYSTWNI asian mp l... mp l....v......l... lla t hq... lla t hq... ng...t m..h......l thl... ng m..h thl... on...v l... ne S.TT...V T......M......YM.V... YTL... ebm..tm...v T....V......Y.M... HR... ebp..tt...v T Y.M... YR... se S.TT...V T I.V... YTL... e S.TT...V T I.V... YTL... ocer..tt...v T....M......I.V... YTL... S.TM...V T YM.V...M... YT... l S.TT...V T YM.V... YTT... l S.TT...V T YM.V... YTT... e S.TM...V T....L...M V... YTM... S.TM...V T....L...Q V... YTL... ypus S.T...V....L...M.....L..M.I..... YTQT... aroo S.TL...V....L..N......M....V...I... Y..K... sum S.TL...V....L..NI......M....V...I... Y..K... ken.t.l...v..t.n.q...l..n......i..... Y..K...T pus.t.m...v... LL..N... L...IY K... S.I...V T....L..NV......IYM... Y..K... h S.I...V....L..NI......Y.... Y..K... t S.I...V...S...L..NI......IYM... Y..K... rey NTL...V M...N..LM.N......IY...I... Y..K...V rchin1.i.l... S....LL.NV.....L...MY... SNKI...V rchin2.inl...v S....LL.NV.....L...MY...L TNKI...V Likelihood curve and its confidence interval ln L Transition / transversion ratio (This is for the 14-species primates data available for download).

11 onstraints on a tree for a clock Likelihood-ratio test of molecular clock onstraints for a clock Mouse v 1 v 2 v 4 v 5 v 1 = v 2 Human himp v 6 v 3 v v 4 = 5 orilla Orang v 8 v 1 + v v 6 = 3 ibbon ovine v 7 v v = v 4 + v 8 log likelihood Without clock parameters 11 With clock ifference χ 2 = df = 5 (This is for this 7-species subset of the primates data). (non significant) Likelihoods, ootstraps and Testing Trees p.41/60 Likelihoods, ootstraps and Testing Trees p.42/60 Likelihood surface for three clocklike trees Two trees to be tested using KHT test 204 Mouse ln Likelihood x x Tree I ovine ibbon Orang orilla himp Human x Mouse Tree II ovine ibbon x Orang (These are profile likelihoods" as they show the largest likelihood for that value of x,maximizingovertheotherbranchlengthinthetree.) orilla himp Human

12 Table of differences in log-likelihood Histogram of those differences Tree I II site ln L iff ifference in log likelihood at site o sign test, or t-test, or similar nonparametric tests. Likelihoods, ootstraps and Testing Trees p.45/60 Likelihoods, ootstraps and Testing Trees p.46/60 ootstrap sampling (with mixtures of normals) estimate of θ (unknown) true value of θ empirical distribution of sample (unknown) true distribution ootstrap replicates ootstrap sampling To infer the error in a quantity, θ, estimatedfromasampleofpoints x 1, x 2,...,x n we can o the following R times (R =1000or so) raw a bootstrap sample" by sampling n times with replacement from the sample. all these x 1, x 2,...,x n.notethatsomeofthe original points are represented more than once in the bootstrap sample, some once, some not at all. stimate θ from each of the bootstrap samples, call these ˆθ k (k = 1, 2,...,R) istribution of estimates of parameters When all R bootstrap samples have been done, the distribution of ˆθ i estimates the distribution one would get if one were able to draw repeated samples of n data points from the unknown true distribution.

13 ootstrap sampling of phylogenies nalyzing bootstraps with phylogenies Original ata sequences sites The sites are assumed to have evolved independently given the tree. They are the entities that are sampled (the x i ). The trees play the role of the parameter. One ends up with a cloud of R sampled trees. ootstrap sample #1 sequences sites sample same number of sites, with replacement stimate of the tree 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. ootstrap sample #2 sites ootstrap estimate of the tree, #1 sequences sample same number of sites, with replacement (and so on) ootstrap estimate of the tree, #2 Likelihoods, ootstraps and Testing Trees p.49/60 Likelihoods, ootstraps and Testing Trees p.50/60 Partitions from branches in an (unrooted) tree The majority-rule consensus tree and so on for all the other external (tip) branches 1

14 The majority-rule consensus tree The majority-rule consensus tree 1 1 Likelihoods, ootstraps and Testing Trees p.52/ Likelihoods, ootstraps and Testing Trees p.52/60 The majority-rule consensus tree The majority-rule consensus tree

15 The majority-rule consensus tree The majority-rule consensus tree Likelihoods, ootstraps and Testing Trees p.52/ Likelihoods, ootstraps and Testing Trees p.52/60 The majority-rule consensus tree The majority-rule consensus tree Majority rule consensus tree of the unrooted trees:

16 ootstrap sampling of a phylogeny ovine Mouse Squir Monk himp Human orilla Orang ibbon Rhesus Mac Jpn Macaq rab.mac 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 overstated Pvaluesarebiased(tooconservative) ootstrapping does not correct biases in phylogeny methods arbmacaq 49 Tarsier Lemur In this example, parsimony was used to infer the tree. Likelihoods, ootstraps and Testing Trees p.53/60 Likelihoods, ootstraps and Testing Trees p.54/60 elete-half jackknife P values diagramoftheparametricbootstrap ovine Mouse computer simulation estimation of tree Squir Monk himp Human orilla Orang ibbon Rhesus Mac Jpn Macaq rab.mac arbmacaq original data estimate of tree data set #1 data set #2 data set #3 data set #100 T 1 T 2 T 3 T Tarsier Lemur In this example, parsimony was used to infer the tree.

17 References Likelihood dwards,. W.. and L. L. avalli-sforza Reconstruction of evolutionary trees. pp in Phenetic and Phylogenetic lassification, ed. V. H. Heywood and J. McNeill. Systematics ssociation Publication No. 6. Systematics ssociation, London. [The founding paper for parsimony and likelihood for phylogenies, using gene frequencies] Jukes, T. H. and. antor volution of protein molecules. pp in Mammalian Protein Metabolism, ed. M. N. Munro. cademic Press, New York. [The Jukes-antor model, in one formula and a couple of sentences] Neyman, J Molecular studies of evolution: a source of novel statistical problems. In Statistical ecision Theory and Related Topics, ed. S. S. upta and J. Yackel, pp New York: cademic Press. [irst paper on likelihood for molecular sequences. Neyman was a famous statistician.] elsenstein, J Maximum-likelihood and minimum-steps methods for estimating evolutionary trees from data on discrete characters. Systematic Zoology 22: [The pruning algorithm, parsimony is not same as likelihood] elsenstein, J volutionary trees from N sequences: a maximum likelihood approach. Journal of Molecular volution 17: [Making likelihood useable for molecular sequences] Likelihoods, ootstraps and Testing Trees p.57/60 (more references) Yang, Z Maximum-likelihood estimation of phylogeny from N sequences when substitution rates differ over sites. Molecular iology and volution 10: [Use of gamma distribution of rate variation in ML phylogenies] Yang, Z Maximum likelihood phylogenetic estimation from N sequences with variable rates over sites: approximate methods. Journal of Molecular volution 39: [pproximating gamma distribution in ML phylogenies by an HMM] Yang, Z space-time process model for the evolution of N sequences. enetics 139: [llowing for autocorrelated rates along the molecule using an HMM for ML phylogenies] elsenstein, J. and.. hurchill Hidden Markov Model approach to variation among sites in rate of evolution Molecular iology and volution 13: [HMM approach to evolutionary rate variation] Thorne, J. L., N. oldman, and. T. Jones ombining protein evolution and secondary structure. Molecular iology and volution [HMM for secondary structure of proteins, with phylogenies] ootstraps etc. fron, ootstrap methods: another look at the jackknife. nnals of Statistics 7: [The original bootstrap paper] Likelihoods, ootstraps and Testing Trees p.58/60 (more references) Margush, T. and. R. McMorris onsensus n-trees. ulletin of Mathematical iology 43: i. [Majority-rule consensus trees] elsenstein, J onfidence limits on phylogenies: an approach using the bootstrap. volution 39: [The bootstrap first applied to phylogenies] Zharkikh,., and W.-H. Li Statistical properties of bootstrap estimation of phylogenetic variability from nucleotide sequences. I. our taxa with a molecular clock. Molecular iology and volution 9: [iscovery and explanation of bias in P values] Künsch, H. R The jackknife and the bootstrap for general stationary observations. nnals of Statistics 17: [The block-bootstrap] Wu,.. J Jackknife, bootstrap and other resampling plans in regression analysis. nnals of Statistics 14: [The delete-half jackknife] fron, ootstrap confidence intervals for a class of parametric problems. iometrika 72: [The parametric bootstrap] Templeton,. R Phylogenetic inference from restriction endonuclease cleavage site maps with particular reference to the evolution of humans and the apes. volution 37: [The first paper on the KHT test] (more references) oldman, N Statistical tests of models of N substitution. Journal of Molecular volution 36: [Parametric bootstrapping for testing models] Shimodaira, H. and M. Hasegawa Multiple comparisons of log-likelihoods with applications to phylogenetic inference. Molecular iology and volution 16: [orrection of KHT test for multiple hypothesis] Prager,. M. and.. Wilson ncient origin of lactalbumin from lysozyme: analysis of N and amino acid sequences. Journal of Molecular volution 27: [winning-sites test] Hasegawa, M. and H. Kishino ccuracies of the simple methods for estimating the bootstrap probability of a maximum-likelihood tree. Molecular iology and volution 11: [RLL probabilities] eneral reading elsenstein, J Inferring Phylogenies. Sinauerssociates, Sunderland, Massachusetts. [ook you and all your friends must rush out and buy] Yang, Z omputational Molecular volution. OxfordUniversityPress, Oxford. [Well-thought-out book on molecular phylogenies] Semple,. and M. Steel Phylogenetics. Oxford University Press,

Bootstraps and testing trees. Alog-likelihoodcurveanditsconfidenceinterval

Bootstraps and testing trees. Alog-likelihoodcurveanditsconfidenceinterval ootstraps and testing trees Joe elsenstein epts. of Genome Sciences and of iology, University of Washington ootstraps and testing trees p.1/20 log-likelihoodcurveanditsconfidenceinterval 2620 2625 ln L