Bootstraps and testing trees. Alog-likelihoodcurveanditsconfidenceinterval

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1 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 ln L Transition / transversion ratio (This is for the 14-species primates data available for download). ootstraps and testing trees p.2/20

2 onstraints on a tree for a clock onstraints for a clock v 1 v 2 v 4 v 5 v 1 = v 2 v 6 v 3 v v 4 = 5 v 8 v 1 + v v 6 = 3 v 7 v v = v 4 + v 8 oes not constrain the branch length on the unrooted tree ootstraps and testing trees p.3/20 Likelihood-ratio test of molecular clock Mouse Human himp Gorilla Orang Gibbon ovine log likelihood Without clock With clock parameters 11 6 ifference χ 2 = df = 5 (This is for this 7-species subset of the primates data). (non significant) ootstraps and testing trees p.4/20

3 Likelihood surface for three clocklike trees 204 ln Likelihood x x x 0.10 bigger x bigger (These are profile likelihoods" as they show the largest likelihood for that value of x,maximizingovertheotherbranchlengthinthetree.) ootstraps and testing trees p.5/20 Two trees to be tested using KHT test Mouse Tree I ovine Gibbon Orang Gorilla himp Human Mouse Tree II ovine Gibbon Orang Gorilla himp Human ootstraps and testing trees p.6/20

4 Table of differences in log-likelihood Tree I II site ln L iff ootstraps and testing trees p.7/20 Histogram of those differences ifference in log likelihood at site o sign test, or t-test, or similar nonparametric tests. ootstraps and testing trees p.8/20

5 ootstrap sampling (with mixtures of normals) estimate of θ (unknown) true value of θ empirical distribution of sample ootstrap replicates (unknown) true distribution istribution of estimates of parameters ootstraps and testing trees p.9/20 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) 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. ootstraps and testing trees p.10/20

6 ootstrap sampling of phylogenies Original ata sites sequences ootstrap sample #1 sites stimate of the tree sequences sample same number of sites, with replacement ootstrap sample #2 sequences sites sample same number of sites, with replacement ootstrap estimate of the tree, #1 (and so on) ootstrap estimate of the tree, #2 ootstraps and testing trees p.11/20 nalyzing bootstraps with phylogenies 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. There are many possible ways. The one I will describe here is the most useful, but not the only way we could go. 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. ootstraps and testing trees p.12/20

7 Partitions from branches in an (unrooted) tree and so on for all the other external (tip) branches ootstraps and testing trees p.13/20 The majority-rule consensus tree Trees: How many times each (non tip) partition of species is found: Majority rule consensus tree of the unrooted trees: ootstraps and testing trees p.14/20

8 ootstrap sampling of a phylogeny ovine Mouse Squir Monk himp Human Gorilla Orang Gibbon Rhesus Mac Jpn Macaq rab.mac arbmacaq Tarsier Lemur In this example, parsimony was used to infer the tree. ootstraps and testing trees p.15/20 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 ootstraps and testing trees p.16/20

9 elete-half jackknife P values ovine Mouse Squir Monk himp Human Gorilla Orang Gibbon Rhesus Mac Jpn Macaq rab.mac arbmacaq 59 Tarsier Lemur In this example, parsimony was used to infer the tree. ootstraps and testing trees p.17/20 diagramoftheparametricbootstrap computer simulation estimation of tree data set #1 T 1 estimate of tree data set #2 T 2 original data data set #3 T 3 data set #100 T 100 ootstraps and testing trees p.18/20

10 References ootstraps etc. fron, ootstrap methods: another look at the jackknife. nnals of Statistics 7: [The original bootstrap paper] 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] ootstraps and testing trees p.19/20 (more references) 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] Goldman, 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] ootstraps and testing trees p.20/20

Lecture 27. Phylogeny methods, part 7 (Bootstraps, etc.) p.1/30

Lecture 27. Phylogeny methods, part 7 (Bootstraps, etc.) Joe Felsenstein Department of Genome Sciences and Department of Biology Lecture 27. Phylogeny methods, part 7 (Bootstraps, etc.) p.1/30 A non-phylogeny