Assessing Phylogenetic Hypotheses and Phylogenetic Data We use numerical phylogenetic methods because most data includes potentially misleading evidence of relationships We should not be content with constructing phylogenetic hypotheses but should also assess what confidence we can place in our hypotheses This is not always simple! (but do not despair!)
Assessing Data Quality We expect (or hope) our data will be well structured and contain strong phylogenetic signal We can test this using randomization tests of explicit null hypotheses The behaviour or some measure of the quality of our real data is contrasted with that of comparable but phylogenetically uninformative data determined by randomization of the data
Random Permutation Random permutation destroys any correlation among characters to that expected by chance alone It preserves number of taxa, characters and character states in each character (and the theoretical maximum and minimum tree lengths) TAXA 1 2 CHARACTERS 3 4 5 6 7 8 R-P R P R P R P R P A-E N-R A N E R A N E R A N E R A N E R D-M D M D M D M D M O-U O U O U O U O U M-T M T M T M T M T L-E L E L E L E L E Y-D Y D Y D Y D Y D TAXA CHARACTERS 1 2 3 4 5 6 7 8 R-P N U D E R T O U A-E R E A P L E A D N-R M R M M A D N P D-M L T R E Y M D R O-U M-T D O E M Y O U T D O E U Y L M T L-E Y-D Y A D P N L D R M N P R M R E E Original structured data with strong correlations among characters Randomly permuted data with any correlation among characters due to chance
Matrix Randomization Tests Compare some measure of data quality/hierarchical structure for the real and many randomly permuted data sets This allows us to define a test statistic for the null hypothesis that the real data are no better structured than randomly permuted and phylogenetically uninformative data A permutation tail probability (PTP) is the proportion of data sets with as good or better measure of quality than the real data
Structure of Randomization Tests Reject null hypothesis if, for example, more than 5% of random permutations have as good or better measure than the real data FAIL TEST Frequency 95% cutoff PASS TEST reject null hypothesis Measure of data quality (e.g. tree length, ML, pairwise incompatibilities) GOOD BAD
Matrix Randomization Tests Measures of data quality include: 1. Tree length for most parsimonious trees - the shorter the tree length the better the data (PAUP*) 2. Numbers of pairwise incompatibilities between characters (pairs of incongruent characters) - the fewer character conflicts the better the data 3. Skewness of the distribution of tree lengths (PAUP)
Matrix Randomization Tests Ciliate SSUrDNA Real data Min = 430 Max = 927 Ochromonas Symbiodinium Prorocentrum Loxodes Tracheloraphis Spirostomum Gruberia Euplotes Tetrahymena 1 MPT L = 618 CI = 0.696 RI = 0.714 PTP = 0.01 PC-PTP = 0.001 Significantly non random Randomly permuted Strict consensus Ochromonas Symbiodinium Prorocentrum Loxodes Tetrahymena Tracheloraphis Spirostomum Euplotes Gruberia 3 MPTs L = 792 CI = 0.543 RI = 0.272 PTP = 0.68 PC-PTP = 0.737 Not significantly different from random
Skewness of Tree Length Distributions NUMBER OF TREES shortest tree Studies with random (and phylogenetically uninformative) data showed that the distribution of tree lengths tends to be normal NUMBER OF TREES shortest tree Tree length Tree length In contrast, phylogenetically informative data is expected to have a strongly skewed distribution with few shortest trees and few trees nearly as short
Skewness of Tree Length Distributions Skewness of tree length distributions can be used as a measure of data quality in randomization tests It is measured with the G 1 statistic in PAUP Significance cut-offs for data sets of up to eight taxa have been published based on randomly generated data (rather than randomly permuted data) PAUP does not perform the more direct randomization test
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796 (17) 797 (30) 798 (33) 799 # (42) 800 # (62) 801 # (91) 802 # (111) 803 ## (134) 804 ## (172) 805 ### (234) 806 #### (292) 807 #### (356) 808 ###### (450) 809 ####### (557) 810 ######## (642) 811 ######### (737) 812 ############ (973) 813 ############## (1130) 814 ################ (1308) 815 #################### (1594) 816 ##################### (1697) 817 ########################## (2097) 818 ############################## (2389) 819 ################################## (2714) 820 ###################################### (3080) 821 ######################################### (3252) 822 ############################################# (3616) 823 ################################################# (3933) 824 ################################################### (4094) 825 ####################################################### (4408) RANDOMLY PERMUTED DATA g1=-0.100478 826 ######################################################### (4574) 827 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Matrix Randomization Tests - use and limitations Can detect very poor data - that provides no good basis for phylogenetic inferences (throw it away!) However, only very little may be needed to reject the null hypothesis (passing test great data) Doesn t indicate location of this structure (more discerning tests are possible) In the skewness test, significance levels for G 1 have been determined for small numbers of taxa only so that this test remains of limited use
Assessing Phylogenetic Hypotheses - groups on trees Several methods have been proposed that attach numerical values to internal branches in trees that are intended to provide some measure of the strength of support for those branches and the corresponding groups These methods include: character resampling methods - the bootstrap and jackknife decay analyses additional randomization tests
Bootstrapping (non-parametric) Bootstrapping is a modern statistical technique that uses computer intensive random resampling of data to determine sampling error or confidence intervals for some estimated parameter
Bootstrapping (non-parametric) Characters are resampled with replacement to create many bootstrap replicate data sets Each bootstrap replicate data set is analysed (e.g. with parsimony, distance, ML) Agreement among the resulting trees is summarized with a majority-rule consensus tree Frequency of occurrence of groups, bootstrap proportions (BPs), is a measure of support for those groups Additional information is given in partition tables
Bootstrapping Original data matrix Resampled data matrix Characters Taxa 1 2 3 4 5 6 7 8 A R R Y Y Y Y Y Y B R R Y Y Y Y Y Y C Y Y Y Y Y R R R D Y Y R R R R R R Outgp R R R R R R R R A B C D 1 8 7 6 5 4 3 2 1 2 Outgroup Characters Taxa 1 2 2 5 5 6 6 8 A R R R Y Y Y Y Y B R R R Y Y Y Y Y C Y Y Y Y Y R R R D Y Y Y R R R R R Outgp R R R R R R R R Randomly resample characters from the original data with replacement to build many bootstrap replicate data sets of the same size as the original - analyse each replicate data set A B C D 8 6 6 5 5 2 2 1 Outgroup Summarise the results of multiple analyses with a majority-rule consensus tree Bootstrap proportions (BPs) are the frequencies with which groups are encountered in analyses of replicate data sets A B C D 96% 66% Outgroup
Bootstrapping - an example Ciliate SSUrDNA - parsimony bootstrap 100 84 96 100 100 100 ajority-rule consensus Ochromonas (1) Symbiodinium (2) Prorocentrum (3) Euplotes (8) Tetrahymena (9) Loxodes (4) Tracheloraphis (5) Spirostomum (6) Gruberia (7) Partition Table 123456789 Freq -----------------.**... 100.00...**... 100.00...**.. 100.00...****.. 100.00...****** 95.50...** 84.33...****.* 11.83...*****. 3.83.*******. 2.50.**...*. 1.00.**...* 1.00
Bootstrapping - random data Partition Table 71 Randomly permuted data - parsimony bootstrap 59 Ochromonas Symbiodinium Prorocentrum Loxodes Tracheloraphis Spirostomumum Euplotes Tetrahymena Gruberia Ochromonas Symbiodinium Prorocentrum Loxodes Spirostomumum Tetrahymena Euplotes Tracheloraphis Gruberia Majority-rule consensus (with minority components) 71 26 16 16 59 21 123456789 Freq -----------------.*****.** 71.17..**... 58.87...*..*. 26.43.*...* 25.67.***.*.** 23.83...*...*. 21.00.*..**.** 18.50...*..* 16.00.*...*..* 15.67.***...* 13.17...**.** 12.67...**.*. 12.00..*...*.. 12.00.**..*..* 11.00.*...*... 10.80...*.** 10.50.***... 10.00
Bootstrap - interpretation Bootstrapping was introduced as a way of establishing confidence intervals for phylogenies This interpretation of bootstrap proportions (BPs) depends on the assumption that the original data is a random sample from a much larger set of independent and identically distributed data However, several things complicate this interpretation - Perhhaps the assumptions are unreasonable - making any statistical interpretation of BPs invalid - Some theoretical work indicates that BPs are very conservative, and may underestimate confidence intervals - problem increases with numbers of taxa - BPs can be high for incongruent relationships in separate analyses - and can therefore be misleading (misleading data -> misleading BPs) - with parsimony it may be highly affected by inclusion or exclusion of only a few characters
Bootstrap - interpretation Bootstrapping is a very valuable and widely used technique - it (or some suitable) alternative is demanded by some journals, but it may require a pragmatic interpretation: BPs depend on two aspects of the support for a group - the numbers of characters supporting a group and the level of support for incongruent groups BPs thus provides an index of the relative support for groups provided by a set of data under whatever interpretation of the data (method of analysis) is used
Bootstrap - interpretation High BPs (e.g. > 85%) is indicative of strong signal in the data Provided we have no evidence of strong misleading signal (e.g. base composition biases, great differences in branch lengths) high BPs are likely to reflect strong phylogenetic signal Low BPs need not mean the relationship is false, only that it is poorly supported Bootstrapping can be viewed as a way of exploring the robustness of phylogenetic inferences to perturbations in the the balance of supporting and conflicting evidence for groups
Jackknifing Jackknifing is very similar to bootstrapping and differs only in the character resampling strategy Some proportion of characters (e.g. 50%) are randomly selected and deleted Replicate data sets are analysed and the results summarised with a majority-rule consensus tree Jackknifing and bootstrapping tend to produce broadly similar results and have similar interpretations
Decay analysis In parsimony analysis, a way to assess support for a group is to see if the group occurs in slightly less parsimonious trees also The length difference between the shortest trees including the group and the shortest trees that exclude the group (the extra steps required to overturn a group) is the decay index or Bremer support Total support (for a tree) is the sum of all clade decay indices - this has been advocated as a measure for an as yet unavailable matrix randomization test
Decay analysis -example Ciliate SSUrDNA data +10 +27 +45 +15 +7 Ochromonas Symbiodinium Prorocentrum Loxodes Tracheloraphis Spirostomum Gruberia Euplotes Tetrahymena Randomly permuted data +8 +1 +1 +3 Ochromonas Symbiodinium Prorocentrum Loxodes Tetrahymena Tracheloraphis Spirostomum Euplotes Gruberia
Decay analyses - in practice Decay indices for each clade can be determined by: - Saving increasingly less parsimonious trees and producing corresponding strict component consensus trees until the consensus is completely unresolved - analyses using reverse topological constraints to determine shortest trees that lack each clade - with the Autodecay or TreeRot programs (in conjunction with PAUP)
Decay indices - interpretation Generally, the higher the decay index the better the relative support for a group Like BPs, decay indices may be misleading if the data is misleading Unlike BPs decay indices are not scaled (0-100) and it is less clear what is an acceptable decay index Magnitude of decay indices and BPs generally correlated (i.e. they tend to agree) Only groups found in all most parsimonious trees have decay indices > zero
Trees are typically complex - they can be thought of as sets of less complex relationships A B C D E Clades AB ABC DE Resolved triplets (AB)C (AC)D (DE)A (AB)D (AC)E (DE)B (AB)E (BC)D (DE)C (AC)E Resolved quartets ABCD ABDE ABCE ACDE BCDE
Extending Support Measures The same measures (BP, JP & DI) that are used for clades/splits can also be determined for triplets and quartets This provides a lot more information because there are more triplets/quartets than there are clades Furthermore...
The Decay Theorem The DI of an hypothesis of relationships is equal to the lowest DI of the resolved triplets that the hypothesis entails This applies equally to BPs and JPs as well as DIs Thus a phylogenetic chain is no stronger than its weakest link! and, measures of clade support may give a very incomplete picture of the distribution of support
Extensions Double decay analysis is the determination of decay indices for all relationships - gives a more comprehensive but potentially very complicated summary of support Majority-rule reduced consensus provides a similarly more comprehensive/complicated summary of bootstrap/jackknife proportions Leaf stability provides support values for the phylogenetic position of particular leaves
Bootstrapping with Reduced Consensus X ABCDE J I H G F X A B C D E F G H I J ABCDEX J I H G F ABCDE J I H G F A 1111100000 B 0111100000 C 0011100000 D 0001100000 E 0000100000 F 0000010000 G 0000011000 H 0000011100 I 0000011110 J 0000011111 X 1111111111 50.5 50.5 50.5 50.5 50.5 A B C D E F G H I J 100 99 98 99 100 98 100 100
Bootstrapping X A B C D E F G H I J A B C D E F G H I J 50.5 50.5 50.5 50.5 50.5 100 99 98 99 100 98 100 100
Leaf Stability Leaf stability is the average of supports of the triplets/quartets containing the leaf 94 100 59 100 95 84 Acanthostega Ichthyostega Greererpeton Crassigyrinus Eucritta (54) Gephyrostegus Whatcheeria Balanerpeton Dendrerpeton Pholiderpeton Proterogyrinus Baphetes (67) Loxomma (67) Megalocephalus (98) (98) (69) (53) (58) (49) (64) (64) (66) (66) (67)
PTP tests of groups A number of randomization tests have been proposed for evaluating particular groups rather than entire data matrices by testing null hypotheses regarding the level of support they receive from the data Randomisation can be of the data or the group These methods have not become widely used both because they are not readily performed and because their properties are still under investigation One type, the topology dependent PTP tests are included in PAUP* but have serious problems
Comparing competing phylogenetic hypotheses - tests of two trees Particularly useful techniques are those designed to allow evaluation of alternative phylogenetic hypotheses Several such tests allow us to determine if one tree is statistically significantly worse than another: Winning sites test, Templeton test, Kishino-Hasegawa test, Shimodaira-Hasegawa test, parametric bootstrapping
Tests of two trees All these tests are of the null hypothesis that the differences between two trees (A and B) are no greater than expected from sampling error The simplest wining sites test sums the number of sites supporting tree A over tree B and vice versa (those having fewer steps on, and better fit to, one of the trees) Under the null hypothesis characters are equally likely to support tree A or tree B and a binomial distribution gives the probability of the observed difference in numbers of winning sites
The Templeton test Templeton s test is a non-parametric Wilcoxon signed ranks test of the differences in fits of characters to two trees It is like the winning sites test but also takes into account the magnitudes of differences in the support of characters for the two trees
Templeton s test - an example Seymouriadae Diadectomorpha 1 Synapsida Parareptilia Captorhinidae Paleothyris Araeoscelidia Claudiosaurus Younginiformes Archosauromorpha Lepidosauriformes 2 Placodus Eosauropterygia Recent studies of the relationships of turtles using morphological data have produced very different results with turtles grouping either within the parareptiles (H1) or within the diapsids (H2) the result depending on the morphologist This suggests there may be: - problems with the data - special problems with turtles - weak support for turtle relationships Parsimony analysis of the most recent data favoured H2 However, analyses constrained by H2 produced trees that required only 3 extra steps (<1% tree length) The Templeton test was used to evaluate the trees and showed that the slightly longer H1 tree found in the constrained analyses was not significantly worse than the unconstrained H2 tree The morphological data do not allow choice between H1 and H2
Kishino-Hasegawa test The Kishino-Hasegawa test is similar in using differences in the support provided by individual sites for two trees to determine if the overall differences between the trees are significantly greater than expected from random sampling error It is a parametric test that depends on assumptions that the characters are independent and identically distributed (the same assumptions underlying the statistical interpretation of bootstrapping) It can be used with parsimony and maximum likelihood - implemented in PHYLIP and PAUP*
Kishino-Hasegawa test Sites favouring tree A Mean Expected Sites favouring tree B 0 Distribution of Step/Likelihood differences at each site Under the null hypothesis the mean of the differences in parsimony steps or likelihoods for each site is expected to be zero, and the distribution normal From observed differences we calculate a standard deviation If the difference between trees (tree lengths or likelihoods) is attributable to sampling error, then characters will randomly support tree A or B and the total difference will be close to zero The observed difference is significantly greater than zero if it is greater than 1.95 standard deviations This allows us to reject the null hypothesis and declare the suboptimal tree significantly worse than the optimal tree (p < 0.05)
Kishino-Hasegawa test - an example Ochromonas Symbiodinium Prorocentrum Sarcocystis Theileria Colpoda Ciliate SSUrDNA Maximum likelihood tree Plagiopyla n Plagiopyla f Trimyema c Trimyema s Cyclidium p Cyclidium g Cyclidium l Glaucoma Colpodinium Tetrahymena Paramecium Discophrya Trithigmostoma Opisthonecta Dasytrichia Entodinium Spathidium Loxophylum Homalozoon Metopus c Metopus p Stylonychia Onychodromous Oxytrichia Loxodes Tracheloraphis Spirostomum Gruberia Blepharisma Parsimonious character optimization of the presence and absence of hydrogenosomes suggests four separate origins of hydrogenosomes within the ciliates Questions - how reliable is this result? - in particular how well supported is the idea of multiple origins? - how many origins can we confidently infer? anaerobic ciliates with hydrogenosomes
Kishino-Hasegawa test - an example Ochromonas 99-100 Symbiodinium 95-100 11 Prorocentrum 7 Sarcocystis 81-86 Theileria 3 96-100 10 69-78 3 18-0 1 100 11-0 3 35-17 3 3 7 41-30 12 78-99 3 89-91 4 45-72 46-26 3 53-45 5 83-82 100 27 Colpoda 100 63 26 100 18 80-50 3 15-0 100-99 23 100 33 Plagiopyla n 100 48 Plagiopyla f 100 27 Trimyema c Trimyema s 69-99 Glaucoma 100 75 6 Colpodinium Tetrahymena Paramecium Cyclidium p Cyclidium g Cyclidium l Discophryal 67-99 50-53100-98 Opisthonecta 100 56 Spathidium 3 3 17 Homalozoon Loxophylum 100 42 Metopus c Metopus p Stylonychia Onychodromous Oxytrichia Ciliate SSUrDNA data Most parsimonious tree Loxodes Tracheloraphis Spirostomum Gruberia Blepharisma Trithigmostoma Dasytrichia Entodinium Parsimony analysis yields a very similar tree - in particular, parsimonious character optimization indicates four separate origins of hydrogenosomes within ciliates Decay indices and BPs for parsimony and distance analyses indicate relative support for clades Differences between the ML, MP and distance trees generally reflect the less well supported relationships
Kishino-Hasegawa test - example Ochromonas Symbiodinium Prorocentrum Sarcocystis Theileria Plagiopyla n Plagiopyla f Trimyema c Trimyema s Cyclidium p Cyclidium g Cyclidium l Dasytrichia Entodinium Loxophylum Homalozoon Spathidium Metopus c Metopus p Loxodes Tracheloraphis Spirostomum Gruberia Blepharisma Discophrya Trithigmostoma Stylonychia Onychodromous Oxytrichia Colpoda Paramecium Glaucoma Colpodinium Tetrahymena Opisthonecta Ochromonas Symbiodinium Prorocentrum Sarcocystis Theileria Plagiopyla n Plagiopyla f Trimyema c Trimyema s Cyclidium p Metopus c Metopus p Dasytrichia Entodinium Cyclidium g Cyclidium l Loxophylum Spathidium Homalozoon Loxodes Tracheloraphis Spirostomum Gruberia Blepharisma Discophrya Trithigmostoma Stylonychia Onychodromous Oxytrichia Colpoda Paramecium Glaucoma Colpodinium Tetrahymena Opisthonecta Parsimony analyse with topological constraints were used to find the shortest trees that forced hydrogenosomal ciliate lineages together and thereby reduced the number of separate origins of hydrogenosomes Each of the constrained parsimony trees were compared to the ML tree and the Kishino-Hasegawa test used to determine which of these trees were significantly worse than the ML tree Two examples of the topological constraint trees
Kishino-Hasegawa test Test summary and results - origins of ciliate hydrogenosomes (simplified) No. Constraint Extra Difference Significantly Origins tree Steps and SD worse? 4 ML +10 - - 4 MP - -13 ± 18 No 3 (cp,pt) +13-21 ± 22 No 3 (cp,rc) +113-337 ± 40 Yes 3 (cp,m) +47-147 ± 36 Yes 3 (pt,rc) +96-279 ± 38 Yes 3 (pt,m) +22-68 ± 29 Yes 3 (rc,m) +63-190 ± 34 Yes 2 (pt,cp,rc) +123-432 ± 40 Yes 2 (pt,rc,m) +100-353 ± 43 Yes 2 (pt,cp,m) +40-140 ± 37 Yes 2 (cp,rc,m) +124-466 ± 49 Yes 2 (pt,cp)(rc,m) +77-222 ± 39 Yes 2 (pt,m)(rc,cp) +131-442 ± 48 Yes 2 (pt,rc)(cp,m) +140-414 ± 50 Yes 1 (pt,cp,m,rc) +131-515 ± 49 Yes Constrained analyses used to find most parsimonious trees with less than four separate origins of hydrogenosomes Tested against ML tree Trees with 2 or 1 origin are all significantly worse than the ML tree We can confidently conclude that there have been at least three separate origins of hydrogenosomes within the sampled ciliates
Shimodaira-Hasegawa Test To be statistically valid, the Kishino-Hasegawa test should be of trees that are selected a priori However, most applications have used trees selected a posteriori on the basis of the phylogenetic analysis Where we test the best tree against some other tree the KH test will be biased towards rejection of the null hypothesis The SH test is a similar but more statistically correct technique in these circumstances and should be preferred
Taxonomic Congruence Trees inferred from different data sets (different genes, morphology) should agree if they are accurate Congruence between trees is best explained by their accuracy Congruence can be investigated using consensus (and supertree) methods Incongruence requires further work to explain or resolve disagreements
Reliability of Phylogenetic Methods Phylogenetic methods (e.g. parsimony, distance, ML) can also be evaluated in terms of their general performance, particularly their: consistency - approach the truth with more data efficiency - how quickly (how much data) robustness - how sensitive to violations of assumptions Studies of these properties can be analytical or by simulation
Reliability of Phylogenetic Methods There have been many arguments that ML methods are best because they have desirable statistical properties, such as consistency However, ML does not always have these properties if the model is wrong/inadequate (fortunately this is testable to some extent) properties not yet demonstrated for complex inference problems such as phylogenetic trees
Reliability of Phylogenetic Methods Simulations show that ML methods generally outperform distance and parsimony methods over a broad range of realistic conditions Whelan et al. 2001 Trends in Genetics 17:262-272 Most simulations are very (unrealistically) simple few taxa (typically just four) few parameters (standard models - JC, K2P etc)
Reliability of Phylogenetic Methods Simulations with four taxa have shown: - Model based methods - distance and maximum likelihood perform well when the model is accurate (not surprising!) - Violations of assumptions can lead to inconsistency for all methods (a Felsenstein zone) when branch lengths or rates are highly unequal - Maximum likelihood methods are quite robust to violations of model assumptions - Weighting can improve the performance of parsimony (reduce the size of the Felsenstein zone)
Reliability of Phylogenetic Methods However: - Generalising from four taxon simulations may be dangerous as conclusions may not hold for more complex cases - A few large scale simulations (many taxa) have suggested that parsimony can be very accurate and efficient - Most methods are accurate in correctly recovering known phylogenies produced in laboratory studies More study of methods is needed to help in choice of method using more realistic simulations