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Department of Zoology, The Natural History Museum Inferring Trees from Trees Consensus and Supertree Methods Mark Wilkinson.

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Presentation on theme: "Department of Zoology, The Natural History Museum Inferring Trees from Trees Consensus and Supertree Methods Mark Wilkinson."— Presentation transcript:

1 Department of Zoology, The Natural History Museum Inferring Trees from Trees Consensus and Supertree Methods Mark Wilkinson

2 CONSENSUS “general or widespread agreement”  Consensus tree – a tree depicting agreement among a set of trees - a representation of a set of trees - a phylogenetic inference from a set of trees  Consensus method – a technique for producing consensus trees (of a particular type)  Consensus index – a measure of the agreement among a set of trees (based on their consensus tree)

3 Uses of Consensus Trees Consensus trees are used to represent (or make inferences from) multiple treesConsensus trees are used to represent (or make inferences from) multiple trees Agreement (conservative) Central tendency (liberal) There are a number of different contexts in which this may be of interest (sets of trees can be obtained in a variety of ways)There are a number of different contexts in which this may be of interest (sets of trees can be obtained in a variety of ways) The ultimate aims may be quite differentThe ultimate aims may be quite different Different methods may be more or less appropriate given the aim/contextDifferent methods may be more or less appropriate given the aim/context

4 Mathematician’s Perspective Subsequently there has been an amazing proliferation of consensus methods and consensus indices): a proliferation stimulated by confusions, disagreements, and uncertainties concerning what consensus methods depict and what consensus indices measure. Thus, for example, consensus indices for trees are understood to measure agreement, balance, information, resolution, shape, similarity, and symmetry. One has the impression that taxonomists do not know (or cannot agree on) what consensus objects should depict or how it should be depicted; they do not know (or cannot agree on) what consensus indices should measure or how it should be quantified. Consequently, taxonomists may not appreciate (or do not articulate) the relationships that might or should exist between consensus method and consensus index. Day and McMorris 1985

5 Strict (Component) CM Uniquely defined in terms of two properties Uniquely defined in terms of two properties Pareto - if a component is present in all the input trees it is in the consensus Pareto - if a component is present in all the input trees it is in the consensus Strict - if a component is in the consensus it is present in all the input trees Strict - if a component is in the consensus it is present in all the input trees Can also be defined in terms of Can also be defined in terms of an algorithm an algorithm an objective function an objective function

6 Strict CM(s) Require complete agreement across all the input trees and show relationships that would be true if any input tree were trueRequire complete agreement across all the input trees and show relationships that would be true if any input tree were true With MPTs they show only those relationships that are unambiguously supported by the parsimonious interpretation of the dataWith MPTs they show only those relationships that are unambiguously supported by the parsimonious interpretation of the data The commonest method focuses on components (clusters, groups, splits, clades or monophyletic groups)The commonest method focuses on components (clusters, groups, splits, clades or monophyletic groups) This method produces a consensus tree that includes all (Pareto) and only (strict) the common clades Other relationships (those in which the input trees disagree) are shown as unresolved polytomiesThis method produces a consensus tree that includes all (Pareto) and only (strict) the common clades Other relationships (those in which the input trees disagree) are shown as unresolved polytomies Component version widely implementedComponent version widely implemented

7 Strict CM(s) ABCDEFG A B C E D FG TWO INPUT TREES A B C D E FG STRICT (COMPONENT) CONSENSUS TREE

8 Interpreting Polytomies Polytomies in trees have alternative interpretations.Polytomies in trees have alternative interpretations. The hard interpretationThe hard interpretation ‘multiple speciation’ ‘multiple speciation’ The soft interpretationThe soft interpretation ‘uncertain resolution’ The soft interpretationThe soft interpretation is appropriate for strict component consensus trees The consensus permits allThe consensus permits all resolutions of the polytomy (i.e. it does not conflict with any resolution) (i.e. it does not conflict with any resolution)

9 Semi-strict CM(s) Semi-strict methods require assertion of a relationship by one or more trees and non-contradiction by any treeSemi-strict methods require assertion of a relationship by one or more trees and non-contradiction by any tree The commonest method focuses on components/cladesThe commonest method focuses on components/clades Produces a tree including all components that are present and uncontradicted in the input trees - all that could be true if any input tree were trueProduces a tree including all components that are present and uncontradicted in the input trees - all that could be true if any input tree were true Generally, similar to the strict but may be more resolved when the input trees include polytomiesGenerally, similar to the strict but may be more resolved when the input trees include polytomies It is based on the soft interpretation of polytomiesIt is based on the soft interpretation of polytomies Other relationships are shown as unresolved polytomiesOther relationships are shown as unresolved polytomies Component version implemented in e.g. PAUPComponent version implemented in e.g. PAUP

10 Semi-strict (component) CM TWO INPUT TREES

11 Properties of Semi-strict CM(s) Tend to produce more resolved consensus treesTend to produce more resolved consensus trees Reasonable when combining trees based on different data setReasonable when combining trees based on different data set With trees based on a single data set extra resolution is of relationships that are not true of all the optimal trees:With trees based on a single data set extra resolution is of relationships that are not true of all the optimal trees: the consensus includes relationships that are not supported by all best interpretations of the data It may include relationships that cannot be simultaneously supported by any parsimonious (or other) interpretation of the data These relationships might reasonably be considered less well supported (if supported at all)

12 Semi-strict (component) CM TWO INPUT TREES SEMI-STRICT (component) CONSENSUS TREE

13 Equally Optimal Trees Many phylogenetic analyses yield multiple equally optimal treesMany phylogenetic analyses yield multiple equally optimal trees Multiple trees are due to either:Multiple trees are due to either: Alternative equally optimal interpretations of conflicting data Missing data Or both We can further select among these trees with additional (secondary) criteria, butWe can further select among these trees with additional (secondary) criteria, but A consensus tree may be needed to represent or draw conclusions about the set of MTPsA consensus tree may be needed to represent or draw conclusions about the set of MTPs Typically phylogeneticists are interested in relationships common to all the optimal trees (they want to know that a relationship in the consensus is in all the trees - STRICT)Typically phylogeneticists are interested in relationships common to all the optimal trees (they want to know that a relationship in the consensus is in all the trees - STRICT)

14 Loss of Resolution Generally, as the number of optimal trees increases the resolution of ‘strict consensus trees’ decreases.Generally, as the number of optimal trees increases the resolution of ‘strict consensus trees’ decreases. In the extreme, the ‘strict consensus tree’ may be completely unresolved/uninformative.In the extreme, the ‘strict consensus tree’ may be completely unresolved/uninformative. This extreme is sometimes met in practice (e.g. fossils).This extreme is sometimes met in practice (e.g. fossils). The ‘consensus tree’ can also be poorly resolved when there are few optimal trees, and.furthermore.The ‘consensus tree’ can also be poorly resolved when there are few optimal trees, and.furthermore. The optimal trees need not differ greatly, thus...The optimal trees need not differ greatly, thus... Lack of resolution in a ‘strict consensus tree’ is not always a good guide to the level of agreement among the optimal trees.Lack of resolution in a ‘strict consensus tree’ is not always a good guide to the level of agreement among the optimal trees.

15 Optimal trees need not differ greatly for ‘the consensus tree’ to be unresolved

16 Adams-2 Consensus Adams (1972) method was defined by a recursive algorithm, which, beginning at the root, identifies common sub-clusters using intersection rules (product partitions) The basal splits in these trees yield the four clusters ABC, EFD, AC & BEFD. Their intersections yield AC, B & EFD and these produce the three branches at the base of the consensus tree. The procedure is repeated for the subtrees induced by E, F and D, which in this case are identical.

17 Adams Consensus and Nesting Adams (1972) described the first consensus methods, only one of his methods is usedAdams (1972) described the first consensus methods, only one of his methods is used Adams (1986) characterised his method in terms of nestingsAdams (1986) characterised his method in terms of nestings A group X (e.g. AB) nests within another Y (e.g. A-D) if the last common ancestor of Y is an ancestor of the last common ancestor of XA group X (e.g. AB) nests within another Y (e.g. A-D) if the last common ancestor of Y is an ancestor of the last common ancestor of X The Adams consensus tree includes all those nestings that are in all the input trees (Pareto) and for all clusters displayed by the Adams tree there is a corresponding nesting in each input tree (strict).The Adams consensus tree includes all those nestings that are in all the input trees (Pareto) and for all clusters displayed by the Adams tree there is a corresponding nesting in each input tree (strict). They show only those nestings that are unambiguously supported by the parsimonious interpretation of the dataThey show only those nestings that are unambiguously supported by the parsimonious interpretation of the data Implemented in e.g. PAUPImplemented in e.g. PAUP

18 Adams Consensus and Nesting (1) taxa A & C are more closely related to each other than either is to taxa D, E, or F; (2) taxa E & F are more closely related to each other than either is to taxa A, C, or D; (3) taxon D is more closely related to E & F than it is to either A or C. Swofford (1991) But - Not quite right!

19 Adams Consensus (1) taxa A & C are more closely related to each other than either is to taxa D, E, or F; In each tree A & C are more closely related to each other than they are to D-F and/or B (AC)D-F and/or (AC)B (2) taxa E & F are more closely related to each other than either is to taxa A, C, or D; (EF)D (3) taxon D is more closely related to E & F than it is to either A or C. (D-F)AC and/or (D-F)B

20 Adams polytomies are cladistically ambiguous What can be inferred from the consensus? (A,B)CD - No (AB)C - No (AB)D - No (AB)C and/or (AB)D INPUT TREES

21 Note on the meaning of cladistic relationship Cladistic relationships are based on recency of common ancestry (& dependent on rooted trees).Cladistic relationships are based on recency of common ancestry (& dependent on rooted trees). Two taxa are more closely related to each other than either is to a third iff they share a more recent common ancestor - e.g. (AB)CDE.Two taxa are more closely related to each other than either is to a third iff they share a more recent common ancestor - e.g. (AB)CDE. Nestings are also based on common ancestry but nestings are ambiguous with respect to cladistic relationships - e.g. {AB}CD = (AB)C and/or (AB)DNestings are also based on common ancestry but nestings are ambiguous with respect to cladistic relationships - e.g. {AB}CD = (AB)C and/or (AB)D

22 Properties of Adams Consensus Trees Adams consensus trees are more topologically sensitive to shared structure in input trees than is the strict component consensus, but...Adams consensus trees are more topologically sensitive to shared structure in input trees than is the strict component consensus, but... Care must be taken in the interpretation of their ‘elastic’ polytomiesCare must be taken in the interpretation of their ‘elastic’ polytomies Adams consensus trees can include groups that don’t occur in any input tree (Rholf Groups)Adams consensus trees can include groups that don’t occur in any input tree (Rholf Groups) It exists only for rooted treesIt exists only for rooted trees

23 Greatest Agreement Subtrees A B C D E FG TWO INPUT TREES GAS/LCP TREE Taxon G is excluded AGBCDEF A B C D E F A B C DE F G Strict component consensus completely unresolved

24 Strict Reduced CM A B C D E FG TWO INPUT TREES STRICT REDUCED CONSENSUS TREE Taxon G is excluded AGBCDEF A B C D E F A B C DE F G Strict component consensus B C D E F A C D E F A B D E F Agreement Subtrees

25 Rhynchosaurs

26 Fossil & Recent Arthropods

27

28 Majority-rule CM(s) Majority-rule consensus methods require agreement across a majority of the input treesMajority-rule consensus methods require agreement across a majority of the input trees The commonest method focuses on components/cladesThe commonest method focuses on components/clades This method produces a consensus tree that includes all and only those clades found in a majority (>50%) of the input treesThis method produces a consensus tree that includes all and only those clades found in a majority (>50%) of the input trees Majority components which are necessarily mutually compatibleMajority components which are necessarily mutually compatible Other relationships are shown as unresolved polytomiesOther relationships are shown as unresolved polytomies Of particular use in bootstrapping, jackknifing, quartet puzzling, Bayesian inference (with e.g. average branch lengths).Of particular use in bootstrapping, jackknifing, quartet puzzling, Bayesian inference (with e.g. average branch lengths). Component version widely implementedComponent version widely implemented

29 Majority-rule (component) CM ABCDEFG A B C E D FG ABCEDFG MAJORITY-RULE (COMPONENT) CONSENSUS A B C E F DG THREE INPUT TREES Numbers indicate frequency of clades in the input trees

30 Properties of Majority-rule Tend to produce more resolved consensus treesTend to produce more resolved consensus trees Extra resolution is of relationships that are not true of all the optimal treesExtra resolution is of relationships that are not true of all the optimal trees In the context of equally optimal trees, this means the consensus includes relationships that are not supported by all the best interpretations of the dataIn the context of equally optimal trees, this means the consensus includes relationships that are not supported by all the best interpretations of the data These relationships might reasonably be considered less well supported (if supported at all)These relationships might reasonably be considered less well supported (if supported at all) Related to the Median consensus (objective function minimises the sum of the symmetric differences between the consensus and input trees)Related to the Median consensus (objective function minimises the sum of the symmetric differences between the consensus and input trees)

31 Adding minority components Further resolution can sometimes be achieved by adding relationships that occur in a minority of trees.Further resolution can sometimes be achieved by adding relationships that occur in a minority of trees. These must be compatible with the majority relationshipsThese must be compatible with the majority relationships Two approachesTwo approaches Greedy (PAUP) Frequency-difference (TNT)

32 Majority-rule

33 Other Consensus methods A variety of other consensus methods have been devised but few implementedA variety of other consensus methods have been devised but few implemented These include:These include: Other intersection rules based on cluster height Nelson, Asymmetric Median and other clique consensus methods Other matrix respresentation methods, e.g. MRP MRP Average consensus

34 A Consensus Classification Consensus trees vary with respect to:Consensus trees vary with respect to: The kind of agreement (components, triplets, nestings, subtrees) The level of agreement (strict, semi-strict, majority- rule, largest clique) The level of agreement (strict, semi-strict, majority- rule, largest clique) Adams Reduced LCP / GAS Full SplitsNestings Splits Subtrees Full SplitsNestings Splits Subtrees StrictYes Yes YesYes Semi-strictYes ? Yes ? Majority-ruleYes ? Yes ? Nelson (clique)Yes ? Yes ?

35 Consensus methods Use strict methods to identify those relationships unambiguously supported by parsimonious interpretation of the dataUse strict methods to identify those relationships unambiguously supported by parsimonious interpretation of the data Use more liberal (semi-strict, majority-rule) consensus methods for taxonomic congruenceUse more liberal (semi-strict, majority-rule) consensus methods for taxonomic congruence Use majority-rule methods in bootstrapping etc.Use majority-rule methods in bootstrapping etc. Use Adams consensus when strict component consensus is poorly resolved - if Adams is better resolved use strict reduced consensusUse Adams consensus when strict component consensus is poorly resolved - if Adams is better resolved use strict reduced consensus Use reduced methods where consensus trees are poorly resolvedUse reduced methods where consensus trees are poorly resolved Avoid over-interpreting results from methods which have ambiguous interpretationsAvoid over-interpreting results from methods which have ambiguous interpretations

36 Input Trees Consensus Trees More or less Conservative More or less Liberal

37 Input Trees More or less Conservative More or less Liberal SuperTrees

38 Biologists want (Big) Trees “Nothing in Biology makes sense except in the light of evolution” Dobzhansky, 1973“Nothing in Biology makes sense except in the light of evolution” Dobzhansky, 1973 The Tree of Life: Holy Grail of SystematicsThe Tree of Life: Holy Grail of Systematics Bigger Trees: more powerful comparative analysesBigger Trees: more powerful comparative analyses –Adaptation –Biogeography –Speciation and diversification –Conservation

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40 When Input Trees Conflict Semi-strictSemi-strict Gene Tree ParsimonyGene Tree Parsimony MinCut (modified Aho)MinCut (modified Aho) Quartet puzzlingQuartet puzzling Matrix representationsMatrix representations Splits (standard MRP, MRC, MRF)Splits (standard MRP, MRC, MRF) Sister groups (Purvis MRP)Sister groups (Purvis MRP) TripletsTriplets QuartetsQuartets Distances (MRD)Distances (MRD) –Analysed with Parsimony (MRP) - , Parsimony (MRP) - ,  Clique (MRC)Clique (MRC) MinFlip (MRF)MinFlip (MRF) Least squaresLeast squares

41 MRP Tree 1 Tree 2 Fecampiida ??11??111 Neodermata ??? 111??? 111?????????? Tricladida ??1???11111 Lecithoepitheliata ????111110?? Polycladida 000 ???00? ???0000?0??0? Kalytporhyncha ?? ?0?0??0 MRP-outgroup Component Purvis Triplet

42 MRP A C B Components unordered - A,B,C irreversible - C Triplets - A Quartets - A & C

43 MRP – an unusual consensus

44 MRP, total evidence and taxonomic congruence

45 Majority- rule Goloboff and Pol (2002), Goloboff (2006) Majority rule supertrees desirable in principle Fundamental problem in generalising frequency of occurrence of groups MRP is a (poor) surrogate

46 Majority-rule Majority-rule consensus is also a median tree for the symmetric differenceMajority-rule consensus is also a median tree for the symmetric difference Alternative basis for generalising beyond consensusAlternative basis for generalising beyond consensus How to compute symmetric difference for trees with different leaf sets?How to compute symmetric difference for trees with different leaf sets? Convert into trees with identical leaf setsConvert into trees with identical leaf sets Prune leaves from supertree Prune leaves from supertree Graft leaves onto supertree Graft leaves onto supertree

47 ML Supertrees

48 ‘Taxonomic Congruence’

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