Treesearches and tests for phylogenetic signal

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Tree-searches and tests for phylogenetic signal
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Many issues glossed over

• What if characters disagree?
• How is the tree score determined?
• How can we root the trees?
• How do we find the optimal tree?
• How can we evaluate the robustness of our
  conclusions?

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Terminology
Polytomy
Binary/dichotomous/fully-resolved
polyotomous/unresolved
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Unrooted trees
Polytomy
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Number of unrooted fully resolved trees for t taxa
i t

• ? (2i-5)

i 3
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How many places can you add another taxon?

• Two taxa
• Three taxa
• Four Taxa
• Five taxa

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What is the number of rooted trees?

• The root is just one more taxon same formula but
  t number of taxa 1

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The number of trees gets big

• Number of binary unrooted trees
• 1
• 3
• 15
• 105
• 2,027,025
• 2.2 x 1020
• 2.8 x 1074
• 1 x 101074

• Number of tips
• 3
• 4
• 5
• 6
• 10
• 20
• 50
• 500

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How do you find the optimal tree?

• Exhaustive (

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How do you find the optimal tree?

• Exhaustive (
• Branch-and-bound (

• Obtain the length of a random tree (initial upper
  bound)
• As trees are built determine length
• If length exceeds upper bound then that tree and
  all its descendant trees are ignored

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How do you find the optimal tree?

• Exhaustive (
• Branch-and-bound (
• Heuristic search (unlimited?)

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Heuristic searches

• Search for optimal trees by finding good trees
  and then rearranging them in the hopes of finding
  an even better tree

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Getting starting trees

• Random tree - not done
• User tree (e.g., a NJ tree)
• Build a tree by adding taxa to the location that
  is optimal
• Can hold more than one tree at each step

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Taxon addition order

• As-is
• In the order of the matrix (not done for
  parsimony)
• Simple taxon addition
• use a distance algorithm to decide order
• Closest taxon addition
• Add the taxon that makes the optimal tree
• Random taxon addition order
• Repeat many times

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Heuristic search
Suboptimal island of trees
Global optimum
Starting trees
Treespace
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Branch swapping

• Nearest-neighbour interchange (NNI)

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Branch swapping

• Subtree pruning and regrafting (SPR)

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Branch swapping

• Tree-bisection reconnection (TBR)

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Many issues glossed over

• What if characters disagree?
• How is the tree score determined?
• How can we root the trees?
• How do we find the optimal tree?
• How can we evaluate the robustness of our
  conclusions?

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Even if the shortest trees is the best estimate
of the true tree - the true tree might not be the
shortest
We should consider suboptimal trees
We should use statistical tests to help us
determine what to actually believe
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Questions we can ask

• Are the data random or do they have signal?
• How much homoplasy is there?
• To what extent are particular elements of the
  trees (clades) supported?
• What alternative results can we reject?

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How can we evaluate the reliability of the
tree(s) we obtain?

• Is there agreement within the data?

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The logic of looking at consistency indices

• If all the characters have the same signal then
  the tree is more trustworthy
• The more agreement there is, the less homoplasy
  (more consistency) the characters will show on
  the most parsimonious tree
• We need statistics to measure consistency

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How much homoplasy is there?

• Taxon 1 A C A T T T A
• Taxon 2 A C G A T T A
• Taxon 3 A G G A T A G
• Taxon 4 G A A A A C ?
• Taxon 5 G A T A ? C G
• ObsL 1 2 3 1 1 2 1

Min L 1 2 2 1 1 2 1
Minimum length overall 10 Length of MP tree 11
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Consistency index

• CI Min L 10 0.91
• Obs L 11

Homoplasy index
HI 1-CI 0.09
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How much homoplasy is there?

• Taxon 1 A C A T T T A
• Taxon 2 A C G A T T A
• Taxon 3 A G G A T A G
• Taxon 4 G A A A A C ?
• Taxon 5 G A T A ? C G
• ObsL 1 2 3 1 1 2 1

Min L 1 2 2 1 1 2 1
CI 1 1 .67 1 1 1 1
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CI is affected by uninformative characters

• Taxon 1 A C A T T T A
• Taxon 2 A C G A T T A
• Taxon 3 A G G A T A G
• Taxon 4 G A A A A C ?
• Taxon 5 G A T A ? C G
• Min L 1 2 2 1 1 2 1

CI 1 1 .67 1 1 1 1
Minimum length overall 8 Length of MP tree
9 CI 0.89
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Retention Index

• Taxon 1 A C A T T T A
• Taxon 2 A C G A T T A
• Taxon 3 A G G A T A G
• Taxon 4 G A A A A C ?
• Taxon 5 G A T A ? C G
• Min L 1 2 2 1 1 2 1
• Max L 2 3 3 1 1 3 2

Maximum length overall 15
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Retention index (RI)

• MaxL - ObsL 5 0.83
• MaxL - MinL 6

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General trends observed with CI/RIs

• Strong negative correlation between taxon number
  and CI/RI
• Data sets with few characters can show unexpected
  high CI/RI

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How can we evaluate the significance of CI/RI?

• CI depends directly on tree length
• We can compare the observed tree length with that
  we would obtain if there were no phylogenetic
  signal

The permutation tail probability (PTP) test
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Permuting data removes phylogenetic signal

• Taxon 1 ACATTTA
• Taxon 2 ACGATTA
• Taxon 3 AGGATAG
• Taxon 4 GAAAAC?
• Taxon 5 GATA?CG

Permuted data sets
Taxon 1 GAAA?AA Taxon 2 ACAATC? Taxon 3
GAGTATG Taxon 4 AGTATCG Taxon 5 ACGATTA
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Example with signal
Number of Tree length replicates
------------------------- 1222 1
1669 1 1671 1
1672 1 1673 1 1674
1 1675 2 1676 2
1678 1 1679 2
1680 4 1681 5 1682
8 1683 4 1684 4
1685 2
Number of Tree length replicates
------------------------- 1686 8
1687 7 1688 6
1689 8 1690 6 1691
3 1692 2 1693
3 1694 3 1695 3
1696 3 1697 2
1699 2 1702 1 1704
2 1705 1
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Example without signal
Number of Tree length replicates
------------------------- 1924
3 1926 1 1927
4 1928 1
1929 2 1930 8
1931 6 1932
5 1933 4 1934
4 1935 5
1936 1 1937 8
1938 11 1939
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Number of Tree length replicates
------------------------- 1940 6
1941 7 1942
4 1943 2 1944
1 1945 1
1946 1 1947 1
1950 3 1952
1 1953 1 1955
1 1958 1
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The PTP test is slow

• Hillis and Huelsenbeck (1991) observed a
  difference between the shape of the tree length
  distribution as a function of phylogenetic signal

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A data set without signal
mean599.182107 sd4.944738 g1-0.150922 582.0000
0 /-----------------------------------------------
------------------------- 583.80000
(5) 585.60000 (25) 587.40000
(71) 589.20000 (209) 591.00000
(161) 592.80000
(521) 594.60000
(883) 596.40000
(1132) 598.20000

(1469) 600.00000

(788) 601.80000

(1631) 603.60000

(1486) 605.40000
(1047) 607.20000
(567) 609.00000
(157) 610.80000
(171) 612.60000 (57) 614.40000
(11) 616.20000 (3) 618.00000 (1)
\-------------------------------------------------
-----------------------
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A data set with signal
mean611.572872 sd31.049455 g1-0.942643 501.000
00 /----------------------------------------------
-------------------------- 508.65000
(15) 516.30000 (60) 523.95000
(84) 531.60000 (135) 539.25000
(21) 546.90000 (26) 554.55000
(96) 562.20000 (166) 569.85000
(290) 577.50000
(737) 585.15000
(1118) 592.80000
(665) 600.45000 (120) 608.10000
(268) 615.75000
(497) 623.40000
(796) 631.05000
(1337) 638.70000

(2031) 646.35000

(1610) 654.00000 (323)
\--------------------------------------------
----------------------------
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Skewness test for phylogenetic signal

• Hillis and Huelsenbeck (1991) generated random
  data for different numbers of taxa/characters to
  find the null distribution of g1 scores
• One can compare observed g1 statistics with this
  null distribution

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Tests for phylogenetic signal (g1 and PTP)

• Are sensitive to any signal in the data
• For example
• g1 of permuted data -0.04 (ns)
• Duplicate one taxon and g1 -1.56
• Useful for identifying truly useless data (very
  rare)
• But otherwise does not tell you much about data
  quality

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Tests of signal

• These methods seek to determine overall data
  quality as a guide to whether we should believe
  particular results
• We can, instead, evaluate particular results
• Clade support measures bootstrap/decay
• Statistical tests of alternative hypotheses