Cladistics
Cladistics 22 (2006) 378–386
www.blackwell-synergy.com
The how and why of branch support and partitioned branch support,
with a new index to assess partition incongruence
Andrew V. Z. Brower
Department. of Zoology, Oregon State University, Corvallis OR 97331, USA
Accepted 19 January 2006
Abstract
An introduction and step-by-step guide to the calculation of branch support (BS) and partitioned branch support (PBS) is
provided, along with a new summary measure of conflict among data partitions, the partition congruence index (PCI). Patterns of
PCI vs. BS are compared for two data matrices from butterflies. PCI is a close mirror of BS unless BY is relatively low and there is a
lot of conflict among data partitions, in which case PCI becomes negative. Some theoretical concerns relating to PBS are noted.
The Willi Hennig Society 2006.
There are a lot of phylogenetic trees around these
days. Everything from cladograms to directly optimized
alignments to phenograms based on distance algorithms
are presented as ‘‘phylogenies’’ of this or that taxon. A
question that most systematists would like to answer,
at least in a heuristic way, is, ‘‘How good is my tree?’’
(e.g., Wenzel, 1997). Two aspects of clade quality that
are frequently mentioned are support and stability.
Support for a given branch in a tree is a measure of the
net amount of evidence that favors the appearance of
that branch in a most-parsimonious topology. Stability
is the persistence of a given branch in the face of the
addition, deletion or reweighting of characters, taxa, or
both from the data matrix (Judd, 1998). Both of these
attributes have proven to be particularly tricky to
measure in a direct manner, due to the complexity of
character interactions in homoplastic data (Goloboff
and Farris, 2001). Nevertheless, having some measure of
the stability of and support for particular groups in an
inferred topology serves as a means to discern groups
that are plausible from those that are dubious, and can
Corresponding author: A. V. Z. Brower
E-mail address: browera@science.oregonstate.edu
The Willi Hennig Society 2006
act as a guide to the generation of additional data to
refine and improve the hypothesis.
There are numerous methods to infer the plausibility of particular branches in a phylogenetic tree,
and the pros and cons of many of these have been
discussed at length in the literature (e.g., Kluge and
Wolf, 1993; Davis, 1995; Sanderson, 1995). The
purpose of this paper is to provide an accessible
review of branch support (BS) and partitioned BS
(PBS), including a simple worked example based on a
contrived data matrix. Towards the end of the paper,
a new index of congruence among data partitions is
discussed and illustrated with contrived and empirical
data sets.
Branch support
An increasingly popular way to measure the support
for individual branches of a cladogram is BS, also
referred to as Bremer support and sometimes as the
‘‘decay index’’ (Bremer, 1988, 1994; Donoghue et al.,
1992). BS is a statistical parameter of a particular data
set, rather than being an estimate based on pseudoreplicated subsamples of the data (like bootstrapping and
jackknifing), and thus is not dependent on the data
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A. V. Z. Brower / Cladistics 22 (2006) 378–386
matching a particular assumed distribution1. Another
useful quality of BS is that it never hits a maximum
value (such as 100%), and continues to increase as
character support for a particular branch in the tree
accumulates.
When it was first developed by Bremer (1988), BS was
determined by discovering the most parsimonious tree(s)
for a given data set, and then examining sets of trees of
increasing length (referred to as the ‘‘tree decay’’
method). If there are multiple equally parsimonious
trees of length L, then the branches that collapse in a
strict consensus tree have BS ¼ 0. Branches that were
resolved in the strict consensus of the L trees but
collapse in the strict consensus of the (L and L + 1)
trees have a BS of 1. Branches present in the (L and
L + 1) consensus that collapse in the (L and L + 1 and
L + 2) consensus have a BS of 2 and so on. The tree
decay approach is relatively fast, as it can determine BS
for multiple branches at the same time. It works well for
small, well-resolved data sets, but large data matrices
with a lot of homoplastic characters (such as DNA data)
tend to produce many thousands of equally parsimonious trees after L + 2 or so, and it becomes increasingly
difficult to perform adequate searches to ensure that all
equally parsimonious trees have been discovered. It is
thus quite easy to overestimate the BS for a given branch
using the tree decay approach.
Another way to infer BS is by the employment of
anticonstraint trees. An anticonstraint tree is a specification to the search algorithm to find the most parsimonious tree(s) that do not contain a particular clade.
The difference between the length of the most parsimonious tree(s) and the length of the most parsimonious
tree(s) not containing clade X is the BS for clade X.
Using the anticonstraint approach requires branches to
be assessed one at a time, which is more time consuming,
but it does not get overwhelmed as easily as tree decay
by accumulation of multiple equally parsimonious trees.
Like the tree decay approach, it is possible to underestimate BS values using anticonstraints if searches are not
performed exhaustively. For large, homoplastic data
sets, it is important to perform anticonstraint searches
with comparable rigor with the original search for the
MP tree(s). There are several computer programs
available (Sorenson, 1999; Eriksson, 2001) that automate this process to some degree.
Partitioned branch support
An important extension of BS was the discovery by
Baker and DeSalle (1997; Baker et al., 1998) that when a
data set is divided into partitions (e.g., multiple gene
regions, or DNA versus morphology), the overall BS for
a given branch is the sum of the BS derived from each of
the data partitions for the most parsimonious tree(s).
Like BS, PBS is a parameter of the data matrix. The
PBS value for a particular branch for a given data
partition is determined by subtracting the length of the
data partition on the MP tree(s) from the length of the
data partition on the MP anticonstraint tree(s) for that
branch. Note that the length of the most parsimonious
tree for the partition analyzed separately does not enter
into the calculation of PBS.
A given partition may contribute positively, be
neutral, or conflict with the weight of the evidence that
supports a particular branch in combined analysis. The
details of PBS have been elaborated in a series of papers
by Gatesy and colleagues (Gatesy et al., 1999; Gatesy
and Arctander, 2000; Gatesy, 2000, 2002; see also Lee
2000; Baker et al., 2001; Lambkin et al., 2002). Lee and
Huggall (2003) have proposed a significance test for PBS
values.
PBS is particularly valuable because it allows
exploration of partition incongruence within a ‘‘total
evidence’’ framework (Kluge, 1989; Brower et al.,
1996; DeSalle and Brower, 1997; Judd, 1998), and on
a branch-by-branch basis, not simply as a unitary
value for the entire tree, like the incongruence length
difference (Mickevich and Farris, 1981; Farris et al.,
1994) or various statistical tests that have been
proposed [Thornton and DeSalle (2000) have proposed
a technique for assessing the significance of incongruence among data partitions for individual branches].
The ability to localize incongruence to a single
partition for a single branch has the potential to
reveal both interesting evolutionary processes, such as
selection on a particular gene, and also allows detection of error by the process of reciprocal illumination
(Hennig, 1966).
A worked example
Consider the following simple data matrix:
1
It should be noted that tree quality measures that rely on
counterfactual perturbations of the data are really measures of stability
not support in the sense defined above, at all: stability is the
recoverability of relationships implied in the optimal tree under
alternate search criteria from those used to obtain the result being
tested. ‘‘Counterfactual perturbations’’ include differential character
weighting, variable gap-change costs, and the resampling methods of
bootstrap and jackknife, which randomly assign zero weights to some
characters (and multiple weights to others, in the case of bootstrap).
1
1234567890
X 0000000000
A 1100011111
B 1111111000
C 0011100111
The shortest tree for this data set, which contains clade
(AB), and the two shortest trees not containing clade
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(a)
5
4
3
C
X
A
B
109
8
5
4
3
8
67
2
1
C (c)
(b)
B
X
7
62
A
1
10
98
5
7
62
34
1
8
10
9
10
9
X
A
C
5
4
B
3
7
6
5
43
10
9
2
8
1
67
2
1
Fig. 1. (a) MP tree for matrix 1, L ¼ 16 steps. (b, c)
Alternative topologies for matrix 1, L ¼ 17 steps.
(AB) are shown in Fig. 1. To determine the BS for
clade (AB), the length of the MP tree is determined
(16 steps, in this case). Then the length of the shortest
tree(s) not containing (AB) is found. In this case, there
are two alternatives, both of which are 17 steps, 1 step
longer than the MP tree. Thus, the BS for (AB) is 1.
Now suppose that the data matrix is split into two
partitions:
X
A
B
C
I
II
12345
00000
11000
11111
00111
1
67890
00000
11111
11000
00111
It will be immediately evident that neither of these
partitions supports clade (AB): partition I supports
(BC), and partition II supports (AC). However, as noted
above, the calculation of PBS depends upon the length of
each partition on the MP tree and on the average length
of the shortest trees not containing the branch of interest
(AB) from the analysis of all the data, not on the lengths
of the MP trees for the partitions analyzed separately.
Figure 2 shows partition I mapped on the three
cladograms from Fig. 1. The partition mapped on the
MP tree (a) is eight steps, and on the two other trees (b)
and (c), its length is seven and 10 steps, respectively. The
average length of these two trees is 8.5 steps. Thus, the
PBS for partition I is (L(b) + L(c)) ⁄ 2) ) L(a) ¼ 0.5.
Figure 3 shows partition II mapped on the three
cladograms from Fig. 1. Because the partitions of the
data in this example are symmetrical, it is a mirror image
of Fig. 2. Again, the length of the partition on MP
cladogram (a) is eight steps, and now the lengths of the
other two trees are 10 and 7. Their average length is 8.5
steps, and the PBS from partition II is also 0.5. Note
that PBSI + PBSII ¼ BS.
Partitions need not be symmetrical or of equal size
(in most instances real data will not be). Here the same
data matrix has been split into two partitions of unequal
size:
X
A
B
C
III
IV
1234
0000
1100
1111
0011
1
567890
000000
011111
111000
100111
As in the first case, neither partition unambiguously
supports the grouping (AB) that is supported by the
entire matrix. Figures 4 and 5 show partitions III and IV
mapped on the three cladograms (different ways of
dividing up the data have no effect on the trees: the MP
tree and shortest trees not containing the node of
interest will always be the same, unless the data matrix
as a whole is changed). Partition III, although smaller,
contains all the support: PBSIII ¼ (L(b) +
L(c)) ⁄ 2) ) L(a) ¼ 1.0; while partition IV contributes
no net support. Again, PBSIII + PBSIV ¼ BS.
If multiple equally most-parsimonious trees are discovered in the search of the entire data set, then the
length of a partition is calculated on each of these. PBS
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(a)
5
4
3
C
X
(b)
X
54
3
B
A
A
B
12
C
(c)
5
4
3
B
2
1
21
34
X
5
A
B
C
8
X
10
98
6
B
(b)
X
A
10
98
7
6
Fig. 3. The trees from Fig. 1, with partition II
mapped on to them. L(a) ¼ 8 steps; L(b) ¼ 10 steps,
L(c) ¼ 7 steps.
is equal to the difference between the average length of
the partition among the MP trees and the average length
of the partition among the shortest tree(s) not containing the node of interest.
The partition congruence index
One difficulty of BS is that the value is a single
number that does not reflect potential underlying
disagreement among constituent data partitions. For
C
7
6
8
12
5
43
2
1
Fig. 2. The trees from Fig. 1, with partition I
mapped on to them. L(a) ¼ 8 steps; L(b) ¼ 7 steps,
L(c) ¼ 10 steps.
(a)
A
C
7
(c)
10
9
10
9
A
C
B
X
6
76
8
7
10
9
example, a given branch may be supported by two
characters and contradicted by none, or supported
by 100 characters and contradicted by 98. Both of
these branches will have a BS of two. Goloboff and
Farris (2001) developed the idea of the relative fit
difference (RFD) to address this ambiguity in terms of
homoplasy within an unpartitioned matrix, but (to my
knowledge) it has not been addressed in the context of
PBS.
The simplest approach to evaluating incongruence
among partitions is simply to present the PBS values for
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A. V. Z. Brower/ Cladistics 22 (2006) 378–386
(a)
A
B
4
3
C
X
4
3
1
(b)
B
A
X
(c)
C
2
2
4
3
B
1
X
21
A
C
4
4
3
2
1
32
1
Fig. 4. The trees from Fig. 1, with partition III
mapped on to them. L(a) ¼ 6 steps; L(b) ¼ 6 steps,
L(c) ¼ 8 steps.
(a)
A
B
C
5
8
10
X
(b)
B
10
9
8
7
6
C
7
A
X
9
8
5
6
8
5
7
(c)
10
9
A
C
5
B
X
6
10
9
7
6
6
5
8
each branch and discuss instances in which there is
apparent disagreement among them (e.g., Wahlberg
et al., 2005). This is not particularly satisfactory, as it
does not provide an objective numerical measure, and
results in qualitative value judgments about ‘‘wellsupported’’ or ‘‘poorly supported’’ clades. A parameter
that took into account both the overall support for a
given branch and the disagreement among data partitions underlying that support value could be a useful
measure.
7
10
9
Fig. 5. The trees from Fig. 1, with partition IV
mapped on to them. L(a) ¼ 10 steps; L(b) ¼ 11
steps, L(c) ¼ 9 steps.
The more disagreement there is among partitions, the
greater will be the difference between positive PBS
values from the partition(s) that provide the BS for a
given branch, and the negative PBS values for partition(s) that contradict that branch. John Gatesy (pers.
comm.) has suggested the value BS ⁄ S|PBS| as a measure
of this disagreement. This value ranges from 1 when no
partition contradicts the branch, to approaching zero,
when there are strongly disagreeing partitions (for
example, +100 and )99 yield a value of 0.005). This is
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A. V. Z. Brower / Cladistics 22 (2006) 378–386
20
10
1
2
3
4
5
6
7
8
9
10
15
20
0
PCI
–10
–20
–30
15
9
–40
PBS1
–19
–18
1
–17
–16
–15
–14
–13
3
–12
PBS2
–11
–10
–9
–8
–7
–5
–6
–4
–3
–1
–2
0
7
5
Fig. 6. PCI values resulting from a range of conflicting PBS values from an artificial data set with two partitions. PBS2 represents the partition that
drives the structure of the data, PBS1 represents the conflicting partition. Conflict in PBS1 ranges from 0 to PBS2-1 for each PBS2 value. The data are
presented in Table 1.
Table 1
Artificial data matrix showing the range of PCI values as a function of various PBS values, as graphically depicted in Fig. 6. Top row represents PBS
for partition 1, left column represents PBS with increasing degrees of conflict in partition 2
0
)1
)2
)3
)4
)5
)6
)7
)8
)9
)10
)11
)12
)13
)14
)15
)16
)17
)18
)19
1
2
3
4
5
6
7
8
9
10
15
20
1
2
)1
3
1
)3
4
2.3
0
)5
5
3.5
1.67
)1
)7
6
4.6
3
1
)2
)9
7
5.67
4.2
2.5
0.33
)3
)11
8
6.71
5.33
3.8
2
)0.33
)4
)13
9
7.75
6.43
5
3.4
1.5
)1
)5
)15
10
8.78
7.50
6.14
4.67
3
1
)1.67
)6
)17
15
13.86
12.69
11.5
10.27
9
7.667
6.25
4.714
3
1
)1.5
)5
)11
)27
20
18.89
17.78
16.65
15.5
14.33
13.14
11.92
10.67
9.364
8
6.556
5
3.286
1.333
)1
)4
)8.333
)16
)37
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A. V. Z. Brower/ Cladistics 22 (2006) 378–386
a simple measure, but it does not reflect the amount of
BS, nor does it increase continuously, like BS values do.
A slightly more complex value that incorporates those
features is the partition congruence index (PCI), which is
defined as follows:
PCI ¼ BS ð
n
X
jPBSi j BSÞ=BSÞ
i¼1
Described in words, the partition congruence index
for a given branch equals the BS minus (the difference
between the sum of the absolute values of the PBS for
each partition and the BS, divided by the BS). The
behavior of the PCI relative to various PBS values for
a made up data set with two partitions is presented in
Fig. 6. The accompanying values are presented in
Table 1. As is evident, when there is no conflict
among partitions, PCI ¼ BS, but as the conflicting
signal between partitions increases, PCI decreases,
until it becomes negative. With lower BS values,
relatively little conflict results in a negative PCI
value, but as BS increases, the degree of conflict must
become increasingly severe to produce a negative PCI
score.
Figure 7 shows actual BS and PCI scores from two
equal-weighted simultaneous analyses of empirical
data sets for different groups of nymphalid butterflies
(Wahlberg et al., 2005; Brower et al. 2006). As noted
A
Ithomiini
140.00
120.00
100.00
80.00
60.00
40.00
20.00
0.00
0
20
40
60
80
100
120
140
50
60
70
–20.00
Branch Support
B
Nymphalinae
70.00
60.00
50.00
40.00
30.00
20.00
10.00
0.00
0
10
20
30
40
–10.00
–20.00
Branch Support
Fig. 7. PCI values as a function of BS values for two
molecular data sets for groups of nymphalid butterflies. (A) Ithomiini, mtDNA COI-COII, Ef-1a and
wingless; (B) Nymphalinae, mtDNA COI, Ef-1a and
wingless.
A. V. Z. Brower / Cladistics 22 (2006) 378–386
above, the PCI is equal to the BS when there is no
conflict among partitions. The impact of conflicting
signal in one or more partitions on the PCI is relatively
small when BS is large, as indicated by the near linearity
of the two graphs when BS is greater than 5. When BS is
relatively low, the PCI is much more sensitive to
incongruence, and many nodes with positive BS values
have negative PCI values. Although the potential exists
for very large negative PCI values when there is strong
conflict among partitions (PCImin ¼ )2BS + 2), no
such values were observed in these data, the lowest PCI
value in the Nymphalinae data set was )13.2, and the
lowest in the Ithomiini data set was )11.
Discussion
The exercise of assessing congruence or incongruence among data partitions is not dependent upon
reification of partitions as distinct functional units
evolving under homogeneous processes (Bull et al.,
1993) or separate parts of organisms (different genes,
molecules versus morphology, etc.) (Miyamoto and
Fitch, 1995). Instead, partitioning data is a potentially
useful way to explore incongruence of signal among
characters from different sources. Doing so in the
context of the weight of the evidence from the data
matrix as a whole using BS and PBS values avoids the
problem of implied non-independence of characters
within a partition (DeSalle and Brower, 1997). Much
of the criticism of support measures (e.g., Kluge and
Wolfe, 1993) is focused upon their employment of
reanalyses of data subsets or partitions as though they
were separate sources of evidence. PBS has the
advantage of being based exclusively on parameters
of the complete data matrix, and PBS may be
calculated for any combination of partitions that the
researcher deems useful, or even for individual characters (referred to as ‘‘character support’’, which is
also additive; see Gatesy et al., 1999).
Pablo Goloboff (pers. comm.) has contrived some
particular cases in which PBS values behave unpredictably. Multiple, incongruent groups of homoplastic
character states in alternate partitions can produce
non-intuitive and seemingly inexplicable results due to
the differential distribution of homoplasy among partitions in the most parsimonious and subparsimonious
trees used to calculate BS and PBS values. PBS also
appears to be sensitive to missing data, and can shift
dramatically among partitions as missing data are filled
into the matrix. Further exploration of the impact of
these problems for the interpretation of PBS values will
be a valuable contribution to the theoretical investigation of clade quality. However, as Goloboff et al. (2003)
have pointed out, no measure of clade quality yet
developed is immune from bizarre behavior in certain
385
conceivable instances. Even parsimony analysis itself
can be ‘‘inconsistent’’ under the wrong circumstances. A
premise of the cladistic approach is that signal from
synapomorphies rises above the random noise of
homoplasy, even when homoplasy is abundant (Farris,
1983). The detection of strong incongruence among data
partitions may imply a violation of this premise, and
negative PCI values should be considered as a sign that
further inquiry may reveal errors or other interesting
phenomena intrinsic to the data.
There are no objective means to set a criterion of
rejection of support or stability for a particular branch
in a particular cladogram. Given the dependence of the
tree on the underlying sample of characters and taxa,
support for the current data does not necessarily imply
that the support will be robust to the addition of taxa
and characters: support today is no guarantee of
stability in the future. For this reason, measurements
like bootstrap values and Bayesian posterior probabilities that imply a confidence interval are potentially
misleading. BS, by contrast, because it has no upper
bound, is a more direct and less sophisticated way to
document the accumulation of character support for a
particular branch in a particular phylogenetic hypothesis. The expectation ⁄ hope is that strongly supported
branches will remain stable against collapse and
perhaps even continue to accumulate further support
as additional data are incorporated. The PCI is a
simple parametric value that largely mirrors BS, but
provides a more refined way of distinguishing between
branches that are weakly supported as a result of a
small amount of inferred character transformation, and
those that are weakly supported as a result of conflict
among partitions of the data that imply different
topologies.
Acknowledgments
Thanks to Niklas Wahlberg and Darlene Judd for
support and stability, and for discussions about them.
John Gatesy and Pablo Goloboff provided helpful
criticisms of an earlier draft of the manuscript. The
work was sponsored by US National Science Foundation DEB 0089886 and the Harold and Leona Rice
Endowment for Systematic Entomology.
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