Overcredibility of molecular phylogenies obtained by Bayesian phylogenetics.
ABSTRACT Bayesian phylogenetics has recently been proposed as a powerful method for inferring molecular phylogenies, and it has been reported that the mammalian and some plant phylogenies were resolved by using this method. The statistical confidence of interior branches as judged by posterior probabilities in Bayesian analysis is generally higher than that as judged by bootstrap probabilities in maximum likelihood analysis, and this difference has been interpreted as an indication that bootstrap support may be too conservative. However, it is possible that the posterior probabilities are too high or too liberal instead. Here, we show by computer simulation that posterior probabilities in Bayesian analysis can be excessively liberal when concatenated gene sequences are used, whereas bootstrap probabilities in neighbor-joining and maximum likelihood analyses are generally slightly conservative. These results indicate that bootstrap probabilities are more suitable for assessing the reliability of phylogenetic trees than posterior probabilities and that the mammalian and plant phylogenies may not have been fully resolved.
- SourceAvailable from: Jonathan Luke Heeney[Show abstract] [Hide abstract]
ABSTRACT: The Simian Immunodeficiency Virus (SIV) mus/mon/gsn lineage is a descendant of one of the precursor viruses to the HIV-1/SIVcpz/gor viral lineage. SIVmus and SIVgsn were sequenced from mustached and greater spot nosed monkeys in Cameroon and SIVmon from mona monkeys in Cameroon and Nigeria. In order to further document the genetic diversity of SIVmus, we analyzed two full-length genomes of new strains identified in Gabon. The whole genomes obtained showed the expected reading frames for gag, pol, vif, vpr, tat, rev, env, nef, and also for a vpu gene. Analyses showed that the Gabonese SIVmus strains were closely related and formed a monophyletic clade within the SIVmus/mon/gsn lineage. Nonetheless, within this lineage, the position of both new SIVmus differed according to the gene analyzed. In pol and nef gene, phylogenetic topologies suggested different evolutions for each of the two new SIVmus strains whereas in the other nucleic fragments studied, their positions fluctuated between SIVmon, SIVmus-1, and SIVgsn. In addition, in C1 domain of env, we identified an insertion of seven amino acids characteristic for the SIVmus/mon/gsn and HIV‑1/SIVcpz/SIVgor lineages. Our results show a high genetic diversity of SIVmus in mustached monkeys and suggest cross-species transmission events and recombination within SIVmus/mon/gsn lineage. Additionally, in Central Africa, hunters continue to be exposed to these simian viruses, and this represents a potential threat to humans.Viruses. 01/2014; 6(7):2880-2898.
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ABSTRACT: A phylogenetic analysis of Hormaphidinae is presented based on a total-evidence approach. Four genes (two mitochondrial, COI and CytB, and two nuclear, EF-1α and LWO) are combined with 65 morphological and seven biological characters. Sixty-three hormaphidine species representing three tribes and 36 genera as well as nine outgroups are included. Parsimony and model-based approaches are used, and several support values and implied weighting schemes are explored to assess clade stability. The monophyly of Hormaphidinae and Nipponaphidini is supported, but Cerataphidini and Hormaphidini are not recovered as monophyletic. Based on the parsimony hypothesis from the total-evidence analysis, the phylogenetic relationships within Hormaphidinae are discussed. Cerataphidini is re-delimited to exclude Doraphis and Tsugaphis, and Hormaphidini is redefined to include Doraphis. Ceratocallis Qiao & Zhang is established as a junior synonym of Ceratoglyphina van der Goot, syn. nov. Lithoaphis quercisucta Qiao, Guo & Zhang is transferred to the genus Neohormaphis Noordam as Neohormaphis quercisucta (Qiao, Guo & Zhang) comb. nov. Galls have evolved independently within three tribes of Hormaphidinae. In Cerataphidini, pseudogalls are ancestral, both single-cavity and multiple-cavity galls have evolved once, and galls appear to have evolved towards greater complexity. Galling on secondary hosts has evolved twice in hormaphidines.Cladistics 04/2013; · 5.04 Impact Factor
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ABSTRACT: The genus Brycon, the largest subunit of the Bryconinae, has 42 valid species distributed from southern Mexico to the La Plata River in Argentina. Henochilus is a monotypic genus, comprising a single species (H. wheatlandii) found in the upper Rio Doce basin. In the present study, partial sequences of the mitochondrial gene 16S were obtained for fifteen species of Brycon and for Henochilus wheatlandii. The results showed that the genus Brycon is paraphyletic, since Henochilus is the sister-group of B. ferox and B. insignis. The most basal species analyzed were the trans-Andean species B. henni, B. petrosus, and B. chagrensis.Genetics and Molecular Biology 01/2008; 31(1). · 0.74 Impact Factor
Overcredibility of molecular phylogenies obtained by
Yoshiyuki Suzuki†, Galina V. Glazko, and Masatoshi Nei‡
Institute of Molecular Evolutionary Genetics and Department of Biology, Pennsylvania State University, 328 Mueller Laboratory, University Park, PA 16802
Contributed by Masatoshi Nei, October 23, 2002
Bayesian phylogenetics has recently been proposed as a powerful
method for inferring molecular phylogenies, and it has been
reported that the mammalian and some plant phylogenies were
branches as judged by posterior probabilities in Bayesian analysis
is generally higher than that as judged by bootstrap probabilities
in maximum likelihood analysis, and this difference has been
interpreted as an indication that bootstrap support may be too
conservative. However, it is possible that the posterior probabili-
ties are too high or too liberal instead. Here, we show by computer
simulation that posterior probabilities in Bayesian analysis can be
excessively liberal when concatenated gene sequences are used,
whereas bootstrap probabilities in neighbor-joining and maximum
likelihood analyses are generally slightly conservative. These re-
sults indicate that bootstrap probabilities are more suitable for
assessing the reliability of phylogenetic trees than posterior prob-
abilities and that the mammalian and plant phylogenies may not
have been fully resolved.
phylogenetic relationships of different species in the post-
genomic era (1), in which it would become a common practice
to construct phylogenetic trees by using concatenated sequences
of a large number of genes (2). In fact, when Murphy et al. (3)
constructed a phylogenetic tree of major mammalian species
using concatenated nucleotide sequences of 22 genes, the reli-
ability of interior branches (or clades) as judged by the posterior
probability in Bayesian analysis was generally higher than that as
judged by the bootstrap probability (4) in maximum likelihood
(ML) analysis (5). Eleven of 27 interior branches were supported
with 95% confidence level only by the posterior probability,
whereas no interior branch was supported by the bootstrap
probability alone. Similarly, when Karol et al. (6) constructed a
phylogenetic tree of some major evolutionary lineages of plants
using concatenated nucleotide sequences of four genes, 12 of 37
interior branches were supported only by the Bayesian posterior
probability, whereas no interior branch was supported by the ML
bootstrap probability alone. Murphy et al. (3) interpreted these
results as an indication that ‘‘bootstrap support may be too
conservative,’’ and the mammalian and plant phylogenies were
claimed to have been resolved. However, it is possible that the
posterior probability was too high or too liberal. The purpose of
this paper is to examine which of these interpretations is more
reasonable by conducting computer simulation.
Theoretically, a phylogenetic tree of genes from different
species should be bifurcating, because the replication of nucle-
otide sequence is a bifurcating process. Therefore, if we con-
struct a gene tree for four species A, B, C, and D, one of three
(B, D)), will be chosen. In reality, however, different genes from
the same set of species may show different topologies because of
the polymorphism, recombination, and homoplasy in ancestral
populations (7). If we concatenate sequences of many genes of
the same size or similar sizes and construct a phylogenetic tree,
the inferred tree is likely to have the topology that is supported
by the largest number of genes in the sequences. It is therefore
ayesian inference by using the Markov chain Monte Carlo
method has been advocated as a powerful tool for inferring
important to use a large number of randomly chosen genes in the
inference of species phylogenies.
In the statistical inference of phylogenetic trees of four
species, the null hypothesis to be tested is that the three different
topologies occur with equal frequency. If a particular topology
is chosen with high statistical confidence, we assume that this
topology is established, although it may be rejected later by some
additional data. If different species diverged during a short
period of evolutionary time, as in the case of divergence of
mammalian orders, it would be difficult to identify the true tree
unless we use a large number of genes. However, even if we use
tree may be identified as though it were the true tree (false
positive) if an excessively liberal statistical method is used. Here,
we examine the frequency of occurrence of this false-positive
result in the Bayesian, neighbor-joining (NJ; ref. 8), and ML
methods under the condition that all of the three topologies
occur with equal frequency. We then discuss statistical proper-
ties of Bayesian posterior and bootstrap probabilities.
Three sets of four nucleotide sequences (a?, b?, c?, and d?), (a?,
b?, c?, and d?), and (a?, b?, c?, and d?) with 5,000 sites were
generated following topologies ((a?, b?), (c?, d?)), (Fig. 1A);
((a?, d?), (b?, c?)), (Fig. 1B); and ((a?, c?), (b?, d?)), (Fig. 1C),
respectively, by using the computer program SEQGEN (version
1.25; ref. 9). The length of all exterior branches (bE) was assumed
to be 0.05 substitutions per site and that of interior branch (bI)
0.005, except for a few cases in which bE? 0.1 and bI? 0.01 were
assumed. These branch lengths were determined on the basis of
the observation that, in a phylogenetic analysis of amino acid
sequences from humans, cows, and rodents with chicken as an
outgroup, different genes supported different topologies but bE
for mammalian lineages was always about 0.05 on average and bI
was about 0.005 (G.V.G. and M.N., unpublished data). In
addition, the rate of nucleotide substitution seems to be similar
to or slightly higher than that of amino acid substitution in
mammals (10). After generating the sequences with a given
model of nucleotide substitution, sequences a?, a?, and a? were
concatenated into a single sequence a. Similarly, b?, b?, and b?;
c?, c?, and c?; and d?, d?, and d? were concatenated into single
sequences b, c, and d, respectively. The number of nucleotide
sites in the concatenated sequences was 15,000, which was close
to that (16,397) used in the phylogenetic analysis of mammalian
species by Murphy et al. (3). Note that sequences a, b, c, and d
are expected to generate three different topologies ((a, b),
(c, d)), (Fig. 1A); ((a, d), (b, c)), (Fig. 1B); and ((a, c), (b, d)),
(Fig. 1C) with equal probability, but, in actual inference, one of
them is chosen because of the stochastic error of nucleotide
substitution. However, the inferred tree should not be supported
with a high posterior or bootstrap probability, because it was
Abbreviations: NJ, neighbor-joining; ML, maximum likelihood; JC, Jukes–Cantor.
†Present address: Center for Information Biology and DNA Data Bank of Japan, National
Institute of Genetics, 1111 Yata, Mishima-shi, Shizuoka-ken 411-8540, Japan.
‡To whom correspondence should be addressed. E-mail: firstname.lastname@example.org.
December 10, 2002 ?
vol. 99 ?
chosen just by chance. Therefore, if it happened to be supported,
the result was judged as a false-positive.
to reveal the major features of Bayesian and bootstrap proba-
bilities. The models used were the Jukes–Cantor (JC) and
Kimura models, with or without rate variation among sites (11).
The phylogenetic tree of sequences a, b, c, and d was inferred
by the Bayesian, NJ, and ML methods. We used the computer
program MRBAYES (version 2.01; ref. 12) for constructing Bayes-
ian trees. One cold and three incrementally heated chains were
run for 2,000,000 generations, with random starting trees and the
temperature parameter value of 0.2. Trees were sampled every
100 generations from the last 1,000,000 generations (well after
the chain reached stationality), and 10,000 sampled trees were
ML trees, respectively. In both methods, we conducted 1,000
bootstrap resamplings. In the case of ML trees, the nucleotide
frequencies were estimated by the observed frequencies (the
default option of PAUP*). Bayesian, NJ, and ML trees were
judged as false-positives when the posterior or bootstrap prob-
ability was ?95%. The entire procedure was repeated 50 times
(replications) for estimating the false-positive rate of these
is 5% for all methods because the confidence level is 95%.
We first analyzed the concatenated sequences generated with
the JC model and the expected branch lengths of bE? 0.05 and
bI? 0.005 in each of the three model trees. Bayesian analysis
produced topologies ((a, b), (c, d)), ((a, d), (b, c)), and ((a, c),
(b, d)) in 16 (32%), 14 (28%), and 20 (40%) replications,
respectively (Table 1). These frequencies were more or less
equal to one another, because the topologies were chosen with
the same probability in each replication. However, the poste-
rior probability was 85% on average and ?95% in 21 repli-
cations (42%). Therefore, the false-positive rate was much
higher than the expected (5%), indicating that the Bayesian
interior branches, respectively. No interior branch exists in D.
Model trees used for generating concatenated sequences (A–C) and completely linked sequences (D). bEand bIindicate the lengths of exterior and
Table 1. Frequencies of false-positives inferred by Bayesian posterior probability and NJ and ML bootstrap probabilities for JC and
Bayesian posterior probabilityNJ bootstrap probability ML bootstrap probability
0.050.005 0.5 (0.5)858 21851012 630000 64
16 14 2050 171320 501614 20 50
0.1 0.010.5 (0.5)677 20851001 640000 65
18201250 16 1915 50 1820 1250
0.05 0.0055 (0.5) 14 139 36 910000 620000 63
19 19 1250 1720 1350 1919 12 50
0.10.015 (0.5) 12 14113795011268001169
18 16 16 5019 1813501816 1650
0.050 5 (0.5)1114 6 31891012 68101266
10.10 5 (0.5)95 69 68
18 1616 5016 1816 5018 1616 50
for phylogenetic inference is given in parentheses).
Suzuki et al.PNAS ?
December 10, 2002 ?
vol. 99 ?
no. 25 ?
posterior probability is excessively high as an indication of
By contrast, the NJ bootstrap probability was 63% on average
and ?95% in two replications (4%; Table 1). The average ML
bootstrap probability was 64%, with no false-positives. These
false-positive rates were similar to or lower than the expected
rate, indicating that the NJ and ML bootstrap probabilities are
In Fig. 2, the Bayesian posterior probability and the NJ and
ML bootstrap probabilities obtained for the same set of se-
quences are plotted as a scattergram to show the relationship
between them. NJ and ML bootstrap probabilities are located
roughly on the diagonal line (Fig. 2C), indicating that they are
similar to each other. However, the posterior probability is much
the latter is 70% or higher (Fig. 2 A and B). Similar results were
obtained when we analyzed the concatenated sequences gener-
ated under the assumption of bE? 0.1 and bI? 0.01 in the model
trees (Table 1; Fig. 2 D–F).
In the above analysis, we used the same (JC) model for
generating and analyzing the sequences. In actual data analysis,
however, we usually do not know the correct model by which the
sequences were generated but assume a simplified model for
analyzing them. To examine the effect of using a simplified
model on the false-positive rate, we generated the sequences
following Kimura’s model (15), with a transition?transversion
ratio (R) of 5 and analyzed them using the JC model. Note that,
in the JC model, R ? 0.5. When we used bE? 0.05 and bI? 0.005
in the model trees, the Bayesian posterior probability was 91%
on average and ?95% in 36 replications (72%), whereas the NJ
and ML bootstrap probabilities were 62% and 63% on average,
respectively, both with no false-positives (Table 1). In the
scattergram, two bootstrap probabilities seem to be similar to
obtained similar results under the assumption of bE? 0.1 and
bI? 0.01 in the model trees (Table 1; Fig. 2 J–L). These results
indicate that the posterior probability is unreasonably high in the
analysis of concatenated sequences whereas the bootstrap prob-
ability is still slightly conservative.
15,000 sites following the star phylogeny (Fig. 1D) using the
to examine the effect of using a simplified model on the
phylogenetic analysis of completely linked sequences. Note that
the star phylogeny is usually used as the null hypothesis tree in
statistical inference of phylogenetic trees of completely linked
sequences (11). When we used bE? 0.05 and bI? 0 in the model
trees, the Bayesian posterior probability was 89% on average and
?95% in 31 replications (62%; Table 1). By contrast, the NJ and
ML bootstrap probabilities were 68% and 66% on average,
respectively, and ?95% in two replications (4%). In the scat-
tergram, NJ and ML bootstrap probabilities are again similar to
when the bootstrap probability is 70% or higher (Fig. 2 M and
N). Similar results were also obtained when we used bE? 0.1
and bI? 0 (Table 1; Fig. 2P–R), indicating that the posterior
probability is excessively high even in the analysis of completely
linked sequences whereas the bootstrap probability is again
In actual DNA sequences the evolutionary rate varies from
nucleotide site to nucleotide site, and this variation is usually
approximated by a gamma (?) distribution (16). We therefore
conducted another simulation using the JC ? ? and the
Kimura ? ? model. First, we generated concatenated se-
quences using the JC ? ? model with a gamma parameter value
(a) of 1 and the expected branch lengths of bE ? 0.05 and
bI? 0.005 and inferred Bayesian, NJ, and ML trees using the
same JC ? ? model. In this case, topologies A, B, and C were
obtained again with nearly the same frequencies in all Bayes-
ian, NJ, and ML analyses (data not shown). In the case of
Bayesian analysis, however, the posterior probability was
?95% in 22 of the 50 replications (44%), and the average
probability value for all replications (P?) was 84% (Table 2). By
contrast, the bootstrap probability was ?95% only in two
replications in NJ analysis and only in one replication in ML.
P?was 63% in NJ and 65% in ML. When we generated sequence
data using the Kimura ? ? model with R ? 5 and a ? 1 and
constructed Bayesian, NJ, and ML trees using the same model,
the results were nearly the same (Table 2). In NJ, the false
positive rate (10%) was higher than that (4%) for the case of
the JC ? ? model probably by chance, but the P?value (66%)
was nearly the same as that for the latter case or the cases
considered in Table 1.
When the sequence data were generated by the JC ? ? model
(R ? 0.5 and a ? 1) but phylogenetic inference was done with
the JC model (R ? 0.5 and a ? ?), the false positive rate was
41?50 or 82% and P?was 95% for Bayesian analysis. By contrast,
the false positive rate was 4% and P?was 64–65% for NJ and ML.
When the sequences were generated by the Kimura ? A ˜model
(R ? 5 and a ? 1) and trees were inferred with either the JC (R ?
0.5 and a ? ?) or the Kimura (R ? 5 and a ? ?) model, the false
positive rate and the P?value were essentially the same as those
for the above case. Therefore, the false-positive rate is too high
in Bayesian analysis, whereas it is close to the expected value
(5%) in NJ and ML. Tables 1 and 2 show that both NJ and ML
bootstrap tests tend to be slightly conservative but that the NJ
test is not always so.
We demonstrated that the posterior probability in Bayesian
phylogenetics can be excessively high in the analysis of concat-
enated sequences even when the same model as that for gener-
ating each gene sequence was used. The false-positive rate
became even higher when a simplified model was used for
phylogenetic inference. Under the same condition, the posterior
probability was also excessively liberal in the analysis of com-
pletely linked sequences. In actual data analysis, we usually do
not know the correct model by which the sequences were
generated but use a simplified model for analyzing them, as
mentioned above. The posterior probability therefore can be
unreasonably high in actual data analysis even when unconcat-
enated sequences are used.
By contrast, the bootstrap probabilities in NJ and ML
analyses were generally slightly conservative regardless of
whether the correct or simplified model was used or whether
the concatenated or completely linked sequences were ana-
lyzed. This is particularly so for ML analysis. The bootstrap
probability therefore seems to be a conservative estimate of
statistical confidence. These results are consistent with the
previous observations that the false-positive rate of bootstrap
probabilities in the NJ and maximum parsimony methods is
lower than the expected rate in phylogenetic analysis of
unconcatenated sequences, as long as these methods are not
inconsistent (17–20). In fact, the bootstrap probability, when
it is close to unity, is theoretically shown to be an underesti-
mate if it is simply interpreted as the probability that the
inferred tree is correct (21). However, a conservative method
should be preferable to an overly liberal method in phyloge-
netic analysis, because we usually draw conclusions only from
statistical analysis without doing any experiments (11). In
addition, it may be possible to modify the resampling proce-
dure for obtaining a less conservative value of bootstrap
probability (22). The bootstrap probability therefore seems to
be more suitable for assessing the reliability of phylogenetic
www.pnas.org?cgi?doi?10.1073?pnas.212646199Suzuki et al.
(B, E, H, K, N, and Q), and two bootstrap probabilities (C, F, I, L, O, and R) obtained for the same set of sequences. The sequences analyzed are (i) concatenated
sequences generated by using the model trees (Fig. 1 A–C) with bE? 0.05, bI? 0.005 and R ? 0.5 (A–C); bE? 0.1, bI? 0.01, and R ? 0.5 (D–F); bE? 0.05, bI?
0.005 and R ? 5 (G–I); and bE? 0.1, bI? 0.01, and R ? 5 (J–L) and (ii) completely linked sequences generated by using the model tree (Fig. 1D) with bE? 0.05
(M–O) and bE? 0.1 (P–R). Diagonal lines indicate the perfect correspondence between two probabilities.
Scattergrams of Bayesian posterior probability and NJ bootstrap probability (A, D, G, J, M, and P), posterior probability and ML bootstrap probability
Suzuki et al. PNAS ?
December 10, 2002 ?
vol. 99 ?
no. 25 ?
trees than the posterior probability, although the theoretical
basis of bootstrap probability is not well understood at present.
A high Bayesian posterior probability for a given interior
branch (or a clade) is obviously due to the appearance of the
same branch or clade in most or all sampled trees used for
constructing a consensus tree. This result is in turn caused by
the fact that the highest ML tree (or set of trees) is visited again
and again in the Markov chain Monte Carlo computation for
the original set of sequences. In the computation of bootstrap
probability, however, the original set of sequences is consid-
ered as a single evolutionary event realized with stochastic
errors, and therefore the original sequences are reshuffled
(bootstrap-resampled) to evaluate the reliability of the original
or consensus tree. In this case, different bootstrap-resampled
sequences may generate different ML or NJ trees, unless the
extent of stochastic errors is small. Therefore, the bootstrap
probability computed for a bootstrap consensus tree is ex-
pected to be lower than the Bayesian posterior probability.
Because original sequences are always subject to stochastic
errors, the reliability of an inferred tree should be evaluated
by considering stochastic errors.
In the phylogenetic analyses of mammals (3) and plants (6),
some interior branches were not supported by ML bootstrap
phylogenetic trees published in these papers may not have been
established yet. Similarly, the reliability of other molecular
phylogenies obtained by Bayesian phylogenetics (e.g., refs. 23–
25) should be reexamined by additional methods and additional
sequence data. It is known that two different topologies for the
same set of mammalian species can both be supported by high
Bayesian probabilities when DNA and protein data were ana-
lyzed separately (K. Misawa and M.N., unpublished work). This
finding also indicates that Bayesian phylogenetics may give
overcredibility of the tree inferred.
After completion of this paper, we came to know a paper in
which a computer simulation was conducted to evaluate the
reliability of Bayesian and bootstrap probabilities as a statis-
tical confidence of interior branches (26). The model tree used
was a maximum likelihood tree of 23 species of snakes
obtained by using the general time reversible (GTR) model of
nucleotide substitution with invariable sites (I) plus a ?
distribution of variable-rate sites (GTR ? I ? ? model) for a
portion of mitochorial DNA. By using this model tree and the
same GTR ? I ? ? model, 120 datasets of 500 nucleotide sites
were randomly generated, and each of the datasets was used
to infer Bayesian and ML trees. This computation generated
(23 ? 3) ? 120 ? 2400 Bayesian posterior probability (PP) and
bootstrap probability (BP) values ranging from 0% to 100%.
These probability values were then classified into 10 bins with
an interval of 10%. At the same time, the proportions of
interior branches that were correctly inferred among 120
reconstructed trees (PBC) were computed for each bin class of
PP and BP values. Comparison of PBC and PP or BP showed
that BP is a clear underestimate of PBC though PP also tends
to be an underestimate. From this observation, the authors
concluded that PP is a better indicator of statistical confidence
This simulation is different from ours in that the same
substitution model as that for sequence generation was used
for phylogenetic inference without concatenation of different
genes that might have evolved differently. In reality, the
substitution model used for phylogenetic inference would
never be the same as the real substitution pattern, and many
different genes are concatenated when a large-scale data
analysis is done. Therefore, we believe our simulation is more
realistic than the one mentioned above. It should also be noted
that the null hypothesis of the statistical tests used in the above
simulation is not clearly defined, though theoretically it should
be the absence (or length 0) of the interior branch under
consideration (11). When a tree with many positive interior
branches is used as a model tree for generating replicate
datasets, the bootstrap test of this null hypothesis is quite
complicated (19, 21). More theoretical studies are needed on
this important problem.
We thank Dan Graur, Xun Gu, Rodney Honeycutt, Junhyong Kim,
Sudhir Kumar, Bill Martin, Mike Miyamoto, Pam Soltis, and Jianzhi
(George) Zhang for their comments on an earlier version of the
manuscript. This work was supported by a grant from the National
Institutes of Health (GM20293) to M.N.
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Table 2. Frequencies of false-positives inferred by Bayesian posterior probability and NJ and ML bootstrap
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The number of false-positive cases (replications) is given before the slash sign, and the total number of replications is given after the
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