全 文 ：Journal of Systematics and Evolution 46 (3): 307–314 (2008) doi: 10.3724/SP.J.1002.2008.08040
(formerly Acta Phytotaxonomica Sinica) http://www.plantsystematics.com
Missing data and the accuracy of Bayesian phylogenetics
John J. WIENS* Daniel S. MOEN
(Department of Ecology and Evolution, Stony Brook University, Stony Brook, NY 11794-5245, USA)
Abstract The effect of missing data on phylogenetic methods is a potentially important issue in our attempts to
reconstruct the Tree of Life. If missing data are truly problematic, then it may be unwise to include species in an
analysis that lack data for some characters (incomplete taxa) or to include characters that lack data for some
species. Given the difficulty of obtaining data from all characters for all taxa (e.g., fossils), missing data might
seriously impede efforts to reconstruct a comprehensive phylogeny that includes all species. Fortunately, recent
simulations and empirical analyses suggest that missing data cells are not themselves problematic, and that in-
complete taxa can be accurately placed as long as the overall number of characters in the analysis is large. How-
ever, these studies have so far only been conducted on parsimony, likelihood, and neighbor-joining methods.
Although Bayesian phylogenetic methods have become widely used in recent years, the effects of missing data on
Bayesian analysis have not been adequately studied. Here, we conduct simulations to test whether Bayesian
analyses can accurately place incomplete taxa despite extensive missing data. In agreement with previous studies
of other methods, we find that Bayesian analyses can accurately reconstruct the position of highly incomplete taxa
(i.e., 95% missing data), as long as the overall number of characters in the analysis is large. These results suggest
that highly incomplete taxa can be safely included in many Bayesian phylogenetic analyses.
Key words accuracy, Bayesian analysis, missing data, phylogenetic analysis.
The impact of missing data is a potentially im-
portant issue in phylogenetic analyses, particularly if
the goal is to reconstruct a comprehensive Tree of Life
that includes both fossil and living taxa. Missing data
are often encountered when combining data from two
or more different genes, when some of the taxa have
sequence data available for one gene but not the other.
If the taxa lacking data for a gene are included in the
combined analysis, then the characters associated with
this gene are typically coded as missing or unknown
(often denoted with a “?”). Similarly, missing data are
often encountered in analyses that include fossil taxa,
when certain taxa must be scored as unknown for
certain characters because the relevant features have
not been adequately preserved.
Concerns about missing data may often deter-
mine what characters and taxa will be included in an
analysis (Wiens, 2006), even if this is not always
stated explicitly by researchers. For example, if
missing data are considered to be problematic, then
one should only include species that have complete
data for all characters or else only include characters
that have complete data for all species. Thus, one may
have to reduce the number of taxa or characters in an
analysis in order to avoid including missing data cells.
Furthermore, it would be difficult (if not impossible)
to combine molecular and morphological character
data from fossil and living taxa in the same analysis,
because the fossil taxa will almost always lack mo-
lecular data.
But are missing data truly problematic? Several
authors have suggested that including taxa with a high
proportion of missing data cells is potentially prob-
lematic for phylogeny reconstruction, based on both
empirical data (e.g., Novacek, 1992; Wiens & Reeder,
1995; Wilkinson, 1995; Kearney, 2002) and computer
simulations (Huelsenbeck, 1991; Hartmann & Vision,
2008). These authors suggested that including highly
incomplete taxa can potentially lead to uncertain
relationships (e.g., Novacek, 1992) and an overall
decrease in the accuracy of the reconstructed trees
(Huelsenbeck, 1991; Hartmann & Vision, 2008). By
accuracy, we mean the frequency with which the true
phylogeny is reconstructed correctly.
Computer simulations, such as those of Huelsen-
beck (1991), can offer important insights into whether
a given phylogenetic method is able to accurately
reconstruct the true phylogeny under a broad range of
conditions (Hillis, 1995; Huelsenbeck, 1995). Simula-
tions are important because in most empirical studies
the true phylogeny of the organisms is unknown. In
contrast, simulations provide a context where the true
phylogeny is known and the conditions that affect the

———————————
Received: 1 April 2008 Accepted: 2 May 2008
* Author for correspondence. E-mail: wiensj@life.bio.sunysb.edu;
Tel.: 631-632-1101.
Journal of Systematics and Evolution Vol. 46 No. 3 2008 308
phylogenetic accuracy of a method can be varied in a
controlled, experimental fashion. However, simula-
tions always require making many simplifying as-
sumptions, and the results may depend entirely on the
set of simulated conditions that were examined (Hillis,
1995; Huelsenbeck, 1995). For example, Huelsenbeck
(1991) examined the effects of missing data on parsi-
mony analysis under a broad range of conditions, but
did not vary the number of characters (only 100
characters were included in the simulations).
Wiens (2003) conducted simulations (in which the
number of characters was varied extensively) which
indicate that missing data are not themselves prob-
lematic. Instead, problems with highly incomplete taxa
arise because there are too few characters in these taxa
to accurately place them on the tree. If the overall
number of characters in the analysis is small, then
overall accuracy may be low when many of the taxa are
incomplete (i.e., if the overall number of characters in
the analysis is only 100, a taxon with 95% missing data
will have data for only 5 characters). Conversely, if the
overall number of characters is large, then even highly
incomplete taxa should have enough characters to
allow them to be accurately placed in the tree (i.e., if
there are 2,000 characters, and a taxon has 95%
missing data, there are still 100 characters that can
allow it to be placed on the tree). Based on these
results, the missing data themselves are irrelevant, and
the more important parameter is the quantity of the
characters that are present. However, these results are
dependent not only on the simulated conditions, but
also on how each phylogenetic method and software
package deals with missing data. Wiens (2003) exam-
ined the performance of parsimony, likelihood and
neighbor-joining as implemented by the widely used
software package PAUP* (Swofford, 2002). Subse-
quently, an empirical molecular study of plant phy-
logeny by Driskell et al. (2004) and a combined
empirical and simulation study of higher-level eu-
karyote molecular phylogeny by Philippe et al. (2004)
also found results suggesting that incomplete taxa can
be accurately placed in phylogenetic analyses. The
results of Driskell et al. (2004) were based on parsi-
mony (using PAUP*), and those of Philippe et al.
(2004) were based on parsimony, likelihood, and
neighbor-joining (using various programs, but espe-
cially PHYML; Guindon & Gascuel, 2003). A recent
simulation study analyzed the effect of missing data in
datasets resembling those from ESTs (Hartmann &
Vision, 2008) and found that replacing complete data
with missing data cells decreased accuracy, especially
when the missing data are randomly distributed among
characters and taxa (but note that if data are simply
replaced with missing data cells, one expects accuracy
to decrease simply because you are decreasing the
overall amount of data).
The papers by Wiens (2003), Driskell et al.
(2004), Philippe et al. (2004), and Hartmann & Vision
(2008) all shared a serious omission, however. Start-
ing around 2002, Bayesian methods have become
widely used for reconstructing phylogenies. Although
the performance of Bayesian phylogenetic methods
has become relatively well understood through a large
number of simulation studies (e.g., Suzuki et al., 2002;
Wilcox et al., 2002; Alfaro et al., 2003; Cummings et
al., 2003; Douady et al., 2003; Erixon et al., 2003;
Huelsenbeck & Rannala, 2004; Lewis et al., 2005),
how Bayesian methods perform when faced with
missing data is largely unknown.
Two previous studies, one empirical and one
based on computer simulations, suggest that missing
data may not be problematic for Bayesian analyses.
First, Wiens et al. (2005) examined relationships
among hylid frog species using Bayesian analysis of
molecular and morphological data. They had rela-
tively complete data for 81 species and incomplete
data (typically with data for only one gene) for an
additional 117. They found that the taxa with only one
gene were placed in the expected clades, despite their
missing data. For example, the eight species in the
analysis that had >90% missing data were each placed
in the clades expected based on their current taxon-
omy (e.g., species of the genus Scinax were placed
with other species of Scinax), and the Bayesian sup-
port for the monophyly of these clades was very high
(posterior probability of all clades=1.00). Further-
more, these authors found no relationship between
levels of completeness (100 - % missing data) in each
species and the level of support (Bayesian posterior
probability) for the placement of that species on the
terminal branches of the tree. Instead, there was a
significant relationship between the level of support in
the combined analysis (including missing data) and
the level of support in the analyses of the gene that
was sequenced in almost all taxa (mitochondrial
ribosomal 12S). In other words, the level of support
seemed to depend on the data that were present, not
the amount of data that was absent. However, it is
important to note that the actual phylogenetic rela-
tionships of the species were unknown, and so this
study did not directly assess the impact of missing
data on phylogenetic accuracy of Bayesian analysis.
In the second study, Wiens (2005) used simula-
tions to test the ability of added taxa to improve
WIENS & MOEN: Missing data and Bayesian phylogenetics

309
phylogenetic accuracy (for the complete taxa) for
various phylogenetic methods when the added taxa had
a high proportion of missing data. These analyses
included Bayesian analyses in addition to parsimony,
likelihood, and neighbor-joining. Taxa were added
under simulated conditions where there was
long-branch attraction (which greatly reduces phy-
logenetic accuracy for most methods; Felsenstein,
1978; Huelsenbeck, 1995) and the added taxa could
subdivide or “break up” these long branches (e.g., Poe,
2003). These analyses showed that incomplete taxa can
successfully subdivide long branches and thereby
increase phylogenetic accuracy, in many cases, as well
as complete taxa can. Although these results are
encouraging about the ability of Bayesian analyses to
cope with missing data, these analyses did not address
the overall accuracy of the trees, only the accuracy of
the relationships among the complete taxa. Thus, it is
theoretically possible that the incomplete taxa were
placed inaccurately. In summary, despite some en-
couraging results from two previous studies, the impact
of missing data on the accuracy of Bayesian analysis is
in need of further study.
In this paper, we use computer simulations to ex-
plicitly examine the accuracy of Bayesian phylogenetic
analysis when many of the taxa are incomplete. These
analyses follow closely the protocols used by Wiens
(2003), in order to make the results easily comparable
to those based on other phylogenetic methods. Indeed,
we find here that the results from Bayesian analyses
mirror those from likelihood, parsimony, and
neighbor-joining. In Bayesian analyses, highly in-
complete taxa can be accurately placed if the overall
number of characters is large. As in previous studies,
we find that the missing data cells themselves do not
appear to be problematic for phylogeny reconstruction.
1 Material and Methods
The general methodology for simulating data and
analyzing these data followed Wiens (2003), and only a
brief explanation is provided here. The overall design
was to test the accuracy of Bayesian analysis when
many of the taxa have different proportions of missing
data, and to test this across different numbers of
characters. Based on previous studies, we anticipated
that analyses including taxa with a high proportion of
missing data would have low accuracy when the
number of characters in the analysis was small, but
relatively high accuracy when the number of characters
was large.
We initially simulated a 16-taxon phylogeny that
was fully asymmetric and had equal branch lengths.
Asymmetric trees are expected to be more common
when all topologies are considered to be equally likely
(e.g., Huelsenbeck & Kirkpatrick, 1996), and previous
studies suggest that tree shape will have little impact on
the results (Wiens, 2003). Characters were simulated
along this phylogeny to create a complete set of
character data for each taxon. We used DNA sequence
data evolving according to the simple Jukes-Cantor
model (Jukes & Cantor, 1969), with equal rates of
change between all substitution types, equal base
frequencies, and equal rates of change between char-
acters. We focused primarily on how Bayesian analy-
ses are affected by missing data, and not other issues
(e.g., how they deal with more complex models of
evolution). Different branch lengths were also ana-
lyzed ranging from 0.05 (i.e., 1 in 20 characters
expected to change from the beginning to the end of the
branch), 0.10, 0.20, to 0.30. These branch lengths span
a broad range of levels of variability and homoplasy,
ranging from conditions where phylogeny reconstruc-
tion is relatively easy to those where it is relatively
difficult. Different numbers of characters were also
analyzed including 100, 500, 1,000, and 2,000. These
total numbers of characters included both sites that
were variable and invariant among taxa.
For a given simulation replicate, 8 taxa were then
randomly chosen to be incomplete. These taxa had a
certain proportion of their characters replaced with
missing data cells (“?”). The same characters were
made incomplete in each incomplete taxon (but see
below). We systematically varied the level of com-
pleteness from 5% (95% of data cells missing), to 10%,
25%, 50%, 75%, and 100% (no missing data). 100
replicates were examined for each combination of level
of completeness, branch length, and number of char-
acters.
Each simulated data set was analyzed using
MrBayes version 3.0b4 (Huelsenbeck & Ronquist,
2001) and the resulting tree was compared to the true,
known phylogeny that was used to generate the data.
Accuracy for a given replicate was measured as the
proportion of nodes from the estimated Bayesian tree
that matched nodes in the true species phylogeny. The
accuracy for a given set of simulated conditions was
the average accuracy across all 100 replicates.
In general, we used default options for MrBayes
in the Bayesian analyses. These default options in-
cluded use of the Jukes-Cantor model with no
among-site rate variation or invariant sites (thus, there
was a close correspondence between the model used
to simulate the data and reconstruct the tree). The
Journal of Systematics and Evolution Vol. 46 No. 3 2008 310
number of generations was set to 50,000. The first
5,000 generations were discarded as burn-in, and the
phylogeny was estimated as the majority-rule consen-
sus tree of the post burn-in trees, following standard
practice. Some readers may be surprised by the low
number of generations used. After all, empirical
analyses typically use several million. However, the
overall number of taxa analyzed in each replicate is
relatively low (16; making thorough searching of tree
space much easier). Furthermore, prior to selecting
this number of generations, we analyzed a subset of
the results using twice as many generations (100,000)
and found no detectable difference in the results. We
also examined the results using 100,000 generations to
determine when stationarity was achieved, and found
that it was consistently reached in less than 5,000
generations (based on a plateau in a plot of likelihood
values against number of generations). Finally, we
found that even when using only 50,000 generations,
Bayesian analyses had an accuracy of 100% (all nodes
correctly reconstructed) under many different condi-
tions. This result demonstrates that, at least under
these conditions, there are no random errors in tree
reconstruction associated with inadequate tree search-
ing.
Programs for simulating the data and compiling
the results were written in C by J.J.W. Analyses were
conducted using PAUP* version 4.0b10 (Swofford,
2002) to make consensus trees from the Bayesian
analyses and to compare the estimated Bayesian trees
to the true trees.
In addition to these basic simulations, we also
performed a smaller set of simulations to test how
robust the results were to changes in different pa-
rameters. First, we examined more complex models of
DNA sequence evolution. We performed analyses that
incorporated unequal base frequencies, a different ratio
of transitions and transversion, and different rates of
change among sites. We assumed a 3:1 transition:
transversion ratio and base frequencies of A=37%,
G=12%, C=24%, and T=27% (parameter values based
on mammalian sequences as reported by Zwickl &
Hillis, 2002). We simulated among-site rate variation
by modeling the data to resemble protein-coding
sequences. Thus, the first two characters of every
three had branch lengths of 0.02 and 0.02, whereas the
third had a branch length of 0.20 (the ten-fold differ-
ence in rates was initially chosen based on protein-
coding genes in salamanders; Wiens, unpubl.). Over-
all, the simulated data corresponded to the HKY
model (Hasegawa et al., 1985). We performed two
sets of analyses under these conditions. In the first, we
analyzed the data using the simple Jukes-Cantor
model, to evaluate the combined effects of missing
data and an oversimplified model. In the second, we
analyzed the data using a more appropriate model
(HKY, with the gamma parameter added to account
for among-site rate variation; Yang, 1993).
We also examined the effects of changing tree
shape. We analyzed a fully symmetric 16-taxon tree
for a limited set of conditions (branch length=0.05,
Jukes-Cantor model of sequence evolution).
Finally, we changed the way in which missing
data were distributed among characters in the incom-
plete taxa. Instead of having the same set of characters
lacking data in all of the incomplete taxa, we ran-
domly distributed missing data cells among charac-
ters. Again, we analyzed a limited set of conditions to
address the effects of changing this parameter (asym-
metric tree with branch lengths=0.05, Jukes-Cantor
model).
2 Results
The main results of the study are summarized in
Fig. 1. These results show that highly incomplete taxa
can be accurately placed in Bayesian analyses as long
as the overall number of characters in the analysis is
large. When the number of characters is low, Bayesian
analyses that include highly incomplete taxa may have
relatively low accuracy. But it is clear that the low
accuracy in analyses with 100 characters and
75%–90% missing data is not directly caused by a
large number or proportion of missing data cells,
because analyses with 2,000 characters and 95%
missing data have relatively high accuracy but a larger
number and higher proportion of missing data cells.
Instead, the low accuracy in analyses with 100 char-
acters is presumably associated with the limited
number of characters that are present and that can
place these highly incomplete taxa on the tree. These
general results are robust across various branch
lengths (Fig. 1), and closely parallel those for parsi-
mony, likelihood, and neighbor-joining under compa-
rable simulation conditions (Wiens, 2003). These
results also appear to robust when analyzing data
evolved under more complex models of sequence
evolution (regardless of whether that complexity is
included in the analysis; Fig. 2: A, B), under different
tree shapes (Fig. 2: C), and different ways of distrib-
uting missing data among characters (Fig. 2: D).
WIENS & MOEN: Missing data and Bayesian phylogenetics

311



Fig. 1. Results of simulations showing the effects of missing data on the accuracy of Bayesian phylogenetic analysis. The results show that highly
incomplete taxa are only problematic when the number of characters is very low. When the number of characters in the analysis is large, even taxa
with 95% missing data can be accurately placed. Each data point is the mean of 100 replicates. A–D refers to different branch lengths on the
simulated phylogeny. Accuracy refers to the percentage of nodes that are correctly reconstructed in the Bayesian majority-rule consensus tree of each
replicate, averaged across the 100 replicates. The percentage of missing data refers to the proportion of missing data cells in each of the 8 taxa that are
randomly selected to be incomplete in each replicate (out of 16 taxa total).


3 Discussion
Our results suggest that highly incomplete taxa
can be accurately placed in Bayesian phylogenetic
analyses, as long as the number of characters in the
analyses is not unusually low. These results further
support those of empirical analyses (Wiens et al.,
2005), simulation analyses with a different design
(Wiens, 2005), and empirical and simulation analyses
based on other methods, such as parsimony and
likelihood (e.g., Wiens, 2003; Phillipe et al., 2004).
Taken together, these results indicate that taxa should
not be excluded from Bayesian analyses merely
because they have many missing data cells. Further-
more, these results suggest considerable promise for
constructing a comprehensive Tree of Life using
Bayesian methods, even though some taxa may be
missing data for many characters. However, a number
of caveats should be noted.
3.1 Simulations versus the real world
Simulations involve many simplifying assump-
tions, and considerable caution must be taken when
using the results of simulation studies to inform
empirical analyses (Hillis, 1995; Huelsenbeck, 1995).
For example, the combination of tree shapes (fully
asymmetric or fully symmetric), branch lengths (all
Journal of Systematics and Evolution Vol. 46 No. 3 2008 312



Fig. 2. Results of simulations showing the effects of missing data on the accuracy of Bayesian phylogenetic analysis, as in Fig. 1. A, Data
simulated under the HKY model with unequal rates of change among characters, but analyzed under the JC model. B, Data simulated under the HKY
model with unequal rates of change among characters, analyzed under the HKY + Γ model. C, Data simulated on a fully symmetric tree, using the JC
model with branch lengths of 0.05 (analyzed using JC model). D, Data simulated using the JC model with branch lengths of 0.05 (analyzed using JC
model), with missing data randomly distributed among characters in the 8 incomplete taxa.


equal), and simple models of sequence evolution that
were simulated here will not be encountered in every
(or perhaps any) empirical data set. Nevertheless, our
analyses here suggest that tree shape per se has little
impact on the results, given that the two most extreme
tree shapes possible gave similar results (Figs. 1, 2).
Our basic results are also supported under a broad
range of equal branch lengths (Fig. 1). In theory, there
might be additional negative consequences of missing
data for Bayesian analysis when analyzing certain
combinations of unequal branch lengths (e.g., Huel-
senbeck, 1995), but these were not apparent in a
previous simulation study (Wiens, 2005). The results
were also robust to increasing the complexity of the
model of sequence evolution, even when the analysis
failed to account for that complexity (i.e., analyzing
data evolved under the HKY model with unequal rates
among sites, but using only the simple JC model).
Although our results were robust under many
different simulated conditions, we acknowledge that it
is theoretically possible that there might be negative
impacts of missing data when combined with other
problems. For example, we assumed that the sets of
missing and non-missing characters were basically
equivalent. We expect that cases of extensive missing
data could be far more problematic if the only charac-
ters that were non-missing were themselves problem-
atic for some reason (e.g., evolving too slowly to be
WIENS & MOEN: Missing data and Bayesian phylogenetics

313
informative or too quickly to be accurate). In such a
situation, we would not expect highly incomplete taxa
to be accurately placed by Bayesian analysis, or by
any other method.
Finally, there are some particular distributions of
missing data that most phylogenetic methods will not
be readily able to deal with. For example, imagine that
there are five species (A–E) having data for four genes
(1–4) and another five species (F–J) having data for a
different set of four genes (5–8). If these data sets are
combined, there will be considerable missing data and
the analysis will not be able to simultaneously resolve
the relationships of all 10 species. However, the
problem is that there is no overlap between the two
data sets, not the amount of missing data per se (i.e.,
the matrix here would have 50% missing data cells, an
amount which is unproblematic under most conditions
we analyzed). Our simulations have focused primarily
on the issue of including some incomplete taxa in
analyses that include some complete taxa, and not that
of combining poorly overlapping datasets with few
taxa or characters in common (e.g., Sanderson et al.,
2003).
3.2 Levels of support
Another issue that we did not examine here is the
relationship between missing data and levels of sup-
port. It is theoretically possible that, under conditions
when accuracy is low, missing data might cause
Bayesian analyses to yield incorrect results with strong
statistical support. Other studies have found that
Bayesian analysis may overestimate support when
relationships are highly uncertain (e.g., Suzuki et al.,
2002), such as when three taxa split almost simulta-
neously ( e.g., Lewis et al., 2005). This topic might be
worthy of additional investigation. However, our
results do suggest that the circumstances under which
extensive missing data would be associated with low
phylogenetic accuracy are quite limited.
3.3 Missing data and analysis time
A final issue that we did not address is the effect
of missing data on the duration of analyses and the time
to reach stationarity. One can imagine that more
missing data cells might slow down a Bayesian analy-
sis. Similarly, one could also imagine that the number
of generations required to achieve stationarity might be
increased by extensive missing data. We briefly tested
these assumptions with our simulations. Under the
conditions where Bayesian analysis has relatively low
accuracy (100 characters, branch length = 0.05), the
duration of the analysis is actually shorter with 95%
missing data (in the 8 taxa) than with no missing data
(mean with 95% missing data=73.4 seconds, range=
62–85; vs. mean with no missing data=102.1, range=
89–113; t-test, P<0.0001). Further, we were unable to
detect any difference in the time to reach stationarity
(i.e., before 5,000 generations in all cases). Although
more analysis of these issues would be welcome, we
have found no evidence so far to suggest that having
extensive missing data leads to slower analyses or
longer times to achieve stationarity.
Acknowledgements We would like to thank
Yin-Long Qiu for helping to organize the Beijing Tree
of Life symposium in 2007 and for inviting Wiens to
participate and contribute this paper. We thank the
U.S. National Science Foundation for financial sup-
port (EF 0334923 to J.J.W. and a Graduate Research
Fellowship to D.S.M.). We are grateful to two
anonymous reviewers for helpful comments on the
manuscript.
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