Comparison of Bayesian and Maximum Likelihood Bootstrap Measures of Phylogenetic Reliability
* Department of Biochemistry and Molecular Biology, Dalhousie University, Halifax, Nova Scotia, Canada:, http://www.100md.com
Laboratoire de Paléontologie, Paléobiologie et Phylogénie, Institut des Sciences de l'Evolution, Université Montpellier II, Montpellier, France:, http://www.100md.com
Abstract:, http://www.100md.com
Owing to the exponential growth of genome databases, phylogenetic trees are now widely used to test a variety of evolutionary hypotheses. Nevertheless, computation time burden limits the application of methods such as maximum likelihood nonparametric bootstrap to assess reliability of evolutionary trees. As an alternative, the much faster Bayesian inference of phylogeny, which expresses branch support as posterior probabilities, has been introduced. However, marked discrepancies exist between nonparametric bootstrap proportions and Bayesian posterior probabilities, leading to difficulties in the interpretation of sometimes strongly conflicting results. As an attempt to reconcile these two indices of node reliability, we apply the nonparametric bootstrap resampling procedure to the Bayesian approach. The correlation between posterior probabilities, bootstrap maximum likelihood percentages, and bootstrapped posterior probabilities was studied for eight highly diverse empirical data sets and were also investigated using experimental simulation. Our results show that the relation between posterior probabilities and bootstrapped maximum likelihood percentages is highly variable but that very strong correlations always exist when Bayesian node support is estimated on bootstrapped character matrices. Moreover, simulations corroborate empirical observations in suggesting that, being more conservative, the bootstrap approach might be less prone to strongly supporting a false phylogenetic hypothesis. Thus, apparent conflicts in topology recovered by the Bayesian approach were reduced after bootstrapping. Both posterior probabilities and bootstrap supports are of great interest to phylogeny as potential upper and lower bounds of node reliability, but they are surely not interchangeable and cannot be directly compared.
Key Words: Bayesian • bootstrap • Markov chain Monte Carlo • maximum likelihood • phylogeny • posterior probability8l10-u1, 百拇医药
Introduction8l10-u1, 百拇医药
Testing evolutionary hypotheses in a phylogenetic context becomes more reliable as reconstruction methods based on more realistic models of molecular evolution are available. However, computing time burden limits the application of model-based methods such as maximum likelihood (ML) when many taxa and/or assessment of reliability via standard—nonparametric—bootstrap methods are involved . Time savings thus account in part for the increasing popularity of Bayesian inference methods (e.g), as implemented in programs like MrBayes . These methods promise computational tractability with large data sets and complex evolutionary models.8l10-u1, 百拇医药
Bayesian inference of phylogeny combines the prior probability of a phylogeny with the tree likelihood to produce a posterior probability distribution on trees . The best estimate of the phylogeny can be selected as the tree with the highest posterior probability (i.e., the MAximum Posterior probability [MAP] tree) . Topologies and branch lengths are not treated as parameters—as in ML methods —but as random variables. Because posterior probabilities cannot be obtained analytically, they are approximated by numerical methods known as Markov chain Monte Carlo (MCMC) or Metropolis coupled MCMC (MCMCMC). These chains are designed to explore the posterior probability surface by integration over the space of model parameters. Trees are sampled at fixed intervals and the posterior probability of a given tree is approximated by the proportion of time that the chains visited it . A consensus tree can be obtained from these sampled trees, and Bayesian posterior probabilities (PP) of individual clades, as expressed by the consensus indices, may be viewed as clade credibility values. Thus, Bayesian analysis of the initial matrix of taxa and characters produces both a MAP tree and estimates of uncertainty of its nodes, directly assessing substitution model, branch length, and topological variables, as well as clade reliability values, all in a reasonable computation time.
Reliability of nodes in phylogenetic trees is classically evaluated in two ways. First, from the initial matrix of characters, a strength of grouping value is measured, that is, the least decrease in log-likelihood associated with the breaking of the clade defined by that node (e.g). The statistical significance of this decrease can be estimated with nonparametric or parametric tests (e.g). With Bayesian methods, reliability of MAP tree nodes derives directly from corresponding posterior probabilities. In the second way, the initial character matrix is redrawn with replacement, and bootstrap percentages (BP) are calculated, for example under the ML criterion (BPML), and interpreted as a measure of experiment repeatability or phylogenetic accuracy .@}^)g1[, 百拇医药
The Bayesian approach is presumed to perform roughly as bootstrapped ML but runs much faster . However, Bayesian phylogenetics has its currently unsolved problems, and "perhaps the most vexing mystery is the observed discrepancy between Bayesian posterior probabilities and nonparametric bootstrap support values" . Recent analyses have aimed at comparing Bayesian and ML supports by studying the correlation between PP and BPML . A compilation of literature values ( Wilcox et al. 2002) reveals that plotting PP as a function of BPML can show significant correlation (P < 0.02), but that the strength of this correlation is highly variable and sometime very low (correlation coefficient r2 between 0.33 and 0.99; median at 0.73). Moreover, the slope (S) of the regression line (S between 0.29 and 1.08; median at 0.79) indicates that BPML values are generally lower than PP values. This trend has already been noticed by in their pioneering work, where PP values appeared systematically higher than resampling estimated log-likelihood (RELL) bootstrap support values.
As more phylogenetic results relying strictly on Bayesian analyses are published ,a better understanding of the relation between PP and BPML becomes essential. Wilcox et al. (2002) explored this relation by performing simulations on their original data set. They conclude that, under the condition of their study, PP and BP are both overconservative measures of phylogenetic accuracy but that Bayesian support values provide closer estimates of the true probabilities of recovering clades. Thus they advocate the preferential use of PP rather than BP (Wilcox et al. 2002). However, cases where conflicting hypotheses are supported by high posterior probabilities have been reported ( Douady et al. in press). This suggests that at least in certain cases, PP put overconfidence on a given phylogenetic hypothesis, and drawing conclusions from this sole measure of support might be misleading.!1, 百拇医药
To better understand the relationship between PP and BP, we applied standard (i.e., nonparametric) bootstrap resampling procedures to the Bayesian approach, studying the correlation between PP, BPML, and BPBay—that is, posterior probabilities estimated after bootstrapping of the data—for eight empirical data sets spanning different kinds of characters, types of sequences, genomic compartments, and taxonomic groups. Even when the correlation between PP and BPML was weak (r2 < 0.52), it became very strong (r2 > 0.96) when Bayesian posterior probabilities are computed on bootstrapped data matrices. Moreover, albeit less clearly, simulation seems to confirm this trend. These simulations also tend to predict that PP overcome BP support for both true and false nodes. We discuss the effect of the bootstrapped approach in the case of apparent conflicts between data sets and consider its practical implications for measuring phylogenetic reliability.
Material and Methods)/5v, 百拇医药
Maximum Likelihood and Standard Bayesian Analyses)/5v, 百拇医药
Eight highly diverse empirical data sets were chosen (see details in ), including two pairs showing conflict (i.e., PP strongly supporting mutually exclusive nodes): mitochondrial versus nuclear markers for 14 cicadas and mitochondrial rRNA markers for 20 chondrichthyans and either one or three outgroup taxa (Douady et al. in press). The model of sequence evolution that best fits each DNA data set and the corresponding GTR substitution rate parameters, shape of the four-categories gamma distribution (4) and fraction of invariable sites (INV) were estimated by Modeltest 3.06 , and then used in PAUP* 4b10 to compute ML bootstrap percentages (BPML) after 100 pseudoreplications with Neighbor-Joining starting trees and Tree Bisection-Reconnection branch swapping. For the amino acid data set, BPML were obtained using PROML version 3.6a2.1 of the PHYLIP package with a JTT substitution matrix provided by E. Tillier (personal communication) combined to a {Gamma} 4 + INV model, and parameters optimized by Puzzle 4.0.2 .
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Table 1 Linear Correlation Between ML Bootstrap Percentages (BPML) and Bayesian Supporta for Eight Highly Diverse Empirical Data Sets.|n%2q+^, 百拇医药
Bayesian posterior probabilities (PP) were computed under the same ML models with MrBayes 2.01 by running four chains for 100,000 MCMCMC generations using the program default priors on model parameters. For all analyses, 1,000 trees were sampled from the posterior probability distribution (one every 100 generations) and a conservative 50% of the trees (500) was systematically discarded as "burn-in" to ensure that the chains have reached stationarity.|n%2q+^, 百拇医药
Bootstrapped Bayesian Analyses|n%2q+^, 百拇医药
We generated 100 bootstrap pseudo-replicates for each of the eight data sets using the program SEQBOOT 3.6a2.1 . For each pseudoreplicate, Bayesian posterior probabilities were estimated as previously described (i.e., the tree sampling and burn-in value were fixed as for the standard Bayesian approach). Bootstrapped Bayesian support was computed for each node into three ways: (1) the bootstrapped posterior probabilities (BPBay) obtained from the consensus of the 500 x 100 = 50,000 trees generated from the 100 bootstrapped pseudoreplicates, (2) the Bayesian bootstrap percentages obtained from the consensus of the 100 MAP trees (i.e., a "consensus of consensus" procedure), and (3) the average of each nodes PP for the 100 MAP trees. Given the tedious aspect of preparing files for bootstrapped Bayesian analyses, a Perl script was custom made and is available upon request.
Simulation Studieshf, http://www.100md.com
We also explored the relation between PP and BP using a simulation design. Monte Carlo simulation of 100 data sets of 1,000 characters for seven taxa each was performed using SEQ-GEN 1.2.5 , under a model topology and associated branch lengths taken from the armadillo subset of VWF xenarthran data. The K2P model of nucleotide substitution was chosen with a transition:transversion ratio of 2.00 and a 8 distribution with {alpha}hf, http://www.100md.com
= 1.00. BPML and PP supports were obtained for these 100 simulated data sets following the same procedure as described above. For computing time reasons (i.e., running 2,500 times MrBayes), BPBay were only computed for the 25 data sets showing the greatest contrast between BPML and PP.hf, http://www.100md.com
Results and Discussionhf, http://www.100md.com
Standard and Bootstrapped Bayesian Posterior Probabilities Versus Maximum Likelihood Bootstrap
For all eight data sets, the scatter plots of PP and BPBay versus BPML were very similar. In all cases, PP versus BPML are characterized by a moderate dispersion but a flattened slope (S column in : 0.18–0.93), while BPBay versus BPML have very little unexplained variation, slopes of correlation lines appearing much steeper and being always very close to 1 (0.93–1.22). illustrates this trend for three individual data sets, showing that the results are independent of the nature of the data analyzed: nucleotide versus amino acid characters, nuclear versus mitochondrial compartments, protein coding versus noncoding markers and different taxonomic groups and levels . Because empirical observations suffer from the difficulty of drawing general conclusions from a limited number of observations, we combined the six strictly independent data sets and confirmed our observations ( and ). Therefore, we are confident that, in empirical data sets, PP and BPML will prove only moderately correlated (r2 = 0.27–0.93; S = 0.18–0.93; P < 0.02), whereas the BPBay and BPML are strongly correlated (r2 = 0.95–0.99; S = 0.93–1.22; P < 10-6).
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FIG. 1. Linear correlation between maximum likelihood bootstrap percentages (BPML) and Bayesian posterior probabilities (PP; circles) or bootstrapped Bayesian posterior probabilities (BPBay; triangles) for empirical data sets. The dotted line represents a slope of 1—with equality of BPML and PP or BPBay—while dashed and plain lines represent PP = f(BPML) and BPBay = f(BPML) regression lines, respectively. All axes represent node support as percentages. See for further information regarding data sets-s79y6c, 百拇医药
We tested several of the assumptions leading to the strong correlation between BPBay and BPML. First, the possibility that the quality of the correlations observed could depend upon random error occurring between independent runs seems to be discarded by the minimal PP variance observed on MCMCMC repeatability plots . It is thus unlikely that the low correlation between PP and BPML reflects a problem of repeatability between independent runs. Second, we a priori removed 50% of the sampled trees as MCMCMC "burn-in." This was done to ensure that all trees sampled before stationarity were discarded, without actually checking Bayesian results of each individual bootstrap pseudoreplicate. To check for potential biases at this stage, we recomputed BPBay, keeping 90% of all sampled trees (i.e., removing 100 instead of 500 trees for each pseudoreplicate). Results indicate that bias is quite unlikely, as the level of BPBay variation is very low (e.g., 1% for the ITS data set). Therefore, "burn-in" threshold seems to be of modest importance as long as it is kept realistic, probably because of the rapid convergence towards stationarity of our data. Third, we looked at the effect of making an overall consensus (i.e., consensus of all 50,000 trees sampled over all 100 pseudoreplicates and after a 50% burn-in) versus making the consensus of the 100 MAP trees or the average of the PP. Compilation of node supports—for example, for both ITS and nuclear data sets—yields high correlations (r2 > 0.95) between BPML and BPBay, "MAP trees consensus," or "PP average." However, it seems that BPBay and "PP average" node supports are closer to Bayesian philosophy, whereas "MAP tree consensus" values are closer to the ML bootstrap approach. Indeed, in the two first cases, the complete collection of trees is considered, while in the last case, a single optimal tree is kept to represent each pseudoreplicate. Given the likely loss of information during the consensus iteration, it seems that using an overall consensus was a better option to calculate bootstrapped Bayesian node support.
Such a correlation between BPML and BPBay seems expectable since the use of uniform priors in the Bayesian analyses involves that the posterior probability density is strongly dependent upon the likelihood function. However, this correlation is not trivial either, because the ML and the MAP trees obtained from each bootstrap pseudoreplicate are not always identical. For example, in the case of the 21 shark and xenarthran data sets, ML and MAP trees are different in 38% and 27% of the replicates, respectively. Therefore, the very high quality correlation between BPBay and BPML (r2 > 0.95) cannot be expected a priori.k, http://www.100md.com
A Simulation Study to Compare Maximum Likelihood and Bayesian Node Reliabilityk, http://www.100md.com
Nonparametric bootstrapping may be an overconservative estimator of node reliability ( Wilcox et al. 2002; but see ), but it remains the most commonly used way to characterize it. From the statistical point of view, posterior probabilities have the advantage to be of straightforward interpretation as they represent the probability that the corresponding clade is true, given the model, the priors, and the data . However, as we showed, they are not tightly correlated with ML bootstrap percentages. Thus, these estimators seem rather different, as PP needs to be calculated on bootstrapped data to behave like BPML supports. Recently, Wilcox et al. (2002), based on a simulation study, concluded that PP and BP are both overconservative measures of node support but that PP provided closer estimates of the true probabilities of recovering clades.
Results from our simulations seem to confirm the fact that PP is less conservative than BP. Indeed, when considering true nodes—those that were present in the model topology—PP are generally higher than BPML and BPBay ( upper right quarter). However, PP is also higher when looking at strong support for false nodes—those that were absent of the model tree ( upper right quarter). Below 50% of PP and BP ( lower left quarter), that is, for values that are usually not interpreted for phylogenetic inference, there is a large dispersion of points with a trend of low BP to overestimate accuracy, as noted by . As a whole these simulation results imply that, at least in certain cases, high PP falsely interpret signal and may end up strongly supporting incorrect phylogenetic relationships. Thus, the more conservative BPML and BPBay seem less subject to the behavior of strongly supporting a node when it is actually false.-)i, 百拇医药
fig.ommitted
FIG. 2. Linear correlation between maximum likelihood bootstrap percentages (BPML) and Bayesian posterior probabilities (PP; circles) or bootstrapped Bayesian posterior probabilities (BPBay; triangles) in 25 simulated data sets. "True nodes" are nodes that were present in the model topology used to simulate the data sets and "false nodes" are nodes that were not in the model topology. The dotted line represents a slope of 1—with equality of BPML and PP or BPBay—while dashed and plain lines represent PP = f(BPML) and BPBay = f(BPML) regression lines, respectively. All axes represent node support as percentages. The phylogram used for the simulation was (A:0.043143,((B:0.027559,(C:0.018247,D:0.024211):0.003601):0.011055,(E:0.005704,(F:0.010024,G:0.006528):0.000708):0.021913):0.003809) with branch lengths issued from the xenarthran data set, and taxa A–G corresponding to Dasypus kappleri, Tolypeutes matacus, Cabassous unicinctus, Priodontes maximus, Euphractus sexcinctus, Chaetophractus villosus, and Zaedyus pichiy, respectively4y4iawt, 百拇医药
Bootstrap Effect on Apparent Topological Conflicts
Bayesian analyses on bootstrapped data were able to eliminate apparent topological conflicts . Two nodes opposed by PP = 0.93/0.94 (Rhodopsalta, sister to either Maoricicada or Kikihia, depending on the choice of mitochondrial or nuclear markers ) and 0.99/0.98 (relative position of Hexanchiformes in sharks' interordinal tree, depending on the choice of the outgroup [Douady et al. in press]), then, respectively, received BPBay = 59/65 and 47/57 after bootstrap resampling. Evidently, some conflicts diagnosed by PP could be biologically explained by differences between gene trees and species trees introduced by horizontal transfer, lineage sorting, and gene duplication and extinction (review in ). In particular, hybridization between taxa might alternatively account for the conflict observed between mitochondrial and nuclear genes in cicadas. However, in the case of sharks, the conflict arose when taxa are added to the outgroup (for the same gene). It appears more than likely that this spurious conflict was the result of the overestimation of node support based on PP and that conclusions based solely on this estimator would have been positively misleading.
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FIG. 3. Illustration of the effect of the bootstrapped Bayesian approach in two cases of apparently conflicting nodes based on Bayesian posterior probabilities. (A) Conflict between maximum likelihood phylograms for mitochondrial (mtDNA) and nuclear (nucDNA) genes of Pacific Cicada genera (original data from ). (B) Conflict about the position of Hexanchiformes within Elasmobranches between maximum likelihood phylograms of the chondrichthyan 12S and 16S rRNA data sets differing only by the number of outgroup taxa (i.e., rooting with and without Petromyzon and Siren [original data from Douady et al. in press]). PP: Bayesian posterior probability; BPML: maximum likelihood bootstrap percentage; BPBay: Bayesian bootstrapped posterior probability. The arrows indicate apparently conflicting nodes for which PP, BPML, and BPBay are given. Thick branches connect taxa (bold names) involved in the apparent topological conflicts7^, 百拇医药
The existence of strong conflicts in empirical data using standard Bayesian inference seems to argue that this approach may be sensitive to small model misspecifications as theoretically anticipated by , subsequently shown by and , and acknowledged by . The question of the potential impact of model adequacy on the Bayesian approach may actually be viewed as a stimulating purpose to encourage the future development of more realistic models of sequence evolution . Obviously, additional studies on these issues would provide better understandings of the origin and nature of the observed differences between maximum likelihood and Bayesian node supports.
Conclusions\0|pzz, 百拇医药
Drawing general conclusions from empirical studies could be problematic because we do not know how representative our example data sets are of phylogenetic problems. However, using "real" data sets does have the advantage of avoiding the simplifying assumptions inherent in simulating DNA data under an idealized model . Furthermore, in our case, analyses based on both empirical and simulated data seem to corroborate each other in suggesting that, being more conservative, BPML and BPBay might be less prone to strongly supporting a false phylogenetic hypothesis. These observations reinforce concerns regarding PP sensitivity to model misspecifications.\0|pzz, 百拇医药
Nevertheless, Bayesian inference—with and without bootstrap—remains a very efficient way to simultaneously estimate substitution model parameters, branch lengths, and topology under complex models of evolutionary change . If we take the chondrichthyan 12S–16S data sets with 23 taxa as an example , a standard Bayesian search—or one Bayesian bootstrap replicate—runs roughly 80 times faster on a 1.80 GHz Pentium 4 than a single PAUP* replicate of BPML with simultaneous estimation of all parameters. Bayesian search on bootstrap data is much faster than ML if the user wants parameters to be estimated as the search goes, and it gives very similar results . However, in the wide majority of cases, an ML (or BPML) search with simultaneous estimation of the parameters is not necessary, and a priori approximations allow the identification of the optimal trees and bootstrap supports. The Bayesian approach also provides a unique way to analyze amino acid data with simultaneous parameters estimation, whereas this option is only available for DNA in popular phylogenetic packages such as PAUP or PHYLIP.
Both PP and bootstrap supports are of great interest to phylogeny as potential upper and lower bound of node support, but they are surely not interchangeable and cannot be directly compared. In that context, users may prefer computing PP and BPBay or BPML to better explore the range of node support estimates, especially when potential conflicts between data sets are explored.e, 百拇医药
Acknowledgementse, 百拇医药
As Molecular Biology and Evolution is no longer publishing Letters to the Editor, this paper has been reformatted into an Original Article. Chris Simon and Thomas R. Buckley kindly made their cicada data sets available. David M. Hillis and Thomas P. Wilcox provided precious enlightenment about their simulations on the snake data set. We are grateful to Alastair G. B. Simpson, the associate editor Mark A. Ragan, and two anonymous reviewers for insightful comments. This work was supported by a grant from the Canadian Institute for Health Research (MOP-42470), a Killam Postdoctoral Fellowship to C.J.D., a doctoral MENRT grant (contract 99075) to F.D., a doctoral research award from the Canadian Institutes of Health Research to Y.B., and by the TMR Network "Mammalian phylogeny" (contract FMRX-CT98-022) of the European Community, the "Génopole Montpellier Languedoc-Roussillon," and the "Action Bioinformatique inter-EPST" of the CNRS to E.J.P.D. This is the contribution ISEM 2002-048 of the Institut des Sciences de l'Evolution de Montpellier (UMR 5554-CNRS).
Note Added in Proof: Suzuki, Glazko, and Nei recently showed by simulation that posterior probabilities in Bayesian analysis can be excessively liberal, whereas bootstrap probabilities in Neighbor-Joining and maximum likelihood analyses are generally slightly conservative (2000, Proc. Natl. Acad. Sci. USA 99:16138–16143).cu, 百拇医药
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Yang, Z., and B. Rannala. 1997. Bayesian phylogenetic inference using DNA sequences: a Markov Chain Monte Carlo method. Mol. Biol. Evol. 14:717-724.7r(;.l|, 百拇医药
Accepted for publication October 10, 2002.(Christophe J. Douady Frédéric Delsuc, Yan Boucher W. Ford Doolittle and Emmanuel J. P. Douzery)
Laboratoire de Paléontologie, Paléobiologie et Phylogénie, Institut des Sciences de l'Evolution, Université Montpellier II, Montpellier, France:, http://www.100md.com
Abstract:, http://www.100md.com
Owing to the exponential growth of genome databases, phylogenetic trees are now widely used to test a variety of evolutionary hypotheses. Nevertheless, computation time burden limits the application of methods such as maximum likelihood nonparametric bootstrap to assess reliability of evolutionary trees. As an alternative, the much faster Bayesian inference of phylogeny, which expresses branch support as posterior probabilities, has been introduced. However, marked discrepancies exist between nonparametric bootstrap proportions and Bayesian posterior probabilities, leading to difficulties in the interpretation of sometimes strongly conflicting results. As an attempt to reconcile these two indices of node reliability, we apply the nonparametric bootstrap resampling procedure to the Bayesian approach. The correlation between posterior probabilities, bootstrap maximum likelihood percentages, and bootstrapped posterior probabilities was studied for eight highly diverse empirical data sets and were also investigated using experimental simulation. Our results show that the relation between posterior probabilities and bootstrapped maximum likelihood percentages is highly variable but that very strong correlations always exist when Bayesian node support is estimated on bootstrapped character matrices. Moreover, simulations corroborate empirical observations in suggesting that, being more conservative, the bootstrap approach might be less prone to strongly supporting a false phylogenetic hypothesis. Thus, apparent conflicts in topology recovered by the Bayesian approach were reduced after bootstrapping. Both posterior probabilities and bootstrap supports are of great interest to phylogeny as potential upper and lower bounds of node reliability, but they are surely not interchangeable and cannot be directly compared.
Key Words: Bayesian • bootstrap • Markov chain Monte Carlo • maximum likelihood • phylogeny • posterior probability8l10-u1, 百拇医药
Introduction8l10-u1, 百拇医药
Testing evolutionary hypotheses in a phylogenetic context becomes more reliable as reconstruction methods based on more realistic models of molecular evolution are available. However, computing time burden limits the application of model-based methods such as maximum likelihood (ML) when many taxa and/or assessment of reliability via standard—nonparametric—bootstrap methods are involved . Time savings thus account in part for the increasing popularity of Bayesian inference methods (e.g), as implemented in programs like MrBayes . These methods promise computational tractability with large data sets and complex evolutionary models.8l10-u1, 百拇医药
Bayesian inference of phylogeny combines the prior probability of a phylogeny with the tree likelihood to produce a posterior probability distribution on trees . The best estimate of the phylogeny can be selected as the tree with the highest posterior probability (i.e., the MAximum Posterior probability [MAP] tree) . Topologies and branch lengths are not treated as parameters—as in ML methods —but as random variables. Because posterior probabilities cannot be obtained analytically, they are approximated by numerical methods known as Markov chain Monte Carlo (MCMC) or Metropolis coupled MCMC (MCMCMC). These chains are designed to explore the posterior probability surface by integration over the space of model parameters. Trees are sampled at fixed intervals and the posterior probability of a given tree is approximated by the proportion of time that the chains visited it . A consensus tree can be obtained from these sampled trees, and Bayesian posterior probabilities (PP) of individual clades, as expressed by the consensus indices, may be viewed as clade credibility values. Thus, Bayesian analysis of the initial matrix of taxa and characters produces both a MAP tree and estimates of uncertainty of its nodes, directly assessing substitution model, branch length, and topological variables, as well as clade reliability values, all in a reasonable computation time.
Reliability of nodes in phylogenetic trees is classically evaluated in two ways. First, from the initial matrix of characters, a strength of grouping value is measured, that is, the least decrease in log-likelihood associated with the breaking of the clade defined by that node (e.g). The statistical significance of this decrease can be estimated with nonparametric or parametric tests (e.g). With Bayesian methods, reliability of MAP tree nodes derives directly from corresponding posterior probabilities. In the second way, the initial character matrix is redrawn with replacement, and bootstrap percentages (BP) are calculated, for example under the ML criterion (BPML), and interpreted as a measure of experiment repeatability or phylogenetic accuracy .@}^)g1[, 百拇医药
The Bayesian approach is presumed to perform roughly as bootstrapped ML but runs much faster . However, Bayesian phylogenetics has its currently unsolved problems, and "perhaps the most vexing mystery is the observed discrepancy between Bayesian posterior probabilities and nonparametric bootstrap support values" . Recent analyses have aimed at comparing Bayesian and ML supports by studying the correlation between PP and BPML . A compilation of literature values ( Wilcox et al. 2002) reveals that plotting PP as a function of BPML can show significant correlation (P < 0.02), but that the strength of this correlation is highly variable and sometime very low (correlation coefficient r2 between 0.33 and 0.99; median at 0.73). Moreover, the slope (S) of the regression line (S between 0.29 and 1.08; median at 0.79) indicates that BPML values are generally lower than PP values. This trend has already been noticed by in their pioneering work, where PP values appeared systematically higher than resampling estimated log-likelihood (RELL) bootstrap support values.
As more phylogenetic results relying strictly on Bayesian analyses are published ,a better understanding of the relation between PP and BPML becomes essential. Wilcox et al. (2002) explored this relation by performing simulations on their original data set. They conclude that, under the condition of their study, PP and BP are both overconservative measures of phylogenetic accuracy but that Bayesian support values provide closer estimates of the true probabilities of recovering clades. Thus they advocate the preferential use of PP rather than BP (Wilcox et al. 2002). However, cases where conflicting hypotheses are supported by high posterior probabilities have been reported ( Douady et al. in press). This suggests that at least in certain cases, PP put overconfidence on a given phylogenetic hypothesis, and drawing conclusions from this sole measure of support might be misleading.!1, 百拇医药
To better understand the relationship between PP and BP, we applied standard (i.e., nonparametric) bootstrap resampling procedures to the Bayesian approach, studying the correlation between PP, BPML, and BPBay—that is, posterior probabilities estimated after bootstrapping of the data—for eight empirical data sets spanning different kinds of characters, types of sequences, genomic compartments, and taxonomic groups. Even when the correlation between PP and BPML was weak (r2 < 0.52), it became very strong (r2 > 0.96) when Bayesian posterior probabilities are computed on bootstrapped data matrices. Moreover, albeit less clearly, simulation seems to confirm this trend. These simulations also tend to predict that PP overcome BP support for both true and false nodes. We discuss the effect of the bootstrapped approach in the case of apparent conflicts between data sets and consider its practical implications for measuring phylogenetic reliability.
Material and Methods)/5v, 百拇医药
Maximum Likelihood and Standard Bayesian Analyses)/5v, 百拇医药
Eight highly diverse empirical data sets were chosen (see details in ), including two pairs showing conflict (i.e., PP strongly supporting mutually exclusive nodes): mitochondrial versus nuclear markers for 14 cicadas and mitochondrial rRNA markers for 20 chondrichthyans and either one or three outgroup taxa (Douady et al. in press). The model of sequence evolution that best fits each DNA data set and the corresponding GTR substitution rate parameters, shape of the four-categories gamma distribution (4) and fraction of invariable sites (INV) were estimated by Modeltest 3.06 , and then used in PAUP* 4b10 to compute ML bootstrap percentages (BPML) after 100 pseudoreplications with Neighbor-Joining starting trees and Tree Bisection-Reconnection branch swapping. For the amino acid data set, BPML were obtained using PROML version 3.6a2.1 of the PHYLIP package with a JTT substitution matrix provided by E. Tillier (personal communication) combined to a {Gamma} 4 + INV model, and parameters optimized by Puzzle 4.0.2 .
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Table 1 Linear Correlation Between ML Bootstrap Percentages (BPML) and Bayesian Supporta for Eight Highly Diverse Empirical Data Sets.|n%2q+^, 百拇医药
Bayesian posterior probabilities (PP) were computed under the same ML models with MrBayes 2.01 by running four chains for 100,000 MCMCMC generations using the program default priors on model parameters. For all analyses, 1,000 trees were sampled from the posterior probability distribution (one every 100 generations) and a conservative 50% of the trees (500) was systematically discarded as "burn-in" to ensure that the chains have reached stationarity.|n%2q+^, 百拇医药
Bootstrapped Bayesian Analyses|n%2q+^, 百拇医药
We generated 100 bootstrap pseudo-replicates for each of the eight data sets using the program SEQBOOT 3.6a2.1 . For each pseudoreplicate, Bayesian posterior probabilities were estimated as previously described (i.e., the tree sampling and burn-in value were fixed as for the standard Bayesian approach). Bootstrapped Bayesian support was computed for each node into three ways: (1) the bootstrapped posterior probabilities (BPBay) obtained from the consensus of the 500 x 100 = 50,000 trees generated from the 100 bootstrapped pseudoreplicates, (2) the Bayesian bootstrap percentages obtained from the consensus of the 100 MAP trees (i.e., a "consensus of consensus" procedure), and (3) the average of each nodes PP for the 100 MAP trees. Given the tedious aspect of preparing files for bootstrapped Bayesian analyses, a Perl script was custom made and is available upon request.
Simulation Studieshf, http://www.100md.com
We also explored the relation between PP and BP using a simulation design. Monte Carlo simulation of 100 data sets of 1,000 characters for seven taxa each was performed using SEQ-GEN 1.2.5 , under a model topology and associated branch lengths taken from the armadillo subset of VWF xenarthran data. The K2P model of nucleotide substitution was chosen with a transition:transversion ratio of 2.00 and a 8 distribution with {alpha}hf, http://www.100md.com
= 1.00. BPML and PP supports were obtained for these 100 simulated data sets following the same procedure as described above. For computing time reasons (i.e., running 2,500 times MrBayes), BPBay were only computed for the 25 data sets showing the greatest contrast between BPML and PP.hf, http://www.100md.com
Results and Discussionhf, http://www.100md.com
Standard and Bootstrapped Bayesian Posterior Probabilities Versus Maximum Likelihood Bootstrap
For all eight data sets, the scatter plots of PP and BPBay versus BPML were very similar. In all cases, PP versus BPML are characterized by a moderate dispersion but a flattened slope (S column in : 0.18–0.93), while BPBay versus BPML have very little unexplained variation, slopes of correlation lines appearing much steeper and being always very close to 1 (0.93–1.22). illustrates this trend for three individual data sets, showing that the results are independent of the nature of the data analyzed: nucleotide versus amino acid characters, nuclear versus mitochondrial compartments, protein coding versus noncoding markers and different taxonomic groups and levels . Because empirical observations suffer from the difficulty of drawing general conclusions from a limited number of observations, we combined the six strictly independent data sets and confirmed our observations ( and ). Therefore, we are confident that, in empirical data sets, PP and BPML will prove only moderately correlated (r2 = 0.27–0.93; S = 0.18–0.93; P < 0.02), whereas the BPBay and BPML are strongly correlated (r2 = 0.95–0.99; S = 0.93–1.22; P < 10-6).
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FIG. 1. Linear correlation between maximum likelihood bootstrap percentages (BPML) and Bayesian posterior probabilities (PP; circles) or bootstrapped Bayesian posterior probabilities (BPBay; triangles) for empirical data sets. The dotted line represents a slope of 1—with equality of BPML and PP or BPBay—while dashed and plain lines represent PP = f(BPML) and BPBay = f(BPML) regression lines, respectively. All axes represent node support as percentages. See for further information regarding data sets-s79y6c, 百拇医药
We tested several of the assumptions leading to the strong correlation between BPBay and BPML. First, the possibility that the quality of the correlations observed could depend upon random error occurring between independent runs seems to be discarded by the minimal PP variance observed on MCMCMC repeatability plots . It is thus unlikely that the low correlation between PP and BPML reflects a problem of repeatability between independent runs. Second, we a priori removed 50% of the sampled trees as MCMCMC "burn-in." This was done to ensure that all trees sampled before stationarity were discarded, without actually checking Bayesian results of each individual bootstrap pseudoreplicate. To check for potential biases at this stage, we recomputed BPBay, keeping 90% of all sampled trees (i.e., removing 100 instead of 500 trees for each pseudoreplicate). Results indicate that bias is quite unlikely, as the level of BPBay variation is very low (e.g., 1% for the ITS data set). Therefore, "burn-in" threshold seems to be of modest importance as long as it is kept realistic, probably because of the rapid convergence towards stationarity of our data. Third, we looked at the effect of making an overall consensus (i.e., consensus of all 50,000 trees sampled over all 100 pseudoreplicates and after a 50% burn-in) versus making the consensus of the 100 MAP trees or the average of the PP. Compilation of node supports—for example, for both ITS and nuclear data sets—yields high correlations (r2 > 0.95) between BPML and BPBay, "MAP trees consensus," or "PP average." However, it seems that BPBay and "PP average" node supports are closer to Bayesian philosophy, whereas "MAP tree consensus" values are closer to the ML bootstrap approach. Indeed, in the two first cases, the complete collection of trees is considered, while in the last case, a single optimal tree is kept to represent each pseudoreplicate. Given the likely loss of information during the consensus iteration, it seems that using an overall consensus was a better option to calculate bootstrapped Bayesian node support.
Such a correlation between BPML and BPBay seems expectable since the use of uniform priors in the Bayesian analyses involves that the posterior probability density is strongly dependent upon the likelihood function. However, this correlation is not trivial either, because the ML and the MAP trees obtained from each bootstrap pseudoreplicate are not always identical. For example, in the case of the 21 shark and xenarthran data sets, ML and MAP trees are different in 38% and 27% of the replicates, respectively. Therefore, the very high quality correlation between BPBay and BPML (r2 > 0.95) cannot be expected a priori.k, http://www.100md.com
A Simulation Study to Compare Maximum Likelihood and Bayesian Node Reliabilityk, http://www.100md.com
Nonparametric bootstrapping may be an overconservative estimator of node reliability ( Wilcox et al. 2002; but see ), but it remains the most commonly used way to characterize it. From the statistical point of view, posterior probabilities have the advantage to be of straightforward interpretation as they represent the probability that the corresponding clade is true, given the model, the priors, and the data . However, as we showed, they are not tightly correlated with ML bootstrap percentages. Thus, these estimators seem rather different, as PP needs to be calculated on bootstrapped data to behave like BPML supports. Recently, Wilcox et al. (2002), based on a simulation study, concluded that PP and BP are both overconservative measures of node support but that PP provided closer estimates of the true probabilities of recovering clades.
Results from our simulations seem to confirm the fact that PP is less conservative than BP. Indeed, when considering true nodes—those that were present in the model topology—PP are generally higher than BPML and BPBay ( upper right quarter). However, PP is also higher when looking at strong support for false nodes—those that were absent of the model tree ( upper right quarter). Below 50% of PP and BP ( lower left quarter), that is, for values that are usually not interpreted for phylogenetic inference, there is a large dispersion of points with a trend of low BP to overestimate accuracy, as noted by . As a whole these simulation results imply that, at least in certain cases, high PP falsely interpret signal and may end up strongly supporting incorrect phylogenetic relationships. Thus, the more conservative BPML and BPBay seem less subject to the behavior of strongly supporting a node when it is actually false.-)i, 百拇医药
fig.ommitted
FIG. 2. Linear correlation between maximum likelihood bootstrap percentages (BPML) and Bayesian posterior probabilities (PP; circles) or bootstrapped Bayesian posterior probabilities (BPBay; triangles) in 25 simulated data sets. "True nodes" are nodes that were present in the model topology used to simulate the data sets and "false nodes" are nodes that were not in the model topology. The dotted line represents a slope of 1—with equality of BPML and PP or BPBay—while dashed and plain lines represent PP = f(BPML) and BPBay = f(BPML) regression lines, respectively. All axes represent node support as percentages. The phylogram used for the simulation was (A:0.043143,((B:0.027559,(C:0.018247,D:0.024211):0.003601):0.011055,(E:0.005704,(F:0.010024,G:0.006528):0.000708):0.021913):0.003809) with branch lengths issued from the xenarthran data set, and taxa A–G corresponding to Dasypus kappleri, Tolypeutes matacus, Cabassous unicinctus, Priodontes maximus, Euphractus sexcinctus, Chaetophractus villosus, and Zaedyus pichiy, respectively4y4iawt, 百拇医药
Bootstrap Effect on Apparent Topological Conflicts
Bayesian analyses on bootstrapped data were able to eliminate apparent topological conflicts . Two nodes opposed by PP = 0.93/0.94 (Rhodopsalta, sister to either Maoricicada or Kikihia, depending on the choice of mitochondrial or nuclear markers ) and 0.99/0.98 (relative position of Hexanchiformes in sharks' interordinal tree, depending on the choice of the outgroup [Douady et al. in press]), then, respectively, received BPBay = 59/65 and 47/57 after bootstrap resampling. Evidently, some conflicts diagnosed by PP could be biologically explained by differences between gene trees and species trees introduced by horizontal transfer, lineage sorting, and gene duplication and extinction (review in ). In particular, hybridization between taxa might alternatively account for the conflict observed between mitochondrial and nuclear genes in cicadas. However, in the case of sharks, the conflict arose when taxa are added to the outgroup (for the same gene). It appears more than likely that this spurious conflict was the result of the overestimation of node support based on PP and that conclusions based solely on this estimator would have been positively misleading.
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FIG. 3. Illustration of the effect of the bootstrapped Bayesian approach in two cases of apparently conflicting nodes based on Bayesian posterior probabilities. (A) Conflict between maximum likelihood phylograms for mitochondrial (mtDNA) and nuclear (nucDNA) genes of Pacific Cicada genera (original data from ). (B) Conflict about the position of Hexanchiformes within Elasmobranches between maximum likelihood phylograms of the chondrichthyan 12S and 16S rRNA data sets differing only by the number of outgroup taxa (i.e., rooting with and without Petromyzon and Siren [original data from Douady et al. in press]). PP: Bayesian posterior probability; BPML: maximum likelihood bootstrap percentage; BPBay: Bayesian bootstrapped posterior probability. The arrows indicate apparently conflicting nodes for which PP, BPML, and BPBay are given. Thick branches connect taxa (bold names) involved in the apparent topological conflicts7^, 百拇医药
The existence of strong conflicts in empirical data using standard Bayesian inference seems to argue that this approach may be sensitive to small model misspecifications as theoretically anticipated by , subsequently shown by and , and acknowledged by . The question of the potential impact of model adequacy on the Bayesian approach may actually be viewed as a stimulating purpose to encourage the future development of more realistic models of sequence evolution . Obviously, additional studies on these issues would provide better understandings of the origin and nature of the observed differences between maximum likelihood and Bayesian node supports.
Conclusions\0|pzz, 百拇医药
Drawing general conclusions from empirical studies could be problematic because we do not know how representative our example data sets are of phylogenetic problems. However, using "real" data sets does have the advantage of avoiding the simplifying assumptions inherent in simulating DNA data under an idealized model . Furthermore, in our case, analyses based on both empirical and simulated data seem to corroborate each other in suggesting that, being more conservative, BPML and BPBay might be less prone to strongly supporting a false phylogenetic hypothesis. These observations reinforce concerns regarding PP sensitivity to model misspecifications.\0|pzz, 百拇医药
Nevertheless, Bayesian inference—with and without bootstrap—remains a very efficient way to simultaneously estimate substitution model parameters, branch lengths, and topology under complex models of evolutionary change . If we take the chondrichthyan 12S–16S data sets with 23 taxa as an example , a standard Bayesian search—or one Bayesian bootstrap replicate—runs roughly 80 times faster on a 1.80 GHz Pentium 4 than a single PAUP* replicate of BPML with simultaneous estimation of all parameters. Bayesian search on bootstrap data is much faster than ML if the user wants parameters to be estimated as the search goes, and it gives very similar results . However, in the wide majority of cases, an ML (or BPML) search with simultaneous estimation of the parameters is not necessary, and a priori approximations allow the identification of the optimal trees and bootstrap supports. The Bayesian approach also provides a unique way to analyze amino acid data with simultaneous parameters estimation, whereas this option is only available for DNA in popular phylogenetic packages such as PAUP or PHYLIP.
Both PP and bootstrap supports are of great interest to phylogeny as potential upper and lower bound of node support, but they are surely not interchangeable and cannot be directly compared. In that context, users may prefer computing PP and BPBay or BPML to better explore the range of node support estimates, especially when potential conflicts between data sets are explored.e, 百拇医药
Acknowledgementse, 百拇医药
As Molecular Biology and Evolution is no longer publishing Letters to the Editor, this paper has been reformatted into an Original Article. Chris Simon and Thomas R. Buckley kindly made their cicada data sets available. David M. Hillis and Thomas P. Wilcox provided precious enlightenment about their simulations on the snake data set. We are grateful to Alastair G. B. Simpson, the associate editor Mark A. Ragan, and two anonymous reviewers for insightful comments. This work was supported by a grant from the Canadian Institute for Health Research (MOP-42470), a Killam Postdoctoral Fellowship to C.J.D., a doctoral MENRT grant (contract 99075) to F.D., a doctoral research award from the Canadian Institutes of Health Research to Y.B., and by the TMR Network "Mammalian phylogeny" (contract FMRX-CT98-022) of the European Community, the "Génopole Montpellier Languedoc-Roussillon," and the "Action Bioinformatique inter-EPST" of the CNRS to E.J.P.D. This is the contribution ISEM 2002-048 of the Institut des Sciences de l'Evolution de Montpellier (UMR 5554-CNRS).
Note Added in Proof: Suzuki, Glazko, and Nei recently showed by simulation that posterior probabilities in Bayesian analysis can be excessively liberal, whereas bootstrap probabilities in Neighbor-Joining and maximum likelihood analyses are generally slightly conservative (2000, Proc. Natl. Acad. Sci. USA 99:16138–16143).cu, 百拇医药
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Accepted for publication October 10, 2002.(Christophe J. Douady Frédéric Delsuc, Yan Boucher W. Ford Doolittle and Emmanuel J. P. Douzery)