Bootstrap method of interior-branch test for phylogenetic trees

Statistical properties of the bootstrap test of interior branch lengths of phylogenetic trees have been studied and compared with those of the standard interior-branch test in computer simulations. Examination of the properties of the tests under the null hypothesis showed that both tests for an interior branch of a predetermined topology are quite reliable when the distribution of the branch length estimate approaches a normal distribution. Unlike the standard interior-branch test, the bootstrap test appears to retain this property even when the substitution rate varies among sites. In this case, the distribution of the branch length estimate deviates from a normal distribution, and the standard interior-branch test gives conservative confidence probability values. A simple correction method was developed for both interior- branch tests to be applied for testing the reliability of tree topologies estimated from sequence data. This correction for the standard interior-branch test appears to be as effective as that obtained in our previous study, though it is much simpler. The bootstrap and standard interior-branch tests for estimated topologies become conservative as the number of sequence groups in a star-like tree increases.      

A test of a mitochondrial gene-based phylogeny of woodpeckers (genus Picoides) using an independent nuclear gene, beta-fibrinogen intron 7

A conservative estimate of the species tree for the woodpecker genus Picoides based on two mitochondrial protein-coding genes is tested using sequences of an independently evolving nuclear intron, beta-fibrinogen intron 7. The mitochondrial gene-based topology and the intron-based topology are concordant, and a partition-homogeneity statistical test did not detect phylogenetic heterogeneity. The intron evolves more slowly than the mitochondrial sequences and tends not to resolve relationships among recently evolved species. However, the intron is superior over mitochondrial genes in resolving older bifurcations in the phylogeny. The two data sets were combined resulting in a robust estimate of the Picoides species tree in which most every node is statistically supported by bootstrap proportions. The Picoides species tree clearly shows that many morphological and behavioral characters used to lump species into this single genus have evolved by convergent evolution. Picoides is considered the largest genus of woodpeckers, but the molecular-based species tree suggests that Picoides is actually a conglomerate of several smaller groups.     

Statistical properties of bootstrap estimation of phylogenetic variability from nucleotide sequences. I. Four taxa with a molecular clock.

The statistical properties of sample estimation and bootstrap estimation of phylogenetic variability from a sample of nucleotide sequences are studied by using model trees of three taxa with an outgroup and by assuming a constant rate of nucleotide substitution. The maximum-parsimony method of tree reconstruction is used. An analytic formula is derived for estimating the sequence length that is required if P, the probability of obtaining the true tree from the sampled sequences, is to be equal to or higher than a given value. Bootstrap estimation is formulated as a two-step sampling procedure: (1) sampling of sequences from the evolutionary process and (2) resampling of the original sequence sample. The probability that a bootstrap resampling of an original sequence sample will support the true tree is found to depend on the model tree, the sequence length, and the probability that a randomly chosen nucleotide site is an informative site. When a trifurcating tree is used as the model tree, the probability that one of the three bifurcating trees will appear in > or = 95% of the bootstrap replicates is < 5%, even if the number of bootstrap replicates is only 50; therefore, the probability of accepting an erroneous tree as the true tree is < 5% if that tree appears in > or = 95% of the bootstrap replicates and if more than 50 bootstrap replications are conducted. However, if a particular bifurcating tree is observed in, say, < 75% of the bootstrap replicates, then it cannot be claimed to be better than the trifurcating tree even if > or = 1,000 bootstrap replications are conducted. When a bifurcating tree is used as the model tree, the bootstrap approach tends to overestimate P when the sequences are very short, but it tends to underestimate that probability when the sequences are long. Moreover, simulation results show that, if a tree is accepted as the true tree only if it has appeared in > or = 95% of the bootstrap replicates, then the probability of failing to accept any bifurcating tree can be as large as 58% even when P = 95%, i.e., even when 95% of the samples from the evolutionary process will support the true tree. Thus, if the rate-constancy assumption holds, bootstrapping is a conservative approach for estimating the reliability of an inferred phylogeny for four taxa.     

Statistical measures of uncertainty for branches in phylogenetic trees inferred from molecular sequences by using model-based methods

In recent years, the emphasis of theoretical work on phylogenetic inference has shifted from the development of new tree inference methods to the development of methods to measure the statistical support for the topologies. This paper reviews 3 approaches to assign support values to branches in trees obtained in the analysis of molecular sequences: the bootstrap, the Bayesian posterior probabilities for clades, and the interior branch tests. In some circumstances, these methods give different answers. It should not be surprising: their assumptions are different. Thus the interior branch tests assume that a given topology is true and only consider if a particular branch length is longer than zero. If a tree is incorrect, a wrong branch (a low bootstrap or Bayesian support may be an indication) may have a non-zero length. If the substitution model is oversimplified, the length of a branch may be overestimated, and the Bayesian support for the branch may be inflated. The bootstrap, on the other hand, approximates the variance of the data under the real model of sequence evolution, because it involves direct resampling from this data. Thus the discrepancy between the Bayesian support and the bootstrap support may signal model inaccuracy. In practical application, use of all 3 methods is recommended, and if discrepancies are observed, then a careful analysis of their potential origins should be made.     

Approximate likelihood-ratio test for branches: A fast, accurate, and powerful alternative

We revisit statistical tests for branches of evolutionary trees reconstructed upon molecular data. A new, fast, approximate likelihood-ratio test (aLRT) for branches is presented here as a competitive alternative to nonparametric bootstrap and Bayesian estimation of branch support. The aLRT is based on the idea of the conventional LRT, with the null hypothesis corresponding to the assumption that the inferred branch has length 0. We show that the LRT statistic is asymptotically distributed as a maximum of three random variables drawn from the chi(0)2 + chi(1)2 distribution. The new aLRT of interior branch uses this distribution for significance testing, but the test statistic is approximated in a slightly conservative but practical way as 2(l1- l2), i.e., double the difference between the maximum log-likelihood values corresponding to the best tree and the second best topological arrangement around the branch of interest. Such a test is fast because the log-likelihood value l2 is computed by optimizing only over the branch of interest and the four adjacent branches, whereas other parameters are fixed at their optimal values corresponding to the best ML tree. The performance of the new test was studied on simulated 4-, 12-, and 100-taxon data sets with sequences of different lengths. The aLRT is shown to be accurate, powerful, and robust to certain violations of model assumptions. The aLRT is implemented within the algorithm used by the recent fast maximum likelihood tree estimation program PHYML (Guindon and Gascuel, 2003).     

Confidence regions and hypothesis tests for topologies using generalized least squares

A confidence region for topologies is a data-dependent set of topologies that, with high probability, can be expected to contain the true topology. Because of the connection between confidence regions and hypothesis tests, implicitly or explicitly, the construction of confidence regions for topologies is a component of many phylogenetic studies. Existing methods for constructing confidence regions, however, often give conflicting results. The Shimodaira-Hasegawa test seems too conservative, including too many topologies, whereas the other commonly used method, the Swofford-Olsen-Waddell-Hillis test, tends to give confidence regions with too few topologies. Confidence regions are constructed here based on a generalized least squares test statistic. The methodology described is computationally inexpensive and broadly applicable to maximum likelihood distances. Assuming the model used to construct the distances is correct, the coverage probabilities are correct with large numbers of sites.     

Model misspecification and probabilistic tests of topology: evidence from empirical data sets

Probabilistic tests of topology offer a powerful means of evaluating competing phylogenetic hypotheses. The performance of the nonparametric Shimodaira-Hasegawa (SH) test, the parametric Swofford-Olsen-Waddell-Hillis (SOWH) test, and Bayesian posterior probabilities were explored for five data sets for which all the phylogenetic relationships are known with a very high degree of certainty. These results are consistent with previous simulation studies that have indicated a tendency for the SOWH test to be prone to generating Type 1 errors because of model misspecification coupled with branch length heterogeneity. These results also suggest that the SOWH test may accord overconfidence in the true topology when the null hypothesis is in fact correct. In contrast, the SH test was observed to be much more conservative, even under high substitution rates and branch length heterogeneity. For some of those data sets where the SOWH test proved misleading, the Bayesian posterior probabilities were also misleading. The results of all tests were strongly influenced by the exact substitution model assumptions. Simple models, especially those that assume rate homogeneity among sites, had a higher Type 1 error rate and were more likely to generate misleading posterior probabilities. For some of these data sets, the commonly used substitution models appear to be inadequate for estimating appropriate levels of uncertainty with the SOWH test and Bayesian methods. Reasons for the differences in statistical power between the two maximum likelihood tests are discussed and are contrasted with the Bayesian approach.     

CONFIDENCE LIMITS ON PHYLOGENIES: THE BOOTSTRAP REVISITED

Abstract— The bootstrap, a non-parametric statistical analysis, can be used to assess confidence limits on phylogcnics. The method most widely used tests the monophyly of individual clades. This paper proposes additional applications of the bootstrap which provide useful information about phylogeny even when many clades are found not to be supported with confidence (as often occurs in practice). In such cases it is still possible to place a constraint on the phylogenetic position of taxa by examining the relative size of the smallest monophyletic groups that contain them. In addition, the taxonomic composition of these larger clades can be determined, as well as the relative likelihood of their occurrence. The distinction between hypotheses about membership in particular clades and hypotheses about entire topologies is also discussed. To investigate the latter, the bootstrap is used to estimate the sampling distribution of tree similarity indices. All methods are illustrated by reference to a large data set on the angiosperm family Asteraccae, selected from the literature.     

An approximately unbiased test of phylogenetic tree selection

An approximately unbiased (AU) test that uses a newly devised multiscale bootstrap technique was developed for general hypothesis testing of regions in an attempt to reduce test bias. It was applied to maximum-likelihood tree selection for obtaining the confidence set of trees. The AU test is based on the theory of Efron et al. (Proc. Natl. Acad. Sci. USA 93:13429-13434; 1996), but the new method provides higher-order accuracy yet simpler implementation. The AU test, like the Shimodaira-Hasegawa (SH) test, adjusts the selection bias overlooked in the standard use of the bootstrap probability and Kishino-Hasegawa tests. The selection bias comes from comparing many trees at the same time and often leads to overconfidence in the wrong trees. The SH test, though safe to use, may exhibit another type of bias such that it appears conservative. Here I show that the AU test is less biased than other methods in typical cases of tree selection. These points are illustrated in a simulation study as well as in the analysis of mammalian mitochondrial protein sequences. The theoretical argument provides a simple formula that covers the bootstrap probability test, the Kishino-Hasegawa test, the AU test, and the Zharkikh-Li test. A practical suggestion is provided as to which test should be used under particular circumstances.     

On the distributions of bootstrap support and posterior distributions for a star tree

Several authors have recently noted that when data are generated from a star topology, posterior probabilities can often be very large, even with arbitrarily large sequence lengths. This is counter to intuition, which suggests convergence to the limit of equal probability for each topology. Here the limiting distributions of bootstrap support and posterior probabilities are obtained for a four-taxon star tree. Theoretical results are given, providing confirmation that this counterintuitive phenomenon holds for both posterior probabilities and bootstrap support. For large samples the limiting results for posterior probabilities are the same regardless of the prior. With equal-length terminal edges, the limiting distribution is similar but not the same across different choices for the lengths of the edges. In contrast to previous results, the case of unequal lengths of terminal edges is considered. With two long edges, the posterior probability of the tree with long edges together tends to be much larger. Using the neighbor-joining algorithm, with equal edge lengths, the distribution of bootstrap support tends to be qualitatively comparable to posterior probabilities. As with posterior probabilities, when two of the edges are long, bootstrap support for the tree with long branches together tends to be large. The bias is less pronounced, however, as the distribution of bootstrap support gets close to uniform for this tree, whereas posterior probabilities are much more likely to be large. Our findings for maximum likelihood estimation are based entirely on simulation and in contrast suggest that bootstrap support tends to be fairly constant across edge-length choices.     

Determining evolutionary distances from highly diverged nucleic acid sequences: Operator metrics

Summary Operator metrics are explicity designed to measure evolutionary distances from nucleic acid sequences when substitution rates differ greatly among the organisms being compared, or when substitutions have been extensive. Unlike lengths calculated by the distance matrix and parsimony methods, in which substitutions in one branch of a tree can alter the measured length of another branch, lengths determined by operator metrics are not affected by substitutions outside the branch.In the method, lengths (operator metrics) corresponding to each of the branches of an unrooted tree are calculated. The metric length of a branch reconstructs the number of (transversion) differences between sequences at a tip and a node (or between nodes) of a tree. The theory is general and is fundamentally independent of differences in substitution rates among the organisms being compared. Mathematically, the independence has been obtained becuase the metrics are eigen vectors of fundamental equations which describe the evolution of all unrooted trees.Even under conditions when both the distance matrix method or a simple parsimony length method are show to indicate lengths than are an order of magnitude too large or too small, the operator metrics are accurate. Examples, using data calculated with evolutionary rates and branchings designed to confuse the measurement of branch lengths and to camouflage the topology of the true tree, demonstrate the validity of operator metrics. The method is robust. Operator metric distances are easy to calculated, can be extended to any number of taxa, and provide a statistical estimate of their variances.The utility of the method is demonstrated by using it to analyze the origins and evolutionary of chloroplasts, mitochondria, and eubacteria.     

Comparison of Bayesian and maximum likelihood bootstrap measures of phylogenetic reliability

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.     

The relative contribution of band number to phylogenetic accuracy in AFLP data sets

We examined the effect of increasing the number of sampled amplified fragment length polymorphism (AFLP) bands to reconstruct an accurate and well-supported AFLP-based phylogeny. In silico AFLP was performed using simulated DNA sequences evolving along balanced and unbalanced model trees with recent, uniform and ancient radiations and average branch lengths (from the most internal node to the tip) ranging from 0.02 to 0.05 substitutions per site. Trees were estimated by minimum evolution (ME) and maximum parsimony (MP) methods from both DNA sequences and virtual AFLP fingerprints. The comparison of the true tree with the estimated AFLP trees suggests that moderate numbers of AFLP bands are necessary to recover the correct topology with high bootstrap support values (i.e. >70%). Fewer numbers of bands are necessary for shorter tree lengths and for balanced than for unbalanced tree topologies. However, branch length estimation was rather unreliable and did not improve substantially after a certain number of bands were sampled. These results hold for different levels of genome coverage and number of taxa analysed. In silico AFLP using bacterial genomic DNA sequences recovered a well-supported tree topology that mirrored an empirical phylogeny based on a set of 31 orthologous gene sequences when as few as 263 AFLP bands were scored. These results suggest that AFLPs may be an efficient alternative to traditional DNA sequencing for accurate topology reconstruction of shallow trees when not very short ancestral branches exist.     

When Being “Most Likely” Is Not Enough: Examining the Performance of Three Uses of the Parametric Bootstrap in Phylogenetics

Abstract I show that three parametric-bootstrap (PB) applications that have been proposed for phylogenetic analysis, can be misleading as currently implemented. First, I show that simulating a topology estimated from preliminary data in order to determine the sequence length that should allow the best tree obtained from more extensive data to be correct with a desired probability, delivers an accurate estimate of this length only in topological situations in which most preliminary trees are expected to be both correct and statistically significant, i.e. when no further analysis would be needed. Otherwise, one obtains strong underestimates of the length or similarly biased values for incorrect trees. Second, I show that PB-based topology tests that use as null hypothesis the most likely tree congruent with a pre-specified topological relationship alternative to the unconstrained most likely tree, and simulate this tree for P value estimation, produce excessive type I error (from 50% to 600% and higher) when they are applied to null data generated by star-shaped or dichotomous four-taxon topologies. Simulating the most likely star topology for P value estimation results instead in correct type-I-error production even when the null data are generated by a dichotomous topology. This is a strong indication that the star topology is the correct default null hypothesis for phylogenies. Third, I show that PB-estimated confidence intervals (CIs) for the length of a tree branch are generally accurate, although in some situations they can be strongly over- or under-estimated relative to the “true” CI. Attempts to identify a biased CI through a further round of simulations were unsuccessful. Tracing the origin and propagation of parameter estimate error through the CI estimation exercise, showed that the sparseness of site-patterns which are crucial to the estimation of pivotal parameters, can allow homoplasy to bias these estimates and ultimately the PB-based CI estimation. Concluding, I stress that statistical techniques that simulate models estimated from limited data need to be carefully calibrated, and I defend the point that pattern-sparseness assessment will be the next frontier in the statistical analysis of phylogenies, an effort that will require taking advantage of the merits of black-box maximum-likelihood approaches and of insights from intuitive, site-pattern-oriented approaches like parsimony.     

New Aspects on Lanosterol 14α-Demethylase and Cytochrome P450 Evolution: Lanosterol/Cycloartenol Diversification and Lateral Transfer

Sterol 14-demethylase (CYP51) is a member of the cytochrome P450 superfamily, widely found in animals, fungi, and plants but present in few prokaryotic groups. CYP51 is currently believed to be the ancestral cytochrome P450 that has been transferred from prokaryotes to eukaryotic kingdoms. We propose an alternate view of CYP51 evolution that has an impact on understanding the evolution of the entire CYP superfamily. Two hundred forty-nine bacterial and four archaeal CYP sequences have been aligned and a bacterial CYP tree designed, showing a separation of two branches. Prokaryotic CYP51s cluster to the minor branch, together with other eukaryote-like CYPs. Mycobacterial and methylococcal CYP51s cluster together (100% bootstrap probability), while Streptomyces CYP51 remains on a distant branch. A CYP51 phylogenetic tree has been constructed from 44 sequences resulting in a ((plant, bacteria),(animal, fungi)) topology (100% bootstrap probability). This is in accordance with the lanosterol/cycloartenol diversification of sterol biosynthesis. The lanosterol branch (nonphotosynthetic lineage) follows the previously proposed topology of animal and fungal orthologues (100% bootstrap probability), while plant and D. discoideum CYP51s belong to the cycloartenol branch (photosynthetic lineage), all in accordance with biochemical data. Bacterial CYP51s cluster within the cycloartenol branch (69% bootstrap probability), which is indicative of a lateral gene transfer of a plant CYP51 to the methylococcal/mycobacterial progenitor, suggesting further that bacterial CYP51s are not the oldest CYP genes. Lateral gene transfer is likely far more important than hitherto thought in the development of the diversified CYP superfamily. Consequently, bacterial CYPs may represent a mixture of genes with prokaryotic and eukaryotic origin.     

Type I error and the power of the s-test: old lessons from a new,analytically justified statistical test for phylogenies

We present a new procedure for assessing the statistical significance of the most likely unrooted dichotomous topology inferrable from four DNA sequences. The procedure calculates directly a P-value for the support given to this topology by the informative sites congruent with it, assuming the most likely star topology as the null hypothesis. Informative sites are crucial in the determination of the maximum likelihood dichotomous topology and are therefore an obvious target for a statistical test of phylogenies. Our P-value is the probability of producing through parallel substitutions on the branches of the star topology at least as much support as that given to the maximum likelihood dichotomous topology by the aforementioned informative sites, for any of the three possible dichotomous topologies. The degree of statistical significance is simply the complement of this P-value. Ours is therefore an a posteriori testing approach, in which no dichotomous topology is specified in advance. We implement the test for the case in which all sites behave identically and the substitution model has a single parameter. Under these conditions, the P-value can be easily calculated on the basis of the probabilities of change on the branches of the most likely star topology, because under these assumptions, each site can become informative independently from every other site; accordingly, the total number of informative sites of each kind is binomially distributed. We explore the test's type I error by applying it to data produced in star topologies having all branches equally long, or having two short and two long branches, and various degrees of homoplasy. The test is conservative but we demonstrate, by means of a discreteness correction and progressively assumption-free calculations of the P-values, that (1) the conservativeness is mostly due to the discrete nature of informative sites and (2) the P-values calculated empirically are moreover mostly quite accurate in absolute terms. Applying the test to data produced in dichotomous topologies with increasing internal branch length shows that, despite the test's "conservativeness," its power is much higher than that of the bootstrap, especially when the relevant informative sites are few.     

Using Confidence Set Heuristics During Topology Search Improves the Robustness of Phylogenetic Inference

We examine the impact of likelihood surface characteristics on phylogenetic inference. Amino acid data sets simulated from topologies with branch length features chosen to represent varying degrees of difficulty for likelihood maximization are analyzed. We present situations where the tree found to achieve the global maximum in likelihood is often not equal to the true tree. We use the program covSEARCH to demonstrate how the use of adaptively sized pools of candidate trees that are updated using confidence tests results in solution sets that are highly likely to contain the true tree. This approach requires more computation than traditional maximum likelihood methods, hence covSEARCH is best suited to small to medium-sized alignments or large alignments with some constrained nodes. The majority rule consensus tree computed from the confidence sets also proves to be different from the generating topology. Although low phylogenetic signal in the input alignment can result in large confidence sets of trees, some biological information can still be obtained based on nodes that exhibit high support within the confidence set. Two real data examples are analyzed: mammal mitochondrial proteins and a small tubulin alignment. We conclude that the technique of confidence set optimization can significantly improve the robustness of phylogenetic inference at a reasonable computational cost. Additionally, when either very short internal branches or very long terminal branches are present, confident resolution of specific bipartitions or subtrees, rather than whole-tree phylogenies, may be the most realistic goal for phylogenetic methods. [Reviewing Editor: Dr. Nicolas Galtier]     

Survey of branch support methods demonstrates accuracy, power, and robustness of fast likelihood-based approximation schemes

Phylogenetic inference and evaluating support for inferred relationships is at the core of many studies testing evolutionary hypotheses. Despite the popularity of nonparametric bootstrap frequencies and Bayesian posterior probabilities, the interpretation of these measures of tree branch support remains a source of discussion. Furthermore, both methods are computationally expensive and become prohibitive for large data sets. Recent fast approximate likelihood-based measures of branch supports (approximate likelihood ratio test [aLRT] and Shimodaira-Hasegawa [SH]-aLRT) provide a compelling alternative to these slower conventional methods, offering not only speed advantages but also excellent levels of accuracy and power. Here we propose an additional method: a Bayesian-like transformation of aLRT (aBayes). Considering both probabilistic and frequentist frameworks, we compare the performance of the three fast likelihood-based methods with the standard bootstrap (SBS), the Bayesian approach, and the recently introduced rapid bootstrap. Our simulations and real data analyses show that with moderate model violations, all tests are sufficiently accurate, but aLRT and aBayes offer the highest statistical power and are very fast. With severe model violations aLRT, aBayes and Bayesian posteriors can produce elevated false-positive rates. With data sets for which such violation can be detected, we recommend using SH-aLRT, the nonparametric version of aLRT based on a procedure similar to the Shimodaira-Hasegawa tree selection. In general, the SBS seems to be excessively conservative and is much slower than our approximate likelihood-based methods.     

Pitfalls of heterogeneous processes for phylogenetic reconstruction

Different genes often have different phylogenetic histories. Even within regions having the same phylogenetic history, the mutation rates often vary. We investigate the prospects of phylogenetic reconstruction when all the characters are generated from the same tree topology, but the branch lengths vary (with possibly different tree shapes). Furthering work of Kolaczkowski and Thornton (2004, Nature 431: 980-984) and Chang (1996, Math. Biosci. 134: 189-216), we show examples where maximum likelihood (under a homogeneous model) is an inconsistent estimator of the tree. We then explore the prospects of phylogenetic inference under a heterogeneous model. In some models, there are examples where phylogenetic inference under any method is impossible - despite the fact that there is a common tree topology. In particular, there are nonidentifiable mixture distributions, i.e., multiple topologies generate identical mixture distributions. We address which evolutionary models have nonidentifiable mixture distributions and prove that the following duality theorem holds for most DNA substitution models. The model has either: (i) nonidentifiability - two different tree topologies can produce identical mixture distributions, and hence distinguishing between the two topologies is impossible; or (ii) linear tests - there exist linear tests which identify the common tree topology for character data generated by a mixture distribution. The theorem holds for models whose transition matrices can be parameterized by open sets, which includes most of the popular models, such as Tamura-Nei and Kimura's 2-parameter model. The duality theorem relies on our notion of linear tests, which are related to Lake's linear invariants.     

Weighted bootstrapping: a correction method for assessing the robustness of phylogenetic trees

Background  

Non-parametric bootstrapping is a widely-used statistical procedure for assessing confidence of model parameters based on the empirical distribution of the observed data [1] and, as such, it has become a common method for assessing tree confidence in phylogenetics [2]. Traditional non-parametric bootstrapping does not weigh each tree inferred from resampled (i.e., pseudo-replicated) sequences. Hence, the quality of these trees is not taken into account when computing bootstrap scores associated with the clades of the original phylogeny. As a consequence, traditionally, the trees with different bootstrap support or those providing a different fit to the corresponding pseudo-replicated sequences (the fit quality can be expressed through the LS, ML or parsimony score) contribute in the same way to the computation of the bootstrap support of the original phylogeny.