Accuracy of estimated phylogenetic trees from molecular data

Summary The accuracies and efficiencies of four different methods for constructing phylogenetic trees from molecular data were examined by using computer simulation. The methods examined are UPGMA, Fitch and Margoliash's (1967) (F/M) method, Farris' (1972) method, and the modified Farris method (Tateno, Nei, and Tajima, this paper). In the computer simulation, eight OTUs (32 OTUs in one case) were assumed to evolve according to a given model tree, and the evolutionary change of a sequence of 300 nucleotides was followed. The nucleotide substitution in this sequence was assumed to occur following the Poisson distribution, negative binomial distribution or a model of temporally varying rate. Estimates of nucleotide substitutions (genetic distances) were then computed for all pairs of the nucleotide sequences that were generated at the end of the evolution considered, and from these estimates a phylogenetic tree was reconstructed and compared with the true model tree. The results of this comparison indicate that when the coefficient of variation of branch length is large the Farris and modified Farris methods tend to be better than UPGMA and the F/M method for obtaining a good topology. For estimating the number of nucleotide substitutions for each branch of the tree, however, the modified Farris method shows a better performance than the Farris method. When the coefficient of variation of branch length is small, however, UPGMA shows the best performance among the four methods examined. Nevertheless, any tree-making method is likely to make errors in obtaining the correct topology with a high probability, unless all branch lengths of the true tree are sufficiently long. It is also shown that the agreement between patristic and observed genetic distances is not a good indicator of the goodness of the tree obtained.     

Genetic distances and ordination: the land snail Helix aspersa in north Africa as a test case

We examined the efficiencies of ordination methods in the treatment of gene frequency data at intraspecific level, using metric and nonmetric distance measures (Nei's and Rogers' genetic distances, chi 2 distance). We assessed initial processes responsible for the geographical distribution of the Mediterranean land snail Helix aspersa. Seventeen enzyme loci from 30 North African snail populations were considered in the present analysis. Five combinations of distance/multivariate analysis were compared: correspondence analysis (CA), nonmetric multidimensional scaling (NMDS) on Nei's, Rogers', and chi 2 distances, and principal coordinates analysis on Rogers' distances. Configuration of the objects resulting from ordination was projected onto three-dimensional graphics with the minimum spanning tree or the relative neighborhood graph superimposed. Pre- and postordination or clustering distance matrices were compared by means of correlation methods. As expected, all combinations led to a clear west versus east pattern of variation. However, the intraregional relationships and degree of connectivity between pairs of operational taxonomic units were not necessarily constant from one method to another. Ordination methods when applied with Nei's and Rogers' distances provided the best fit, with original distances (r = 0.98) compared with UPGMA clustering (r approximately 0.75). The Nei/NMDS combination seems to be a good compromise (distortion index dt = 10%) between Rogers/NMDS, which produces a more confusing pattern of differentiation (dt = 24%), and chi 2/CA, which tends to distort large distances (dt = 31%). NMDS obviously provides a powerful method to summarize relationships between populations, when neither hierarchical structure nor phylogenetic inference are required. These findings led the discussion on the good performance of NMDS, the appropriate distances to be used, and the potential application of this method to other types of allelic data (such as microsatellite loci) or data on nucleotide sequences of genes.     

The neighbor-joining method: a new method for reconstructing phylogenetic trees

A new method called the neighbor-joining method is proposed for reconstructing phylogenetic trees from evolutionary distance data. The principle of this method is to find pairs of operational taxonomic units (OTUs [= neighbors]) that minimize the total branch length at each stage of clustering of OTUs starting with a starlike tree. The branch lengths as well as the topology of a parsimonious tree can quickly be obtained by using this method. Using computer simulation, we studied the efficiency of this method in obtaining the correct unrooted tree in comparison with that of five other tree-making methods: the unweighted pair group method of analysis, Farris's method, Sattath and Tversky's method, Li's method, and Tateno et al.'s modified Farris method. The new, neighbor-joining method and Sattath and Tversky's method are shown to be generally better than the other methods.      

Accuracy of phylogenetic trees estimated from DNA sequence data

The relative merits of four different tree-making methods in obtaining the correct topology were studied by using computer simulation. The methods studied were the unweighted pair-group method with arithmetic mean (UPGMA), Fitch and Margoliash's (FM) method, thd distance Wagner (DW) method, and Tateno et al.'s modified Farris (MF) method. An ancestral DNA sequence was assumed to evolve into eight sequences following a given model tree. Both constant and varying rates of nucleotide substitution were considered. Once the DNA sequences for the eight extant species were obtained, phylogenetic trees were constructed by using corrected (d) and uncorrected (p) nucleotide substitutions per site. The topologies of the trees obtained were then compared with that of the model tree. The results obtained can be summarized as follows: (1) The probability of obtaining the correct rooted or unrooted tree is low unless a large number of nucleotide differences exists between different sequences. (2) When the number of nucleotide substitutions per sequence is small or moderately large, the FM, DW, and MF methods show a better performance than UPGMA in recovering the correct topology. The former group of methods is particularly good for obtaining the correct unrooted tree. (3) When the number of substitutions per sequence is large, UPGMA is at least as good as the other methods, particularly for obtaining the correct rooted tree. (4) When the rate of nucleotide substitution varies with evolutionary lineage, the FM, DW, and MF methods show a better performance in obtaining the correct topology than UPGMA, except when a rooted tree is to be produced from data with a large number of nucleotide substitutions per sequence.(ABSTRACT TRUNCATED AT 250 WORDS)      

Statistical properties of molecular tree construction methods under the neutral mutation model

Summary The statistical properties of three molecular tree construction methods—the unweighted pair-group arithmetic average clustering (UPG), Farris, and modified Farris methods—are examined under the neutral mutation model of evolution. The methods are compared for accuracy in construction of the topology and estimation of the branch lengths, using statistics of these two aspects. The distribution of the statistic concerning topological construction is shown to be as important as its mean and variance for the comparison.Of the three methods, the UPG method constructs the tree topology with the least variation. The modified Farris method, however, gives the best performance when the two aspects are considered simultaneously. It is also shown that a topology based on two genes is much more accurate than that based on one gene.There is a tendency to accept published molecular trees, but uncritical acceptance may lead one to spurious conclusions. It should always be kept in mind that a tree is a statistical result that is affected strongly by the stochastic error of nucleotide substitution and the error intrinsic to the tree construction method itself.     

Relative efficiencies of the maximum-likelihood, neighbor-joining, and maximum-parsimony methods when substitution rate varies with site

The relative efficiencies of the maximum-likelihood (ML), neighbor- joining (NJ), and maximum-parsimony (MP) methods in obtaining the correct topology and in estimating the branch lengths for the case of four DNA sequences were studied by computer simulation, under the assumption either that there is variation in substitution rate among different nucleotide sites or that there is no variation. For the NJ method, several different distance measures (Jukes-Cantor, Kimura two- parameter, and gamma distances) were used, whereas for the ML method three different transition/transversion ratios (R) were used. For the MP method, both the standard unweighted parsimony and the dynamically weighted parsimony methods were used. The results obtained are as follows: (1) When the R value is high, dynamically weighted parsimony is more efficient than unweighted parsimony in obtaining the correct topology. (2) However, both weighted and unweighted parsimony methods are generally less efficient than the NJ and ML methods even in the case where the MP method gives a consistent tree. (3) When all the assumptions of the ML method are satisfied, this method is slightly more efficient than the NJ method. However, when the assumptions are not satisfied, the NJ method with gamma distances is slightly better in obtaining the correct topology than is the ML method. In general, the two methods show more or less the same performance. The NJ method may give a correct topology even when the distance measures used are not unbiased estimators of nucleotide substitutions. (4) Branch length estimates of a tree with the correct topology are affected more easily than topology by violation of the assumptions of the mathematical model used, for both the ML and the NJ methods. Under certain conditions, branch lengths are seriously overestimated or underestimated. The MP method often gives serious underestimates for certain branches. (5) Distance measures that generate the correct topology, with high probability, do not necessarily give good estimates of branch lengths. (6) The likelihood-ratio test and the confidence-limit test, in Felsenstein's DNAML, for examining the statistical of branch length estimates are quite sensitive to violation of the assumptions and are generally too liberal to be used for actual data. Rzhetsky and Nei's branch length test is less sensitive to violation of the assumptions than is Felsenstein's test. (7) When the extent of sequence divergence is < or = 5% and when > or = 1,000 nucleotides are used, all three methods show essentially the same efficiency in obtaining the correct topology and in estimating branch lengths.(ABSTRACT TRUNCATED AT 400 WORDS)      

Nonuniformity of nucleotide substitution rates in molecular evolution: Computer simulation and analysis of 5S ribosomal RNA sequences

Summary The effects of temporal (among different branches of a phylogeny) and spatial (among different nucleotide sites within a gene) nonuniformities of nucleotide substitution rates on the construction of phylogenetic trees from nucleotide sequences are addressed. Spatial nonuniformity may be estimated by using Shannon's (1948) entropy formula to measure the Relative Nucleotide Variability (RNV) at each nucleotide site in an aligned set of sequences; this is demonstrated by a comparative analysis of 5S rRNAs. New methods of constructing phylogenetic trees are proposed that augment the Unweighted Pair-Group Using Arithmetic Averages (UPGMA) algorithm by estimating and compensating for both spatial and temporal nonuniformity in substitution rates. These methods are evaluated by computer simulations of 5S rRNA evolution that include both kinds of nonuniformities. It was found that the proposed Reference Ratio Method improved both the ability to reconstruct the correct topology of a tree and also the estimation of branch lengths as compared to UPGMA. A previous method (Farris et al. 1970; Klotz et al. 1979; Li 1981) was found to be less successful in reconstructing topologies when there is high probability of multiple mutations at some sites. Phylogenetic analyses of 5S rRNA sequences support the endosymbiotic origins of both chloroplasts and mitochondria, even though the latter exhibit an accelerated rate of nucleotide substitution. Phylogenetic trees also reveal an adaptive radiation within the eubacteria and another within the eukaryotes for the origins of most major phyla within each group during the Precambrian era.     

Phylogenetic analysis and intraspecific variation: performance of parsimony,likelihood, and distance methods

Intraspecific variation is abundant in all types of systematic characters but is rarely addressed in simulation studies of phylogenetic method performance. We compared the accuracy of 15 phylogenetic methods using simulations to (1) determine the most accurate method(s) for analyzing polymorphic data (under simplified conditions) and (2) test if generalizations about the performance of phylogenetic methods based on previous simulations of fixed (nonpolymorphic) characters are robust to a very different evolutionary model that explicitly includes intraspecific variation. Simulated data sets consisted of allele frequencies that evolved by genetic drift. The phylogenetic methods included eight parsimony coding methods, continuous maximum likelihood, and three distance methods (UPGMA, neighbor joining, and Fitch-Margoliash) applied to two genetic distance measures (Nei's and the modified Cavalli-Sforza and Edwards chord distance). Two sets of simulations were performed. The first examined the effects of different branch lengths, sample sizes (individuals sampled per species), numbers of characters, and numbers of alleles per locus in the eight-taxon case. The second examined more extensively the effects of branch length in the four-taxon, two-allele case. Overall, the most accurate methods were likelihood, the additive distance methods (neighbor joining and Fitch-Margoliash), and the frequency parsimony method. Despite the use of a very different evolutionary model in the present article, many of the results are similar to those from simulations of fixed characters. Similarities include the presence of the "Felsenstein zone," where methods often fail, which suggests that long-branch attraction may occur among closely related species through genetic drift. Differences between the results of fixed and polymorphic data simulations include the following: (1) UPGMA is as accurate or more accurate than nonfrequency parsimony methods across nearly all combinations of branch lengths, and (2) likelihood and the additive distance methods are not positively misled under any combination of branch lengths tested (even when the assumptions of the methods are violated and few characters are sampled). We found that sample size is an important determinant of accuracy and affects the relative success of methods (i.e., distance and likelihood methods outperform parsimony at small sample sizes). Attempts to generalize about the behavior of phylogenetic methods should consider the extreme examples offered by fixed-mutation models of DNA sequence data and genetic-drift models of allele frequencies.     

Nucleotide substitution models and estimation of phylogeny

The nucleotide substitution matrix inferred from avian data sets using cytochrome b differs considerably from the models commonly used in phylogenetic analyses. To analyze the possible effects of this particular pattern of change in phylogeny estimation we performed a computer simulation in which we started with a real sequence and used the inferred model of change to produce a tree of 10 species. Maximum parsimony (MP), maximum likelihood (ML), and various distance methods were then used to recover the topology and the branch lengths. We used two kinds of data with varying levels of variation. In addition, we tested with the removal of third positions and different weighting schemes. At low levels of variation, MP was outstanding in recovering the topology (90% correct), while unweighted pair-group method, arithmetic average (UPGMA), regardless of distances used, was poor (40%). At the higher level, most methods had a chance of around 40%-58% of finding the true tree. However, in most cases, the trees found were only slightly wrong, with only one or a few branches misplaced. On the other hand, the use of a "wrong" model had serious effects on the estimation of branch lengths (distances). Although precision was high, accuracy was poor with most methods, giving branch lengths that were biased downward. When seeded with the true distance matrix, Fitch and NJ always found the true tree, while UPGMA frequently failed to do so. The effect of removing third positions was dramatic at low levels of variation, because only one MP program was able to find a true tree at all, albeit rarely, while none of the others ever did so. At higher levels, the situation was better, but still much worse than with the whole data set.     

DNA似近距离及进化时间的估算

在似近分析和Ｎｅｉ氏遗传距离的基础上,给出了ＤＮＡ似近距离计算公式,并以ＤＮＡ似近距离估算类群间的分歧时间（进化时间）,应用１０种限制内切酶对猕猴属（ｇｅｎｕｓ Ｍａｃａｃａ）内５个种ｍｔＤＮＡ的切点数据计算了这５个种的ＤＮＡ似近距离和进化时间,比较由ＤＮＡ似近距,遗传距离构建的歧化树和Ｆｏｏｄｅｎ及Ｄｅｌｓｏｎ的形态歧化树表明,除遗传距离的歧化树外,其它三种歧化树都有一个共同点,就是熊猴（Ｍ．ａ     

The K tree score: quantification of differences in the relative branch length and topology of phylogenetic trees

SUMMARY: We introduce a new phylogenetic comparison method that measures overall differences in the relative branch length and topology of two phylogenetic trees. To do this, the algorithm first scales one of the trees to have a global divergence as similar as possible to the other tree. Then, the branch length distance, which takes differences in topology and branch lengths into account, is applied to the two trees. We thus obtain the minimum branch length distance or K tree score. Two trees with very different relative branch lengths get a high K score whereas two trees that follow a similar among-lineage rate variation get a low score, regardless of the overall rates in both trees. There are several applications of the K tree score, two of which are explained here in more detail. First, this score allows the evaluation of the performance of phylogenetic algorithms, not only with respect to their topological accuracy, but also with respect to the reproduction of a given branch length variation. In a second example, we show how the K score allows the selection of orthologous genes by choosing those that better follow the overall shape of a given reference tree. AVAILABILITY: http://molevol.ibmb.csic.es/Ktreedist.html     

Relative efficiencies of the maximum-parsimony and distance-matrix methods of phylogeny construction for restriction data.

The relative efficiencies of the maximum-parsimony (MP), UPGMA, and neighbor-joining (NJ) methods in obtaining the correct tree (topology) for restriction-site and restriction-fragment data were studied by computer simulation. In this simulation, six DNA sequences of 16,000 nucleotides were assumed to evolve following a given model tree. The recognition sequences of 20 different six-base restriction enzymes were used to identify the restriction sites of the DNA sequences generated. The restriction-site data and restriction-fragment data thus obtained were used to reconstruct a phylogenetic tree, and the tree obtained was compared with the model tree. This process was repeated 300 times. The results obtained indicate that when the rate of nucleotide substitution is constant the probability of obtaining the correct tree (Pc) is generally higher in the NJ method than in the MP method. However, if we use the average topological deviation from the model tree (dT) as the criterion of comparison, the NJ and MP methods are nearly equally efficient. When the rate of nucleotide substitution varies with evolutionary lineage, the NJ method is better than the MP method, whether Pc or dT is used as the criterion of comparison. With 500 nucleotides and when the number of nucleotide substitutions per site was very small, restriction-site data were, contrary to our expectation, more useful than sequence data. Restriction-fragment data were less useful than restriction-site data, except when the sequence divergence was very small. UPGMA seems to be useful only when the rate of nucleotide substitution is constant and sequence divergence is high.     

EVALUATION OF THE RESTRICTED MAXIMUM-LIKELIHOOD METHOD FOR ESTIMATING PHYLOGENETIC TREES USING SIMULATED ALLELE-FREQUENCY DATA

Comparisons are made of the accuracy of the restricted maximum-likelihood, Wagner parsimony, and UPGMA (unweighted pair-group method using arithmetic averages) clustering methods to estimate phylogenetic trees. Data matrices were generated by constructing simulated stochastic evolution in a multidimensional gene-frequency space using a simple genetic-drift model (Brownian-motion, random-walk) with constant rates of divergence in all lineages. Ten differentphylogenetic tree topologies of 20 operational taxonomic units (OTU's), representing a range of tree shapes, were used. Felsenstein's restricted maximum-likelihood method, Wagner parsimony, and UPGMA clustering were used to construct trees from the resulting data matrices. The computations for the restricted maximum-likelihood method were performed on a Cray-1 supercomputer since the required calculations (especially when optimized for the vector hardware) are performed substantially faster than on more conventional computing systems. The overall level of accuracy of tree reconstruction depends on the topology of the true phylogenetic tree. The UPGMA clustering method, especially when genetic-distance coefficients are used, gives the most accurate estimates of the true phylogeny (for our model with constant evolutionary rates). For large numbers of loci, all methods give similar results, but trends in the results imply that the restricted maximum-likelihood method would produce the most accurate trees if sample sizes were large enough.     

iGLASS: an improvement to the GLASS method for estimating species trees from gene trees

Several methods have been designed to infer species trees from gene trees while taking into account gene tree/species tree discordance. Although some of these methods provide consistent species tree topology estimates under a standard model, most either do not estimate branch lengths or are computationally slow. An exception, the GLASS method of Mossel and Roch, is consistent for the species tree topology, estimates branch lengths, and is computationally fast. However, GLASS systematically overestimates divergence times, leading to biased estimates of species tree branch lengths. By assuming a multispecies coalescent model in which multiple lineages are sampled from each of two taxa at L independent loci, we derive the distribution of the waiting time until the first interspecific coalescence occurs between the two taxa, considering all loci and measuring from the divergence time. We then use the mean of this distribution to derive a correction to the GLASS estimator of pairwise divergence times. We show that our improved estimator, which we call iGLASS, consistently estimates the divergence time between a pair of taxa as the number of loci approaches infinity, and that it is an unbiased estimator of divergence times when one lineage is sampled per taxon. We also show that many commonly used clustering methods can be combined with the iGLASS estimator of pairwise divergence times to produce a consistent estimator of the species tree topology. Through simulations, we show that iGLASS can greatly reduce the bias and mean squared error in obtaining estimates of divergence times in a species tree.     

微卫星DNA标记探讨镜鲤的种群结构与遗传变异

采用30个微卫星分子标记, 对5个镜鲤群体的观测杂合度(^H_o)、期望杂合度(^H_e)、多态信息含量(PIC)和有效等位基因数(^A_e)等进行了遗传检测, 根据基因频率计算遗传相似系数和Nei氏标准遗传距离, 以c2检验估计Hardy-Weinberg平衡, 以近交系数(FST)和基因流(Nm)分析群体的遗传分化。同时, 使用PHYLIP3.63软件绘制基于Nei氏标准遗传距离的UPGMA聚类图, 并进行bootstrap自举检验验证进化树的可靠性。在德国镜鲤选育系(Scattered Cyprinus carpio L.)和来自4个不同养殖场(松浦、东岗、奉城和辽中)的德国镜鲤群体中共检测到7 083个扩增片段, 长度在102 ~ 446 bp之间, 在群体内扩增出等位基因1~16个不等, 共计356个等位基因。结果表明: (1)5个群体检测的有效等位基因数在1.07~12.30个不等, 平均多态信息含量为0.74、0.74、0.69、0.75和0.75, 无偏期望杂合度的平均值为0.74、0.78、0.70、0.76和0.78, 说明这几个群体属于高度多态, 遗传多样性水平较高。(2)群体间相似系数在0.52以上, 相似性较高。聚类分析显示, 东岗、奉城和辽中3个养殖场的德国镜鲤群体聚类成一个分支, 而德国镜鲤选育系与松浦群体聚类成另一分支。聚类的先后与它们在地理分布上距离远近有一定的相关性。(3)在与功能基因相关的多个微卫星基因座位上, 扩增产物呈现不同程度的缺失现象, 这些无效等位基因的产生可能与结构基因在育种中受到人工选择的影响较大有关。     

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.     

Construction of phylogenetic trees for proteins and nucleic acids: Empirical evaluation of alternative matrix methods

Summary The methods of Fitch and Margoliash and of Farris for the construction of phylogenetic trees were compared. A phenetic clustering technique - the UPGMA method — was also considered.The three methods were applied to difference matrices obtained from comparison of macromolecules by immunological, DNA hybridization, electrophoretic, and amino acid sequencing techniques. To evaluate the results, we used the goodness-of-fit criterion. In some instances, the F-M and Farris methods gave a comparably good fit of the output to the input data, though in most cases the F-M procedure gave a much better fit. By the fit criterion, the UPGMA procedure was on the average better than the Farris method but not as good as the F-M procedure.On the basis of the results given in this report and the goodness-of-fit criterion, it is suggested that where input data are likely to include overestimates as well as true estimates and underestimates of the actual distances between taxonomic units, the F-M method is the most reasonable to use for constructing phylogenies from distance matrices. Immunological, DNA hybridization, and electrophoretic data fall into this category. By contrast, where it is known that each input datum is indeed either a true estimate or an underestimate of the actual distance between 2 taxonomic units, the Farris procedure appears, on theoretical grounds, to be the matrix method of choice. Amino acid and nucleotide sequence data are in this category.The following abbreviations are used in this work F-M Fitch-Margoliash - UPGMA unweighted pair-group method using arithmetic averages - SD percent standard deviation     

Genetic relationship among Japanese sentinel crabs (Decapoda: Ocypodidae: genus Macrophthalmus)

Seven species (eight populations) of sentinel crabs (genus Macrophthalmus) from the Japan coast and Uca vocans and Ocypode ceratophthalma, were examined electrophoretically for genetic variations in 13 enzymatic and one non-enzymatic protein comprising 17 loci. Most species were highly differentiated from each other (Nei's genetic distance, 0.29-1.63). The least genetic distance was found between M. japonicus and M. banzai, the genetic distinctiveness of the two taxa being supported by three divergent loci with no common allele. The genetic relationships among Macrophthalmus species differed greatly from those inferred from morphological features, with a UPGMA tree suggesting that the sub-genus Macrophthalmus is polyphyletic.     

A two-stage pruning algorithm for likelihood computation for a population tree

We have developed a pruning algorithm for likelihood estimation of a tree of populations. This algorithm enables us to compute the likelihood for large trees. Thus, it gives an efficient way of obtaining the maximum-likelihood estimate (MLE) for a given tree topology. Our method utilizes the differences accumulated by random genetic drift in allele count data from single-nucleotide polymorphisms (SNPs), ignoring the effect of mutation after divergence from the common ancestral population. The computation of the maximum-likelihood tree involves both maximizing likelihood over branch lengths of a given topology and comparing the maximum-likelihood across topologies. Here our focus is the maximization of likelihood over branch lengths of a given topology. The pruning algorithm computes arrays of probabilities at the root of the tree from the data at the tips of the tree; at the root, the arrays determine the likelihood. The arrays consist of probabilities related to the number of coalescences and allele counts for the partially coalesced lineages. Computing these probabilities requires an unusual two-stage algorithm. Our computation is exact and avoids time-consuming Monte Carlo methods. We can also correct for ascertainment bias.     

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.