An examination of the monophyly of morning glory taxa using Bayesian phylogenetic inference

The objective of this study was to obtain a quantitative assessment of the monophyly of morning glory taxa, specifically the genus Ipomoea and the tribe Argyreieae. Previous systematic studies of morning glories intimated the paraphyly of Ipomoea by suggesting that the genera within the tribe Argyreieae are derived from within Ipomoea; however, no quantitative estimates of statistical support were developed to address these questions. We applied a Bayesian analysis to provide quantitative estimates of monophyly in an investigation of morning glory relationships using DNA sequence data. We also explored various approaches for examining convergence of the Markov chain Monte Carlo (MCMC) simulation of the Bayesian analysis by running 18 separate analyses varying in length. We found convergence of the important components of the phylogenetic model (the tree with the maximum posterior probability, branch lengths, the parameter values from the DNA substitution model, and the posterior probabilities for clade support) for these data after one million generations of the MCMC simulations. In the process, we identified a run where the parameter values obtained were often outside the range of values obtained from the other runs, suggesting an aberrant result. In addition, we compared the Bayesian method of phylogenetic analysis to maximum likelihood and maximum parsimony. The results from the Bayesian analysis and the maximum likelihood analysis were similar for topology, branch lengths, and parameters of the DNA substitution model. Topologies also were similar in the comparison between the Bayesian analysis and maximum parsimony, although the posterior probabilities and the bootstrap proportions exhibited some striking differences. In a Bayesian analysis of three data sets (ITS sequences, waxy sequences, and ITS + waxy sequences) no supoort for the monophyly of the genus Ipomoea, or for the tribe Argyreieae, was observed, with the estimate of the probability of the monophyly of these taxa being less than 3.4 x 10(-7).     

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.     

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.     

Does choice in model selection affect maximum likelihood analysis?

In order to have confidence in model-based phylogenetic analysis, the model of nucleotide substitution adopted must be selected in a statistically rigorous manner. Several model-selection methods are applicable to maximum likelihood (ML) analysis, including the hierarchical likelihood-ratio test (hLRT), Akaike information criterion (AIC), Bayesian information criterion (BIC), and decision theory (DT), but their performance relative to empirical data has not been investigated thoroughly. In this study, we use 250 phylogenetic data sets obtained from TreeBASE to examine the effects that choice in model selection has on ML estimation of phylogeny, with an emphasis on optimal topology, bootstrap support, and hypothesis testing. We show that the use of different methods leads to the selection of two or more models for approximately 80% of the data sets and that the AIC typically selects more complex models than alternative approaches. Although ML estimation with different best-fit models results in incongruent tree topologies approximately 50% of the time, these differences are primarily attributable to alternative resolutions of poorly supported nodes. Furthermore, topologies and bootstrap values estimated with ML using alternative statistically supported models are more similar to each other than to topologies and bootstrap values estimated with ML under the Kimura two-parameter (K2P) model or maximum parsimony (MP). In addition, Swofford-Olsen-Waddell-Hillis (SOWH) tests indicate that ML trees estimated with alternative best-fit models are usually not significantly different from each other when evaluated with the same model. However, ML trees estimated with statistically supported models are often significantly suboptimal to ML trees made with the K2P model when both are evaluated with K2P, indicating that not all models perform in an equivalent manner. Nevertheless, the use of alternative statistically supported models generally does not affect tests of monophyletic relationships under either the Shimodaira-Hasegawa (S-H) or SOWH methods. Our results suggest that although choice in model selection has a strong impact on optimal tree topology, it rarely affects evolutionary inferences drawn from the data because differences are mainly confined to poorly supported nodes. Moreover, since ML with alternative best-fit models tends to produce more similar estimates of phylogeny than ML under the K2P model or MP, the use of any statistically based model-selection method is vastly preferable to forgoing the model-selection process altogether.     

Robustness of compound Dirichlet priors for Bayesian inference of branch lengths

We modified the phylogenetic program MrBayes 3.1.2 to incorporate the compound Dirichlet priors for branch lengths proposed recently by Rannala, Zhu, and Yang (2012. Tail paradox, partial identifiability and influential priors in Bayesian branch length inference. Mol. Biol. Evol. 29:325-335.) as a solution to the problem of branch-length overestimation in Bayesian phylogenetic inference. The compound Dirichlet prior specifies a fairly diffuse prior on the tree length (the sum of branch lengths) and uses a Dirichlet distribution to partition the tree length into branch lengths. Six problematic data sets originally analyzed by Brown, Hedtke, Lemmon, and Lemmon (2010. When trees grow too long: investigating the causes of highly inaccurate Bayesian branch-length estimates. Syst. Biol. 59:145-161) are reanalyzed using the modified version of MrBayes to investigate properties of Bayesian branch-length estimation using the new priors. While the default exponential priors for branch lengths produced extremely long trees, the compound Dirichlet priors produced posterior estimates that are much closer to the maximum likelihood estimates. Furthermore, the posterior tree lengths were quite robust to changes in the parameter values in the compound Dirichlet priors, for example, when the prior mean of tree length changed over several orders of magnitude. Our results suggest that the compound Dirichlet priors may be useful for correcting branch-length overestimation in phylogenetic analyses of empirical data sets.     

Success of maximum likelihood phylogeny inference in the four-taxon case

We used simulated data to investigate a number of properties of maximum- likelihood (ML) phylogenetic tree estimation for the case of four taxa. Simulated data were generated under a broad range of conditions, including wide variation in branch lengths, differences in the ratio of transition and transversion substitutions, and the absence of presence of gamma-distributed site-to-site rate variation. Data were analyzed in the ML framework with two different substitution models, and we compared the ability of the two models to reconstruct the correct topology. Although both models were inconsistent for some branch-length combinations in the presence of site-to-site variation, the models were efficient predictors of topology under most simulation conditions. We also examined the performance of the likelihood ratio (LR) test for significant positive interior branch length. This test was found to be misleading under many simulation conditions, rejecting too often under some simulation conditions. Under the null hypothesis of zero length internal branch, LR statistics are assumed to be asymptotically distributed chi 2(1); with limited data, the distribution of LR statistics under the null hypothesis varies from chi 2(1).      

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.     

Fair-balance paradox, star-tree paradox, and Bayesian phylogenetics

The star-tree paradox refers to the conjecture that the posterior probabilities for the three unrooted trees for four species (or the three rooted trees for three species if the molecular clock is assumed) do not approach 1/3 when the data are generated using the star tree and when the amount of data approaches infinity. It reflects the more general phenomenon of high and presumably spurious posterior probabilities for trees or clades produced by the Bayesian method of phylogenetic reconstruction, and it is perceived to be a manifestation of the deeper problem of the extreme sensitivity of Bayesian model selection to the prior on parameters. Analysis of the star-tree paradox has been hampered by the intractability of the integrals involved. In this article, I use Laplacian expansion to approximate the posterior probabilities for the three rooted trees for three species using binary characters evolving at a constant rate. The approximation enables calculation of posterior tree probabilities for arbitrarily large data sets. Both theoretical analysis of the analogous fair-coin and fair-balance problems and computer simulation for the tree problem confirmed the existence of the star-tree paradox. When the data size n --> infinity, the posterior tree probabilities do not converge to 1/3 each, but they vary among data sets according to a statistical distribution. This distribution is characterized. Two strategies for resolving the star-tree paradox are explored: (1) a nonzero prior probability for the degenerate star tree and (2) an increasingly informative prior forcing the internal branch length toward zero. Both appear to be effective in resolving the paradox, but the latter is simpler to implement. The posterior tree probabilities are found to be very sensitive to the prior.     

Calculating the evolutionary rates of different genes: a fast, accurate estimator with applications to maximum likelihood phylogenetic analysis

In phylogenetic analyses with combined multigene or multiprotein data sets, accounting for differing evolutionary dynamics at different loci is essential for accurate tree prediction. Existing maximum likelihood (ML) and Bayesian approaches are computationally intensive. We present an alternative approach that is orders of magnitude faster. The method, Distance Rates (DistR), estimates rates based upon distances derived from gene/protein sequence data. Simulation studies indicate that this technique is accurate compared with other methods and robust to missing sequence data. The DistR method was applied to a fungal mitochondrial data set, and the rate estimates compared well to those obtained using existing ML and Bayesian approaches. Inclusion of the protein rates estimated from the DistR method into the ML calculation of trees as a branch length multiplier resulted in a significantly improved fit as measured by the Akaike Information Criterion (AIC). Furthermore, bootstrap support for the ML topology was significantly greater when protein rates were used, and some evident errors in the concatenated ML tree topology (i.e., without protein rates) were corrected. [Bayesian credible intervals; DistR method; multigene phylogeny; PHYML; rate heterogeneity.].     

Branch-length prior influences Bayesian posterior probability of phylogeny

The Bayesian method for estimating species phylogenies from molecular sequence data provides an attractive alternative to maximum likelihood with nonparametric bootstrap due to the easy interpretation of posterior probabilities for trees and to availability of efficient computational algorithms. However, for many data sets it produces extremely high posterior probabilities, sometimes for apparently incorrect clades. Here we use both computer simulation and empirical data analysis to examine the effect of the prior model for internal branch lengths. We found that posterior probabilities for trees and clades are sensitive to the prior for internal branch lengths, and priors assuming long internal branches cause high posterior probabilities for trees. In particular, uniform priors with high upper bounds bias Bayesian clade probabilities in favor of extreme values. We discuss possible remedies to the problem, including empirical and full Bayesian methods and subjective procedures suggested in Bayesian hypothesis testing. Our results also suggest that the bootstrap proportion and Bayesian posterior probability are different measures of accuracy, and that the bootstrap proportion, if interpreted as the probability that the clade is true, can be either too liberal or too conservative.     

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.      

Guided tree topology proposals for Bayesian phylogenetic inference

Increasingly, large data sets pose a challenge for computationally intensive phylogenetic methods such as Bayesian Markov chain Monte Carlo (MCMC). Here, we investigate the performance of common MCMC proposal distributions in terms of median and variance of run time to convergence on 11 data sets. We introduce two new Metropolized Gibbs Samplers for moving through "tree space." MCMC simulation using these new proposals shows faster average run time and dramatically improved predictability in performance, with a 20-fold reduction in the variance of the time to estimate the posterior distribution to a given accuracy. We also introduce conditional clade probabilities and demonstrate that they provide a superior means of approximating tree topology posterior probabilities from samples recorded during MCMC.     

Bias and conflict in phylogenetic inference of myco-heterotrophic plants: a case study in Thismiaceae

Due to morphological reduction and absence of amplifiable plastid genes, the identification of photosynthetic relatives of heterotrophic plants is problematic. Although nuclear and mitochondrial gene sequences may offer a welcome alternative source of phylogenetic markers, the presence of rate heterogeneity in these genes may introduce bias/systematic error in phylogenetic analyses. We examine the phylogenetic position of Thismiaceae based on nuclear 18S rDNA and mitochondrial atpA DNA sequence data, as well as using parsimony, likelihood and Bayesian inference methods. Significant differences in evolutionary rates of these genes between closely related taxa lead to conflicting results: while parsimony analyses of 18S rDNA and combined data strongly support the monophyly of Thismiaceae, Bayesian inference, with and without a relaxed molecular clock, as well as the Swofford–Olsen–Waddell–Hillis (SOWH) test confidently reject this hypothesis. We show that rate heterogeneity in our data leads to long-branch attraction artifacts in parsimony analysis. However, using model-based inference methods the question of whether Thismiaceae are monophyletic remains elusive. On the one hand maximum likelihood nonparametric bootstrapping and parametric hypothesis tests fail to support a paraphyletic Thismiaceae, on the other hand Bayesian inference methods (both without and with a relaxed clock) significantly reject a monophyletic Thismiaceae. These results show that an adequate sampling, the use of rate homogeneous data, and the application of different inference methods are important factors for developing phylogenetic hypotheses of myco-heterotrophic plants. © The Willi Hennig Society 2009.     

The relative performance of Bayesian and parsimony approaches when sampling characters evolving under homogeneous and heterogeneous sets of parameters

We tested whether it is beneficial for the accuracy of phylogenetic inference to sample characters that are evolving under different sets of parameters, using both Bayesian MCMC (Markov chain Monte Carlo) and parsimony approaches. We examined differential rates of evolution among characters, differential character-state frequencies and character-state space, and differential relative branch lengths among characters. We also compared the relative performance of parsimony and Bayesian analyses by progressively incorporating more of these heterogeneous parameters and progressively increasing the severity of this heterogeneity. Bayesian analyses performed better than parsimony when heterogeneous simulation parameters were incorporated into the substitution model. However, parsimony outperformed Bayesian MCMC when heterogeneous simulation parameters were not incorporated into the Bayesian substitution model. The higher the rate of evolution simulated, the better parsimony performed relative to Bayesian analyses. Bayesian and parsimony analyses converged in their performance as the number of simulated heterogeneous model parameters increased. Up to a point, rate heterogeneity among sites was generally advantageous for phylogenetic inference using both approaches. In contrast, branch-length heterogeneity was generally disadvantageous for phylogenetic inference using both parsimony and Bayesian approaches. Parsimony was found to be more conservative than Bayesian analyses, in that it resolved fewer incorrect clades.
© The Willi Hennig Society 2006.     

The importance of proper model assumption in bayesian phylogenetics

We studied the importance of proper model assumption in the context of Bayesian phylogenetics by examining >5,000 Bayesian analyses and six nested models of nucleotide substitution. Model misspecification can strongly bias bipartition posterior probability estimates. These biases were most pronounced when rate heterogeneity was ignored. The type of bias seen at a particular bipartition appeared to be strongly influenced by the lengths of the branches surrounding that bipartition. In the Felsenstein zone, posterior probability estimates of bipartitions were biased when the assumed model was underparameterized but were unbiased when the assumed model was overparameterized. For the inverse Felsenstein zone, however, both underparameterization and overparameterization led to biased bipartition posterior probabilities, although the bias caused by overparameterization was less pronounced and disappeared with increased sequence length. Model parameter estimates were also affected by model misspecification. Underparameterization caused a bias in some parameter estimates, such as branch lengths and the gamma shape parameter, whereas overparameterization caused a decrease in the precision of some parameter estimates. We caution researchers to assure that the most appropriate model is assumed by employing both a priori model choice methods and a posteriori model adequacy tests.     

The comparison of the confidence regions in phylogeny

In this paper, several different procedures for constructing confidence regions for the true evolutionary tree are evaluated both in terms of coverage and size without considering model misspecification. The regions are constructed on the basis of tests of hypothesis using six existing tests: Shimodaira Hasegawa (SH), SOWH, star form of SOWH (SSOWH), approximately unbiased (AU), likelihood weight (LW), generalized least squares, plus two new tests proposed in this paper: single distribution nonparametric bootstrap (SDNB) and single distribution parametric bootstrap (SDPB). The procedures are evaluated on simulated trees both with small and large number of taxa. Overall, the SH, SSOWH, AU, and LW tests led to regions with higher coverage than the nominal level at the price of including large numbers of trees. Under the specified model, the SOWH test gives accurate coverage and relatively small regions. The SDNB and SDPB tests led to the small regions with occasional undercoverage. These two procedures have a substantial computational advantage over the SOWH test. Finally, the cutoff levels for the SDNB test are shown to be more variable than those for the SDPB test.     

Tail paradox, partial identifiability, and influential priors in Bayesian branch length inference

Recent studies have observed that Bayesian analyses of sequence data sets using the program MrBayes sometimes generate extremely large branch lengths, with posterior credibility intervals for the tree length (sum of branch lengths) excluding the maximum likelihood estimates. Suggested explanations for this phenomenon include the existence of multiple local peaks in the posterior, lack of convergence of the chain in the tail of the posterior, mixing problems, and misspecified priors on branch lengths. Here, we analyze the behavior of Bayesian Markov chain Monte Carlo algorithms when the chain is in the tail of the posterior distribution and note that all these phenomena can occur. In Bayesian phylogenetics, the likelihood function approaches a constant instead of zero when the branch lengths increase to infinity. The flat tail of the likelihood can cause poor mixing and undue influence of the prior. We suggest that the main cause of the extreme branch length estimates produced in many Bayesian analyses is the poor choice of a default prior on branch lengths in current Bayesian phylogenetic programs. The default prior in MrBayes assigns independent and identical distributions to branch lengths, imposing strong (and unreasonable) assumptions about the tree length. The problem is exacerbated by the strong correlation between the branch lengths and parameters in models of variable rates among sites or among site partitions. To resolve the problem, we suggest two multivariate priors for the branch lengths (called compound Dirichlet priors) that are fairly diffuse and demonstrate their utility in the special case of branch length estimation on a star phylogeny. Our analysis highlights the need for careful thought in the specification of high-dimensional priors in Bayesian analyses.     

Effects of branch length uncertainty on Bayesian posterior probabilities for phylogenetic hypotheses

In Bayesian phylogenetics, confidence in evolutionary relationships is expressed as posterior probability--the probability that a tree or clade is true given the data, evolutionary model, and prior assumptions about model parameters. Model parameters, such as branch lengths, are never known in advance; Bayesian methods incorporate this uncertainty by integrating over a range of plausible values given an assumed prior probability distribution for each parameter. Little is known about the effects of integrating over branch length uncertainty on posterior probabilities when different priors are assumed. Here, we show that integrating over uncertainty using a wide range of typical prior assumptions strongly affects posterior probabilities, causing them to deviate from those that would be inferred if branch lengths were known in advance; only when there is no uncertainty to integrate over does the average posterior probability of a group of trees accurately predict the proportion of correct trees in the group. The pattern of branch lengths on the true tree determines whether integrating over uncertainty pushes posterior probabilities upward or downward. The magnitude of the effect depends on the specific prior distributions used and the length of the sequences analyzed. Under realistic conditions, however, even extraordinarily long sequences are not enough to prevent frequent inference of incorrect clades with strong support. We found that across a range of conditions, diffuse priors--either flat or exponential distributions with moderate to large means--provide more reliable inferences than small-mean exponential priors. An empirical Bayes approach that fixes branch lengths at their maximum likelihood estimates yields posterior probabilities that more closely match those that would be inferred if the true branch lengths were known in advance and reduces the rate of strongly supported false inferences compared with fully Bayesian integration.     

Utility of the dentin matrix protein 1 (DMP1) gene for resolving mammalian intraordinal phylogenetic relationships

We sequenced exon 6 of the nuclear dentin matrix protein 1 (DMP1) gene from 19 species of bats (order Chiroptera) to assess the utility of this gene for higher-level phylogenetic studies. Bayesian analysis revealed high support (posterior probabilities >/=0.95) for monophyly of Noctilionoidea (Phyllostomidae, Noctilionidae, and Mormoopidae), all genera and most families examined. Comparison of the phylogenetic information present in DMP1 with mitochondrial rDNA and nuclear RAG2 genes indicated no significant heterogeneity. Thus, we concatenated these three data sets into a single "total evidence" phylogenetic analysis. Combined analysis was congruent with study of RAG2 and combined RAG2 and mtrDNA sequences, but improved support (Bayesian posterior probabilities) for many nodes. Our results indicate that exon 6 of DMP1 is rapidly evolving, able to tolerate non-frame shifting insertion and deletion events, is more variable than RAG2, and provides phylogenetic resolution from the interfamilial to infraclass levels in mammals.     

Exploring among-site rate variation models in a maximum likelihood framework using empirical data: effects of model assumptions on estimates of topology,branch lengths,and bootstrap support

We have investigated the effects of different among-site rate variation models on the estimation of substitution model parameters, branch lengths, topology, and bootstrap proportions under minimum evolution (ME) and maximum likelihood (ML). Specifically, we examined equal rates, invariable sites, gamma-distributed rates, and site-specific rates (SSR) models, using mitochondrial DNA sequence data from three protein-coding genes and one tRNA gene from species of the New Zealand cicada genus Maoricicada. Estimates of topology were relatively insensitive to the substitution model used; however, estimates of bootstrap support, branch lengths, and R-matrices (underlying relative substitution rate matrix) were strongly influenced by the assumptions of the substitution model. We identified one situation where ME and ML tree building became inaccurate when implemented with an inappropriate among-site rate variation model. Despite the fact the SSR models often have a better fit to the data than do invariable sites and gamma rates models, SSR models have some serious weaknesses. First, SSR rate parameters are not comparable across data sets, unlike the proportion of invariable sites or the alpha shape parameter of the gamma distribution. Second, the extreme among-site rate variation within codon positions is problematic for SSR models, which explicitly assume rate homogeneity within each rate class. Third, the SSR models appear to give severe underestimates of R-matrices and branch lengths relative to invariable sites and gamma rates models in this example. We recommend performing phylogenetic analyses under a range of substitution models to test the effects of model assumptions not only on estimates of topology but also on estimates of branch length and nodal support.