Web supplement: Nine easy steps for constructing reliable trees from published phylogenetic analyses

R.  H. Zander

Res Botanica, a Missouri Botanical Garden Web Site
August 6, 2007

WEB SUPPLEMENT:

NINE EASY STEPS FOR CONSTRUCTING RELIABLE TREES FROM PUBLISHED PHYLOGENETIC ANALYSES

 

Richard H. Zander

Missouri Botanical Garden

PO Box 299

St. Louis, MO 63166-0299

richard.zander@mobot.org

 

Res Botanica, a Missouri Botanical Garden Web Site, August 6, 2007

 

INTRODUCTION

 

Extended discussions, including additional calculations, are presented here in support of a nine-step procedure (Zander, 2007a) recommended for extracting reliable conclusions of monophyly from published manuscripts, and developing hypotheses of evolutionary relationship involving, when appropriate, selectionist and paraphyletic concepts of taxa.

 

This electronic document is organized along the lines of the nine-step procedure, and complements the discussions of the hardcopy publications. Certain repetitions from the original article are made for clarity. See corrections to the original hardcopy paper at end of this Supplement.

 

The Nine Easy Steps are:

 

STEP ONE: Change support measures to PBAs.

STEP TWO: Begin evaluation of published conclusions of monophyly.

STEP THREE: Preselect conclusions of monophyly to test, prioritized to break ties.

STEP FOUR: Impose a penalty for unaccounted assumptions.

STEP FIVE: Combine two or more contiguous branch arrangements of PBAs of 0.50 to 0.94 into one of 0.95 or higher using the formula for implied reliable internodes (FIRI).

STEP SIX: Make corrections for multiple tests.

STEP SEVEN: With molecular data alone, reliable demonstration of paraphyly will split a taxonomic group, i.e., distinguish two groups as warranting separate names.

STEP EIGHT: Use Bayes’ Formula to combine separately published branch arrangements based on different data.

STEP NINE: Examine the potential value of recognizing paraphyletic taxa by tempering strict monophyly with evidence for taxonomic unity based on inferred functional evolution and non-dichotomous evolutionary processes.

 

Summary of abbreviations: AB represents a posteriori evolutionary support indicating shared ancestry of taxa A and B as measured by branch length (steps or state changes in maximum parsimony); (AB) is a lineage of two taxa of shared ancestry, in which A and B are in this paper always the terminal sister taxa of the optimal arrangement among three taxa A, B and C; BPP is Bayesian posterior probability, calculated via the well-known Bayes' Formula or as given in the publication being analyzed; PBA is the probability of branch arrangement, and is here used as equivalent to the Bayesian credible interval and is a general term for both the BCI and BPP; BP is nonparametric bootstrap proportion; JP is jackknife proportion; DI is decay index or Bremer support; BCI is binomial credible interval, essentially 1 minus the chance by chance alone of support of AB steps or better out of all support for A or B in all NNI combinations of branches in a 4-taxon dataset; NNI is nearest neighbor interchange, or switching of each of two terminal lineages with the next lower branch, e.g. (AB)C, (AC)B, and (BC)A;  IRI is implied reliable internode, a single reliable internode representing the pooled less-then-reliable support of two or more chained internodal branch arrangements later collapsed into one reliable internode, and an IRCI would be the credible interval for that internode; FIRI is the formula for calculating the probability of an implied reliable internode; α (alpha) in hypothesis testing is probability of a Type I Error, or in a nonparametric context the chance of obtaining these data by chance alone, where BCI is the complement or 1 minus α.

 

Although cladograms are often highly detailed, various analyses of the same taxa often produce contradictory arrangements, and fine resolution may be viewed suspiciously in some cases as merely precise, not accurate. Molecular data may conflict among sources, or molecular data and morphological data, or be simply incorrect in certain situations (Avise, 1994: 314; Marshall, 1997; Philippe et al., 1996; Seberg et al., 1997; Sites et al., 1996).  Nei et al. (1998) wrote: “We suggest that more attention should be given to testing the statistical reliability of an estimated tree rather than to finding the optimal tree with excessive efforts.” A possible explanation of why poorly supported arrangements are tolerated or accepted is “statistical relevance” (Salmon, 1971: 11).  Statistical relevance is the philosophy-of-science version of the Bayes Factor, recently much promoted by Bayesian statisticians (e.g. Aris-Brosou & Yang, 2002; Suchard et al., 2002). The prior understanding, in this case, is that there is no or equal support for a particular hypothesis, and this is replaced after analysis by some statistical support which demonstrates what appears to be a relatively great and perhaps significant increase in support. This, however, is only apparent; a particular absolute level of support is required for the arrangement to be accepted as due to shared ancestry. As Huelsenbeck et al. (2002) have pointed out, although the Bayes Factor has applications of value, such as model selection, it is the posterior probability that genuinely reflects the chance of an arrangement being correct. Also, a similar attitude known as “clinical relevance” (Hopkins 2001, 2003) is valuable in practice when an effect is demonstrated as not entirely reliable (e.g. p-values of 0.80 or 0.90) but the chance that it is helpful far outweighs the risk, e.g. of using a harmless drug to treat a dread illness. In betting our science, however, the loss on failure far outweighs any benefit on success. The fact that cladograms based on molecular data commonly match morphology-based intuitive or cladistic groupings of organisms is no longer a marvel, and DNA work is now expected to advance science in resolving previously problematic groups or decisively deciding between two or more equally and often well-supported but contradictory arrangements.

 

Following are the Nine Easy Steps with discussions extending those of the hardcopy (Zander, 2007a).

 

STEP ONE: Change support measures to PBAs.

 

confidence intervals and Bayesian philosophy

There are two contending statistical schools (excluding the more narrowly applicable hypothesis testers). Both rely on a physical probability phenomenon in physics described by the Central Limit Theorem. Frequentists analyze a series of past events due to some process, and predict continuing events with a measure of continued occurrence within a stated range, the confidence interval. Since phylogenetics deals with retrodiction of single, one-time events, frequentist techniques must be modified to apply. Bayesian statisticians compute the probability of single events in light of previous knowledge in the context of perceived risk, and the measure of expectation is the credibility interval, or, in phylogenetics, the probability of shared ancestry being correct. Bayesian methods have been criticized as too subjective in regard to evaluation of prior knowledge, and that one-time bets are not statistically describable or predictable. But the two statistical methods, as far as a user taxonomist is concerned, are much alike in that both rely on the Central Limit Theorem, while Bayesian techniques can (1) reject use of notional or hunch priors in favor of prior empirical data, and (2) the success of one-time bets based on Bayesian techniques is well established because betting Bayesians generally win in the long run, given judicious estimates of probabilities with reasonable priors in light of acceptable risk. For this reason, in practice and particularly in this phylogenetics paper, a confidence interval is taken as essentially the same measure of reliability as a Bayesian credible interval.

 

Cladograms have been published over the past 30 years with internodes of relatively low or merely moderate nonparametric bootstrap (BP) scores, low decay indexes (DI) or low Bayesian posterior probabilities (BPP) or even without any measures of reliability. For critical distinctions, a credible interval of at least 0.95 should be the minimum acceptable (in the nonparametric context, the level when only one or fewer out of 20 optimal internode branch arrangements can be expected to be generated by chance alone). In some cases, 0.99 may be required, as in cases where software has demonstrated high error rates (Popp & Oxelman, 2004); this is a cost of fully 4/5 of the reliability window of 0.05.

 

In Bayesian phylogenetic analysis (e.g., Randle et al., 2005) a credible interval is the probability that a hypothesis is correct given the data, model and a set of prior probabilities. The over-arching Bayesian philosophy (Bernardo & Smith, 1994), however, is that, using the Bayesian formula together with all relevant information, a posterior probability is derived such that an intelligent bet can be made (or declined to be made); this is opposed to statistical analysis as pure inference (Winkler, 1972: 393). By bettable, one means a theory presented as suitable for use in other scientific research, such as biogeography or biodiversity analysis and biodiversity triage, on which additional new hypotheses, some of which may themselves not be fully certain, depend.

 

The Bayesian bet is always made in the environment of a particular acknowledged risk. Bayesians are not adverse to considering the relevance of any probability level if an acceptable bet can be made in light of a given risk.

 

The rationale, for the user taxonomist, for 0.95 (or 1 in 20 results being due to chance alone in a hypothesis-testing framework) as a minimum confidence interval, is that one wishes sufficient certainty to act confidently (to bet). Ninety-nine percent (or 1 in a 100 results being due to chance alone) is impressively reliable yet phylogenetic research commonly provides results with a range of confidence measurements, implying a less rigorous standard. The difference is, however, extreme because 0.99 implies one in 100 results by chance alone, 0.98 one in 50, 0.97 one in 33, 0.96 one in 25, 0.95 one in 20, and 0.94 is one in 17. Ninety-nine percent is wanted, yet 0.95 is commonly tolerated in spite of being nowhere near relative certainty. In any case, in the literature 0.95 as confidence or credible interval commonly characterizes what is expected of research reliability in the entire field (e.g. psychology or ecology).

 

Do we really have 0.95 confidence or chance of being correct when phylogenetic Bayesian analyses produce that measure? Or when there are equivalent high bootstrapping scores? There are a number of problems afflicting phylogenetic analysis that violate the Bayesian goal of a bettable or actionable result given the risk.

 

Bayesian posterior probabilities may be increased by (1) additional information, e.g., total evidence (Allard & Carpenter 1996; Eernisse & Kluge 1993; Nixon & Carpenter 1996) from combined multiple data sets where all evidence is derived from the same evolutionary process, (2) using a previous analysis as a source of priors with Bayes' Formula, or (3) increasing confidence scores by combining them (see below). Posterior probabilities as published may be decreased, however, by unaccounted assumptions that introduce sufficient uncertainty that reliability values produced by software must be reduced by the probability that an assumption when wrong affects an arrangement of interest. This paper uses the above extended Bayesian philosophy to gauge multi-internode reliability, incorporate Bayesian uncertainty, and deal with traditional classifications.

 

The binomial confidence interval was presented by Zander (2001a, 2004) as a way to standardize the older confidence measures BP, DI, JP, and BPPs in a phylogenetic context. Here it is renamed the binomial credible interval inasmuch as it is basically a simple probability. Essentially, of any three nearest neighbor taxa (lineages) A, B and C, there are three combinations of pairs of taxa, (AB), (AC) and (BC) that may uniquely share advanced traits (i.e. features not in closely neighboring taxa and therefore recently evolved, these traits usually provided by an outgroup taxon, D). If the number of advanced traits (as steps in maximum parsimony) shared by one lineage, say (AB), are statistically significantly greater than expected among all three pairs, then that combination is inferred as sharing an ancestor, and the traits shared by each of the other two pairs, (AC) or (BC), are inferred as parallelisms.

 

If the support for the optimal arrangement is not significantly greater than that which may be obtained by chance alone at a particular confidence level, then all shared pairs of traits should be considered generated randomly, and no branch arrangements or even a star can be inferred. Although the null hypothesis is a star, not rejecting the null does not mean a star is correct, only that one chooses not to decide that one or the other of the three pair combinations represents shared ancestry. Just as the Tree of Life (Delsuc et al., 2005; Salamin et al. 2005) is developed in parts, it is fully proper to analyze separately phylogenetically largely isolated portions of a cladogram or larger group.

 

The BCI is a binomial (chi-squared with one degree of freedom) analysis because the null hypothesis (in the 4-taxon case) is AB = AC = BC, all variation from equality being random. The original analytic method of Zander (2004) was problematic because homoplasy outside the 4-taxon NNI was ignored. This is here investigated, and a table for evaluating published BP and BPP values is advanced, which may be used by those interpreting the literature.

 

A chi-squared test with more than one degree of freedom, which evaluates the chance of data from a number of categories, each of data independent and uniformly distributed, being due to chance alone, is appropriate when one suspects that sampled categories are associated with different probabilities of generating data. Although molecular data, for instance, consists of separate categories of different probabilistic distributions, the maximum parsimony analysis in effect standardizes the branch lengths, where a branch length of 2 is evolutionarily twice the length of a branch length of 1. The one-tailed binomial test is a special case of the multinomial (i.e. one degree of freedom) where “spread” of the contrary data is unimportant and is used here for the BCI.

 

As for the DI and homoplasy, assuming that support for (AB)C is unexpectedly larger than that for the other arrangements involving A or B if randomly generated, a cladogram with more combinations than the two possible via NNI would involve support values for contrary arrangements never larger than the support contrary to (AB)C as measured by the DI. Because the DI measures support for the optimal subclade against contrary arrangements anywhere in the cladogram this is true whether values for AC or BC are large or small. But from the published data one does not know if there are any more than two contrary combinations (AC and BC) in the cladogram. A binomial calculation involving AB = 5 steps, AC = 3 and BC = 3 (with a DI in the 4-taxon case of 2) gives an α of 0.29, and thus a BCI (1 minus α) of 0.71. With additional alternative combinations with branches other than C of A or B supported at (a maximum of) 3 steps, the probability of (AB) being correct changes from 1/3 to 1/4 to 1/5, etc., the chance of the data by chance alone decreases and the BCI goes to 0.74, 0.76, 0.77, etc. In the worst case (worst meaning most different from the BCI as calculated from the 4-taxon tree), there are very many combinations each with as much support for a contrary arrangement as the highest possible estimated from a 4-taxon exemplar, and a Poisson distribution (mean and variance near each other) will demonstrate a higher chance of the optimal support being non-random. For instance, a branch arrangement with DI of 2 for branch length 5 (implying a 4-taxon AB:AC:BC support ratio of 5:3:3) increases the 4-taxon BCI from 0.71 to a multi-taxon BCI of 0.81 for 100 combinations (AB plus 99 others each supported at 3 steps), probability 1/100 via VassarStat's Poisson calculator (Lowry, 2005). Thus, BCI percentages beyond that for 4-taxon AB:AC:BC DI simulations with additional homoplastic combinations increase and are probably not large for real data where AC and BC are somewhat imbalanced in support levels. The 4-taxon BCI equivalent of DI is thus fail-safe, being always less than that which is possible but not known by the user of phylogenetic literature to actually occur.

 

The situation with nonparametric bootstrapping and homoplasy is different and more problematic. The branch arrangements alternative to those of NNI with the optimal branch arrangement involved in BP analysis are from the universe of compatible trees generated by maximum parsimony during the BP analysis. The BP is often cited as a measure of repeatability (Kolaczkowski, B. & J. W. Thornton, 2004) with other, additional data, and the results can therefore be treated as independent. Bootstrapping is similar to Monte Carlo sampling, which is itself a replacement for the chi-squared test. Bootstrapping differs in the simplicity of the algorithm, but oversampling converges to a form depending not only on the data and the test statistic but on the resampling subset size, and generally does not converge to the exact test as is the case with Monte Carlo resampling. The BPs generated under maximum parsimony, however, are not the same as probabilities generated by, for instance, Lowry's (2005) Monte Carlo (MC) sampling module. For example, a distribution ratio AB:AC:BC of 5:3:3 in three categories at probability 1/3 yields MC CI = 0.35, BP = 0.65, and BCI = 0.71, that of 5:2:2 yields MC CI = 0.62, BP = 0.80, and BCI = 0.85, and that of 5:1:1 yields MC CI = 0.87, BP = 0.95, and BCI = 0.95, based in part on contrived data sets as described by Zander (2001a, 2004). Whether or not the BP is generated to some extent with a multinomial calculation in software, the results can be converted to a binomial calculation for estimation of reliability because the data involved in the BP are ultimately the branch lengths.

 

The BP may be intuitively re-scaled in reflecting reliability, for instance to the extent accepted classifications are supported or problematic arrangements are resolved in such a way that other data remains consilient (e.g. when results of analysis of morphological data support those of molecular data), or by means of complex formulae (e.g., Zharkikh & Li, 1995), or here with simulations that result in a table of BP-comparable BCI values. As noted by Zander (2004), with the BP, the difference between AB and the next highest contrary support value (AC in 4-taxon trees) is not given in published literature (as it is when DI is given). The BCI cannot directly deal with strong imbalance between non-optimal contrary support values such that one or more of them (say AC) is nearly as great as AB (and BC is much smaller). In the case of strong imbalance between AC and  BC, there are two (or more) implied phylogenetic signals affecting the 4-taxon arrangement, which is theoretically nonsense. Lack of information about AC, BC and other imbalance is, however, not a problem with the BP because the BP (as noted by Zander, 2004) responds strongly to the difference between AB and the next highest contrary support value (AC in 4-taxon trees), but not the third. Thus, the BCI, as calculated from the BP with 4-taxon trees with AC ~ BC, properly converts the BP to a low BCI when AB or BC is approached in size by the next highest contrary support value. But when there is strong imbalance the BCI translated from BP is fail-safe.

 

stochastic artifacts of random data

Strict cladism, in which a tree of maximum parsimony is presented as sufficient because scientists should accept the simplest hypothesis or that the tree probably converges on the true tree with additional data, has now been generally superseded by analyses that provide various measures of reliability for the arrangements of branches on internodes (Wilkinson et al., 2003; Zander, 2001a). There are tests (reviewed by, e.g. Felsenstein, 2004) that demonstrate that whole trees based on real data are well endowed with phylogenetic signal, but evaluating the reliability of internal branch arrangements has been a knotty problem in statistics.

 

Resolution alone does not provide support for arrangements because fully random data sets (Goloboff, 1991; Zander, 2003) usually generate fully resolved cladograms of maximum parsimony and Bayesian MCMC (Zander, 2004 and see below). Therefore, reliability measures on branch arrangements are important in distinguishing relationships based on shared ancestry (at some confidence level) from those generated by chance alone.

 

four-taxon confidence interval

The BCI proved to be more of a judicious approximation than an exact conversion from the nonparametric bootstrap in that interactions between data supporting various branch arrangements in a lineage are commonly complex except for lineages of very short branch lengths. For instance, particular support in numbers of steps of an arrangement contrary to a particular optimal branch arrangement often support an even more distal arrangement in the optimal tree. The simulation results indicate, however, that the tables for the conversion of BP into BCI based on four taxa at various branch lengths may be used to gauge reliability better than was heretofore possible for many-taxon cladograms because (1) the BP and 4-taxon BCI vary approximately with the BPP whatever the size of the cladogram, with the BCI being higher than the BP, which is commonly considered somewhat too low a measure of reliability, and lower than the BPP, commonly considered too high a measure at least with short branch lengths, (2) simple homoplasy distant in the lineage is ignored by BP and BCI as shown below, and (3) there is no better alternative for user taxonomists in weighing the reliability of the BP values of published studies, other than intuitive guesswork or complex formulae.

 

Hillis and Bull (1993) demonstrated with simulations that BP's are generally lower than equivalent CIs, with reliable (0.95 and above) CIs reached by BP values of 0.70 and above (assuming rates of evolution are not high and disparity of rates between lineages is not great); but see objections of Zander (2004). Marvaldi et al. (2002) considered nodes with BP of > 0.50 to be “well supported.” La Farge et al. (2002) used as robust support BP’s of ³ 0.70, moderate as < 0.70 and ³ 0.50, and weak as < 0.50. Fishbein et al. (2001) used BP values of < 0.70 for weak support, 0.70--0.84 for moderate support, and ³ 0.85 as strong support. Generally in the literature, the empirical ranges of BP support accepted as reliable approximately match the ranges of variation in observed BP support in authors' cladograms, with authors publishing mostly well-supported cladograms more inclined to rigorous requirements for acceptable BP's. Correction formulae have been proposed (Efron et al., 1996; J. Farris in Salamin et al., 2002; Rodrigo, 1993; Sanderson and Wojciechowski, 2000; Zharkikh and Li, 1995) that purport to provide the equivalent in CIs for BP values. Most are methodologically complex, often requiring special software or complex data treatment, and are not easy for end-user taxonomists, or often impossible without reanalysis. The 4-taxon BP to BCI conversion tables provided here consider 0.95 reliability to be reached with BPs of at least 0.95 for branch lengths of ca. 5 steps, and BPs of at least 0.90 for branch lengths of about 10 steps or greater.

 

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Table 3. Data set for 25 steps at each internode but immediate NNI contrary support of 19 + 19 steps, not overlapping, for 7 terminal branches, simplified format.

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Taxon Numbers of advanced shared traits

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0      0  0  0  0  0  0  0  0  0  0  0  0  0  0  0(outgroup)

A     25 19  0 25 19  0 25 19  0 25 19  0 25 19  0

B     25  0 19 25 19  0 25 19  0 25 19  0 25 19  0

C      0 19 19 25  0 19 25 19  0 25 19  0 25 19  0

D      0  0  0  0 19 19 25  0 19 25 19  0 25 19  0

E      0  0  0  0  0  0  0 19 19 25  0 19 25 19  0

F      0  0  0  0  0  0  0  0  0  0 19 19 25  0 19

G      0  0  0  0  0  0  0  0  0  0  0  0  0 19 19

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Table 4. Data set for 25 steps at each internode but immediate NNI contrary support of 19 + 19 steps, overlapping, for 7 terminal branches, simplified format.

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Taxon       Numbers of advanced shared traits

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0      0  0  0  0  0  0  0  0  0  0  0 (outgroup)

A     25 19  0 25  0 25  0 25  0 25  0

B     25  0 19 25  0 25  0 25  0 25  0

C      0 19 19 25 19 25  0 25  0 25  0

D     19  0  0  0 19 25 19 25  0 25  0

E      0  0  0 19  0  0 19 25 19 25  0

F      0  0  0  0  0 19  0  0 19 25 19

G      0  0  0  0  0  0  0 19  0  0 19

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In ladderized cladograms, support for and against particular branch arrangements interact. There are two forms of interactions that involve an expected nearly equal NNI support pro and con an optimal arrangement (i.e. where AC ≈ BC), (1) where AC and BC are distinct, and (2) where one overlaps (i.e. AC includes some of the traits supporting a branch higher in the cladogram).

 

The extremes are shown in Tables 3 and 4. The data set of Table 3 was contrived of 8 taxa (5 internodes) each internode of 25:19:19, that is, 25 traits supporting AB (or ABC or ABCD, etc.), plus 19 additional traits each supporting the two contrary NNI arrangements, AC (or ABD or ABCE, etc.), and BC (or CD or DE, etc.). The BP for 25:19:19 of a 4 taxon data set is 0.71, BL = 25, BCI = 0.83, BPP = 0.82. The data set resulted in BPs (10000 reps) (AB)C first of: 0.77, 0.81, 0.84, 0.81, and 0.77. The BPs when translated via the table to BCIs yield 0.79, 0.89, 0.92, 0.89, and 0.79, which may seem to overestimate the expected CI but simply reflect the support for distal branch arrangements by support contrary to more basal arrangements. Branch lengths were 9, 17, 17, 17, and 13 steps. BPPs are all 1.00.

 

The contrived data set of Table 4 also had individual branch arrangements of the form AB:AC:BC = 25:19:19, but the 19 traits of AC were included as part of the 25 traits of AB in the next higher branch arrangement. The results also showed interaction among the data for the individual branch arrangements. The parsimony optimal tree was (((((AC)B)D)(EF))G,O), with BPs of 0.36, 0.82, 0.87, 0.35 (for EF), and 0.31; respective BLs were 19, 50, 25, 19, and 50. Given the BLs, the BCIs were, respectively, 0.40, 0.93, 0.94, 0.40, and 0.33, which are acceptable given the data interaction. The Bayesian MCMC optimal tree was different, ((((((BC)A)D)E)F)G,O), and BPPs for the branch arrangements were, respectively, 0.50, 1.00, 1.00, 1.00, 0.99. There is clearly a disconnect between the calculated BPs and BPPs that is not satisfactory, but too complex to be pursued in this paper.

 

The BP is that of the 4-taxon BCI equivalent when the arrangement is abutted at both ends by reliable arrangements or the top or base of a tree. For instance, one branch arrangement of 25:19:19 bracketed on both ends by arrangements of 20:5:5 (these with BPs of 1.00) has an expected, correct BP (8000 reps) of 0.71. Thus, intra-lineage interactions between support both pro and con for particular branch arrangements are not out of line with expected reliability values at particular branch lengths, but one must be aware that some data generated randomly (as contrary to the true branch arrangement) as noise may be automatically treated by the software as supportive of more distal arrangements. Optimal branch arrangements with very high CIs or being at the top of the tree isolate these interactions, however. Thus (1) isolated or few branch arrangements with low CIs approximate the 4-taxon BCI, and (2) several contiguous branch arrangements with low CIs may be evaluated with the 4-taxon BCI table because the values are proportionate, not being out of line and reflecting the longer effective branch lengths. Two less-than-reliable internodes have only moderately increased BPs. In the common situation of reliable internodes scattered among unreliable ones, the BP may be expected to reflect in most cases the 4-taxon NNI case or a value reasonably higher given interactive data.

 

A totally random sequence of 800 A, C, G, T characters generated by RandSet was duplicated three times to form 4 identical artificial sequences, and two As were substituted for taxa A and B at a single site of all Ts. Thus, there was only one phylogenetically informative character (for maximum parsimony) for the entire 800 sites supporting the tree (AB)C,D. The BPP with Mr. Bayes ver. 3 with all models without correction for invariant sites was always 1.00; but the BPP with correction for invariants (lset rates = propinvar or invgamma) was 0.34. Clearly, when all 800 sites may vary, MCMC gave a correct answer of 1.00 BPP because the second informative “A” occurred at that site and at no other, and was not over-credible in spite of the paucity of data. With correction for invariance, assuming invariance of all non-varying sites, the result of 0.33 BPP is undercredible, compared to a BCI, which also assumes invariants, of 0.68, which is about the same as the BP, which likewise assumes invariants, of 0.64. Thus, MCMC analysis correctly increases CI when there are variable sites that are not parsimoniously informative.

 

A 8-taxon data set of 05:04:04 (not overlapping) with 0 represented by T and 1 by A, mimicking a DNA dataset with no uninformative sites, yielded the same BPs as digital binary data, namely 0.64, 0.72, 0.82, 0.72, 0.62, at BL 9, 17, 17, 17, 13. BCIs from the table are: 0.75, 0.85, 0.92, 0.85, 0.74, also reflecting branch support for more distal branch arrangements by data that are contrary to medial arrangements.  A data set with 65 additional characters all of T, gave the same BPs; but for BPP, without propinvar, 0.95, 1.00, 0.99, 0.99, 0.99, and with lset nst = 1 and rates = propinvar BPP = 0.95, 1.00, 0.99, 0.99, 0.99. This is apparently too high, and may be a problem evaluating branch arrangements with short branch lengths. A data set of 588 extra DNA characters corresponding to that of Grimmia pulvinata from Werner et al. (2004) but identical for all 8 taxa was added. The BPPs from MrBayes 3.1 with lset nst = 1 (simplest model), rates = equal were 1.00, 1.00, 1.00, 1.00 and 0.99. With rates = propvar (corrected for invariants) the BPPs were 0.84, 1.00, 1.00, 1.00 and 0.87. Thus, in general, variable sites that are phylogenetically non-informative correctly do increase the BPP, but in this case such sites seem unimportant or inapplicable because the AB:AC:BC ratio of 5:4:4 is nearly totally equivocal as support for any branch arrangement.

 

Raw BPs in a chain of contested-group (doubtfully arranged) taxa of CI 0.95 and above reflect, in a fail-safe manner, BCIs that are truly 0.95 and above. They represent the equivalent of accepted arrangements. It is also clear that particular support for AB isolated in a large, uninformative data set with all sites potentially variable usually must be reliable when traits only appear at one site for A and B and nowhere else; but in the case of the existence of support for arrangements contrary to AB, e.g. AC or BC, such support also occurs nowhere else and is therefore equally enhanced. Invariant sites are the key to several problems with models. Maximum parsimony ignores all non-phylogenetically informative DNA sites, but MCMC may ignore them or not depending on the model. Thus, for exons and strongly conserved non-coding sequences, not correcting for invariants will incorrectly increase BPP. DNA analysis with correction for invariant sites is important for any data with support for contrary arrangements.

 

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Table 5. Effects of homoplasy in a lineage: BP, BL, and BCI for internodes beginning with the most distal (AB) for data set yielding a ten-taxon tree with non-overlapping support ratio AB:AC:BC of 25:19:19 at each internode, and with 19 additional homoplastic traits added as shared by AC, AD, AE, etc. Roman lightface = BPs of internodes after maximum parsimony analysis with PAUP*, bandb, bootstrap nreps = 4000, search = heuristic; BL is branch length calculated by PAUP* for that internode. Roman boldface = table 4-taxon BCI equivalents of BPs.

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Without homoplasy:

BP  0.57, 0.74, 0.82, 0.71, 0.78, 0.74, 0.63;

BL  39, 77, 77, 77, 77, 77, 58;

BCI 0.77, 0.89, 0.93, 0.85, 0.90, 0.89, 0.79

 

AD sharing 19 additional traits (AC now optimal):

BP 0.92, 0.97, 0.98, 0.78, 0.82, 0.82, 0.78;

BL 38, 88, 63, 82, 82, 82, 63;

BCI 0.97, 0.99, 0.99, 0.90, 0.92, 0.92, 0.90

 

AE sharing 19 additional traits (AB is optimal):

BP 0.73, 0.78, 0.84, 0.83, 0.83, 0.81, 0.78;

BL 44, 82, 82, 82, 82, 82, 63;

BCI 0.88, 0.91, 0.93, 0.93, 0.93, 0.92, 0.88

 

AF sharing 19 additional traits (AB is optimal):

BP 0.76, 0.81, 0.85, 0.82, 0.83, 0.81, 0.78;

BL 44, 82, 82, 82, 82, 63;

BCI 0.90, 0.92, 0.93, 0.92, 0.93, 0.92, 0.91

 

AG, AH, AI sharing 19 traits all with similar results.

___________________________________________________________

 

 

Several ten-taxon data sets with support ratio AB:AC:BC of 25:19:19 at each internode but with 19 additional homoplastic traits added and shared by AC, AD, AE, etc. were contrived. Thus, each analysis was done with the 19-step homoplasy, but with taxon A slid farther down the tree from (AB). Results (Table 5) showed that the data set for 25:19:19 with AD sharing an additional 19 traits caused a switch from (AB)C to (AC)B, and high BPs for (AC) and (ABC). Moving the homoplasy two or more internodes from A, however, dropped the BPs and BCIs to normal and there was no change in branch length. One can conclude that even moderately distant simple homoplasy is ignored by the BP analysis. The 4-taxon BCI values from the table are therefore approximately correct.

 

 

STEP TWO: Begin evaluation of published conclusions of monophyly.

 

See Step Six below and original paper for comments.

 

 

STEP THREE: Preselect conclusions of monophyly to test, prioritized to break ties.

 

See Step Six below and original paper for comments.

 

 

STEP FOUR: Impose a penalty for unaccounted assumptions.

 

unaccounted assumptions

There are many assumptions having to do with regularity and sample error that may significantly affect the reliability of phylogenetic analysis, even purportedly Bayesian in nature, that are commonly ignored or incorrectly passed off as trivial in the speculative literature. The tree itself is a branching series of nested sets (e.g. the set of all taxa exhibiting certain state changes). A set may appear to be more definite a concept than a sample, yet it cannot be any better than the samples included in it, and a set is itself a sample. As pointed out by Posada and Buckley (2004), statistical analysis commonly makes three fairly straight-forward but not easily checked assumptions:  that data sets are drawn from the same underlying process, that sample size is large enough to obtain meaningful results, and a multivariate normal distribution is involved. An even more basic assumption is that there is only one alternative hypothesis to expectations of independent and random probabilistic distribution of data given a model, namely shared ancestry (see discussion of homoiology below). Huelsenbeck and Rannala (2004), however, echo a common viewpoint in the literature downplaying the effect of many possible additional assumptions by stating that “the posterior probability of a tree is the probability that the tree is correct (assuming that the model is correct).” The likelihood principle, which states that the likelihood function contains all the information from the sample that is relevant for inferential and decision-making purposes (Winkler, 1972), is in this manner misused. Some authors detail how their own study is robust to variation in certain major assumptions, but this is usually restricted to model selection and sequence alignment, while “robust” is never quantified probabilistically. Some recent papers (e.g. Engstrom et al., 2004) attempt to explore various dimensions of uncertainty aside from the analytic algorithm, but these papers are few, the analysis is complex, and probabilities of less than 1.00 certainty generally are not precisely calculated and made to affect reliability measures given on the cladogram.

As few as six external factors that affect the reliability of internode branch arrangements each at 0.99 chance of being correct will reduce confidence in each branch arrangement to a maximum of 0.94 probability (as the product). Only if morphology agrees with the arrangement and is “uncontested” can it be used as a prior (here done at an arbitrarily assigned 0.95) that will ensure reliability of at least 0.95 via Bayes' Formula (if the reliability of the agreeing molecularly based branch arrangement is less than 0.95 and greater than 0.50). Below is a list of presuppositions (variously discussed in general by, among others, Avise, 1994; Felsenstein, 2004; Huelsenbeck et al., 1994; Jenner, 2004; Kolaczkowski & Thornton, 2004; Lipscomb et al., 2003; Lyons-Weiler & Milinkovitch, 1997; Maddison, 1996; Naylor & Adams, 2003; Philippe et al., 1996; Pickett & Randle, 2005; Rokas et al., 2003; Ronquist, 2004; Ruedas et al., 2000; Sites et al., 1996; Templeton, 1986; Wendel & Doyle, 1998; Wilcox, et al., 2002). These can be important but are commonly not factored in, and this is especially true in the older literature. Some are obvious and major problems, and some are cryptic to the non-adept, or merely minor, or inapplicable to particular loci. There are doubtless other factors, and each may affect the reliability of a branch arrangement of interest as the product of the confidence interval assigned to the internode times the probability that each and every particular assumption is correct. It is doubtless possible to assign particular probabilities to at least some if not many of the assumptions below for particular data sets, but that task is beyond the scope of this paper. Commonly unaccounted (unfactored) assumptions or problems that could require reduction in branch arrangement reliability are included in the following several categories:

 

1. alternative alignments of DNA sequence data, including alignment by eye or computerized optimization for best fit; mistakes in assignment of homology of morphological characters (Hickson et al., 2000; Page, 2004; Wheeler, 1994, 1999);

2. avoiding using introns or especially emphasizing them for sometimes conflicting technical reasons (Pons et al., 2004; Engstrom et al., 2004);

3. BPP not lowered in a second study when reliability values in a previous study of less than 0.50 for the same lineages could be used as priors (as per discussion below);

4. hybridization or reticulate evolution, unbalanced gene flow during introgression, gene conversion, chloroplast capture, paralogy or gene duplication (occasionally between organelles), conflation with orthology, recombination, heteroplasmy, haplotype polymorphism (Doyle et al., 2004; Holder et al., 2001; Jackson, 2005; Mason-Gamer, 2004; Popp & Oxelman, 2004; van Oppen et al., 2001l Wolfe & Randle, 2004);

5. clade probablilities not equal a priori (Pickett & Randle, 2005; Randle et al., 2005);

6. clocklike behavior or lack thereof, use of optimal model parameters in likelihood ratio test of molecular clock, use of nonparametric rate smoothing (Bromham & Woolfit, 2004; Sanderson, 1997);

7. concerted evolution (Nei et al., 2000; Popp & Oxelman, 2004);

8. convergence due to environmental selection on morphology or exons, assumed “neutral” mutations influenced by evolutionary pressures (Caporale, 2003; Doebley & Lukens, 1998; Rodriquez-Trelles et al., 2004; Zang & Kumar, 1997);

9. differences between consensus trees, best trees and true trees (Pagel et al., 2004);

10. differences between results of total evidence (combined data sets) and repeatability of results using separate gene and morphology evaluations, novel clades, using or not using different gene data at different levels in the tree; accepting data sets as compatible if non-corresponding clades lack a BP value >50% for each of two data set); random generation of traits shared by two sister lineages that is difficult to distinguish statistically from similar parallelism between each of the two sister lineages and the nearest neighbor lineage (Ané & Sanderson, 2005; Benton, 1999; Buckley et al., 2002; Chen et al., 2003; Eernisse & Kluge 1993; Engstrom et al., 2004; Johnson & Soltis, 1998; Nixon & Carpenter 1996; Nylander et al., 2004; Olmstead & Scotland, 2005; Scotland et al., 2003)

11. different results from different iterations, generations and replications of analysis processes, including Dollo or transversion parsimony, and ordered or unordered states, insufficient mixing and convergence of MCMC chains (Randle et al., 2005);

12. different results from parsimony, neighbor joining, maximum likelihood and Bayesian methods, or from the many different phylogenetic analytic software packages commonly used in the past 20 years, including ability to find shortest trees or proper trees in 0.95 credible interval, limited available selections of models or weighting (Douady et al., 2003; Felsenstein, 1978; Mindell & Thacker, 1996; Randle et al., 2005; Sober, 2004);

13. differential lineage sorting, i.e., different gene histories (Doyle, 1992, 1993; Hudson, 1992);

14. effect of uncertainty contributed by more or fewer taxa included in the data set, or the use of exemplar taxa to represent larger taxonomic units with presumably insignificant variation in traits among taxa, or the effect of inclusion or exclusion of problematic taxa, or selection of different or multiple outgroup(s); data per taxon sample size (Funk et al., 2004; Graybeal, 1998);

15. effect of under- and over-credibility of Bayesian analysis, Bayesian priors over-determining results with small data sets, extended lineages that are represented in a cladogram simply by an outer node may render the analysis imprecise because these sequences are unknown (Alfaro et al., 2003; Bininda-Emonds, 1996; Bollback, 2004; Churchill et al., 1992; Lewis et al., 2005);

16. genomic problems including differences between nucleotide- and amino acid-based analyses; codon bias in exons; reversal of asymmetric mutational constraints of strand nucleotide composition bias in mtDNA; possible strong selection pressure on strongly conserved non-coding sequences and persistent pseudogenes; limitations on congruence of orthologues; re-expression of pseudogenes; regulator- or promoter-switched deep homology masquerading as homoplasy (convergence); endogenous retroviruses causing portions of genome to appear to have a different evolutionary history (Bapteste et al., 2005; Barbulescu et al., 2001; Christianson, 2005; Collin & Cipriani, 2003; Hall, 2003; Hassanin et al., 2005; Hollyoake et al., 2005; Inagaki et al., 2004; Lockwood & Fleagle, 1999; Rohwer & Rudolph, 2005; Rokas et al., 2003).

17. heterogeneity of models among sites, heterogeneous evolutionary processes over phylogenetic history, nucleotide composition not constant over time (Goldman, 1993; Kolaczkowski & Thornton, 2004; Pagel & Meade, 2004; Tuffley & Steel, 1997);

18. inclusion or exclusion of fossil evidence (Smith & Turner, 2005);

19. incongruence, sometimes well supported, between mitrochondrial, chloroplast and nuclear data sets (Cronn et al., 2002; Des Marais & Mishler, 2002; Sang & Zhong, 2000; Shaw, 2002; Steppan et al., 2004; Wendel & Doyle, 1998);

20. inconsistent method leading to high bootstrap support for an incorrect clade (Cummings et al., 2003);

21. method of incorporation of indels and the effect on arrangements of interest, different gap costs (e.g., Pons et al., 2004; Simmons & Ochoterena, 2000);

22. model selection choice type and procedures, including amino acid and secondary structure, homogeneous versus heterogeneous models, choice between Bayesian or Akaike information criteria, too few data to ensure accuracy of likelihood ratio test, i.e. likelihood curve not shaped like a normal distribution, using 0.95 as significant for LRTs (Bollback, 2002; Buckley, 2002; Buckley et al., 2002; Pol, 2004; Posada & Buckley, 2004; Randle et al., 2005);

23. multiple test problems, e.g., one branch arrangement contrary to tradition among 20 arrangements each at 0.95 probability (Felsenstein, 2004);

24. possibility of horizontal gene transfer (Davis & Wurdack, 2004; Nickrent et al., 2004);

25. rates other than gamma-distributed (Felsenstein, 2004; Pagel et al., 2004);

26. reliability values differing by method or only comparable between similarly sized clades (Picket & Randle, 2005; Sanjuán & Wróbel, 2005);

27. results affected by inclusion or exclusion of 3rd nucleotide position, high evolutionary rates making sequences unreliable, saturation, compositional heterogeneities, among-lineage and among-site heterogeneities, invariant sites, covariation, non-independence of characters, self-correction of flawed DNA; AFLP markers limited by unequal gain-loss probability, possible lack of independence, possible lack of homology (Engstrom et al., 2004; Ho & Jermiin, 2004; Koopman, 2005; Steppan et al., 2004; Sullivan & Swofford, 1997);

28. sample error, including misidentifications, uncertainty due to lack of vouchers, reagent contaminants, unreliable primers, laboratory mistakes, capture of data, software bugs, confirming DNA sequences by analysis of both forward and reverse strands or two different reactions from same individual; confirmation bias (the tendency to selectively notice and focus on evidence that supports a theory rather than on facts that might disprove it) (Bridge et al., 2003; Engstrom et al., 2004; Funk et al., 2005; Popp & Oxelman, 2004; Steppan et al., 2004; Vilgalys, 2003);

29. sample size of DNA sites;

30. serial extinctions of sister groups or strong anagenetic change modifying ancestral characters, variation in speed of molecular evolution or speciation versus variation in generation times;

31. uncertainty contributed by conflicting morphological results, statistical rejection of  morphological alternative topologies by the molecular and vice versa (Collard & Wood, 2000; Kirchoff et al., 2004; Steppan et al., 2004);

32. uncertainty introduced by choice of ACCTRAN and DELTRAN with PAUP* or rejection of both with MacClade (Donoghue & Ackerly, 1996; Maddison & Maddison, 1992; Swofford, 1998);

33. under- or overspecification or parameterization of the model, limitation of Metropolis coupling (Alfaro et al., 2003; Ericksson et al., 2003; Huelsenbeck & Rannala, 2004; Pagel et al., 2004);

34. unexpected stochastic effects, such as bad luck in exemplar choice, long-branch attraction, unusual noise (Hillis, 1991);

35. weighting inappropriately or variously, doubt in any rescaling or re-optimization, mistakes in use of statistics, use or non-use of “weeded” parsimony; trees not derived independently of the data sets used for testing; two separately published cladograms agreeing but the data relevant to a particular branch arrangement may have been taken from the same DNA database (Engstrom et al., 2004; Goldman et al., 2000; Koopman, 2005; Milinkovitch et al., 1996; Engstrom et al., 2004).

 

The reader conversant with phylogenetic literature is doubtless aware that additional caveats and problems are published periodically. For instance, Syring et al. (2007) recently documented “widespread genealogical nonmonophyly” in pine species. A glance at a recent issue of the journal Systematic Biology February 2007 reveals critical articles on “inconsistency of phylogenetic estimates from concatenated data under coalescence,” “artifactual phylogenies caused by correlated distribution of substitution rages among sites and lineages,” “base-compositional heterogeneity . . .,” “pitfalls of heterogeneous processes for phylogenetic reconstruction,” and, particularly, “opinions on multiple sequence alignment, and an empirical comparison of repeatability and accuracy between POY and structural alignment.” Most offer only stopgap or hypothetical solutions to the problems discussed. One must expect many additional papers addressing uncertainty in phylogenetic assessments because the field includes many analytic methods and philosophies, and after 30 years of increasing popularity of phylogenetic analysis the problems and potential problems have burgeoned, not yet stabilized to kind of constant, simple flux of methodological change in other fields.

 

Only a proportion of these assumptions affect any one study, yet even one problem can contribute significantly to uncertainty in any molecular analysis. The user taxonomist should determine, to the extent possible, which assumptions are relevant, and how robust to each assumption are the published results, i.e. that there is either no change in branch arrangements of interest or, if so, whether the change is at a probability high enough to make the published arrangement unreliable. Commonly, insufficient data is provided in the original paper to even begin to do this adequately. The present contribution uses a general correction factor as a way around this problem.

 

Of the many unaccounted assumptions listed above, at least 10 in any study are here viewed as introducing uncertainty that may affect (change) any of the internode branch arrangements in published cladograms at 1 chance out of a thousand, requiring a correction by total reduction by 0.01 in the probability (the 4-taxon BCI translated from the BP or DI, or the original BPP). This is not insignificant, being 0.20 of the potential range of reliability (0.95 and above). Specific problems may, in addition, be identified for particular arrangements or lineages that call for increased concern for reliability. Calculating the probability of each assumption is beyond the present paper, but is an eventual and necessary goal in phylogenetics.

 

GENERAL COMMENTS

It should be noted that morphological analysis has its own set of assumptions and contingencies. What is does not have is representation of a taxon by an exemplar, but by a large sample whenever possible.

 

According to Graham et al. (2002), “Incorrectly rooted trees may result in profoundly misleading evolutionary and taxonomic inferences, and this may be a relatively widespread phenomenon in phylogenetic studies.”

 

Álvarez. and Wendel (2003), discussing ribosomal ITS sequences,cited  involved with the use of ITS sequences for phylogenetic analysis. Homoplasy is shown to be higher in ITS than in other DNA sequence data sets, most likely because of orthology/paralogy conflation, compensatory base changes, problems in alignment due to indel accumulation, sequencing errors, or some combination of these phenomena. Despite the common use of ITS sequence data in phylogenetic studies, the complex and unpredictable evolutionary behavior reduce its usefulness for analysis. The authors suggested that better results may emerge from the use of single-copy or low-copy nuclear genes.

 

The possible capture of bryophyte mitochondrial genes by the apparently basal vascular plant Amborella is discussed by Bergthorsson et al. (2004).  Stace (2005) indicated that “different sequences contributed by different parents of allopolyploids, e.g. ITS sequences representing only the female parent” and wrote that APG needs to be reexamined; hybridization (including chloroplast capture) contributes to horizontal gene transfer, is common, and contributes to speciation”; and Stace reviewed horizontal transfer of mitochondrial genes between angiosperms, between angiosperms and gymnosperms, and angiosperms and bryophyta (possible with Amborella). Apparently 50% of angiosperms are polyploids. ITS and chloroplast genes are not reliable crosschecks (should use single- or few-copy sequences).

 

Genes are thought to provide informative characters, mainly through variation of synonymous codons (signaling the same amino acid), and such traits generally mutate slower than those of non-coding sites. Synonymous codons, however, are subject to codon bias (differential selection from a pool of transfer RNAs) and this must be checked for (Kellogg & Juliano, 1997). One might consider, however, the fact that phylogenies inferred from non-coding genes apparently agree well with those from protein coding genes, and that variable sites in coding genes mutate slower. Both might be explained by selection, on the non-coding genes (if indirectly) for the first instance, and on variable sites in genes in the second.

 

Stationarity (which implies that the marginal probabilities of the four nucleotides remain constant over all nodes of some one tree, i.e. they are all equally probable all the time), homogeneity (which implies that the instantaneous rate matrix is constant over a branch or the entire tree), and reversibility (which implies that the probability of sampling a given nucleotide from the stationary distribution and transitioning to another nucleotide is the same when reversed) are assumptions often violated by data, and result in increased uncertainty (Ho & Jermiin 2004).

 

Possible reasonable alternative arrangements introduced by research program assumptions often not fully accounted in the study must be addressed by multiplying the posterior probability of a branch arrangement by 1 minus the chance of being incorrect if the assumption is wrong. In the present analysis, ten one-in-a-thousand chances of being wrong are assumed likely and together require a 0.01 reduction in reliability for each BPP or BCI. Other researchers needing to interpret published cladograms may be less generous in assessing this penalty when dealing with studies heavily laden with assumptions.

 

The way to reassure the users of the results of molecular phylogenetic analysis that results are sufficiently reliable to solve problems that were identified in previous morphological study or with other loci is to address all the possibly relevant assumptions and conditional features of the analysis, and give the rationale for which particular assumptions are so definitely correct that they are essentially 100% reliable, and to state when an assumption is not definite and how that affects the ultimate probability of the branch arrangement being correct. Otherwise the onus of guesswork is laid upon the secondary user taxonomist or biogeographer or biodiversity triagist, not the analytic phylogeneticist.

 

reanalysis

This paper deals mainly with results that can be inferred from the data as originally published. Sometimes re-analysis allows additional evaluation of the degree of uncertainty involved. The Werner et al. (2004) Bayesian tree is somewhat different when re-analyzed (the Nexus file kindly supplied by O. Werner) with additional replications of the MCMC chain. The original analysis was done with 400,000 replications (discarding the first 100,000 for burn-in). A second analysis with the same data set and under the same conditions (again with MrBayes ver. 2, rates equal, no correction for invariants) but with 1 million generations (discarding the first 100,000) found 12 differences in branch arrangements among the 73 taxa, or a 17% additional uncertainty factor. These differences were not the same as Werner et al. (2004) found between the original cladogram (of 400,000 replications) and his maximum parsimony analysis with PAUP. One might just discount the original cladogram as published and chose the new, apparently superceding cladogram, but any reanalysis is a new study that requires considerable sophistication, including re-examination of the model and other assumptions, and is inappropriate for a taxonomist merely trying to find reliable guidance from the literature. As an aside, however, any conclusions regarding the phylogenetic position of Timmiella, Ephemerum, Rhabdoweisiaceae, and Hypodontium are not overturned by the re-analysis.

 

Problems introduced by sample error, ambiguous alignment, wrong evolutionary model, and other technical difficulties are complex and often can be addressed only by the molecular phylogeneticist. For instance, the practice of using 0.95 confidence limits in establishing one or more parameters during model selection seldom if ever details the loss in reliability (BPP) if wrong; up to 0.05 of the time the parameter is wrong and may or may not affect a branch arrangement of interest. On the other hand, a taxonomist can re-identify suspect species used in analyses (see review of this problem by Holst-Jensen et al., 2004), and can look for assurance that alignments were unambiguous such that branch arrangements of interest were not affected by slight shifts in alignment or different gap costs.

 

Phylogenetic analysis may in a proportion of cases be using wrong, incremental models of evolution that treat convergence or parallelism based on identical or nearly identical reactivated gene complexes as homoplastic and as not providing information about evolution. Patterns of homoplasy may indicate in fact underlying developmental similarities or homologies that reflect the shared heritage of a particular clade (Lockwood & Fleagle, 1999).

 

Not all non-coding DNA (“junk”) may be the source of neutral mutations that allow a null of random fixation of sequence changes tracked by phylogenetic analysis because there is apparently considerable selection pressure on non-coding but apparently regulatory or promoter sequences (e.g., strongly conserved sequences), or on apparent pseudogenes that are actually functional because of repair by RNA editing (Hollyoake et al., 2005), and so on. In some cases, coding and non-coding data support different topologies, possibly due to secondary structure (Rohwer & Rudolph, 2005). Non-coding DNA apparently evolves faster than exon DNA and may not reflect deep phylogenetic relationships, but coding DNA is subject to codon bias (Christianson, 2005), which must affect new methods of analysis using protein motifs (Moses et al., 2003; Rice et al. 2005).

 

STEP FIVE: Combine two or more contiguous branch arrangements of PBAs of 0.50 to 0.94 into one of 0.95 or higher using the formula for implied reliable internodes (FIRI).

 

increasing reliability by combining internodes

Branch arrangements on internodes in published cladograms often do not attain high reliability, but lower reliability measures or even none at all in an optimal tree are commonly considered informative. The rationale is that lineages at a distance across several internodes, each perhaps only weakly supported, may be intuitively evaluated as evolutionarily distant, that is, that at least one reliable or true internode exists between the lineages against the null hypothesis of a star (no internode).

 

An implied reliable credible interval (IRCI) measure may be devised that evaluates the chance that there are one or more correct internode branch arrangements (on the same cladogram) between two lineages connected by two or more internodes each of probability (BCI or BPP) less than 0.95. The formula concerned is a direct and simple probability calculation (Ash, 1993: 45, 291) replacing the overly complex method of Zander (2003). It is the chance of one or more successes in many probabilistic trials with each trial at a different probability of success.

 

The chance of at least one die showing 1 pip at least once among two throws is one minus the product of the chance both throws will not show 1 pip, or 1 minus the product of 5/6 times 5/6. If the BCI or BPP is the chance that an internode branch arrangement is correct, then 1 minus BCI or BPP is the chance it is incorrect (Q). Thus the formula

          IRCI = P exists = 1 – Qx

 

represents the probability that at least one correct internode exists among several contiguous internodes of only moderate probability, where there are two or more internodes i through j each with a BCI or BPP of 1 minus their probability Q of being incorrect (i.e. random generation). The P exists is here termed the implied reliable credible interval (IRCI). For example, with the above formula, two chained internodes at a BCI of 0.78 will reach or surpass 0.95 (actually IRCI = 1 minus the product of 0.22 × 0.22 = 0.952); three internodes at 0.64 each will reach or surpass 0.95 (IRCI = 0.953); and four internodes at 0.53 reach or surpass 0.95 (IRCI = 0.951). For two or more internodes to be true at once, which is necessary to establish reliable distance in terms of two or more internodes, the product of the BCIs or IRCIs must reach or surpass 0.95. A composite internode with IRCI of 0.95 or higher is here termed an IRI or implied reliable internode.

 

At the Res Botanical Web site (http://www.mobot.org/plantscience/resbot/index.htm), a “Silk Purse” MS Excel spreadsheet is available for downloading that will allow the quick calculation of IRCI values from multiple chained internodes. If morphological arrangements agree or even if they differ, either the morphological or the molecular internode probability may be used as a prior for the other in Bayes' Formula (use the Silk Purse spreadsheet also for easy Bayes' Formula calculation), which gives a posterior probability. Use 1 minus the probability of being correct when an arrangement differs from another as the maximum possible probability that it agrees.

 

With the IRCI measure, exact evaluation of support for internal branch arrangements may be extended across the cladogram. For evaluating published data, previously published measures of reliability of internal branch support, given as the BP or DI, should be translated to BCI values (e.g. as calculated for 0.95 BCI by Zander, 2004), while the BPP can be used directly. With more broadly based charts of such values (Tables 1 and 2), taxonomists can review published phylogenetic trees and evaluate the reliability of lineages separated by internodes of <0.95 BCI or BPP. The frequentist attitude of ignoring arrangements with CIs less than some particular significance level wastes probabilistic information.

 

Sometimes multiple-test problems can be ignored, as when only a small portion of a cladogram is evaluated and phylogenetic distance is not important. But, consider 20 branch arrangements at 0.95 CI each. About one of every 20 arrangements is then determined by chance alone, and the remainder are correct but one cannot tell which internode arrangement is incorrect. The chance that all 20 are correct at once is 0.9520 or 0.36.

 

In the frequentist context, any requirement that 2 or more arrangements have the equivalent of 0.95 CI at the same time requires Bonferroni correction (the α is divided by the number of tests then subtracted from 1 to yield the minimum CI required) or, an equivalent, such as requiring individual probabilities to have a product reaching 0.95. Correction for multiple tests would be needed for biodiversity analyses that evaluate patristic distance as number of nodes (e.g., Webb, 2000) with a view to distinguishing evolutionarily close and distant taxa at a 0.95 CI. For two internode arrangements to have the equivalent of a 0.95 CI at the same time, they must each have a raw 0.975 CI (the product is 0.95). For 20 internode arrangements to each be equivalent to 0.95 BCI at the same time, a Bonferroni correction dividing the α (chance by chance alone) of 0.05 by 20 yields a required 0.9975 probability for each internode (also, 0.997520 = 0.9512). In the Bayesian context, however, the chance of a set being true at the same time is the simple product of the likelihoods of each element of the set, or the joint probability.

 

The use of the IRCI formula to generate a single composite reliable internode from two or more less-than-reliable internodes between two lineages of interest is restricted to those branch arrangements that reflect a demonstrable non-random degree of shared ancestry, i.e. above a BCI, BP or BPP of 0.50. Use the IRI formula to combine two or more internodes that deal with different taxa, but instead use Bayes' Formula to combine two or more internodes with the same taxa (e.g. from morphological or studies of different loci).

 

Cladograms constructed with BCIs and IRCIs are serial representations of branching patterns, and some internodes with less-than-reliable probabilities may not be represented in them. For example, four contiguous internodes associated with ((((AB)C)D)E)F,G of 0.98, 0.85, 0.98 and 0.85 BCI would be represented on a corrected cladogram of these branch arrangements with two reliable internodes, yet when two lineages of interest are separated by those four internodes, the two 0.85 BCI can be combined probabilistically into a third IRI of 0.98, but that internode cannot be placed anywhere in the cladogram (it is isolated from the remainder of the cladogram) and serves only to reliably help separate lineages (AB) and (FG). Therefore, critical evaluations of patristic distance must use the reliability values as published after corrections for uncertainty but before IRCIs are calculated when representing the whole tree.

 

STEP SIX: Make corrections for multiple tests.

 

Savolainen et al. (2000) split the Flacourtiaceae into Salicaceae and Kiggelariaceae (Fig. 1), which results in a multiple test problem entirely different from that of the Euphorbiaceae (detailed in the hardcopy of Nine Easy Steps). Examination of Figure 2 indicates that the Salicaceae and Kiggelariaceae, though widely separated in the original cladogram (Fig. 1), are doubtfully more than sister groups. Molecular coherence of large internal portions of these families (joint probability of 0.95) plus phytochemical distinctions combine, however, in rather reliably distinguishing them.

 

Figure 1. Savolainen et al. (2000) cladogram of Malpighiales.

 

 

Figure 2. Operative transform of above cladogram, multiple tests restricted to Flacourtiaceae problem. Joint probability of 0.98, 0.97, 1, 1 and 1 yields 0.95, but none is more than a basal branch of any other lineage.

 

 

Felsenstein (2004: 299) asked whether selecting for the most probable branch arrangement might not involve multiple tests. There are at least two situations that seem to assert this problem. (1) Most cladograms have less than 0.95 BPs and these imply the presence of support for alternative branch arrangements, even when BPPs for the same data set are very high. Also, my simulations show that BPPs grow quite high because of sites that vary but do not have phylogenetically informative data. This is true even when the sites that do not vary at all in the whole data set are ignored with propinvar, because any subset has some sites that vary for other data but not for the subset. The problem is that although the optimal branch arrangement may be supported by data that are almost impossible by chance alone because data is paired at those sites and not at empty sites, it is also true that contrary data also is paired at those site and not at empty sites. Thus, in the simulations with 5:4:4, the sites with 4 pairs of shared traits are all almost impossible by chance alone with many empty sites, yet the sites with 5 pairs of shared traits are chosen by MCMC correctly as much less probable by chance alone, so much less that the arrangement can be 1.00 BPP. In truth, there are three strong phylogenetic signals and the reliability of the optimal arrangement is probably better represented by maximum parsimony and (double) bootstrapping.

 

This is a multiple test problem because the data strongly supporting 4:4 are thrown out as so much less probable, when in fact they pass any test for probabilistic support of the less than optimal branch arrangements. (2) Although sequence data may show considerable support for some branch arrangements, the arrangements that have < 0.50 BP or BPP are ignored. These arrangements may be incompatible with the cladogram and may be much affected by random data, and as such cannot support any particular arrangement. On the other hand, random data is the null hypothesis and data that cannot be distinguished from random cannot be dismissed as random merely because it is convenient to do so. It is quite possible that a hard polytomy or very short branch is the correct result, and there is truly less than 0.50 CI support for the optimal branch arrangement. Because the uncertainty cannot be eliminated by accepting the null hypothesis as true, any branch arrangement with > 0.50 BP or BPP must be evaluated in light of any previous study of data that do reflect signal for at least some branch arrangements but which support that branch arrangement at less than 0.50 CI. Thus, one minus the BP or BPP of a prior contrary branch arrangement might be the maximum support for a particular new arrangement, or an average of 0.25 BP or BPP might be assigned as the prior support for a new branch arrangement by a prior study with the same arrangement but less than 0.50 BP or BPP.

 

Using the Bayes’ Formula to find BPPs for two or more studies is improper if BPPs of < 0.50 are all eliminated. Consider a coin suspected of being loaded on tails. We flip the coin 10 times, then again 10 times until with have 10 sets of 10 flips each. If we eliminate all data sets that show tails coming up 5 times or less, and use Bayes’ Formula, we quickly demonstrate that the coin is loaded. In the same manner, the chi-squared distribution of reliability measures of branch arrangements from randomized phylogenetic data sets requires that testing multiple studies with Bayes’ Formula or any other analysis no allow rejection of data sets that result in < 0.50 CI.

 

Multiple test problems may occur when selection is made on the chance of being correct, effectively rejecting data sets that support lower probabilities for a given result. For example, flipping many coins many times to determine if loaded will result in several coins coming up heads several times in a row even if all coins are fair. In the context of the number of coins analyzed, this is to be expected. But selecting only that group of coins with the data that generates a high reliability measure and reanalyzing will falsely show high reliability out of context, and the high possibility of this being random data is hidden. In phylogenetic analysis, preselecting taxa for study based on morphological analysis can associate taxa with random changes in gene traits that are reflected in morphology, and reliability measures associated with molecular analysis of just that group are impacted. The same is true with all molecular data in that both maximum parsimony and likelihood analysis will group taxa, sometimes on the basis of imbalanced but random data, and each group is a preselection for succeeding studies. If the first molecular analysis resulted in a BPP of 0.95, then the second analysis based on just this group to reach 0.95 BPP requires Bonferroni correction (i.e. a BPP of 0.975 is needed), such that both analyses are correct at the same time. Any preselection of taxa is a candidate for examination for introduction of multiple test problems.

 

In addition, for A and B the same out of 800 variable sites, note that the taxa for A and B the same are preselected, and therefore there are multiple test problems. Even if the software finds taxa A and B are close enough for the two matching sites to matter, this may be because the two sites are randomly identical out of many taxa (note birthday problem with 1 out of 23 persons with same birthday in a group). The use of maximum parsimony or MCMC with propinvar is therefore recommended. A data set of 50 taxa and 50 random 2-state characters was contrived with RANDSET. Analysis with PAUP* under maximum parsimony (hs with 20 random sequence additions) produced 159 equally parsimonious trees and a largely unresolved strict consensus tree with distinct two lineages (A,B) and ((C,D)E). An analysis of a subset of these 5 taxa, under maximum parsimony (bandb), produced one lineage of ((C,D)E with 0.58 BP support for (C,D) and 0.81 BP for (C,D,E). Bayesian MCMC analysis (MrBayes 3.1, datatype = standard, ngen = 500000) of the 5 taxon data set provided 0.74 BPP and 0.94 BPP for the same groups. Clearly, preselection of a subset on the basis of a reliability measure for further analysis introduces multiple test problems.

 

In addition to the references given in the hardcopy of Nine Easy Steps (Zander, 2007a), a more complete mathematical explanation of statistical rationales for dealing with multiple test (multiple comparison) problems is given by Hogg & Craig (1978). Their orientation to the problem is frequentist, and further biased in favor of statistical power, not reliability, e.g., “After all, it is still a very strong statement to say that the probability is 0.75 that all these events occur.” This may be compelling in situations in which only a majority of events are needed for a conclusion, but not in a situation wherein all events must take place (or results be correct). Note that 0.77 is the probability that five branches, each at 0.95 probability, are all correct at once; six correct at once is 0.74. Thus, a declaration that five or six of such reliable branches are all reciprocally monophyletic will be wrong about one out of four times.

 

STEP SEVEN: With molecular data alone, reliable demonstration of paraphyly will split a taxonomic group, i.e., distinguish two groups as warranting separate names.

 

See hardcopy paper for discussion.

 

STEP EIGHT: Use Bayes’ Formula to combine separately published branch arrangements based on different data.

 

traditional and uncontested arrangements

One cannot have a purely molecular cladogram because morphology is used to select a related set of OTUs. Thus, a morphological prior of apparent close monophyly (close enough that characters can be compared and the outgroup is meaningful) is assumed and this must add to certainty or reliability in some lineages. Much has been written about combining data sets or otherwise dealing with combining information on phylogeny. Zander (2001a), following a rationale that each gene tree is a single character providing equal evidence of species phylogeny (Doyle, 1992; Slowinski & Page, 1999), suggested that three genes are the minimum necessary for agreeing branch arrangements to reach 0.95 CI when differential lineage sorting is a significant possibility. Scotland et al. (2003) and Olmstead and Scotland (2005) have downplayed the utility of morphology in phylogenetic estimation, yet groupings accepted on traditional bases are commonly supported by molecular results, are character-rich (Lee, 2004), may be unique if of palaeontological origin (Smith & Turner, 2005), and thus should play a part in evaluating reliability, while additional characters can increase BP support (Wiens, 2004). Morphological characters are not limited to those in the data set, but, unlike the case with molecular data, include all other characters used to phylogenetically isolate the taxa of the data set; there may be hundreds of characters and thousands of species involved. Although morphological traits have been mapped on cladograms obtained from exon sequences, one might wonder why the opposite might not be heuristic (see Page, 1998), given that both morphology and exons are affected by selection pressure. If branch arrangements based on exon sequences are different from uncontested morphology-based branch arrangements of the same taxa, then exon sequences should not be used to clarify problematic branch arrangements unless natural selection on gene sequences is somehow taken into account.

 

Chen et al. (2003) promoted the idea that repeatability of clades from separate analyses should be the primary criterion for establishing reliability, instead of bootstrap proportions from a combined total evidence data matrix. This is limited by the fact that the use of Bayes' Formula will increase the probability of a particular arrangement only if two results agree each at more than 0.50 probability because any one result (e.g. the prior) with less than 0.50 probability decreases the BPP. In fact, two repeated agreeing branch arrangements from different studies, each with 0.40 probability of being correct will, by Bayes' Formula, together refute the arrangement and the posterior probability is reduced to 0.31. Two cladograms merely agreeing in topology do not corroborate branch arrangements unless like arrangements from both cladograms have probabilities greater than 0.50. With morphology-based data sets, BPs of probabilistic utility may be obtained by partitioning the data set into small, presumed monophyletic groups that are thus stripped of much irrelevant homoplasy. Multiple-test problems may occur if a decision to use the results of combined, as opposed to separate, data sets is not made beforehand; and, it is improper to simply choose the gene data set combination that results in the highest confidence intervals, or, as is common, to ignore results with comparatively low confidence intervals.

 

Traditional, uncontested groupings based on either intuitive or cladistic morphological analysis for which there are no reasonable alternatives are here assigned a high probability of being correct, namely 0.95. When molecular results match such groupings inclusively, the degree of certainty of traditional arrangements serve as a Bayesian prior to the probability of the molecularly based arrangement being correct as BCI or BPP. Any agreement with branch arrangement probability greater than 0.50, thus, of the molecular results with 0.95 CI traditional results gives at least a total 0.95 probability by Bayes' Formula. If morphological and molecular monophyly does not match exactly (the usual case), and are doubtful and problematic, the traditional arrangement is assigned a “diffuse” or “uninformative” prior of 0.50 probability and the posterior probability is then necessarily (by Bayes' Formula) the same as the reliability value of the molecular study. If the uncontested morphological grouping is different from the molecular result but other morphological characters are found that support an alternative arrangement that agrees with the molecular result, there are two probabilities (0.05, the maximum support from the contrary arrangement, and 0.95 from the agreeing arrangement) that lead to a 0.50 probability by Bayes' Formula, which as a prior itself does not change the probability of the molecular result. If the traditional, uncontested arrangement directly contradicts the molecular result (e.g. two clearly closely related well-supported lineages are separated by one or more, clearly more distantly related lineages) and no morphological characters support the molecular result, the BPP can be little larger than 0.5. When morphology at the supraspecific level contradicts molecularly based branch arrangements (e.g. when “heterophyletic”), this is noted on the cladogram but molecular results are here tentatively followed.

 

Given that morphological results commonly agree with molecular results, and given a general lack of confidence in details of branch arrangements, at least at the species level, in morphological studies, direct contradiction where contrary morphological and molecular arrangements are both well supported is not common at least in the present examples. These admittedly broad assignments of probability to morphology-based branch arrangements are more than simply subjective Bayesian priors since they are founded on many, close empirical observations of morphological data. It is possible that one may calculate probabilities for internodes between uncontested morphological groups as 0.95 or 1.00 divided by the number of internodes, but since any prior probability less than 0.50 reduces (never increases) the BPP with Bayes' Formula, the effort is useless.

 

Some papers do different analyses of the same data or of data slightly modified, and report “better resolution.” This is generally not better resolution but simply selecting the highest CI from a range of CIs, which is a multiple test problem. This is the case even when there is consensus, because all CIs apply to that branch arrangement. Even with different data the selection of studies with “better resolution” can be misleading because previous data may indicate low support, and any support less than 0.50 CI by Bayes’ Formula must reduce the posterior probability.

 

The posterior probability of a branch arrangement can be modified by using the posterior probability of one result as the prior in Bayes’ Formula with the posterior probability of another study. Many studies will eventually overwhelm false or chance posterior probabilities, as noted in the Step 8 (Zander, 2007a). Another way of a correct result overwhelming an incorrect one is if the posterior probability of the correct branch arrangement is higher that 0.99, while the posterior probability for the same arrangement in the incorrect study one is low but not very low. It is a shame that very high posterior probabilities are commonly reported as 0.99+ rather than the actual value to, say, four decimal points, because that extra certainty exerts amazing influence through Bayes’ Formula. For instance, using Bayes’ Formula, combining posterior probabilities from different studies of 0.05 and 0.99 for a particular branch arrangement yields 0.84. But 0.05 and 0.999 yields 0.98. Again, 0.05 and 0.9999 yields 0.9981.

 

combining data

Regarding total evidence versus comparison of individual genes: total evidence combining molecular data sets in some cases may be better because one arrangement at high reliability in BPP by chance alone will never be refuted by a few additional studies that show ca. 0.50 BPP because no matter how many 0.50 priors there are, the high score does not change. Only when a low BPP by chance alone occurs can the high score be refuted, and therefore many studies must be made and low scores retained (not discarded if below 0.50). With total evidence, in the 4-taxon case with random data, doubling the data will chance the 0.98 BCI reliability of a chosen randomly occurring branch length of 7 where AB+AC+BC = 10 down to 0.96 in a branch length of 11 with AB+AC+BC = 20. This assumes the additional data is random but shared about equally by AB, AC and BC. Thus, for branch lengths of about 10 steps, rather highly reliable scores that may occur by chance alone are not corrected until the data set is more than doubled in size.

 

Combining data sets is liable to the problem of novel clades, being resultant clades that conflict (for instance, because they do not occur in optimal cladograms from any of the individual data sets). For instance, if support for pairs of three terminal taxa A, B, and C (AB:AC:BC) was 5:0:3 resulting in ((AB)C,D) being optimal, and a second study resulted in 0:5:3 for the same taxa resulting in ((AC)B,D) being optimal, combining the two data sets would give 5:5:6 or the novel clade BC in the newly optimal tree ((BC)A, D). The support for all three clades is equivocal, but consider the AB:AC:BC ratios 9:0:8 and 0:9:8, when combined resulting in 9:9:16. The two data sets separately produce poorly supported clades, but the novel clade is supported by 16 out of 34 at probability 1/3, or a BCI of 0.93. This must be, if possible, identified and avoided.

 

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Table 6. Data set demonstrating different results for maximum parsimony and Bayesian analysis, simplified format.

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Taxon  Numbers of advanced shared traits

___________________________________________________________

O    0  0  0  0

A   25  0 19  0

B   25  0 19  0

C    0 19 19 19

D   19  0  0 19

___________________________________________________________

 

Different software can create novel clades from the same data, for instance, the contrived data set (simplified) in Table 6, will result under maximum parsimony in PAUP* in ((AB)C)D,O with BP of 0.71 at both internodes, but under MCMC with MrBayes (datatype = standard, 1 million generations,  burnin = 1000) ((BD)A)C,O with BPP of 0.92 for (BD) and 0.54 for ((BDA)C. Paucity of data is not an explanation for this, nor is deltran/acctran problems. The tree of maximum parsimony is, also, the same as the BP consensus tree.

 

The practice of combining data should be restricted to data demonstrably generated by the same processes. In the present paper, published combined data sets were usually accepted but with the additional, doubtless justified, assumption that novel clades depending on strong imbalances in presumably randomly generated data (parallelisms) are rare.

 

One may note here that the exemplar OTU enters the analysis as a traditional taxon, with a morphological circumscription, and perhaps reproductive and environmental unique character. It leaves the analysis as (1) with morphological data sets: a lineage of parsimoniously arranged traits, at best forming a phyletic constraint or bound on evolution but no environmentally governed selection is involved, or (2) with nuclear loci: a genealogy of populations that are genetically isolated (though species may include many genetically isolated populations), and the exemplar represents a genetically isolated population, or (3) with matrilineally inherited loci (mitochondria and chloroplasts): a pedigree of individuals, and the exemplar post hoc represents an individual. Combining all these as “total evidence” mixes many different concepts of descent with modification; with the last two, neutral loci are what is modified. This may matter if the results contradict “accepted” classifications.

 

Given good data and no particular problems such as those detailed above, preliminary study indicates that, mathematically, total evidence (pooled data) analysis of contrived data sets gives about the same probability of branch arrangement as separate analysis of the data sets and later combination of results with Bayes’ Formula. This applies, however, only to data sets that are about the same size.

 

Combining data sets of different sizes may involve a variant of Simpson’s Paradox (Simpson, 1951; Wagner, 1982; Yule, 1903). The creation of a novel clade exemplified above is an example. The Paradox is a well-known feature of conditional probability that occurs when two data sets are combined and the reverse of that provided by each set alone is demonstrated, generally because some hidden factor not important in analysis of any single data set comes into play. For instance, in comparing the batting averages of two baseball players, when two years of baseball batting averages are combined, the player with the best averages for both years alone may have a lower two-year batting average. This is because other features of the data are involved when the data are combined, in this case different numbers of times at bat for each player in each year (Ross, 2004).

 

When morphological and molecular data are combined, the larger data set commonly swamps any data in the smaller data set supporting contrary branch arrangements. If the data is simply additive, this does not matter. But if the two different data sets represent two quite different evolutionary processes, then such swamping does matter and a wrong data set (e.g. different gene history) can overwhelm the smaller data set that represents true phylogenetic history. Simpson’s Paradox can be avoided by analyzing all data sets meant to be merely consilient, not additive (including nuclear and organellar DNA, which represent different processes), and using Baye’s Formula as in Step Eight.

 

An second-best alternative to separate analysis of data sets would be the standardization of the smaller data set by duplication as many times as needed to reach or almost match the number of parsimoniously informative traits as the larger data set. Results of such standardization, though making the smaller data set as important as the larger, affects the whole cladogram, and Step Eight can be focused on only one branch arrangement. Simpson’s Paradox has not been dealt with in phylogenetics, though much discussed in evolutionary ecology when data sets are combined. A paper on the phenomenon and its implilcations is being prepared (Zander, in prep.).

 

STEP NINE: Examine the potential value of recognizing paraphyletic taxa by tempering strict monophyly with evidence for taxonomic unity based on inferred functional evolution and non-dichotomous evolutionary processes.

 

According to Blomberg (1987), there are two approaches to historical criticism: evidentialism and presuppositionalism. The evidentialist applies accepted historical criteria to demonstrate reliability. The presuppositionalist first assumes reliability of data and method, then attempts to show that the data and method generate a consistent whole, confirming the presuppositions. Blomberg pointed out, regarding the latter method, that by definition a presuppositionalist cannot counter an argument from a different starting point. Present-day statistical phylogeneticists, in spite of assurances of abundant molecular data, are firmly in the presupositionalist camp, especially in rejection of morphology-based or especially traditional evolutionary hypothesis (which have a different starting point), and in developing detailed inferences from assumptions commonly not demonstrably valid at the necessary levels of probability though the resultant cladogram is exhibited as a consistent whole.

 

In the meta-analysis by Satta et al. (2000) of 39 hominoid loci, 23 supported the ((Homo Pan) Gorilla) gene tree, 8 supported ((Homo Gorilla) Pan), and 8 supported ((Gorilla Pan) Homo). Thus, 40% of the sampled Homo loci matched that of Pan, and 20% matched that of Gorilla. Many of these genes were exons, and subject to evolutionary pressure either alone or in concert with other genes, ortholog or paralog. It is possible that the reason Homo has many of the morphological traits of Pongo, the orangutan (Schwartz, 1988), a taxon surely distantly related, as far as gene data demonstrating inferred tree-like patterns of reproductive isolation do demonstrate, may well be the ease of (not re-evolving mutation by mutation but) desuppressing the silenced but now valuable adaptive throwback traits of a recent but non-sister member of the lineage. Thus, evolutionary sister lineages that reflect speciation may be different from the nearest neighbor on a gene or phylogenetic tree. If so, the evolutionary tree would be more similar to a patchy network than a neatly dichotomous gene tree or inferred optimal phylogenetic tree, and morphological convergence among at least fairly closely related species may be far less a problem than a point of illumination of true descent with modification rather than just descent.

 

DISCUSSION

An alternative to simply mapping species traits onto gene trees is the alternative process of creating reconciled trees by mapping gene traits onto species trees with, e.g. GeneTree (Page, 1998).

 

Examination of the Werner et al. (2004) data set shows that Anoectangium aestivum and Gymnostomum viridulum differ by seven sites (2 first position, 2 second, 3 third), but the two specimens of Splachnobryum obtusum differ by 21 sites (4 first position, 7 second, 10 third). I have examined the voucher specimens and they are not misidentifications. At first take, the two former species must be closely related, but the differences between the two exemplars of Splachnobryum obtusum indicate that variation within species can be heavy, which throws confusion into the evaluation of what exactly is the significance of patristic distance in classification.  Erthryphyllopsis fuscula and Erythrophyllastrum andinum differ by only six sites (3 first position, 1 second, 3 third), and, given the uncomfortable similarities of morphology, this second look probably requires the synonymy of Erythrophyllastrum R. H. Zander as a genus with Erythrophyllopsis Broth.

 

Position of Hypodontium

 

The number of internodes between Hypodontium dregei and the Pottiaceae subclade in the Werner et al. (2004) Bayesian cladogram is six, with five internodes higher than 0.50 (1.00, 0.65, 0.95, 0.78 and 1.00 BPP). This reduces to three reliable internodes each at 0.99 probability. All three internodes may be treated as correct at once because the product of the probabilities is 0.97. The number of internodes thus demonstrable as reliable are three, and Hypodontium is thus in this study reasonably extracted from the Pottiaceae, given strict monophyly.

 

The molecular analyses of Hedderson et al. (2004) based on the same gene provided both Bayesian and maximum parsimony analyses. Both species of Hypodontium were included and their joint lineage is treated here as a single taxon. In the maximum parsimony analysis, for the nine internodes between Hypodontium and the Pottiaceae subclade, the reliability values given were translated to 4-taxon BCIs with the tables. There were two JPs given: 0.92 and 0.74, taken as equivalent to BPs and translated to 0.97 and 0.83 BCI. These, corrected to 0.96 and 0.82 BCI allow only one reliable IRI, of 0.99 probability. Inference is possible only of a basal sister group lineage for Hypodontium with Pottiaceae. In the Bayesian analysis, seven internodes stood between Hypodontium and the Pottiaceae subclade, of which two, of BPP 1.00 and 0.69, were higher than 0.50. Again, in this study, only one IRI can be inferred, of 0.99 probability. The difference between the two studies lies in the recognition of a second reliable internode in the first study (needed to infer an intermediate family).

 

Zander's (1993) treatment of Hypodontium  (see detailed descriptions of species by Magill, 1981) accepted it as Pottiaceae, with some qualifications and provisos. Traditionally, the genus Hypodontium has been placed in the Calymperaceae. Reese et al. (1986), in a review of the Calymperaceae, suggested this genus might be pottiaceous. Edwards (1980) accepted Hypodontium as Calymperaceae, and did not discuss it extensively. A literature review and examination of a series of collections of H. dregei and H. pomiformis (Zander, 1993) indicated that Hypodontium lacks many of the central characters of Calymperaceae.

 

Hypodontium is unusual in both the Pottiaceae and Calymperaceae by the inner perichaetial leaves sheathing proximally but narrowly subulate or awned apically. The genus is distinctive by a combination of uncommon traits. It is molecularly closely related to the morphologically totally different genus Fissidens (Fissidentaceae), is patristically distant from the Pottiaceae, and thus may deserve family recognition. The Australian genus Calymperastrum may also belong in this relationship though now placed with the Pottiaceae, as it, too, seems transitional morphologically between the Pottiaceae and the Calymperaceae. There is no reliable support from molecular analyses, however, that distinguish, at the required two internode distance, Hypodontium from the Calymperaceae. Given the many problems with molecular taxonomy (separately reviewed, for instance, by Jenner, 2004), however, taxonomists hoping for guidance from molecular studies should settle for nothing less than highly reliable results in these simple a posteriori calculations. Thus, proposing here a new family to contain Hypodontium would be premature, and the genus must provisionally be considered returned to Calymperaceae.

 

Position of Ephemerum

 

The deep nesting of Ephemeraceae species within a large clade of Pottiaceae species is certainly cause to re-evaluate the traditional distinction of the family from the Pottiaceae. Nomenclatural recognition of paraphyletic taxa (Nordal & Stedje, 2005) is in my opinion perfectly acceptable as long as characters are available that support a new or strong evolutionary direction. Such traits do not characterize the Ephemeraceae, however, which may easily be viewed as strongly reduced members of the Pottiaceae, and a priori nesting is acceptable. As an exercise, however, the statistical reliability of this nesting in the Werner et al. (2004) study may be examined. For Ephemeraceae to be a member of the Pottiaceae, all that need be demonstrated is at most finding one reliable internode (or creating at least one IRI) between the apparently nested Ephemeraceae and the branch below the lowest branch of the Pottiaceae subclade (such that the position of Ephemerum is distinguishable from that of the basalmost taxon, Scopelophila, at the 0.95 CI level). If reliable internodes can be demonstrated to show that Ephemerum is more highly nested than being a basal branch, all the better. A priori, there is no contrary hypothesis based on morphology to be strongly refuted. The Bayesian posterior probabilities larger than 0.50 supporting the nesting of the single species Ephemerum spinosum in the Pottiaceae subclade of the Werner et al. (2004) study are: 1.00, 0.57, 0.97, 0.53 and 0.66.  Reducing each by 0.01 for unaccounted assumptions yields the series 0.99, 0.56, 0.96, 0.52 and 0.65. Concerning only this one gene study, then, the nesting is supported by two reliable internodes, there being one IRI of 0.99 probability and another IRI internode at 0.99 (combining 0.56, 0.96, 0.52 and 0.65 by the IRCI formula). For the two internodes to be true at the same time, the product is needed, and that provides a 0.98 final probability. This analysis, conditional on the data set and with considerable correction, shows that Ephemeraceae represented by the exemplar Ephemerum spinosum is reliably imbedded in the Pottiaceae by at least two internodes.

 

In the Werner et al. (2004) maximum parsimony analysis of the same data set, Ephemerum was near the bottom of the Pottiaceae clade, just above Scopelophila, which held the lowest branch. No BP was given for the internode between Scopelophila and Ephemerum, and the two branches are thus interchangeable in position such that Ephemerum would be just as probable to occur outside the Pottiaceae subclade, at least as a basal branch. The internode just below Ephemerum had a BP of 0.67, which translates to a BCI of 0.77 at that branch length. In the maximum parsimony cladogram, one 0.95 IRCI composite internode can be inferred between Ephemerum and 40 of the 51 Pottiaceae taxa, but 11 pottiaceous taxa plus an additional 18 non-pottiaceous taxa (most of the more basal members of the cladogram) are statistically indistinguishable from Ephemerum in regards to monophyly. The maximum parsimony study is therefore ambiguous, and offers no reliable guidance as to the position of this family. Ephemeraceae, however, may be considered a synonym of Pottiaceae with no major conflict of empiric evidence. Goffinet and Cox (2000) discussed the position of Ephemeraceae and commented that its genera could very well be transferred to the Pottiaceae based on morphological data; this translates to a 0.50 (ignorance) prior chance that Ephemeraceae is (by morphology) embedded in Pottiaceae rather than not, which as a Bayesian prior does not affect the posterior probability.

 

According to Lynch and Force (2000),  after gene duplication, “one member of the pair will usually become silenced by degenerative mutation. Such nonfunctionalization is expected to occur within a few million generations because the rate of mutation to null alleles in on the order of 10-6 per generation . . . .” “It is now known that most eukaryotic genomes harbor large numbers of functional gene duplicates, many of which originated tens to hundreds of millions of years ago . . . .” “The high rate of duplicate-gene preservation observed for genome duplication events (commonly on the order of 20--50%) for such long periods of time suggests that some type of positive selection must be offsetting the high rate of production of null alleles . . . .” Note that they ignore methylation silencing and DNA repair. Lynch and Force continue that for small populations, “we expect silencing of one member of a duplicate pair to occur in less that a million generations or so, whereas extremely large populations may harbor active pairs of gene duplicates for tens of millions of generations.” Subfunctionalization, in sum, preserves gene duplicates. “Subfunctionalization is most likely to occur when the effective population size and the coding null mutation rate is low enough that a new loss-of-function mutation arises in the population every five generation or less . . . .” “Some evidence suggests that gain-of-function mutations may be quite common, perhaps as common as loss-of-function mutations . . . gain-of-function mutations may prolong the life of gene duplicates by resurrecting previously impaired copies.”

 

According to Trifonov (1999), “there are sequence designs that promote evolution. One such design suitable for fast adaptation is the tandem repetition of identical sequences, so that their copy numbers in the repeat arrays would modulate (tune) the expression of nearby genes. The tandem repeat expansion diseases illustrate this mechanism in a dramatic way: overtuning of the respective gene expression leads to the disease.” The italics are mine, and here Trifonov suggests that tandem repeat expansion produces evolutionary tuning knobs.

 

THE FUTURE

Phylogenetics excises the environmental aspect of taxonomy, and taxonomy is poorer for it, therefore we should regenerate some form of the environmental species concept and use inferred genealogies as simply additional information.

 

Evolution asserts a process, and the null hypothesis would be no process. Not falsifying the null does not support neutral evolution (drift and exons, or accumulating neutral base changes), but the ecological species concept might help with falsification of the null to maximize how taxonomy reflects process (e.g. selection) to the extent there is process. One might project an eventual classification based on comparative evolutionary ecology, which would fit well with the species concepts commonly used in journals devoted to biodiversity analysis. The basic unit of taxonomy should match the basic unit of biodiversity. Phylogenetics, focusing on inferred genealogies of isolated populations (nuclear loci subject to recombination) or individuals (haplotypes), is only a part of the picture.

 

It has been said that if you define a problem to suit yourself, then the solution will also suit you; and the reverse is true that if you have a solution, it may be pleasing to define a problem to suit that solution. Thus it is that, with the development of the idea (meme) of the cladogram (which presents an internally consistent and logical conclusion based), evolution has been redefined as any changes in hereditable traits and then focus made on non-coding loci (selection and even drift of expressed traits ignored, or perhaps “mapped”), speciation now consists of dichotomous splits in mutating loci, the artificial stricture of monophyly based on such splits is used for classification without regard for a paraphyletic representation of evolution when there is no inferred change in expressed traits, and so on in the new paradigm that now constitutes a systematics out of touch with other fields (biodiversity study, biogeography, evolution), which continue to focus on expressed traits (Zander, 2007b, 2007c).

 

SUMMARY

It is hoped that the present demonstration of uncertainty in molecular analysis, in spite of (1) the use of IRCI to combine internodes to create fewer but more reliable ones and (2) using Bayes' Formula to combine morphological and molecular results into reliable branch arrangements, will encourage the use of morphological traits as a rigorous part of phylogenetic analysis and associated studies, as is promoted by, e.g., various contributors to “Deep Morphology” (Stuessy et al., 2004), Jenner (2004), Zander (2001a), and others. Given the many unaccounted assumptions, it no longer is a problem that morphological analyses do not have the apparent reliability of molecular analyses, but that molecularly based analyses may be little more reliable than the morphological. A recursive (reciprocal illumination) method employing both morphological and molecular data is warranted.

 

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Corrections to hardcopy paper in Annals of Missouri Botanical Garden 94: 691--709:

Page 693: The problem with Bayesian Posterior Probabilities are that they are too high at short branch lengths, not too low as indicated in lower right of page.

 

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