Frequent Inconsistency of Parsimony Under a Simple Model of Cladogenesis

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1 Syst. Biol. 52(5): , 2003 Copyright c Society of Systematic Biologists ISSN: print / X online DOI: / Frequent Inconsistency of Parsimony Under a Simple Model of Cladogenesis JOHN P. HUELSENBECK 1 AND KATHERINE M. LANDER 2 1 Section of Ecology, Behavior and Evolution, Division of Biological Sciences, University of California San Diego, La Jolla, California , USA; johnh@biomail.ucsd.edu 2 Department of Biology, University of Rochester, Rochester, New York 14627, USA Abstract. Although the conditions under which the parsimony method becomes inconsistent have been studied for almost two decades, the probability that the parsimony method would encounter conditions causing inconsistency under simple models of cladogenesis is unknown. Here, we examine the statistical behavior of the parsimony method under a birth death model of cladogenesis, when the molecular clock holds. The parsimony method can become inconsistent a high proportion of the time even under this simple model of cladogenesis. When taxon sampling is poor or rates of evolution are high, the probability that parsimony will become inconsistent increases. [Birth death process; Felsenstein zone; inconsistency; parsimony.] We know what the conditions are for inconsistency of parsimony for some particular four- and five-species models, and these suggest that the problem may extend well beyond those cases. Is this reason for complacency on the part of users of parsimony methods? J. Felsenstein (1988:534) A phylogenetic method is inconsistent if it converges to an incorrect tree as more character data are included in the analysis. Usually, the term inconsistency is applied to the situation in which a method fails to converge to the correct parameter value when the assumptions of the analysis are satisfied. Many methods of phylogenetic inference, including maximum likelihood, are consistent in this sense. However, in the phylogenetics literature, the term has been applied more broadly to include cases in which the assumptions of the analysis are not satisfied. This broader use has been applied probably because it is difficult to identify the assumptions of some phylogenetic methods, such as the parsimony method. Statistical inconsistency under a wide range of conditions is one of the more distressing properties a phylogenetic method can possess. All current phylogenetic methods can become inconsistent when their assumptions are severely violated (Felsenstein, 1978; Hendy and Penny, 1989; DeBry, 1992; Huelsenbeck and Hillis, 1993; Yang, 1994; Huelsenbeck, 1995). However, some phylogenetic methods appear to be inconsistent under a wider range of conditions than are other methods. For example, it is known that the parsimony method can converge to an incorrect tree for the four-taxon case in which two of the branches are much longer (in terms of expected number of substitutions per site) than the remaining branches (Felsenstein, 1978). For the four-taxon case, parsimony is inconsistent only when the molecular clock is violated. However, with more than four taxa, Hendy and Penny (1989) identified cases in which parsimony will be inconsistent even under clock-like conditions. The five-taxon case found by Hendy and Penny (1989) was further explored by Zharkikh and Li (1993). Finally, Kim (1996) identified cases in which parsimony is inconsistent when all of the branches of the true tree are equal in length. Although the conditions (i.e., branch lengths, model of character change, and tree topology) under which parsimony becomes inconsistent are well characterized for some specific cases (Zharkikh and Li, 1993; Takezaki and Nei, 1994; Huelsenbeck, 1995), it is generally unknown how often these conditions might arise. All of the examples of inconsistency of parsimony (or any other method) consist of specific combinations of parameter values or ranges of parameter values that lead to the method converging on the wrong tree. We have no information on how often phylogenetic methods are inconsistent under even the simplest process of cladogenesis. Ideally, of course, one would like to know how often phylogenetic methods are inconsistent for real data. Unfortunately, this information is impossible to obtain, because the actual phylogenetic history is unknown in all but a few of the simplest cases (see Fitch and Atchley, 1985, 1987; Atchley and Fitch, 1991; Hillis et al., 1992). However, one can ask how often a phylogenetic method is inconsistent for trees that are generated under a process of cladogenesis. Although such a study cannot show how often a method might become inconsistent for real data, it can provide insights into how easily conditions might arise that would cause a phylogenetic method to become inconsistent. Among the simplest models of cladogenesis is the linear birth death process (Kendall, 1948). Under this process, speciation and extinction occur at constant rates. Obviously, the birth death process is unrealistic in several ways. For example, we know that speciation and extinction rates are not constant through time or across lineages. However, generating trees under the birth death process has several advantages. First, the process is stochastic; one can generate many different realizations of the process under the same parameter values and investigate the average performance of a phylogenetic method for the combination of parameter values explored. This means that the results of the consistency or inconsistency of a method do not depend upon any single tree or set of branch lengths, having been averaged over many trees generated under the cladogenetic model. Second, the birth death process can produce trees that vary systematically in a number of important features. Probably the most important feature is the bushiness of the trees produced under the process. For example, the 641

2 642 SYSTEMATIC BIOLOGY VOL. 52 fraction of the available species included in the analysis is known to influence the lengths of the terminal branches relative to the internal branches (Rannala et al., 1998). Small taxon-sampling fractions generally lead to trees with long terminal branches relative to the internal branches trees that are typically more difficult for phylogenetic methods to accurately reconstruct. The important point is this: despite its limitations as an accurate representation of the actual process of cladogenesis, the birth death process does allow the exploration of the fraction of the time a phylogenetic method will be inconsistent; this information cannot be gathered from observation of actual data sets or by the types of analyses of phylogenetic consistency that have been performed to date. In this article, we examine the statistical properties of the parsimony method for a simple model of cladogenesis with the object of characterizing the probability that the method will produce inconsistent estimates of phylogeny. We explored the performance of only the parsimony method for a couple of reasons. First, the method is fast. It would be very difficult to examine the consistency of the more computationally expensive methods, such as maximum likelihood or Bayesian inference. Second, several studies have suggested that parsimony is more liable to be inconsistent than methods that attempt, however poorly, to correct for multiple substitutions that occur along single branches of the tree (Huelsenbeck, 1995). It is likely, then, that an analysis of the behavior of the parsimony method under a simple model of cladogenesis will provide the bleakest possible picture of the frequency of the inconsistency problem. If parsimony is only rarely, if ever, inconsistent under a simple model of cladogenesis, then other methods such as maximum likelihood are unlikely to be inconsistent too. Indeed, the claim that the consistency issue is a red herring (Siddall and Kluge, 1997) would be well supported; why worry about the consistency of the parsimony method (or any other method, for that matter) if it is difficult to create conditions under simple models that lead to its inconsistency? If, on the other hand, parsimony is found to be frequently inconsistent under a simple model of cladogenesis, it might motivate a more systematic study of the behavior of parsimony and other phylogenetic methods under more realistic processes of cladogenesis. per site, and (3) a mechanism of change for the characters along the branches of the tree. In this study, we generated the branch lengths and topology of the tree simultaneously. The birth death process of cladogenesis with taxon sampling (Kendall, 1948; Nee et al., 1994), a very simple model of lineage splitting, determined the times between splitting events on the model tree. Under the birth death model, the probability of a speciation event along any branch in a small time interval, dt,is λ dt, and the probability of an extinction event is µdt. The probability of two or more events occurring in an instant of time is of order o(dt). We used a variant of the birth death process in which only a random fraction ρ of the species alive at the present are sampled (Yang and Rannala, 1997). The s sampled species (of a total of S species; ρ = s/s) are related through s 1 speciation events that occurred at times t = (t 1,...,t s 1 ). We condition on the first speciation event (t 1 ) occurring at t 1 = 1. The remaining speciation events (t 2,...,t s 1 ), then, occur at times on the interval (0, 1). Yang and Rannala (1997) derived the distribution of speciation times t 2,...,t s 1 under the birth death process with species sampling conditional on t 1 and s, the number of species in the sample. They presented a simple method for generating realizations under the birth death process of cladogenesis. First, one generates a randomly branching tree (τ), keeping track of the order of the speciation events on the tree. The nodes on the tree corresponding to the speciation events leading to the s sampled species are assigned speciation times as follows: Sample s 2 times from the probability distribution where h(y) = λp 1(y) v 1, (1) p 1 (t) = 1 ρ P(0,t)2 e (µ λ)t (2) is the probability that a lineage that arose at time t leaves exactly one descendant, METHODS The general strategy used in this study was to generate randomly bifurcating trees under a birth death process of cladogenesis, calculate the probability of all possible character patterns for each tree, and then estimate phylogeny under the parsimony criterion assuming that the possible character patterns occurred in their expected frequencies. Generation of Randomly Bifurcating Trees A phylogenetic model consists of three parts: (1) the topology of the tree, (2) the lengths of the branches of the tree in terms of expected number of substitutions and P(0,t) = v 1 = 1 1 ρ P(0, 1)2 e (µ λ), (3) ρ(λ µ) ρλ + (λ(1 ρ) µ)e (µ λ)t (4) is the probability that a lineage that arose at time t leaves at least one descendant. We used the transformation from a uniform random variable on the interval [0, 1] to a realization from h(y) derived by Yang and Rannala (1997). We ordered the resulting speciation times from largest to

3 2003 HUELSENBECK AND LANDER PARSIMONY CONSISTENCY 643 FIGURE 1. Realizations of the birth death process of cladogenesis for seven species when the speciation rate is twice the extinction rate and the sampling fraction (ρ) is 1.00 (a), 0.10 (b), and 0.01 (c).

4 644 SYSTEMATIC BIOLOGY VOL. 52 smallest and assigned these times to the nodes on the tree in their order of occurrence. The advantages of using this method for generating realizations under the birth death model of cladogenesis include its computational speed and ability to explore situations in which taxon sampling is poor. Figure 1 shows several realizations of trees of seven species under the birth death process when the speciation rate is twice the extinction rate (λ = 2µ) and the taxon sampling fraction (ρ) is 1 (Fig. 1a), 0.1 (Fig. 1b), and 0.01 (Fig. 1c). The trees of s species generated by the birth death process have tips at time 0.0 and root at time t 2 = 1.0. We converted these branch times into branch lengths that were in terms of expected number of substitutions per site by multiplying each branch by a factor m. The parameter m is referred to as the tree height and is interpreted as the number of substitutions expected to occur per site on a single lineage that reaches from the root of the tree to the tips. All of the trees generated in this study satisfy the molecular clock assumption. There are a number of different ways that trees can be produced under the birth death process. For example, the forward simulation approach, implemented in programs such as Bi-De (Rambaut et al., 1996), simulates speciation and extinction events explicitly. However, this strategy is computationally expensive. For one, if the extinction rate is large, many of the simulations will die off before the desired number of species is produced. For another, it is difficult to incorporate taxon sampling in the forward simulation approach. For example, to sample s = 10 species when the taxon sampling fraction is ρ = 10 6, one would need to simulate a total of S = 10 million species and randomly sample only s = 10 of them (ρ = s/s). Yang and Rannala (1997) presented a method that allows one to generate valid samples under the birth death process conditioned on any combination of λ, µ, and ρ. The only disadvantage of the Yang and Rannala (1997) method is that it is possible to pick very unrealistic combinations of parameter values and still generate trees; the trees may be very improbable if generated under the forward simulation scheme. For example, if λ = 10, µ = 1, and ρ = 1, on average s = species would result using the forward simulation method. Very few trees with only s = 10 tips would be generated under this combination of parameter values. Yet, the method of Yang and Rannala (1997) would allow simulations of any number of tips (s) under the same parameter values. In this study, we adjusted λ and µ (always with the constraint that λ = 2µ) such that the expected number of species that would be generated under the forward simulation approach with taxon sampling would equal the number of desired tip species, s. We simulated trees of s = 5, 6, 7, and 8 species and varied the tree height, m, from 0.05 to 1.0. For each combination of tree height and sampling fraction examined, we simulated a total of 10,000 realizations of the birth death process, except for the eight-species case, where we simulated 4,000 realizations. Other models of cladogenesis may produce trees with different branch lengths on average from those produced using the birth death model. The birth death model, however, provides a reasonable and simple starting point for examining the behavior of the parsimony method. Calculation of Character Pattern Frequencies After generating a randomly bifurcating tree (τ) ofs species under the birth death model of cladogenesis, we calculated the expected frequency of all 4 s possible DNA sequence site patterns for that tree. As an example, consider a tree of s = 3 species. There are 4 3 = 64 possible site patterns (Fig. 2), denoted x 1 ={A, A, A}, x 2 ={A, A, C}, x 3 ={A, A, G},..., x 63 ={T, T, G}, x 64 ={T, T, T}. We assume that the process generating the sites follows the Jukes Cantor model of DNA substitution (Jukes and Cantor, 1969). This model is a continuous-time Markov process in which the rates of change among different nucleotide states are equal. The transition probabilities from nucleotide i to nucleotide j for the Jukes Cantor model are 1 p ij (v) = e 4 3 v, i j , 4 e 4 3 v, i = j where v is the length of the branch in terms of expected number of substitutions per site. We assume that the rate of substitution across sites is equal. The probability of observing the ith site pattern (x i )is a sum over all possible assignments of nucleotide states to the interior nodes of the tree. Let the branch whose child is node k and whose parent is σ (k) have a length of v k expected substitutions per site. Also, let y be a data vector of nucleotide states at the interior nodes of the tree. The probability of observing the ith site pattern is then f (x i τ, t) = y ( )( ) 1 s 2s 2 p yσ 4 (k) x k (v k ) p yσ (k) y k (v k ). k=1 k=s+1 (5) This approach for exploring the consistency of a phylogenetic method was first applied by DeBry (1992) and Yang (1994). Species 1 AAAAAAAAAAAAAAAACCCCCCCCCCCCCCCCGGGGGGGGGGGGGGGGTTTTTTTTTTTTTTTT Species 2 AAAACCCCGGGGTTTTAAAACCCCGGGGTTTTAAAACCCCGGGGTTTTAAAACCCCGGGGTTTT Species 3 ACGTACGTACGTACGTACGTACGTACGTACGTACGTACGTACGTACGTACGTACGTACGTACGT FIGURE 2. The 64 site patterns possible for s = 3 species.

5 2003 HUELSENBECK AND LANDER PARSIMONY CONSISTENCY 645 Determining the Consistency of Parsimony We wrote a computer program to generate trees under the birth death process of cladogenesis, using the technique described above. We used the program PAUP (Swofford, 2002) to calculate the pattern probabilities and to find the most-parsimonious tree. For a particular realization of the birth death process (a tree with branch lengths), the consistency of the parsimony method was determined by using the expected site pattern frequencies of the 4 s site patterns as the observations. An exact search was then performed among all possible trees, and the parsimony score for each was recorded. Unfortunately, we could not calculate the tree lengths with infinite precision and did not want the consistency of the parsimony method to be influenced by limitations in accurately representing real numbers on a computer. Hence, we decided upon a conservative scheme for scoring the parsimony method as consistent or inconsistent for each realization of the birth death process. We calculated the expected lengths of all trees, assuming that the tree lengths were calculated on a data matrix of 5,000 characters (this was done by simply multiplying each tree length calculated using the pattern probabilities by 5,000). If the best parsimony tree was the same as the tree used to generate the site pattern probabilities (the true tree) or the best parsimony tree was within one step (out of 5,000 characters), then the method was considered consistent for that particular realization of the birth death process. If, on the other hand, the-maximum parsimony tree was more than one expected step (out of 5,000 characters) worse than the true tree, we considered the method to be inconsistent. Note that at two extremes all branches zero in length or all branches infinite in length the parsimony method would always be consistent under this criterion. We are mainly interested in the intermediate cases where the amount of evolution is moderate. RESULTS The birth death model of cladogenesis used in this study produced trees with a variety of branch lengths (Fig. 1). As noted previously (Rannala et al., 1998), the terminal branches were longer on average when the sampling fraction, ρ, was small. In other words, for small values of ρ the trees are more starlike, with terminal branches radiating nearer the base of the tree. One consequence of this result is that the expected number of changes per site over the entire tree also changes with the sampling fraction parameter, ρ. As ρ decreases (as more of the time can be found in the tips), the overall time on the tree increases and the average number of changes per site increases. Parsimony is often inconsistent even under the simple model of cladogenesis examined here. Figure 3 shows the probability that parsimony will provide consistent estimates of phylogeny for different values of the taxon sampling fraction, ρ, and for different numbers of FIGURE 3. The probability that parsimony is inconsistent under a birth death model of cladogenesis when the taxon sampling fraction (ρ) varies from 1.0 to 0.01.

6 646 SYSTEMATIC BIOLOGY VOL. 52 taxa. For low rates of evolution (m < 0.2), parsimony is inconsistent only rarely (<1% of the time). However, we found conditions of higher rates where parsimony was inconsistent as much as 13% of the time. The number of taxa included in the analysis also affects the probability that parsimony will provide inconsistent estimates of phylogeny. Figure 4 shows the probability of inconsistent estimates for 5, 6, 7, and 8 taxa. As the number of taxa increases, the probability that parsimony will become inconsistent also increases. This is true despite the fact that the average length of the branches of trees with more taxa is smaller than that for few taxa. DISCUSSION Why is parsimony inconsistent for the model of evolution chosen in this study? One of the difficulties of the parsimony method is that the assumptions are not explicitly stated. It is difficult, if not impossible, to evaluate FIGURE 4. from5to8. The probability that parsimony is inconsistent under a birth death model of cladogenesis when the number of species (s) is varied

7 2003 HUELSENBECK AND LANDER PARSIMONY CONSISTENCY 647 the performance of the method for cases in which all of its assumptions are satisfied. However, it is clear operationally how the parsimony procedure works: reconstruct changes on a tree and treat that tree with the minimum number of reconstructed changes as the best estimate of phylogeny. The manner in which characters are reconstructed on the tree (in this case, assuming that characters are unordered) underestimates the actual number of changes. In our study, this underestimate could be quite drastic. The underestimate of the number of changes becomes progressively worse as rates of evolution increase. The results presented here are based on a very simple model of cladogenesis and character evolution. It is fair, then, to ask whether these results generalize to other conditions. We suspect that the general result that parsimony is inconsistent a small fraction of the time will hold true if more complicated models of cladogenesis or character evolution are considered. However, future studies could be designed to examine the importance of additional factors. These factors could include (1) mass extinction, (2) more complicated models of substitution, (3) violation of the molecular clock, and (4) increased numbers of species. In particular, it might be useful to examine the effect that mass extinction has on the probability that parsimony is inconsistent. The simple birth death model of cladogenesis used here is unrealistic in the sense that it assumes that speciation and extinction rates are constant through time. It is also unknown how the results of this study will scale for larger numbers of species. Even the eight-species analyses took months of computer time, prohibiting an analysis of nine species where there are 135,135 possible trees and 4 9 = 262, 144 possible site patterns. Studies of more than 9 or 10 species will probably require the development of new methods for determining the consistency or inconsistency of a method. For example, one could explore the consistency of distance methods by concentrating on only the distance matrix that is expected to result from a particular tree. This focus would eliminate the problem of using all of the site patterns but for large phylogenetic problems would only allow the investigation of optimality criteria using a heuristic search or methods that rely on stepwise addition algorithms to build up trees. It is clear that the consistency of a method often is relevant, especially for difficult phylogenetic problems. In nature, difficult phylogenetic problems probably involve short internal branches and are the most likely to resist resolution. The most commonly applied, and of course reasonable, approach to difficult phylogenetic problems is to collect additional data with the hope that an analysis of the additional data, in combination with previously collected data, can resolve the phylogeny. There are numerous examples of difficult phylogenetic problems in the literature (e.g., Huelsenbeck, 1998), and the consistency of a method is relevant here, as one would hope that the problem would be solvable, in principal at least, if sufficient data were collected. Here, we have shown that a very simple model of cladogenesis and character evolution can produce phylogenetic problems for which parsimony will converge on an incorrect tree. This was true even though the simulation conditions were generous in the sense that the molecular clock holds and the process of character evolution was the simplest time-homogeneous model of DNA substitution available. ACKNOWLEDGMENTS J.P.H. was supported by NSF grants DEB and MCB and by a Wenner-Gren Foundation scholarship. REFERENCES ATCHLEY, W. R., AND W. M. FITCH Gene trees and the origins of inbred strains of mice. Science 254: DEBRY, R. W The consistency of several phylogeny-inference methods under varying evolutionary rates. Mol. Biol. Evol. 9: FELSENSTEIN, J Cases in which parsimony and compatibility methods will be positively misleading. Syst. Zool. 27: FELSENSTEIN, J Phylogenies from molecular sequences: Inference and reliability. Annu. Rev. Genet. 22: FITCH, W. M., AND W. R. ATCHLEY Evolution in inbred strains of mice appears rapid. Science 228: FITCH,W.M.,AND W. R. ATCHLEY Divergence in inbred strains of mice: A comparison of three different types of data. 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