10643070
Philosophy and phylogenetic inference: a comparison of likelihood and parsimony methods in the context of Karl Popper's writings on corroboration.
Advocates of cladistic parsimony methods have invoked the philosophy of Karl Popper in an attempt to argue for the superiority of those methods over phylogenetic methods based on Ronald Fisher's statistical principle of likelihood. We argue that the concept of likelihood in general, and its application to problems of phylogenetic inference in particular, are highly compatible with Popper's philosophy. Examination of Popper's writings reveals that his concept of corroboration is, in fact, based on likelihood. Moreover, because probabilistic assumptions are necessary for calculating the probabilities that define Popper's corroboration, likelihood methods of phylogenetic inference--with their explicit probabilistic basis--are easily reconciled with his concept. In contrast, cladistic parsimony methods, at least as described by certain advocates of those methods, are less easily reconciled with Popper's concept of corroboration. If those methods are interpreted as lacking probabilistic assumptions, then they are incompatible with corroboration. Conversely, if parsimony methods are to be considered compatible with corroboration, then they must be interpreted as carrying implicit probabilistic assumptions. Thus, the non-probabilistic interpretation of cladistic parsimony favored by some advocates of those methods is contradicted by an attempt by the same authors to justify parsimony methods in terms of Popper's concept of corroboration. In addition to being compatible with Popperian corroboration, the likelihood approach to phylogenetic inference permits researchers to test the assumptions of their analytical methods (models) in a way that is consistent with Popper's ideas about the provisional nature of background knowledge.
29354147
Statistical tests of models of DNA substitution
SummaryPenny et al. have written that “The most fundamental criterion for a scientific method is that the data must, in principle, be able to reject the model. Hardly any [phylogenetic] tree-reconstruction methods meet this simple requirement.” The ability to reject models is of such great importance because the results of all phylogenetic analyses depend on their underlying models—to have confidence in the inferences, it is necessary to have confidence in the models. In this paper, a test statistics suggested by Cox is employed to test the adequacy of some statistical models of DNA sequence evolution used in the phylogenetic inference method introduced by Felsentein. Monte Carlo simulations are used to assess significance levels. The resulting statistical tests provide an objective and very general assessment of all the components of a DNA substitution model; more specific versions of the test are devised to test individual components of a model. In all cases, the new analyses have the additional advantage that values of phylogenetic parameters do not have to be assumed in order to perform the tests.
13569003
Probabilism and Phylogenetic Inference
The maximum likelihood approach to phylogenetics rests on frequency probability theory. This stands in stark contrast to the logical probability of corroboration‐based cladistic parsimony. History is particular and cannot be described in terms of universal statements about abstract generalities, the task of the historical sciences being one of explanation, not prediction. Thus, frequency probability methods of estimation are inappropriate for making historical inferences. Maximum likelihood estimation procedures are deconstructed from numerous perspectives in spite of their supposed impressive technicalities. Charges of parsimony's inconsistency are rendered mute, because its justification lies elsewhere, yet maximum likelihood is still subject to Wald's dilemma if realism is of any interest. Although all epistemologies make assumptions, the models employed by maximum likelihood are problematic and deterministic, as opposed to the unproblematic background knowledge characteristic of cladistics. Apart from issues of logical and sampling dependencies, the requirements of frequency probability theory are non‐trivial and the maximum likelihood estimation of phylogeny can neither escape, nor satisfy the tenets of calculus independence (e.g. i.i.d.) inherent in the multiplicative relations of the method. If phylogeneticists are to maintain a rational foundation for their epistemology, neo‐justificationist appeals to some metaphysical truth must be abandoned in favour of the realism of sophisticated falsification.
85016424
Approaches for Assessing Phylogenetic Accuracy
?Accuracy of phylogenetic methods may be assessed in terms of consistency, efficiency, and robustness. Four principal methods have been used for assessing phylogenetic accuracy: simulation, known phylogenies, statistical analyses, and congruence studies. Simulation studies are useful for studying accuracy of methods under idealized conditions and can be used to make general predic? tions about the behavior of methods if the limitations of the models are taken into account. Studies of known phylogenies can be used to test predictions from simulation studies, thus providing a check on the robustness of the models (and possibly suggesting refinements for future simulations). Statistical analyses allow general predictions to be applied to specific results, facilitate assessments as to whether or not sufficient data have been collected to formulate a robust conclusion, and indicate whether a given data set is any more structured than random noise. Finally, congruence studies of multiple data sets can be used to assess the degree to which independent results agree and thus the minimum proportion of the findings that can be attributed to an underlying phylogeny. These differ? ent methods of assessing phylogenetic accuracy are largely complementary, and the results are consis? tent in identifying a large class of problems that are amenable to phylogenetic reconstruction. [Phy? logeny; accuracy; simulations; experimental evolution; statistics; congruence; consistency; efficiency; robustness.] Phylogenetic analyses have become com? monplace throughout the biological disci? plines during the past few decades. This in? creased emphasis on evolutionary history is a direct result of the realization of the im? portance of understanding phylogenetic background as a prerequisite to interpreting virtually any biological system in a compar? ative context. However, the increased uti? lization of phylogenetic approaches has been driven at least as much by technologi? cal and methodological advances as by con? ceptual advances. In particular, advances in algorithm development, computer technol? ogy, and molecular biology have made phy? logenetic analyses feasible for almost any problem involving biological lineages, from viral epidemics in extant human popula? tions (e.g., Ou et al., 1992) to the origins of the earliest lineages of life (e.g., Olsen, 1987). Phylogenetic applications depend on accurate reconstructions of phylogenetic trees; therefore, it is natural that systematists should wonder about the accuracy of their reconstructed trees. This issue of Sys? tematic Biology contains reviews of the four approaches that systematists have explored to examine phylogenetic accuracy and as1 E-mail: hillis@bull.zo.utexas.edu. sess confidence in their results: evolution? ary simulations (Huelsenbeck, 1995), explo? ration of known (observed) phylogenies (briefly reviewed here), statistical evalua? tions (Li and Zharkikh, 1995), and congru? ence studies (Miyamoto and Fitch, 1995). I evaluating phylogenetic accuracy, there are two common goals: one may ask about general properties of phylogenetic methods or about a specific phylogenetic es? timate. A systematist may address how well a given method works under different cir? cumstances (e.g., different evolutionary conditions, different amounts of informa? tion, different types of trees, etc.). Such studies may ask about phylogenetic perfor? mance of a single method or may compare several methods. This approach primarily involves numerical simulations and investi? gation of known phylogenies. On the other hand, statistical and congruence studies tend to address specific questions of phylo? genetic accuracy, i.e., how much confidence can be placed in a specific phylogenetic re? sult? Of course, the distinction is not always clear because simulations can be used to ad? dress confidence in a particular empirical result (e.g., Hillis et al., 1994a) and general conclusions about the relative accuracy of phylogenetic methods can come from statis? tical or congruence studies (e.g., Penny et
30718549
Testability and the Refutation and Corroboration of Cladistic Hypotheses
Both refutationist and verificationist philosophies and practices are becoming increasingly evident in phylogenetic inference. Refutation and verification are fundamentally different epistemologies, and it seems unlikely that they can continue to coexist as the basis for inferring species history. The refutationist nature of cladistics is explored in terms of Popperian testability, in order to understand better the alternatives and to predict the outcome of the expected revolution. Testability concerns the logical relationship between a hypothesis (h, such as a cladogram), evidence (e, such as synapomorphy), and background knowledge (b). Of particular importance is the direct relationship between the logical improbability ofhand its potential to be tested, because foreto corroborateh,eshould be improbable givenbalone. Simplicity and boldness, amount of empirical content, and logical improbability all refer to the potential to be tested. Thathmust be testable by severe tests is the same as saying that those tests have greater probability of failing, given onlyb. Descent with modification is sufficient as background knowledge (b) in phylogenetic inference, and such a minimal assumption explains the generality of cladistics. Also of interest to the refutationist position is total evidence. In terms of testability, a statement describing the results of multiple tests is less probable than a statement describing only some of the tests, the multiple test result being more improbable, and accordingly more severe, than its component tests. All other cladistic principles and practices considered in this review are also understandable in terms of Popperian testability, refutation and corroboration. These include minimizing ad hoc hypotheses of homoplasy and minimizing explanatory power, and choosing tentatively among cladograms according to their degree of corroboration (support). Differential character weighting is determined to be unacceptable in terms of testability. Also, testability does not provide a basis for assessing the accuracy of hypotheses, but then that is of no consequence to cladists, because they are not preoccupied with knowing the absolute truth, unlike verificationists.
84998418
CONJECTURES AND REFUTATIONS
Phylogeneticists familiar with Popper's ideas have long regarded mostparsimonious trees as bestcorroborated hypotheses of relationship, but Faith (1992) feels that those trees are leastcorroborated instead. Not that he recommends least-parsimonious trees, rather (p. 268) "Popperian corroboration is properly applied to a higher-level cladistic hypothesis." That hypothesis being that analyzing the data has revealed something of phylogeny. And, conveniently, his permutation tail probability (PTP) establishes both the content (boldness) and the degree of corroboration of that hypothesis. But in fact no one quantity can measure both content and corroboration; and content, unlike PTP, is not calculated from presently available evidence. In trying to connect PTP to corroboration, Faith confuses pointwith cumulative probabilities, evidence with the hypothesis tested, and null models with accepted theories. In arriving at his view of most-parsimonious trees he repeats most of the same mistakes, as will be seen here.
84889341
PTP is Meaningless, T‐PTP is Contradictory: A Reply to Trueman
Abstract — The T‐PTP test for monophyly can attribute significance to entirely unsupported groups and even to both of two contradictory alternatives. The method of evaluating “support” after replacing selected groups of terminals with reconstructed ancestors has similar drawbacks. The proposed placement of Onychophora among Arthropoda is unsupported by 12S data, and strongly refuted by other evidence. Attempts to justify T‐PTP on Popperian grounds rest entirely on misunderstanding Popper's ideas. The PTP test assesses neither Popperian corroboration nor statistical confidence of phylogenetic conclusions. It can attribute high significance to data that support no resolved grouping. Efforts to salvage PTP by proposing new interpretations share the weakness that none of the proposed interpretations generally holds. None of these methods seems useful in phylogenetic
20957422
A METHOD FOR DEDUCING BRANCHING SEQUENCES IN PHYLOGENY
With the advent of relatively objective classifications, such as the phenetic classifications produced by the operational techniques of numerical taxonomy (Sokal and Sneath, 1963), it was inevitable that biologists would wonder what phylogenetic conclusions could be drawn from them and with what reliability. If these phenetic taxonomies did not reflect all of the elements of phyletics (Sokal and Camin, 1965), could techniques be devised for deducing the latter? For example, could operational methods be devised for deducing the cladistic relationships among taxa, so that, given the same initial information, different investigators would obtain the same results? By cladistic relationships we mean the evolutionary branching sequences among taxonomic units without regard to phenetic similarities among them or to an absolute time scale. There is no question that phylogenies could probably be reconstructed without error for any taxonomic group if complete fossil sequences for that group were available. However, can cladistic reconstructions be carried out with any degree of
123235286
Methods for Computing Wagner Trees
Abstract Farris, J. S. (Biol. Sci., State Univ., Stony Brook, N. Y.) 1970. Methods for computing Wagner Trees. Syst. Zool., 19:8342.-The article derives some properties of Wagner Trees and Networks and describes computational procedures for Prim Networks, the Wagner Method, Rootless Wagner Method and optimization of hypothetical intermediates ( HTUs). The Wagner Ground Plan Analysis method for estimating evolutionary trees has been widely employed in botanical studies (see references in Wagner, 1961) and has more recently been employed in zoological evolutionary taxonomy (Kluge, 1966; Kluge and Farris, 1969). Wagner Trees are one possible generalization of the most parsimonious trees of Camin and Sokal (1965). The Wagner technique is of con-siderable interest for quantitative evolution- ary taxonomists because it is readily pro- grammable and because the type of tree produced can tractably be extended to ap- plications in a variety of novel quantitative phyletic techniques. In this paper I shall formalize the concept of a Wagner Network and discuss a number of algorithms for calculating such networks. The rationale for the methods described will not be treated extensively here, as it is published elsewhere (Kluge and Farris, 1969).
122433656
A Numerical Approach to Phylogenetic Systematics
Abstract Farris, J. S. (Biol. Sci., State Unia., Stony Brook, New York, 11790), Kluge, A. G., and Eckardt, Jl. J. (Zool., Univ. Alichigan, Ann Arbor 48104) 1970. A Numerical approach to phylogenetic systematics. Syst. Zool., 19:172-191.-Principles abstracted from Hennig (1966) are used as axioms to form a quantitative analog of phylogenetic systematics. A close connection is demonstrated betneen phylogenetics and most parsinlonious trees. The compatibility of some existing clustering methods nith the principles is discussed, and a nen clustering technique, the Weighted Invariant Step Strategy ( WISS) is described. Generalization of the axioms to the case \there direction of evolution is not assumed is examined, and it is shown that the Wagner Method for estimating evolutionary trees is consistent nith the generalized phylogenetic axioms. The taxonomic philosophy propounded in Hennig's ( 1966) Plz ylogenetic Systematics
61075564
Realism and the aim of science
Editor's Foreword Acknowledgements Introduction, 1982 REALISM AND THE AIM OF SCIENCE Preface, 1956 Part I: The Critical Approach Chapter I Induction Chapter II Demarcation Chapter III Metaphysics: Sense of Nonsense? Chapter IV Corroboration Part II: The Propensity Interpretation of Probability Chapter I Objective and Subjective Probabilities Chapter II Criticism of Probabilistic Induction Chapter III Remarks on the Objective Theories of Probability Index
9034760
Methods for inferring phylogenies from nucleic acid sequence data by using maximum likelihood and linear invariants.
Likelihood methods and methods using invariants are procedures for inferring the evolutionary relationships among species through statistical analysis of nucleic acid sequences. A likelihood-ratio test may be used to determine the feasibility of any tree for which the maximum likelihood can be computed. The method of linear invariants described by Cavender, which includes Lake's method of evolutionary parsimony as a special case, is essentially a form of the likelihood-ratio method. In the case of a small number of species (four or five), these methods may be used to find a confidence set for the correct tree. An exact version of Lake's asymptotic chi 2 test has been mentioned by Holmquist et al. Under very general assumptions, a one-sided exact test is appropriate, which greatly increases power.
13965672
Links between maximum likelihood and maximum parsimony under a simple model of site substitution.
Stochastic models of nucleotide substitution are playing an increasingly important role in phylogenetic reconstruction through such methods as maximum likelihood. Here, we examine the behaviour of a simple substitution model, and establish some links between the methods of maximum parsimony and maximum likelihood under this model.
83923059
A likelihood approach to character weighting and what it tells us about parsimony and compatibility
The statistical framework of maximum likelihood estimation is used to examine character weighting in inferring phylogenies. A simple probabilistic model of evolution is used, in which each character evolves independently among two states, and different lineages evolve independently. When different characters have different known probabilities of change, all sufficiently small, the proper maximum likelihood method of estimating phylogenies is a weighted parsimony method in which the weights are logarithmically related to the rates of change. When rates of change are taken extremely small, the weights become more equal and unweighted parsimony methods are obtained. 
 
 
 
When it is known that a few characters have very high rates of change and the rest very low rates, but it is not known which characters are the ones having the high rates, the maximum likelihood criterion supports use of compatibility methods. By varying the fraction of characters believed to have high rates of change one obtains a ‘threshold method’ whose behavior depends on the value of a parameter. By altering this parameter the method changes smoothly from being a parsimony method to being a compatibility method. This provides us with a spectrum of intermediates between these methods. These intermediate methods may be of use in analysing real data.
86020740
Maximum Likelihood and Minimum-Steps Methods for Estimating Evolutionary Trees from Data on Discrete Characters
Felsenstein, J. (Department of Genetics SK-50, University of Washington, Seattle, Washington 98195). 1973. Maximum likelihood and minimum-steps methods for estimating evolutionary trees from data on discrete characters. Syst. Zool. 22:240-249.The general maximum likelihood approach to the statistical estimation of phylogenies is outlined, for data in which there are a number of discrete states for each character. The details of the maximum likelihood method will depend on the details of the probabilistic model of evolution assumed. There are a very large number of possible models of evolution. For a few of the simpler models, the calculation of the likelihood of an evolutionary tree is outlined. For these models, the maximum likelihood tree will be the same as the "most parsimonious" (or minimum-steps) tree if the probability of change during the evolution of the group is assumed a priori to be very small. However, most sets of data require too many assumed state changes per character to be compatible with this assumption. Farris (1973) has argued that maximum likelihood and parsimony methods are identical under a much less restrictive set of assumptions. It is argued that the present methods are preferable to his, and a counterexample to his argument is presented. An algorithm which enables rapid calculation of the likelihood of a phylogeny is described. [Evolutionary trees: maximum likelihood.] The first systematic attempt to apply standard statistical inference procedures to the estimation of evolutionary trees was the work of Edwards and Cavalli-Sforza (1964; see also Cavalli-Sforza and Edwards, 1967). At about the same time, the "parsimony' or minimum evolutionary steps method of Camin and Sokal (1965) gave a great impetus to the development of welldefined procedures for obtaining evolutionary trees. Edwards and Cavalli-Sforza concerned themselves with data from continuous variables such as gene frequencies and quantitative characters. The CaminSokal approach, on the other hand, was developed for characters which are recorded as a series of discrete states. Although some taxonomists have declared that the problem of guessing phylogenies should be viewed as a problem of statistical inference (Farris, 1967, 1968; Throckmorton, 1968), until recently there have been no attempts to explore the relationship between the statistical inference and minimum-steps approaches. Recently, Farris (1973) has presented a detailed argument that, under certain reasonable assumptions, the maximum-likelihood method of statistical inference appropriate to discrete-character data is precisely the parsimony method of Camin and Sokal. In this paper, I will examine the application of maximum likelihood methods to discrete characters, and will show that parsimony methods are not maximum likelihood methods under the assumptions made by Farris. They are maximum likelihood methods under considerably more restrictive assumptions about evolution. METHODS OF MAXIMUM LIKELIHOOD Suppose that we want to estimate the evolutionary tree, T, which is to be specified by the topological form of the tree and the times of branching. We are given a set of data, D, and a model of evolution, M, which incorporates not only the evolutionary processes, but also the processes of sampling by which we obtained the data. This model will usually be probabilistic, involving random events such as changes of the environment, occurrence of favorable
124046083
MAXIMUM LIKELIHOOD INFERENCE OF PHYLOGENETIC TREES, WITH SPECIAL REFERENCE TO A POISSON PROCESS MODEL OF DNA SUBSTITUTION AND TO PARSIMONY ANALYSES
Maximum likelihood inference is discussed, and some of its advantages and dis- advantages are noted. The application of maximum likelihood inference to phylogenetics is examined, and a simple Poisson process model of DNA substitution is used as one example. Further examples follow from the clarification of implicit models underlying traditional "par- simony" and "compatibility" analyses. From the elucidation of these models and analyses, it is seen that Poisson process analysis gives a statistically consistent estimate of phylogeny, and that parsimony methods do indeed have a maximum likelihood foundation but give potentially incorrect estimates of phylogeny. The maximum likelihood formulation provides a common framework within which these analyses are discussed and compared. (Phylogenetic inference; maximum likelihood inference; Poisson process model; parsimony analysis; DNA substitution.)
85792416
Parsimony in Systematics: Biological and Statistical Issues
Quite recently "parsimony" has become the favored method for inferring phylogenies (evolutionary trees). The accompanying article by Elliott Sober discusses the philosophical issues relating to the status of parsimony from a somewhat different perspective than that adopted here. This review will discuss parsimony, its origins, its major variants, and its biological assumptions. For a review of a wider range of methods of inferring phylogenies, the reader may wish to consult Felsenstein (22). The present discussion does not cover the use of parsimony criteria as measures of how simply information is represented in a classification. Methods of classification and methods of inferring phylogeny are of course separate issues and only the latter are discussed here.
15895455
Estimating the pattern of nucleotide substitution
Knowledge of the pattern of nucleotide substitution is important both to our understanding of molecular sequence evolution and to reliable estimation of phylogenetic relationships. The method of parsimony analysis, which has been used to estimate substitution patterns in real sequences, has serious drawbacks and leads to results difficult to interpret. In this paper a model-based maximum likelihood approach is proposed for estimating substitution patterns in real sequences. Nucleotide substitution is assumed to follow a homogeneous Markov process, and the general reversible process model (REV) and the unrestricted model without the reversibility assumption are used. These models are also applied to examine the adequacy of the model of Hasegawa et al. (J. Mol. Evol. 1985;22:160–174) (HKY85). Two data sets are analyzed. For the Ψν-globin pseudogenes of six primate species, the REV model fits the data much better than HKY85, while, for a segment of mtDNA sequences from nine primates, REV cannot provide a significantly better fit than HKY85 when rate variation over sites is taken into account in the models. It is concluded that the use of the REV model in phylogenetic analysis can be recommended, especially for large data sets or for sequences with extreme substitution patterns, while HKY85 may be expected to provide a good approximation. The use of the unrestricted model does not appear to be worthwhile.
12660243
Maximum-likelihood models for combined analyses of multiple sequence data
Models of nucleotide substitution were constructed for combined analyses of heterogeneous sequence data (such as those of multiple genes) from the same set of species. The models account for different aspects of the heterogeneity in the evolutionary process of different genes, such as differences in nucleotide frequencies, in substitution rate bias (for example, the transition/transversion rate bias), and in the extent of rate variation across sites. Model parameters were estimated by maximum likelihood and the likelihood ratio test was used to test hypotheses concerning sequence evolution, such as rate constancy among lineages (the assumption of a molecular clock) and proportionality of branch lengths for different genes. The example data from a segment of the mitochondrial genome of six hominoid species (human, common and pygmy chimpanzees, gorilla, orangutan, and siamang) were analyzed. Nucleotides at the three codon positions in the protein-coding regions and from the tRNA-coding regions were considered heterogeneous data sets. Statistical tests showed that the amount of evolution in the sequence data reflected in the estimated branch lengths can be explained by the codon-position effect and lineage effect of substitution rates. The assumption of a molecular clock could not be rejected when the data were analyzed separately or when the rate variation among sites was ignored. However, significant differences in substitution rate among lineages were found when the data sets were combined and when the rate variation among sites was accounted for in the models. Under the assumption that the orangutan and African apes diverged 13 million years ago, the combined analysis of the sequence data estimated the times for the human-chimpanzee separation and for the separation of the gorilla as 4.3 and 6.8 million years ago, respectively.
26640644
BEST‐FIT MAXIMUM‐LIKELIHOOD MODELS FOR PHYLOGENETIC INFERENCE: EMPIRICAL TESTS WITH KNOWN PHYLOGENIES
Despite the proliferation of increasingly sophisticated models of DNA sequence evolution, choosing among models remains a major problem in phylogenetic reconstruction. The choice of appropriate models is thought to be especially important when there is large variation among branch lengths. We evaluated the ability of nested models to reconstruct experimentally generated, known phylogenies of bacteriophage T7 as we varied the terminal branch lengths. Then, for each phylogeny we determined the best‐fit model by progressively adding parameters to simpler models. We found that in several cases the choice of best‐fit model was affected by the parameter addition sequence. In terms of phylogenetic performance, there was little difference between models when the ratio of short: long terminal branches was 1:3 or less. However, under conditions of extreme terminal branch‐length variation, there were not only dramatic differences among models, but best‐fit models were always among the best at overcoming long‐branch attraction. The performance of minimum‐evolution‐distance methods was generally lower than that of discrete maximum‐likelihood methods, even if maximum‐likelihood methods were used to generate distance matrices. Correcting for among‐site rate variation was especially important for overcoming long‐branch attraction. The generality of our conclusions is supported by earlier simulation studies and by a preliminary analysis of mitochondrial and nuclear sequences from a well‐supported four‐taxon amniote phylogeny.
35606155
Sophisticated falsification and research cycles: Consequences for differential character weighting in phylogenetic systematics
Practicing phylogenetic systematics as a sophisticated falsification research program provides a basis for claiming increased knowledge of sister species relationships and synapomorphies as evidence for those cladistic propositions. Research in phylogenetic systematics is necessarily cyclic, and the place where the positive shift in understanding occurs is subsequent to discovering the most parsimonious cladogram(s). A priori differential character weighting is inconsistent with seeking the maximally corroborated cladogram (sensu Popper), because weighting adds to background knowledge, the evidence being then less improbable than it would be otherwise. Also, estimating weights from character state frequencies on a cladogram is inconsistent with the view that history is unique. Sophisticated falsification provides the place in the cycle of phylogenetic systematic research where weight of evidence can be evaluated and these inconsistencies do not apply. On balance, phylogenetic systematics appears to achieve greater coherence and generality as a result of focusing on the foundations for claiming increased knowledge and avoiding efforts to differentially weight characters.
84908002
Probability, parsimony, and Popper
ture (Archie, 1989a, 1989b; Faith, 1990; Faith and Cranston, 1991) compare the mini? mum length of the tree found for the orig? inal character data to the length of the tree achieved for corresponding randomized data sets. Each randomized data set is pro? duced under a random character covaria? tion null model in which the character states, within each character, are reas? signed at random to the individual taxa. The corresponding null hypothesis is that the observed degree of cladistic structure (as indicated by tree length) can be re? garded as the product of random data. In practice, the proportion of all data sets (ob? served and random) having a tree length as short or shorter than that of the ob?