Whole-Tree Methods for Detecting Differential Diversification Rates

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Syst. Biol. 51(6):855–865, 2002 DOI: 10.1080/10635150290155836

Whole-Tree Methods for Detecting Differential Diversification Rates K AI M. A. CHAN1,∗ AND B RIAN R. M OORE2 1

Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey 08544-1003, USA 2 Department of Ecology and Evolutionary Biology, Yale University, New Haven, Connecticut 06520, USA Abstract.—Prolific cladogenesis, adaptive radiation, species selection, key innovations, and mass extinctions are a few examples of biological phenomena that lead to differential diversification among lineages. Central to the study of differential diversification rates is the ability to distinguish chance variation from that which requires deterministic explanation. To detect diversification rate variation among lineages, we propose a number of methods that incorporate information on the topological distribution of species diversity from all internal nodes of a phylogenetic tree. These whole-tree methods (M5 , M6 , and MR ) are explicitly connected to a null model of random diversification—the equal-rates Markov (ERM) random branching model—and an alternative model of differential diversification: M5 is based on the product of individual nodal ERM probabilities; M6 is based on the sum of individual nodal ERM probabilities, and MR is based on a transformation of ERM probabilities that corresponds to a formalized system that orders trees by their relative symmetry. These methods have been implemented in a freely available computer program, SYMMETREE, to detect clades with variable diversification rates, thereby allowing the study of biological processes correlated with and possibly causal to shifts in diversification rate. Application of these methods to several published phylogenies demonstrates their ability to contend with relatively large, incompletely resolved trees. These topology-based methods do not require estimates of relative branch lengths, which should facilitate the analysis of phylogenies, such as supertrees, for which such data are unreliable or unavailable. [Cladogenesis; differential diversification rates; equal-rates Markov random branching model; supertrees; tree balance indices; tree shape.]

The use of phylogenies to detect diversification rate variation dates to the inception of modern phylogenetic systematics. Hennig (1966:227) was the first to explicitly recognize that any difference in diversity between sister groups, which are by definition of equal age, must necessarily reflect different rates of diversification (speciation − extinction) in those groups. However, other researchers were quick to caution against overly deterministic interpretations of such differences: even if the underlying probability of diversification were the same in all lineages, some degree of difference in species diversity would be expected to arise because of the inherently stochastic nature of the process (see seminal articles by the Woods Hole group; e.g., Raup et al., 1973; Gould et al., 1977). Central to the study of differential diversification rates, therefore, is the ability to distinguish chance variation from that which requires deterministic explanation. Methods developed to detect significant diversification rate variation (associated with episodes of prolific cladogenesis, adaptive radiation, species selection, key innovations, mass extinction, etc.) include both temporal and topology-based approaches (reviewed by Sanderson and Donoghue,

1996; Mooers and Heard, 1997; Barraclough and Nee, 2001). Temporal methods utilize branch-length data to estimate the (absolute or relative) timing of speciation events and compare the observed distribution of speciation events through time with that expected under a null model of cladogenesis (e.g., Harvey et al., 1991, 1994a, 1994b; Hey, 1992; Nee et al., 1992, 1994a, 1994b, 1996; Harvey and Nee, 1993, 1994; Sanderson and Bharathan, 1993; Sanderson, 1994; Kubo and Iwasa, 1995; Paradis, 1997, 1998a, 1998b; Pybus and Harvey, 2000; Nee, 2001). In contrast, topology-based methods effectively ignore branch-length data, instead focusing on the relative species diversity of two (or more) groups descended from a common node (i.e., the so-called balance or symmetry of phylogenies). Here, we deal exclusively with topology-based methods. Two general types of topology-based methods have been developed: single-node tests and tree-shape indices. Single-node tests (e.g., Slowinski and Guyer, 1989a, 1989b; Slowinski, 1990) compare the observed difference in sister-group diversity with that predicted by a plausible (albeit simplistic) model of random diversification, permitting detection of significant diversification rate variation among sister lineages. In contrast, ∗ The order of authorship was determined randomly. tree-shape indices incorporate information 855

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on the relative diversity of nodes throughout an entire tree and variously summarize whole-tree balance as a single number (e.g., Shao and Sokal, 1990; Kirkpatrick and Slatkin, 1993). By looking beyond the balance of single nodes, the sensitivity of these methods to variation in diversification rate is substantially increased. However, these indices are not explicitly derived from any model of diversification, so the biological meaning of significant imbalance under these tests is unclear. With the advent of methods for estimating exceptionally large, complete specieslevel supertrees (e.g., Sanderson et al., 1998), topology-based methods warrant further consideration. What if we could develop topology-based methods that combined the power of whole-tree indices with the ease of interpretation of single-node tests? Here, we present three new, biologically meaningful, and explicitly model-based whole-tree methods for detecting significant diversification rate variation among lineages. We then discuss some aspects of their differential sensitivity to rate variation at different phylogenetic scales and demonstrate their ability to contend with large, incompletely resolved (supertree) phylogenies. We conclude by considering how these methods might be used to explore a range of evolutionary questions. CALCULATING WHOLE-T REE ASYMMETRY PROBABILITIES In a pioneering series of articles, Slowinski and Guyer (1989a, 1989b; Slowinski, 1990) developed a method for detecting significant diversification rate variation at individual nodes. Under their approach, an observed difference in sister-group diversity is compared to a null distribution of such differences generated under a simple model of random cladogenesis, the so-called equal-rates Markov (ERM) random branching model (e.g., Yule, 1924; Harding, 1971). The ERM probability of observing a difference in sistergroup diversity as extreme or more extreme than that of the node in question is given by the equation 2l (1) P= (n − 1) (unless l = n/2, in which case P = 1), where l is the number of species in the less diverse of the two sister groups and n is the

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combined diversity of both sister groups. Significant asymmetry in sister-group diversity constitutes rejection of the ERM null model and therefore suggests that the sister pair has experienced nonrandom diversification (Slowinski and Guyer, 1989a, 1989b; Slowinski, 1990). Generalization of this single-node approach to summarize the relative imbalance of all internal nodes of a tree would provide a much more powerful and (by virtue of being based on an explicit model of cladogenesis) biologically meaningful test of diversification rate variation among lineages. The development of such whole-tree methods might be achieved by combining individual ERM nodal probabilities on a node-bynode basis over all internal nodes of a given phylogeny (Kirkpatrick and Slatkin [1993] attributed this proposal to a personal communication by J. Slowinski). But how should individual nodal probabilities be combined? A subsequent development by Slowinski and Guyer (1993) suggests a possible solution. They proposed a method for combining individual ERM probabilities from single-node comparisons from many different trees using Fisher’s combined probability test (FCPT; Fisher, 1932). It would seem relatively straightforward to modify the FCPT protocol to combine probabilities from many nodes within the same tree (Fig. 1). Although intuitive, further consideration reveals this proposal to be problematic. The combination of nodal probabilities under the FCPT is extremely biased. This bias stems from violation of the underlying assumptions of omnibus statistics (i.e., statistics, such as the FCPT and ECPT, that reflect the combined significance of several independent tests of a common hypothesis). The FCPT statistic is calculated by estimating the compound probability that a set of probabilities (in this case, the set of nodal ERM probabilities derived with Eq. 1) will have a product equal to or smaller than that of the observed set (Fisher, 1932). A less common but equally valid combined probability test statistic was proposed by Edgington (ECPT; 1972a, 1972b) and takes the sum rather than the product of individual probabilities. Both the FCPT and ECPT assume that the individual probabilities to be combined are independent and can realize any value on the interval (0, 1]. However, nodal probabilities are interdependent to the extent

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CHAN AND MOORE—WHOLE-TREE TESTS OF DIVERSIFICATION RATE VARIATION

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FIGURE 1. Combining nodal probabilities. (A) The probability that an observed difference in sister group diversity arose by random cladogenesis (i.e., under the ERM model) can be derived using Equation 1. Individual ERM nodal probabilities, each from a different tree, can then be combined using FCPT, which takes the product of the individual nodal probabilities (Slowinski and Guyer, 1993). Alternatively, the nodal probabilities may be combined using ECPT, which takes the sum of the individual nodal probabilities. This type of procedure has been used to test the cumulative effect of a putative key innovation on rates of diversification in the various groups in which the innovation independently evolved (indicated by the asterisks). (B) We might consider modifying this approach to develop whole-tree tests of diversification rate variation by using FCPT or ECPT to combine the individual ERM nodal probabilities from many nodes within the same tree (e.g., P1 –P10 ). However, the FCPT and ECPT (like all omnibus statistics) assume that the individual probabilities to be combined are independent and can each realize any value between 0 and 1. Nodal probabilities, however, are both nonindependent (to the extent that they are phylogenetically nested, e.g., P4 and P5 are nested within P3 ) and discretely valued (they are derived from the comparison of discretely valued species numbers), precluding their combination with standard omnibus statistics. Nevertheless, approximate solutions may be devised that allow for the combination of nodal probabilities (as either their product, M5 , or sum, M6 ) by using Monte Carlo simulation to estimate the appropriate distribution of the test statistics.

that they are derived from phylogenetically nested nodes, and these probabilities can realize only a finite number of discrete values for the simple reason that they are derived (using Eq. 1) from the comparison of species diversities, which necessarily occur as whole numbers (i.e., 1, 2, 3, etc.). This discreteness problem is known to cause a discrepancy between the assumed and realizable probability space (Wallis, 1942; Edgington and Haller, 1984) such that the combination of individual nodal probabilities under the FCPT or ECPT will assume a concave function of the true cumulative probabilities (Moore and Chan, in prep.). Combining Individual Nodal Probabilities: M5 and M6 Conventional omnibus statistics do not provide a viable basis for deriving whole-tree

tests of diversification rate variation. However, it is possible to develop a nonanalytical solution that avoids the discreteness and interdependence problems while emulating the logic of the FCPT and ECPT omnibus statistics. We developed two whole-tree tests of diversification rate variation based upon the cumulative ERM probability by the product (M5 ) and sum (M6 ) of individual nodal probabilities. Conceptually, these tests involve mapping the sample space that can be realized by discretely valued, interdependent nodal probabilities. Specifically, we used Monte Carlo simulation to estimate the underlying distribution of topologies that can be realized for a tree of a given size under the ERM model of cladogenesis. These tests are implemented with one of two algorithms depending upon the size of the tree in question. For smaller trees, the

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appropriate ERM sample space can be mapped exactly by applying the small-tree algorithm as follows: (1) calculate the product (or sum) of all ERM nodal probabilities (derived from Eq. 1) in the observed tree; (2) generate all possible topologies for a tree with the same number of species as the observed tree, and for each topology calculate the product (or sum) of its nodal probabilities and its point probability under the ERM model; and (3) sum the point probabilities of all topologies with nodal probability products (or sums) less than or equal to that of the observed tree. This last sum represents the cumulative, whole-tree probability based on the nodal probability product, M5 (or the nodal probability sum, M6 ). For larger trees, the appropriate ERM sample space must be approximated owing to the vast number of possible topologies (e.g., only 46 for 9 species but 105,061,603,969 for 35 species; Stone and Repka, 1998). The sample space for trees with >20 species is estimated by applying the large-tree algorithm as follows: (1) as in the small-tree protocol, first calculate the product (or sum) of ERM nodal probabilities in the observed tree; (2) using the ERM model of cladogenesis, generate a large, random subset of possible topologies for a tree with the same number of species as the observed tree; and (3) count the number of simulated trees with a nodal probability product (or sum) less than or equal to that of the observed tree, and divide by the total number of simulated trees. This quotient is an unbiased estimate of the probability corresponding to M5 (or M6 ). These methods provide a practical alternative to an analytical solution for deriving whole-tree tests of diversification rate variation. Analytical expressions have been derived for the expected values and variances of several other tree shape indices (e.g., Page, 1991; Heard, 1992; Rogers, 1993, 1994, 1996; McKenzie and Steel, 2000), providing more convenient solutions to certain problems. However, the capacity of these analytical expressions to derive whole-tree probabilities for even moderately large trees is severely limited by the nonnormal shape of the underlying probability distributions (with the exception of cherries, which are approximately normally distributed for large trees; McKenzie and Steel, 2000). For this reason, expected dis-

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tributions of various test statistics are frequently estimated by Monte Carlo simulation (e.g., Shao and Sokal, 1990; Heard, 1992; Kirkpatrick and Slatkin, 1993; Fusco and Cronk, 1995; Mooers, 1995). Although approximate solutions are often relatively cumbersome in application, our whole-tree methods have been implemented in a computer program, SYMMETREE, which permits probabilities for arbitrarily large phylogenies to be conveniently calculated. Executables have been compiled for Macintosh, Windows, and UNIX operating systems and are freely available at www.phylodiversity.org/brian/ or www.eeb.princeton.edu/∼kaichan/ or by e-mailing the authors. Cumulative ERM Left-Light Rooting Probability: MR An alternative whole-tree test of diversification rate variation was first described by Page (1993) and subsequently implemented by Kirkpatrick and Slatkin (1993). This whole-tree balance index, R, refers to the position of a given tree in an ordered series of all possible topologies for trees of that size; these topologies are ordered from least to most symmetric according to the left-light rooting (LLR) protocol of Furnas (1984). We extended this approach to calculate the probability of an R value less than or equal to that of the tree in question under the ERM model. This statistic, which we refer to as MR , is therefore analogous to M5 and M6 but summarizes R values rather than products or sums of nodal probabilities. Unlike M5 and M6 , however, MR can be calculated explicitly from a hierarchical combination of individual nodal ERM probabilities that gives priority to deeper nodes. To order trees by LLR, we first swivel nodes in the trees such that the smaller subclade is to the left side of every node. We then rank all of the resulting trees according to their balance at the root node. The relative balance of a node is decided by the following recursive rules: node A is less symmetric than node B if (1) A is less symmetric than B at the local root (A’s smaller [i.e., left] subtree has fewer species than B’s); (2) A and B are equally symmetric at the local root, but A’s left subtree is less symmetric than B’s; and (3) A and B are equally symmetric at the local root and in their left subtrees, but A’s right

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FIGURE 2. The 11 distinct topologies for seven-species trees rendered in LLR order from least to most asymmetric. The probabilities under the ERM model are indicated below each tree: the point probability for each topology was calculated with Harding’s (1971) recursive formula (Pe ), and the cumulative probabilities of equally or more asymmetric topologies were calculated with the ERM LLR formula (MR ). The cumulative MR probabilities can be verified by summing the individual probabilities for all the topologies as asymmetric or more asymmetric than the tree in question.

subtree is less symmetric than B’s. Figure 2 illustrates the LLR ordering of the 11 possible topologies for seven-species trees. The resulting ordered series of trees is then used to calculate the cumulative probability of observing trees as unbalanced or more unbalanced under the LLR rules. The MR probabilities are equivalent to the sum of the individual ERM probabilities of topologies (derived with the recursive equation of Harding, 1971) that are equally asymmetric or more asymmetric than the tree in question. The root node plays a predominant role in dictating the MR value of a given tree; this is a consequence of the hierarchical procedure by which LLR orders trees (i.e., first by their balance at the root and then by their balance at more nested nodes within the left and right subtrees). The MR statistic has also been implemented in SYMMETREE. Derivation of the necessary expressions for its calculation and the details of the algorithms in-

volved are documented elsewhere (Chan and Moore, 2002, in prep.). PROBLEMS WITH POLYTOMIES A major impetus in developing the topology-based whole-tree methods is to exploit recent and anticipated advances in supertree estimation, which promise to provide exceptionally large and complete phylogenies for the study of differential diversification rates. Despite their obvious merits, supertrees often contain a considerable proportion of unresolved nodes (although perhaps no more so than trees estimated by primary analysis: Bininda-Emonds and Sanderson, 2001). Because methods for detecting diversification rate variation typically require strictly dichotomous trees, the empirical reality of polytomies is a serious impediment to their application. Accordingly, SYMMETREE has a number of facilities for

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analyzing trees that are incompletely resolved. For example, a suite of algorithms allow polytomies to be randomly (and repeatedly) resolved under conditions conforming to an ERM model of cladogenesis, providing an estimate of the confidence intervals on the whole-tree probabilities obtained under the various statistics. A formal account of the various algorithms involved in treating polytomies is beyond the scope of the present paper: details may be found in Chan and Moore (2002, in prep.). CHARACTERIZATION: T HE R ELATIVE S ENSITIVITY OF WHOLE-T REE M ETHODS TO D IVERSIFICATION R ATE V ARIATION AT D IFFERENT PHYLOGENETIC S CALES Our motivation for developing the wholetree methods was to increase the statistical power of tests to detect diversification rate variation. Power is the ability of a test to reject a null hypothesis when it is false. Nodal ERM probabilities are the most appropriate measure for tests of differential diversification at individual nodes. Accordingly, we expected the combination of these values—as implemented by the M statistics, M5 , M6 , and MR —to provide tests of the ERM model that would be exceptionally sensitive to diversification rate variation within whole trees. The power of a test is contingent on the nature of the particular alternative hypothesis under consideration. Because there are innumerable possible alternative hypotheses (e.g., infrequent rate shifts dispersed throughout the tree, frequent rate shifts occurring only at the base of the tree), it is unrealistic to expect any single statistic to be maximally powerful in all scenarios involving differential diversification rates. Given the multitude of possible and biologically relevant alternatives to equal-rates cladogenesis, several different statistics are required. The M statistics are intended to provide differential sensitivity to asymmetry arising at different phylogenetic scales (i.e., the relative nodal depth of asymmetry in the tree), permitting their application to a corresponding range of associated evolutionary processes. The manner in which each statistic summarizes information from individual nodes (e.g., ERM probabilities) will determine the type of diversification rate variation (i.e., the alternative hypothesis) to which it is most sensitive. By considering how the dif-

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ferent M statistics differentially summarize ERM nodal probabilities, we can theoretically characterize their differential sensitivity to different patterns of diversification rate shifts, even without performing the simulations necessary for a complete characterization of relative power. The MR statistic is extremely responsive to large-scale asymmetry (i.e., near the root) and is correspondingly insensitive to smallscale asymmetry (i.e., near the tips). This exceptional sensitivity to large-scale asymmetry stems from the fact that under the LLR ordering protocol, upon which MR is based, the root node largely determines the overall whole-tree symmetry. When comparing the relative symmetry of two trees, the symmetry of nodes above the root will only be considered if the two trees are equally symmetric at the root node. Consider, for example, two n-species trees, A and B. The root of A is subtended by two equally diverse (i.e., n/2) but maximally asymmetric subtrees. The root of B is subtended by two subtrees with slightly subequal diversity, i.e., (n/2) − 1 and (n/2) + 1, but maximally symmetric topologies. By focusing on the relative asymmetry at the root (to the exclusion of the relative asymmetry at more nested nodes), MR will identify tree B as more asymmetric than tree A. In contrast to MR , both M5 and M6 always consider the relative asymmetry of all internal nodes. Nevertheless, M5 and M6 exhibit differential sensitivity to large-scale asymmetry. To understand the reason for the different behavior of these statistics, recall that the potential magnitude of differences in species diversity is greater at more inclusive nodes. Consider, for example, that the maximum asymmetry in an n-species tree is a split of 1:(n − 1), which can only be realized at the root. The next most extreme split, 2:(n − 2), can only be realized at the root or at the node just above the root, and so on. Accordingly, the most extreme nodal probabilities (i.e., the smallest) can only be generated by large-scale asymmetry. These extreme probabilities will have a relatively large effect on M5 because calculation of the statistic involves their multiplication. In contrast, M6 combines nodal probabilities additively, such that the impact of such extreme probabilities is greatly diminished, allowing nodal probabilities associated with smallscale asymmetry to make a more equable contribution to the whole-tree probability.

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FIGURE 3. The differential sensitivity of the whole-tree tests to diversification rate variation manifest at different phylogenetic scales. Trees A and B exhibit substantial differences in large-scale phylogenetic asymmetry. A has a basal split of 1:100 (P1 = 0.02, lnP1 = −3.91) versus a 25:76 split in B (P1 = 0.5, lnP1 = −0.69). Now imagine that the only other difference in asymmetry between the two trees is restricted to a five-species subtree and that this subtree has a 2:3 split in A (P2 = 1.0, lnP2 = 0) and a 1:4 split in B (P2 = 0.5, lnP2 = −0.69). M6 would identify B as more asymmetric overall (P1A + P2A = 1.02; P1B + P2B = 1.0), whereas M5 would require five or more differences in smallscale asymmetry of the same magnitude before indicating that B is more asymmetric than A (lnP1A − lnP1B = −3.22, lnP2A − lnP2B = 0.69). This example illustrates that M6 is substantially more sensitive to small-scale asymmetry than is M5 . For comparison, IC (Colless, 1982; Heard, 1992) would require ≥25 equivalent five-species subtree differences before indicating that B is more asymmetric than A, whereas the modified version of IC , I2 (Mooers and Heard, 1997) would identify B as far more asymmetric than A with the addition of a single five-species subtree. No amount of nested small-scale asymmetry within B would be sufficient to render it more asymmetric than A by MR . Thus, the sensitivity of the whole-tree statistics to small-scale diversification rate variation is MR < IC < M5 < M6 < I2 .

Thus, the relative sensitivity of the wholetree statistics to large-scale diversification rate variation is M6 < M5 < MR (Fig. 3). D EMONSTRATION: APPLYING WHOLE-T REE M ETHODS TO EMPIRICAL D ATA The whole-tree methods described here were used to assess diversification rate variation in a number of complete species-level phylogenies spanning a range of tree size (14–916 species) and degree of resolution (3– 53% unresolved nodes). Because these data were analyzed for illustrative purposes only, we made no attempt to estimate confidence in the phylogenies or the sensitivity of our findings associated with phylogenetic uncertainty (e.g., Donoghue and Ackerly, 1996; Huelsenbeck et al., 2000). For ease of retrieval, the phylogenies have been deposited in the TreeBASE archive (www.treebase.org). Probabilities derived with our whole-tree methods were compared with those calculated for the two most commonly used topology-based balance indices: IC (Colless, 1982; Heard, 1992) and B1 (Shao and Sokal, 1990). All analyses were performed with SYMMETREE; the relevant details and results are summarized in Table 1. Analysis of these data, particularly the bat tree, illustrates the ability of our wholetree methods (and their implementation in SYMMETREE) to contend with relatively large and incompletely resolved trees. Our

inspection of the data supports the predicted behavior of the whole-tree methods with respect to their differential sensitivity to diversification rate variation at different phylogenetic scales. For example, probabilities for M5 were most significant for trees exhibiting large-scale asymmetry, whereas probabilities for M6 were most significant for trees displaying small-scale asymmetry. The relative significance of the various methods suggests that several of the groups examined (lagomorphs, apes, Old World monkeys, carnivores, and bats) have experienced shifts in diversification rate. In all cases, the lower whole-tree probability values for M6 and M5 relative to that for MR suggest that major diversification rate shifts have not occurred close to the roots of these trees. The inference of power from single P values based on the analysis of empirical data is not warranted because the processes that generated those data are unknown. Accordingly, although these findings are intriguing, a comprehensive simulation study is needed to properly estimate and characterize the power of these whole-tree methods (Chan and Moore, in prep.). I MPLICATIONS : T HE UTILITY OF WHOLE-T REE T ESTS OF D IVERSIFICATION R ATE V ARIATION Different people have different interests in the study of differential diversification rates.

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916

Bats

0, 0

0, 0.00173

0.00057, 0.02553

M6 (high, low)

0, 0

0, 0.00369

0.00157, 0.09216

7.00E-05, 0.00031 0.00395, 0.22078 0.0050702, 0.0383274 0.1238, 0.67358 0.32427, 0.97545

MR (high, low)

0.346373, 0.346373

0.820088, 0.820088

0.696175, 0.696175

0.633019, 0.633019 0.383857, 0.383868 0.615547, 0.628775 0.769535, 0.770662 0.704613, 0.704891

IC (high, low)

0.00029, 0.00147

0.15664, 0.35922

0.13324, 0.30096

0.00104, 0.00117 0.21646, 0.32791 0.051228, 0.186296 0.67334, 0.89347 0.79086, 0.95886

B1 (high, low)

0, 0

0, 0.00446

0.00158, 0.18689

0.00827, 0.02954 0.00019, 0.10908 0.0007377, 0.0148069 0.1451, 0.7755 0.01519, 0.73075

Source

Bininda-Emonds et al., 1999 Jones et al., 2002

Purvis, 1995

Stoner et al., 2003 Purvis, 1995 Purvis, 1995 Purvis, 1995 Purvis, 1995

Values reported above are probabilities derived by Monte Carlo simulation of the null distribution for each statistic. All results were obtained using the SYMMETREE program with the following user options. The null distribution for each statistic was generated with a sample of 100,000 ERM topologies for each tree size. Probability values of zero reflect a failure to find a single simulated tree as or more asymmetric than the observed tree. Uncertainty associated with polytomies was assessed by generating 100,000 random resolutions under the size-sensitive taxon addition ERM algorithm, providing the upper and lower bounds of the confidence interval. These high and low bounds (associated with high and low symmetry) correspond to the tail probabilities for the 0.025 and 0.975 frequentiles, respectively. b Percent resolution was calculated as k/(n − 1), where k is the number of nodes in a tree of n species. This value implicitly assumes that the underlying phylogeny is strictly dichotomous (i.e., that all polytomies are soft). c Our evaluation of the various constituent primate clades (apes, lemurs, Old and New World monkeys) corresponds to the analysis by Purvis et al. (1995) to aid comparison with their findings. These results should be interpreted cautiously, however, because diversification rate variation is interdependent. Shifts within more nested clades will influence estimates obtained for more inclusive clades. Because these analyses are intended to be illustrative, no attempt was made to assess phylogenetic uncertainty or its effect on the results obtained.

47

81

80

M5 (high, low)

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Lagomorphs All primatesc Apes Lemurs New World monkeys Old World monkeys Carnivores

Taxon

TABLE 1. Probability valuesa corresponding to tests of ERM cladogenesis in several example phylogenies.

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The whole-tree methods presented here are intended to provide more decisive, biologically meaningful answers to the question of whether the branches of a tree have experienced differential diversification rates. Accordingly, these methods should find useful application to a range of problems (outlined below) but will nevertheless be ill-suited to the investigation of other equally valid and interesting questions. For example, having established that a tree has experienced significant diversification rate variation, we might naturally wonder where those shifts in diversification rate have occurred. Temporal methods are available for and appropriate to this question (e.g., Nee et al., 1992, 1996; Sanderson, 1994; Purvis et al., 1995; Paradis, 1998a), and we are currently exploring the possibility of extending our whole-tree methods to provide a complementary topologybased approach to this problem. Additionally, we might want to estimate parameters associated with the diversification process (e.g., net speciation rate) or test whether diversification rates have changed significantly through time. These questions will require information on the (relative) timing of diversification and will necessarily involve the use of temporal methods (e.g., Harvey et al., 1991, 1994a, 1994b; Nee et al., 1992, 1994a, 1994b; Harvey and Nee, 1993, 1994; Kubo and Iwasa, 1995; Paradis, 1997, 1998b; Nee, 2001). In such applications of temporal methods, however, it is necessary to first establish that there has not been significant diversification rate variation among lineages of the phylogeny. This assumption can readily be affirmed (or refuted) with our whole-tree tests, again emphasizing the inherent complementarity of the temporal and topology-based methods. The whole-tree methods presented here have practical implications for both data exploration and hypothesis testing. Whole-tree surveys for significant diversification rate variation may provide an effective discovery method for generating causal hypotheses of factors that have caused, are caused by, or are correlated with differential diversification rates. For example, the discovery that asymmetric clades are frequently polymorphic for growth form (i.e., woody/herbaceous) may lead us to hypothesize that shifts in growth form are affecting diversification rates (e.g., Eriksson and Bremer, 1991, 1992; Bremer and

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Eriksson, 1992; Judd et al., 1994; Ricklefs and Renner, 1994; Tiffney and Mazer, 1995; Dodd et al., 1999). The ability to detect clades with variable diversification rates will also help identify the data relevant to studies of phenomena that are hypothesized to be correlated with diversification rate variation. For example, application of the whole-tree tests could identify the data necessary to evaluate the hypothesized correlation between rates of nucleotide substitution and rates of cladogenesis (e.g., Mindell et al., 1989; Barraclough et al., 1996; Mindell and Thacker, 1996; Savolainen and Goudet, 1998; Barraclough and Savolainen, 2001). A number of evolutionary processes may entail hypotheses that predict multiple diversification rate shifts dispersed throughout whole clades rather than single shifts concentrated at particular nodes, including the effect of various coevolutionary associations on rates of diversification (e.g., the reciprocal radiations predicted for some insect/plant associations; e.g., Farrell, 1998; Farrell and Mitter, 1998; Kelly and Farrell, 1998) and the effect of relative refractory periods associated with age-biased cladogenesis (Hey, 1992; Harvey and Nee, 1993; Losos and Adler, 1995; Chan and Moore, 1999). We are optimistic that the M statistics developed in this study will enhance our ability to detect diversification rate variation and, therefore, to better understand a number of important evolutionary processes. ACKNOWLEDGMENTS We are grateful to Steve Heard and Arne Mooers for inviting us to contribute to this collection of articles on phylogenetic tree shape. We thank Andy Purvis and Olaf Bininda-Emonds for sending published primate and carnivore supertree data, respectively. We are particularly indebted to Olaf Bininda-Emonds for generously sharing unpublished lagomorph and bat supertree data. We give special thanks to Mary Walsh for stimulating discussion and for assistance with the illustrations and to Junling Ma for programming advice. For offering perceptive comments on earlier drafts of this paper, we thank Michael Donoghue, Steve Heard, Simon Levin, Rick Ree, Michael Sanderson, Chris Simon, Mike Steel, Mary Walsh, and two anonymous reviewers. Although this article has benefited greatly from their suggestions, we accept sole responsibility for the remaining errors that it may contain. Funding for this research was made possible through Natural Science and Engineering Research Council (NSERC) postgraduate scholarships to KC and BRM and additional support from a Heinz Foundation Environmental Scholarship and Julie Payette-NSERC research scholarship to KC.

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