Ratio of explanatory power (REP): A new measure of group support

Molecular Phylogenetics and Evolution 44 (2007) 483 487 Short communication Ratio of explanatory power (REP): A new measure of group support Taran Grant a, *, Arnold G. Kluge b a Division of Vertebrate Zoology, Herpetology, American Museum of Natural History, New York, NY 10024, USA b 3140 Dolph Drive, Ann Arbor, MI 48103, USA Received 23 July 2006; revised 17 November 2006; accepted 29 November 2006 Available online 9 December 2006 www.elsevier.com/locate/ympev Bremer support (BS; Bremer, 1988; Källersjö et al., 1992) is frequently employed as a measure of group support in parsimony-based phylogenetic analysis. For a given group present in the most-parsimonious tree, Bremer support is defined as S 0 S; where S denotes the length of the most-parsimonious tree (Kluge and Farris, 1969) and S 0 is the length of the mostparsimonious tree that lacks that group. In other words, BS measures the difference in the explanatory power of the two hypotheses (Kluge and Grant, 2007), and is therefore a measure of objective support (Grant and Kluge, 2003). (For simplicity, we refer only to comparisons of pairs of hypotheses, but the same arguments also apply when sets of optimal hypotheses are compared to sets of equally suboptimal hypotheses.) The interpretation of BS as indicating the amount of contradictory evidence required to refute the optimal hypothesis of group monophyly is simple and unproblematic. However, the fact that BS does not scale between 0 and 1 has been seen as a deficiency. For example, one of the advantages Goloboff and Farris (2001, S31) claimed for their Relative Fit Difference (RFD; relative Bremer support of Goloboff et al., 2003) over BS is that it varies between 0 and 1, and it has been employed in empirical studies (e.g., Sánchez-Villagra et al., 2006). Goloboff and Farris did not discuss their reasons for considering the 0 1 range to be an advantage, but the criticism generally seems to be that BS values are not directly comparable across datasets. For example, a BS value of 5 indicates the amount of contradictory evidence required to overturn * Corresponding author. Fax: +1 212 769 5031. E-mail addresses: grant@amnh.org (T. Grant), akluge@umich.edu (A.G. Kluge). the optimal hypothesis, but, as a report of objective support, it would be advantageous to distinguish between BS values of 5 given datasets of different sizes and tree lengths. Grant and Kluge (submitted for publication) identified a number of defects in the RFD, leaving open the problem of defining a measure of relative explanatory power that varies between 0 and 1. Instead of calculating support as the difference in the explanatory power of competing hypotheses, as done by BS, it may be calculated as the ratio of explanatory power (REP). Where the assumptions of probabilistic inference are met, this is given by the familiar likelihood ratio, which has long been considered an objective measure of support in statistics. Indeed, in many ways Grant and Kluge s (2003) explication of the logic of support in ideographic inference parallels Hacking s (1965) explication of the logic of support in statistical inference. Given that explanatory power in phylogenetics is operationalized as tree length (Kluge and Grant, 2006), the ratio of the tree lengths of competing hypotheses offers an analogous measure of objective support that varies between 0 and 1 and may be compared across datasets. Explanatory power is the same epistemological maxim for both REP and the likelihood ratio; however, REP appeals to severity of test whereas the likelihood ratio relies on inductive arguments (contra de Queiroz, 2004). Two considerations complicate the measurement of REP support. First, by convention, the likelihood ratio is calculated as a fraction with the better hypothesis as the numerator, and it is appropriate to do the same here. This gives S=S 0 : However, because the better explanation has the shorter tree length, this formulation would have the undesirable effect of scoring unsupported hypotheses as 1. This is remedied by taking the complement of the ratio. A simple 1055-7903/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.ympev.2006.11.027

484 T. Grant, A.G. Kluge / Molecular Phylogenetics and Evolution 44 (2007) 483 487 measure of REP support for a group present in the mostparsimonious tree would then be given by 1 S=S 0 : Here, the maximum is attained only in the trivial case in which no transformations are required to explain the observed character-states ðs ¼ 0Þ or as S 0 approaches infinity, which is the second complication. S 0 cannot exceed G, the maximum number of steps required to explain aligned character-states (Farris, 1989), or, in the more general case in which no alignment is assumed, X, the number of steps required to explain each unaligned character-state (e.g., each unaligned nucleotide) as uniquely evolved. Consequently, the maximum value for real datasets would be considerably less than 1, and the theoretical maximum would vary across datasets. However, recognizing that S 6 S 0 6 X (for character-states aligned in static matrices replace X with G) allows an index of REP support that varies between 0 and 1 for all datasets to be defined. We define the REP support for a particular group present in the most-parsimonious tree as REP ¼ 1 ððx S 0 Þ=ðX SÞÞ ¼ ðs 0 SÞ=ðX SÞ: The numerator of the latter expression is the BS, so the REP support for a group is equal to its BS divided by the difference in length between the least-parsimonious tree (X or G) and the most-parsimonious tree (S). If S 0 ¼ S, then the REP support value is 0 (as is the BS). If S 0 ¼ X (or G for static matrices), then the REP support value is 1. For low BS values and high tree lengths, raw REP support values may be too small to be conveniently reported, so it may be preferable to use scientific notation or otherwise multiply values by some constant. Of course, there are other ways to scale BS. For example, considering that the BS for a given group cannot exceed its branch length in the most-parsimonious tree, L, an alternative would be ðs 0 SÞ=L: This would report the BS relative to the maximum possible for the group in question. A similar approach was taken by DeBry (2001) in his effort to derive a statistical interpretation of BS. However, although scaling BS relative to L may be appropriate for other problems, for our purposes it fails for a number of reasons. Most importantly, it does not rank hypotheses according to their relative explanatory power. For example, two groups on the same tree may have identical BS (and, therefore, identical relative explanatory power) but scale differently due to their different branch lengths. Additional complications obtain from the difficulty in defining L, which may differ across equally parsimonious trees and may vary for a single tree due to ambiguous optimizations, and it is unclear if L should be taken as the maximum branch length, minimum branch length, or number of unambiguous changes for a single tree or shared across trees. REP support is not afflicted by any of these problems; in particular, it succeeds in ranking hypotheses according to their explanatory power. To demonstrate the comparison of REP support across datasets, we calculated the values for three published datasets: Sparks and Smith s (2004a) cichlid fish dataset, Sparks and Smith s (2004b) rainbowfish dataset, and Sánchez-Villagra et al. s (2006) morphology-only talpid mole dataset (Table 1). Although the Sparks and Smith datasets were analyzed under direct optimization (e.g., Wheeler, 1996, 2003; see also Kluge and Grant, 2006) and therefore did not assume an alignment prior to analysis, those studies calculated BS values using the implied alignments, and we therefore calculated REP support using G instead of X. The BS and REP support values (multiplied by 1000 for convenience) for the cichlid dataset are shown on the tree in Fig. 1. BS and REP support values for all datasets are summarized in Table 2. BS and REP support values give the same ranking of groups within a dataset, but they may differ across datasets. Despite the different numbers of terminals and aligned characters, G, and optimal tree lengths of the two fish studies (Table 1), their BS values are approximately equivalent. For example, a BS of 5 is equal to a REP support of 6.2 10 4 for both datasets. However, a BS of 5 for the morphology-only mole dataset is equal to a REP support value of 9.8 10 3, which is equivalent to a BS value of 79 for the rainbowfish dataset and is greater than the maximum observed BS for the cichlid dataset. Both BS and REP support provide a measure of the relative explanatory power of hypotheses and are heuristic sensu Grant and Kluge (2003). However, their interpretations differ somewhat, and we recommend reporting both values in empirical studies. BS measures the difference in explanatory power among groups within a dataset, which is heuristic in the sense of Grant and Kluge (2003) in that it indicates the absolute amount of evidence required to overturn the optimal hypothesis. By calculating support Table 1 Summary of three datasets Taxa Evidence No. terminals No. aligned characters G Tree length Source Cichlid fishes DNA sequences 89 2222 16,268 8247 Sparks and Smith, 2004a Rainbowfishes DNA sequences + morphology 63 4394 17,909 9820 Sparks and Smith, 2004b Talpid moles Morphology 21 157 985 473 Sánchez-Villagra et al., 2006 G, the maximum number of steps required to explain the aligned character-states (Farris, 1989).

T. Grant, A.G. Kluge / Molecular Phylogenetics and Evolution 44 (2007) 483 487 485 Fig. 1. Strict consensus of 81 most-parsimonious trees from Sparks and Smith s (2004a) study of cichlid fishes showing Bremer support (above branches) and REP support (below branches) values. For convenience, REP support values are multiplied by 1000.

486 T. Grant, A.G. Kluge / Molecular Phylogenetics and Evolution 44 (2007) 483 487 Table 2 Bremer support (BS) values and the respective REP support values for three datasets (see Table 1 for details) BS REP: cichlid fishes REP: rainbowfishes REP: talpid moles 1 1:2 10 4 1:2 10 4 2:0 10 3 2 2:5 10 4 2:5 10 4 3:9 10 3 3 3:7 10 4 3:7 10 4 5:9 10 3 4 5:0 10 4 4:9 10 4 7:8 10 3 5 6:2 10 4 6:2 10 4 9:8 10 3 6 7:5 10 4 7:4 10 4 1:2 10 2 7 8:7 10 4 8:7 10 4 1:4 10 2 8 1:0 10 3 9:9 10 4 1:6 10 2 9 1:1 10 3 1:1 10 3 1:8 10 2 10 1:2 10 3 1:2 10 3 2:0 10 2 11 1:4 10 3 1:4 10 3 2:1 10 2 12 1:5 10 3 1:5 10 3 2:3 10 2 13 1:6 10 3 1:6 10 3 2:5 10 2 14 1:7 10 3 1:7 10 3 2:7 10 2 15 1:9 10 3 1:9 10 3 2:9 10 2 16 2:0 10 3 2:0 10 3 3:1 10 2 17 2:1 10 3 2:1 10 3 3:3 10 2 18 2:2 10 3 2:2 10 3 3:5 10 2 19 2:4 10 3 2:3 10 3 20 2:5 10 3 2:5 10 3 21 2:6 10 3 2:6 10 3 22 2:7 10 3 2:7 10 3 23 2:9 10 3 2:8 10 3 24 3:0 10 3 3:0 10 3 25 3:1 10 3 3:1 10 3 26 3:2 10 3 3:2 10 3 27 3:4 10 3 3:3 10 3 28 3:5 10 3 3:5 10 3 29 3:6 10 3 3:6 10 3 30 3:7 10 3 3:7 10 3 31 3:9 10 3 3:8 10 3 32 4:0 10 3 4:0 10 3 33 4:1 10 3 4:1 10 3 34 4:2 10 3 4:2 10 3 35 4:4 10 3 4:3 10 3 36 4:5 10 3 4:5 10 3 37 4:6 10 3 4:6 10 3 38 4:7 10 3 4:7 10 3 39 4:9 10 3 4:8 10 3 40 5:0 10 3 4:9 10 3 41 5:1 10 3 5:1 10 3 42 5:2 10 3 5:2 10 3 43 5:4 10 3 5:3 10 3 44 5:5 10 3 5:4 10 3 45 5:6 10 3 5:6 10 3 46 5:7 10 3 5:7 10 3 47 5:9 10 3 5:8 10 3 48 6:0 10 3 5:9 10 3 49 6:1 10 3 6:1 10 3 50 6:2 10 3 6:2 10 3 51 6:4 10 3 6:3 10 3 52 6:5 10 3 6:4 10 3 53 6:6 10 3 6:6 10 3 54 6:7 10 3 6:7 10 3 55 6:9 10 3 6:8 10 3 56 7:0 10 3 6:9 10 3 57 7:1 10 3 7:0 10 3 58 7:2 10 3 7:2 10 3 59 7:4 10 3 7:3 10 3 60 7:5 10 3 7:4 10 3 61 7:6 10 3 7:5 10 3 62 7:7 10 3 7:7 10 3 63 7:9 10 3 7:8 10 3 Table 2 (continued) BS REP: cichlid fishes REP: rainbowfishes REP: talpid moles 64 8:0 10 3 7:9 10 3 65 8:0 10 3 66 8:2 10 3 67 8:3 10 3 68 8:4 10 3 69 8:5 10 3 70 8:7 10 3 71 8:8 10 3 72 8:9 10 3 73 9:0 10 3 74 9:1 10 3 75 9:3 10 3 76 9:4 10 3 77 9:5 10 3 78 9:6 10 3 79 9:8 10 3 80 9:9 10 3 81 1:0 10 2 82 1:0 10 2 83 1:0 10 2 84 1:0 10 2 85 1:1 10 2 For each dataset REP support values are given up to the maximum observed BS. as the ratio of explanatory power, thereby scaling the relative explanatory power between 0 and 1, REP support provides standardized values that allow support to be meaningfully compared across studies of different sets of terminals. The heurism of REP support as a means of comparing cladograms is readily appreciated by considering that, as a report on the relative explanatory power of competing hypotheses, REP support is also a report on the relative strength of refutation of those hypotheses, and evaluating strength of refutation is key in testing derivative hypotheses. For example, given competing biogeographic scenarios, each of which is corroborated by a phylogenetic study of different taxa, REP support provides a means of assessing the relative degree to which each dataset refutes the alternative scenarios. Similarly, REP support may be useful in constructing supertrees (Baum, 1992; Ragan, 1992), where overlapping hypotheses are differentially corroborated by different datasets and there is concern that the pseudocharacters in the group inclusion matrix (Farris, 1973) should be weighted in proportion to the support for the original groups (Bininda-Emonds and Sanderson, 2001). REP support may be similarly useful in testing adaptive scenarios, coevolutionary hypotheses, and any problem in which phylogenetic hypotheses derived from studies of different sets of taxa provide the basis for testing competing theories. We caution that we do not advocate interpreting REP support as an indicator of the reliability, probability, or accuracy of alternative hypotheses, but rather their relative degree of refutation, which is the primary consideration in a progressive research program (Lakatos, 1978).

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