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Using phylogenies to test hypotheses of adaptation: a critique of some current proposals.

Investigations of the adaptive design of traits and the phylogenetic relationships of taxa have traditionally focused on very different sets of questions. Consequently, relatively few linkages have been developed between these subfields of evolutionary biology. This has recently begun to change, however, particularly with the publication of several proposed methods for using data from reconstructed phylogenies (Hennig 1966; Wiley 1981) to test hypotheses of adaptation. Two principal approaches may be distinguished. First, several statistical techniques have been devised to incorporate phylogenetic data into comparative tests of hypotheses predicting an adaptive association between two characters or between a character and an environmental variable (e.g., Ridley 1989; Felsenstein 1985a; Grafen 1992; Harvey and Pagel 1991; Pagel 1992). These techniques use information from reconstructed phylogenetic relationships in efforts to minimize the problem that characters shared among related taxa cannot be treated as statistically independent if they are shared through common ancestry rather than adaptive convergence.

Second, several authors have begun using reconstructed phylogenies to directly test hypotheses of adaptive character evolution. The general procedure entails mapping characters of interest onto independently derived cladograms to assess the timing and direction of historical transitions between character states. Phylogenetic tests using this procedure have been developed to (1) determine whether a character originally spread in ancestral populations through natural selection for its current function (e.g., Coddington 1988, 1990; Baum and Larson 1991), (2) assess whether a character's presence in an extant taxon results from a "phylogenetic constraint" on adaptive evolution (e.g., McLennan et al. 1988; Brooks and McLennan 1991), and (3) assess whether the evolution of one character was facilitated by the presence of another by determining whether these characters arose in a predicted historical sequence (e.g., Sillen-Tullberg 1988; Carpenter 1989, 1991; Donoghue 1989; Prum 1990; McLennan and Brooks 1991; Sillen-Tullberg and Moller 1993).

The prospect of using reconstructed phylogenies to test hypotheses of adaptation is exciting and merits serious appraisal by both systematists and functional biologists. Our objective in this paper is to argue, however, that direct phylogenetic tests of hypotheses of adaptive character evolution have a more limited explanatory power than has been generally acknowledged. Our central points are that homoplasies (character reversals and parallel evolution in related taxa) and temporal variation in selective environments can substantially constrain the reliability with which ancestral character states and selective environments can be inferred. These points are by no means novel: the occurrence of homoplasies, for example, has been extensively discussed as a problem for phylogenetic inference in general (e.g., Sober 1988). However, their consideration in the specific context of testing hypotheses of adaptive character evolution leads us to conclude that reconstructed phylogenies cannot reliably test hypotheses of types (1) and (2) described above but can, with careful application, test hypotheses of type (3). Although Donoghue (1989) suggests that the assumption that characters have undergone a minimum number of state changes may similarly bias the outcome of statistical tests of hypothesized adaptive character associations, we will not pursue this issue here.

We begin by considering the general procedure of character mapping and then examine in turn proposed phylogenetic tests of each of the three hypotheses of adaptive character evolution described above. We take as a starting point that the reconstructed phylogenies upon which characters are mapped closely correspond to true phylogenies, leaving aside the problems of interpretation arising when they are mapped onto trees of imperfect or uncertain accuracy (see Felsenstein 1985b; Armbruster 1992).

Inferring Ancestral Character States

A fundamental assumption underlying phylogenetic tests of hypotheses of adaptive character evolution is that the mapping of a character onto an independently generated cladogram will provide an accurate picture of the historical sequence of transitions between character states. Such a mapping is achieved through "character state optimization" and follows the general application of the principle of parsimony to phylogenetic reconstruction (Sober 1988; Carpenter 1989; Armbruster 1992). Consider, for example, a character that exhibits two or more discrete and heritable states. Ancestral character states are inferred by assigning to each branch point, or node, of a cladogram the state that minimizes the number of transitions needed to explain the observed patterning of the character among descendant taxa. In a simple case, if two sister species who are the sole descendants of a common ancestor each exhibit character state A, their ancestor is inferred to have also exhibited A.

Character state optimization will accurately reveal the timing and direction of historical transitions between character states only if the rate of character change within lineages is low relative to the rate of taxonomic divergence (cladogenesis). If, in particular, transitions between character states are sufficiently frequent within a species lineage, the phylogenetic "memory" of these traits will decay such that their mapping onto a cladogram may bear little relation to the actual sequence of character state transitions between ancestral and derived taxa (see Appendix). That both quantitative and meristic characters commonly exhibit heritable variation and a marked response to artificial selection demonstrates a potential for microevolutionary change that must reduce the reliability with which ancestral character states can be inferred (Falconer 1981; Charlesworth et al. 1982; Endler 1986). Indeed, the occurrence of such reversals is a primary reason why multiple (and presumably conservative) characters are used to build cladograms. Whereas the reconstruction of phylogenies is thus somewhat shielded from this problem (Felsenstein 1983, 1985b), the mapping of single (and potentially labile) characters onto them is not.

We can generalize this problem as follows. Suppose that we wish to trace the evolutionary history of a behavioral character Y that exhibits two states, A and B. Consider the two sister species described above; the sole descendants of a common ancestor, each possesses character state A. If there are nonzero probabilities of within-lineage transitions from A to B and B to A and there has been sufficient time for even a few transitions to occur, then the probabilities of seeing A or B currently in these species can be essentially independent of their starting states, that is, the character state of their most recent common ancestor. Thus, recording A in both extant species may yield little or no information about the ancestral character state. This argument can be quantified by modeling within-lineage transitions as a Markov chain (fig. 1; see Appendix).

In some cases, this problem may be detected readily if relatively frequent character-state transitions within lineages produce a highly variable distribution of character states among extant sister species. Hence, no single parsimonious character mapping would be possible. However, it would not be so detectable if closely related species have also experienced a similar series of selective environments and undergone, therefore, a positively correlated sequence of character-state transitions (see Appendix). Cases of roughly parallel character evolution have been described in related fossil lineages (e.g., Simpson 1953, chap. 8) and might be expected to be fairly common because closely related species tend to share similar niche preferences (Harvey and Pagel 1991) and responses to selection. Thus, particular character states may be maintained by selection and be widely shared among related taxa but be uniquely derived in each of them.

The reliability of an estimated ancestral character state will depend largely on the relative rate of within-lineage transitions to cladogenesis and on the number of descendant taxa from which the ancestral state is inferred. Carpenter (1989), for example, used the optimization procedure of Farris (1970) to deduce that nest sharing is ancestral for the social wasps of the subfamily Stenogastrinae because wasps in six stenogastrine genera commonly share nests with related conspecifics. However, if only a few within-lineage transitions between nest sharing and independent nest founding have occurred, inferring that nest sharing is the ancestral character state from this number of taxa is extremely error prone. Even as the number of sister taxa sharing a common character state increases, there may remain a substantial probability that their common ancestor possessed an alternative character state (fig. 2; see Appendix).

The reliability of an inferred ancestral character state may be enhanced when the number of descendent taxa is large and speciation rates have been relatively rapid. Given the well-documented potential for within-lineage character evolution, parsimonious inferences of the ancestral states of a given character may often not be robust and should be interpreted with caution.

Testing Hypotheses of Adaptive

Historical Origin

Gould and Vrba (1982) have argued that a trait should be considered an adaptation for a current function only if it has been clearly shaped by natural selection for that function (cf. Williams 1966; Bock 1980; Sober 1984; Reeve and Sherman 1993), How can adaptations by this definition be recognized? Demonstrating that a trait serves a particular adaptive function in current environments is clearly not conclusive evidence upon which to conclude that it originally spread through natural selection because it served that function, a problem recognized by Darwin (1859) and reemphasized by many others (e.g., Tinbergen 1964; Williams 1966; Gould and Vrba 1982). Consider, for example, the observation that brooding black-headed gulls (Larus ribidinus) remove broken egg shells from their nests shortly after the chicks hatch. Through a series of simple field experiments, Tinbergen et al. (1962) demonstrated that egg-shell removal is adaptive in that it decreases the visibility of nests to aerial predators and thereby enhances the probability that nestlings survive. It is conceivable, however, that these predators may have been a less significant selective force in ancestral environments and that egg-shell removal initially spread for a completely different reason, perhaps because it served a different adaptive function (such as removing the shell's moist substrate for mold and bacterial growth), or because it was closely associated through pleiotropy or linkage disequilibrium with another directionally selected trait. As Tinbergen (1964, p. 428) noted, "When one finds that a certain characteristic has survival value... one has demonstrated beyond doubt a selection pressure which prevents the species in its present state from deviating... However, the conclusion that this same selection pressure must have been responsible in the past for the moulding of the character studied is speculative, however probable it often is" (original emphasis). We have always lacked a general method for determining whether a trait's current function is the reason that it initially spread within ancestral populations.

Coddington (1988, 1990) and Baum and Larson (1991) have recently argued that the reconstruction of accurate phylogenies now allows hypotheses of adaptive historical origin to be rigorously tested. Defining adaptation in cladistic terms as "apomorphic function promoted by natural selection as compared with plesiomorphic function" (p. 3; apomorphic refers to a derived character state and plesiomorphic to its ancestral counterpart), Coddington (1988) proposed that one could test the hypothesis of adaptive origin for a trait of interest simply by mapping the character states of sister taxa onto an independently derived cladogram. According to his general model (fig. 3), the hypothesis that a particular character state (M1) initially spread because it conferred on its bearers its current adaptive function (F1) is supported if (1) both M1 and F1 arose at the same internode of the cladogram, and (2) as a consequence of its function F1, M1 is advantageous relative to the plesiomorphic character state M0, whose function is F0. That is, taxa bearing the plesiomorphic function F0 of M0 must be "in some definable and measurable sense, ecologically or functionally less well adapted than (taxa bearing the apomorphic) function F1 of M1" (p. 8).

How might this model be applied? Consider, as does Coddington (1988), the problem of determining whether the mimicry of coral snakes by snakes in nonpoisonous taxa is an adaptation to decrease their risk of predation (Greene and McDiarmid 1981). A biologist concerned with current function might be inclined to experimentally manipulate the color patterns of mimics to assess whether they suffer an increased predation rate relative to their unmanipulated conspecifics. Such an assessment of current utility, however, would not address the hypothesis of adaptive origin. To do so, Coddington (1988) argues, the appropriate test would be to assess whether the mimic "suffers less predation due to its visibility than do species retaining the primitive color pattern of its ancestors" (p. 17). If, for example, the plesiomorphic color pattern is cryptic, one would need to compare the relative predation rate upon snakes in cryptic and mimic sister taxa.

Coddington's (1988) proposed test relies upon the central assumption that the apomorphic and plesiomorphic characters borne by sister taxa in current environments have equivalent selective consequences relative to their counterparts in the ancestral population in which the apomorphy first arose. Thus, in addition to the parsimonious assumption of character stasis discussed above, this test further requires that each character state's selective environment has remained essentially unchanged; otherwise, an assessment of whether an apomorphy is now functionally superior to its plesiomorphic counterpart would not shed light on whether it evolved through natural selection for current function. In their modification of Coddington's (1988) model, Baum and Larson (1991) make this assumption explicit: "When direct historical inference cannot be applied... selective regimes may be superimposed on cladograms using the principle of parsimony. In this case, the underlying assumption is that the rate at which lineages move between selective regimes is low relative to the rate of lineage branching" (p. 11).

We suggest that these assumptions are very unlikely to hold for many and perhaps most traits typically studied by evolutionary ecologists. The biotic and abiotic components of selective environments commonly vary on time scales that are far shorter than the intervals between taxonomic divergence, producing both substantial geographic variation in the strength and direction of selection and microevolutionary change (Ford 1975; Charlesworth et al. 1982; Endlet 1986). The assumption, therefore, of invariant selective environments is not clearly superior to one that admits that selective environments may have changed since the apomorphy arose. Moreover, even if the character states themselves have remained unchanged, an apomorphic character might (because of a varying selective environment) have a higher current fitness than its plesiomorphic counterpart even if it did not initially spread through natural selection for current function. Conversely, a plesiomorphic character may have a greater current fitness even if selection had favored the evolution of apomorphic function in ancestral environments. Cryptic snakes, for example, may be currently more adept at avoiding predators than are coral snake mimics in sister taxa even if the mimics had a selective advantage when they first arose in ancestral populations. This example underscores the difficulty of comparing "adaptedness" between different taxa, which may have diverged in multiple attributes in addition to the character of interest. Such extraneous differences can render meaningless comparisons of a particular character's effect on survival or reproductive success. Without independent evidence for character and environmental stasis and controls for the confounding effects of changes in other characters, the mapping of character states onto reconstructed phylogenies does not resolve the problem of determining whether natural selection favoring a character's current function was indeed responsible for its origin.

Testing Hypotheses of Current

Adaptive Function

Several authors also have proposed that phylogenetic analyses can inform hypotheses of current function. The strongest assertion has been put forward by McClennan et al. (1988). Their central claim is that the demonstration that a given trait is plesiomorphic or synapomorphic renders the study of its current function superfluous, since its presence is simply the result of "phylogenetic constraint" (p. 2181) or "phylogenetic inertia" (p. 2187). Consider, for example, their discussion of clutch size in gulls (p. 2187):

"... Graves et al. (1984) asked 'Why does the herring gull lay three eggs?' Based on experimental manipulation of colonies, the authors proposed that parents are hedging their bets by producing a third egg as insurance against loss of the first or second. Phylogenetic analysis of the Charadriformes reveals that (i) all members of Larus lay three eggs and (ii) the ancestral clutch size is four eggs and not three... consequently for researchers interested in selectionist or costbenefit types of questions, the appropriate question is not 'Why three eggs?' but rather 'Why not four eggs?'. The answer to this question involves a comparison of relatively plesiomorphic ... members of Larus with outgroup species having four-egg clutches. It is at this level that the evolutionary transition from four-egg to three-egg clutches occurred. Subsequent to that event, the answer to 'Why three-egg clutches?' among Larus species is 'Because they are descended from an ancestor that produced three-egg clutches'."

This line of argument is extended and formalized by Brooks and McClennan (1991) by reference to a hypothetical cladogram of fishes in which female-only parental care is an autapomorphy and biparental care the ancestral condition. They argue that because biparental care in species C is plesiomorphic, "The answer to the question, Why does C show biparental care? is, thus, because its ancestor did," and that "From an adaptationist perspective, the pertinent question for this clade is, Why does species F show female care only? Since female care represents the evolution of a derived character state in conjunction with current environmental conditions, cost/benefit analyses in this case would address the question of character origin..." (p. 144-145, original emphasis).

We view these claims as seriously flawed for two closely aligned reasons. First, cost-benefit analyses of adaptive function in extant taxa do not address hypotheses of character origin but rather are inherently studies of selective maintenance (Tinbergen 1964; Sherman 1988). As discussed above, the

demonstration that an autapomorphic character is currently adaptive does not indicate that it arose under a selective environment equivalent to that which it now experiences. Second, the observation that a character of interest is either plesiomorphic or shared by sister taxa does not constitute evidence against stabilizing selection as a possible reason for its persistence (see Charlesworth et al. 1982; Miles and Dunham 1992; Reeve and Sherman 1993). Three eggs may well be an optimal clutch size for herring gulls and the hypothesis can be tested legitimately through experimental manipulation. Similarly, the observation that biparental care is symplesiomorphic in a clade of fishes would not invalidate the hypothesis of its selective maintenance in any given species within the clade.

Some authors have suggested that phylogenetic analyses can elucidate the relative responsiveness to selection of different characters, with those widely shared through common descent likely being less responsive than those exhibiting substantial variation among sister taxa. Ross and Carpenter (1990), for example, demonstrated that the number of queens per colony is less variable among taxa of social bees than among ants and social wasps and from this pattern concluded that "colony reproductive structure among ants may be most responsive to ... selection pressures whereas apid bees are least malleable in this regard" (p. 127). This may be the case, but it is worth emphasizing that the lack of interspecific phenotypic variation for a character does not demonstrate a lack of intraspecific genetic variation upon which selection might act. Stabilizing selection may also favor the retention of ancestral characters in sister taxa if these taxa tend to occupy similar selective environments. Moreover, the critical environmental features favoring a conserved trait may be shared even among related species occupying very different habitats. The occurrence of cooperative nesting in birds, for example, is in part dependent upon the probability that offspring can successfully nest independently (Emlen 1984); a low probability may be characteristic of disparate habitats with very different ensembles of competitors and predators.

Clearly, closely related species may share many traits because of a common lack of genetic variation upon which selection might act; this is one reason why they should not be assumed independent in comparative tests of hypotheses of adaptation (Felsenstein 1985a; Ridley 1989). Demonstrating, however, that a character is shared through common ancestry is not equivalent to demonstrating a lack of genetic variation for that character.

How, then, can the reconstruction of accurate phylogenies inform hypotheses of current function? One straightforward application is in the construction of optimality models for the selective maintenance of phenotypic characters. A first step in constructing an optimality model is to select a "strategy set"--a set of alternative traits that reasonably may be expected to have competed with the observed trait during its evolutionary history (Maynard Smith 1982; Reeve and Sherman 1993). The character states of sister taxa clearly should be included in the strategy set. For example, an optimality analysis of beak shape in birds might use engineering criteria to determine whether the morphology of a given species allows a more efficient extraction of energy from the food items in that species' diet than would the morphologies present in sister taxa.

Testing Hypotheses Predicting that Character States have Evolved in a Particular

Historical Sequence

Hypotheses of adaptation often implicitly or explicitly predict that the character states of different traits have arisen in a particular sequence and the character mapping procedures described earlier have been recently applied to test them (e.g., Sillen-Tullberg 1988; Donoghue 1989; Prum 1990;

Sillen-Tullberg and Moller 1993). In a hypothesis imbedded in inclusive-fitness theory, for example, West-Eberhard (1978) proposed that the evolution of a sterile worker caste in vespid wasps was mediated by nest sharing among closely related females of a single generation rather than by associations of mothers and daughters or of unrelated conspecifics. Observations of extant taxa reveal that confounding wasps do commonly reap inclusive-fitness benefits by associating with sisters (reviewed by Reeve 1991), but these studies cannot directly address West-Eberhard's (1978) historical hypothesis, that is, that nesting associations of same-generation kin arose prior to the evolution of worker sterility. In a phylogenetic test, Carpenter (1989, 1991) mapped several social characters of vespid wasps onto independently derived cladograms and concluded that such associations are in fact plesiomorphic to the evolution of worker sterility, whereas associations of mothers and daughters or unrelated conspecifics are not.

Phylogenetic tests can provide substantial insight into hypotheses of adaptive sequences of character-state transitions. However, two principal difficulties confront their use and interpretation. First, the problem discussed earlier of reliably inferring the timing and direction of historical transitions between character states becomes compounded when ancestral states need be inferred accurately for multiple, potentially labile, characters. This problem may be minimized when a hypothesized transition sequence can be evaluated in multiple independent lineages. Prum (1990), for example, parsimoniously mapped the plumage and behavioral display characters of male manikins onto independently derived cladograms and found that novel displays appear to have arisen prior to their associated plumage features in several monophyletic groups. Thus, the hypothesis that novel plumage characters drive the evolution of male displays may be rejected with some confidence. In the absence of evidence of equivalent patterns from multiple independent lineages, however, inferred historical sequences should be interpreted with caution.

Second, phylogenetic analyses are likely to shed light on hypothesized historical sequences only when the selective associations between characters are relatively weak and the rate of cladogenesis relatively high (Armbruster 1992). A phylogenetic test may either reject a particular hypothesis (indicating that characters under study do not appear to have arisen in a predicted sequence) or corroborate it (indicating that the character predicted to have arisen first is, in fact, typically plesiomorphic to the other). When the selective association between traits is strong, however, both characters will tend to appear at the same branch point in the cladogram and their order of appearance remain unresolved.

To summarize, the mapping of characters onto reconstructed phylogenies is in our view a valuable technique for testing whether the character states of multiple traits have evolved in a predicted historical sequence, provided that the test can be carried out on multiple independent lineages and the selective association between traits is relatively weak. In contrast, we argue that phylogenetic analysis cannot be used to determine whether a character arose in ancestral populations through natural selection for its current function or whether its presence in an extant taxon results from a "phylogenetic constraint" on adaptive evolution.

PETER C. FRUMHOFF[1] AND H. KERN REEVE[2,3]

[1] Department of Zoology, University of Maryland, College Park, Maryland 20742-4415

[2]Museum of Comparative Zoology, Harvard University, Cambridge,

Massachusetts 02138

[3] Present address: Section of Neurobiology and Behavior, Division of Biological Sciences, Seeley G. Mudd Hall, Cornell University, Ithaca, NY 14853-2702.

ACKNOWLEDGMENTS

We thank J. M. Carpenter, J. Felsenstein, K. Fischer, L. Keller, M. D. Pagel, H. B. Shaffer, G. J. Vermeij, W. Jackson, P.S. Ward, and G. S. Wilkinson for discussion and thoughtful comments on drafts of this manuscript. P. C. Frumhoff was supported by a Science and Diplomacy Fellowship from the American Association for the Advancement of Science and H. K. Reeve was supported by a Junior Fellowship from the Harvard University Society of Fellows.

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Corresponding Editor: H. B. Shaffer

APPENDIX

Consider a trait that has two phenotypic character states A and B. Two sister species S1 and S2 exhibit state A (formally, S1, S2 = A). We can then ask the following: What is the probability that their most recent common ancestor X exhibited character state B when these species diverged (i.e., X = B) given that the probabilities of intraspecific transitions from A to B and B to A within a given period of time are both greater than zero and that t time periods have elapsed since their divergence? That is, we seek the conditional probability Z = Pr(X = B/S 1 = A and S2 = A), because this gives us the probability of incorrectly ascribing state A to the ancestor. By Bayes' Theorem, Z is equal to

[mathematical expression omitted] where Pr(X = B) is the a priori probability that the ancestral species exhibited character B at the time of speeiation. Now the calculation of Pr(S 1 = A and S2 = A/X = B) can be obtained from an analysis of the Markov chain describing the above transitions. Let the transition probability matrix of the associated Markov chain be as follows:

[mathematical expression omitted] where a is the probability of transition from character state B to A and b is the probability of transition from A to B in one time period. For simplicity, we assume that these probabilities are identical for both descendant species. In general, the probability that a species will exhibit state A t time periods after beginning in B is given by

[MATHEMATICAL EXPRESSION OMITTED]

(Taylor and Karlin 1984). Thus, Pr(SI - A and S2 = A/X = B) is simply the square of this expression. The probability, therefore, of ending up in state A given that one starts in state B depends upon the values of a, b, and t, and can be substantial.

The probability that a species will exhibit state A t time units after beginning in A is equal to

[MATHEMATICAL EXPRESSION OMITTED]

If, therefore, t is sufficiently large, two sister species will exhibit character state A with probability [a/(a + b)]2, independently of the starting state; in such a case, Z is simply equal to Pr(X = B), because Pr(S 1 = A and S2 = A/X = B) becomes equivalent to Pt(S1 = A and S2 = A) [see formula (1)]. In this case, therefore, the character state shared by the sister species yields no information about the character state of their common ancestor.

We can use these results to obtain the error probability Z for all t. We need calculate only Pr(SI = A and S2 = A), which, from above, is equal to Pr(X = B)U, U2 + [1 - Pr(X = B)]V V2, where the subscripts refer to S 1 and S2. Thus, by (1), [MATHEMATICAL EXPRESSION OMITTED]

We can now examine Z for the special case in which there is no prior knowledge of the ancestor's state, that is, Pr(X = B) = %. In this case, Z = [U.sub.1], [U.sub.2]/[U.sub.1][U.sub.2] + [V.sub.1][v.sub.2]). As can be seen in figure 1, the probability of erroneously assigning the character state of two descendant species to a common ancestor can be substantial, particularly when t is large and when a is greater than b (e.g., when selection favors trait A in the environments of both descendant species).

This error probability would clearly decrease as the number of descendant species sharing a common character state increases. However, a generalization of the above model to the case of Ndescendant species shows that, even when a few within-species transitions occur, the error probability can be large even for many descendant species sharing the trait. Suppose for simplicity that the N descendant species simultaneously branch off from their common ancestor and follow independent evolutionary trajectories, each characterized by the same transition probability matrix. The error probability Z in our example is equal to Z = [U.sup.N]/ ([U.sup.N] +[V.sup.n]). Figure 2 show that even two expected transitions within a lineage leads to a very high error probability (even when transitions from a to b and b to a are equiprobable) unless the number of descendant taxa is very large. For example, an error probability of 0.20 or less requires 40 descendant taxa when there are two expected transitions per lineage and requires 2241 descendant taxa when there are only four expected transitions per lineage. Thus, even when many descendant taxa share a character state, the inference through cladistic optimization that their common ancestor must have exhibited that character state is potentially extremely error prone when within-taxa transitions are probable.

Now suppose that character state transitions among descendant lineages are perfectly positively correlated (see text). The probability that all descendant taxa will exhibit state A t time units after beginning in B is simply equal to U. Because U/(U + V)>[U.sup.N]/([U.sup.N] + V.sup.N) for N> 2, if U< V (as will occur if a < 1 - b), positively correlated transition sequences among related taxa can inflate the probability of erroneously inferring the ancestral state from parsimony.
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Author:Frumhoff, Peter C.; Reeve, H. Kerne
Publication:Evolution
Date:Feb 1, 1994
Words:5767
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