rates likelihood methods requires To assess the effect of a character on diversification rates using that a group of extant species

Transcription

1 Syst. Biol. 56(5): , 2007 Copyright (?) Society Systematic Biologists?SSN: print / X online DOI: / Estimating a Binary Character's Effect on Speciation and Extinction Wayne P. Maddison,1'23-4 Peter E. Midford,1 and Sarah P. Otto12 department Zoology, University British Columbia, Vancouver, BC V6T 1Z4, Canada; wmaddisn@interchange.ubc.ca (W.P.M.) 2 Biodiversity Research Centre, University British Columbia, Vancouver, BC V6T 1Z4, Canada ^Department Botany, University British Columbia, Vancouver, BC V6T1Z4, 4Wissenschaftskolleg zu Berlin, Wallotstra?e 19, Berlin 14193, Germany Canada Abstract.?Determining wher speciation and extinction on depend state a particular character has been long-standing interest to evolutionary biologists. To assess effect a character on diversification using likelihood methods requires that we be able to calculate probability that a group extant species would have evolved as observed, a given particular model character's effect. Here we describe how to calculate this probability for a phylogenetic tree and a two-state (binary) character under a simple model evolution ( "BiSSE" model, binary-state speciation and extinction). The model involves six parameters, two specifying speciation (rate when lineage is in state 0; rate when in state 1), two extinction (when in state 0; when in state 1), and two character state change (from 0 to 1, and from 1 to 0). Using se probability calculations, we can do maximum likelihood inference to estimate model's parameters and perform hyposis tests (e.g., is rate speciation elevated for one character state over or?). We demonstrate application method using simulated data with known parameter values. [Birth-death process; branching process; cladogenesis; extinction; key innovation; macroevolution; phylogeny; speciation; speciose; statistical inference.] The pattern branching a phylogenetic tree con tains information about processes speciation and extinction (Nee et al, 1994b; Barraclough and Nee, 2001). For instance, extinction may be revealed by an upturn near present in a plot species lineages through time (Nee et al., 1994a). Of special interest is wher phy logenetic trees can be used to demonstrate that certain characteristics a lineage, such as ecological niche or mating system, affect rate speciation or extinction (Mitter et al, 1988; Barraclough et al., 1998; Gittleman and Purvis, 1998). Often used to answer se questions are sister-clade analyses (Mitter et al. 1988; Farrell et al. 1991; Barraclough et al., 1998; Vamosi and Vamosi 2005). For example, Mitter et al. (1988) showed that herbivorous clades beetles were more speciose than ir carniv orous sister clades; this pattern indicates that herbivory confers eir a higher speciation and/or a lower extinc tion rate. Comparison sister clades is a simple and rel atively nonparametric approach (Slowinski and Guyer, 1993; Barraclough et al., 1996) and has had a broad im pact on macroevolutionary studies. However, it has some drawbacks that prompt us to explore alternatives. Sister clade comparisons cannot distinguish differential speci ation from differential extinction (Barraclough and Nee, 2001). Also, when character interest is a simple cat egorical variable, clades with mixed states cannot easily participate in test. Then, choice clades can be arbitrary, and information is discarded when collapsing phylogenetic tree into a set clade pairs. In princi ple it should be possible to find a method considering whole tree that is more powerful than those selecting a subset clade pairs (Chan and Moore, 2002; Ree, 2005; Paradis, 2006). Ideally, we would like a likelihood-based approach to estimate effect a character on diversification. The underlying likelihood model would allow specia tion and extinction to on depend character state a lineage at each point in time and allow character state to change. Inferences about speciation and extinc tion as a function character state could n be made based on ir likelihood: probability ob serving data ( phylogeny and current charac ter states) given proposed values for rate parameters. In this paper, we describe a method for calculating this likelihood when character controlling diversification has two states (i.e., is binary). We show how this calcula tion leads to new methods for parameter estimation and hyposis testing about a binary character's effect on di versification. The calculations can also be readily applied to a broader set questions about character evolution or indeed about tree inference itself, we although will not address se questions here. Likelihood calculations for models involving specia tion and extinction (Nee et al., 1994b, Moore et al., 2004) and character state change (Pagel, 1994) have been described separately, but have not yet been fully integrated. Notable contributions to this effort have been made by Pagel (1997), Paradis (2005), and Ree (2005). PageTs (1997) model permits different charac ter states to confer different speciation, but it assumes no extinction and that character oc change curs only at speciation events. Paradis (1995) and Ree (2005) present likelihood-based methods that use re constructed ancestral states to compare speciation between states (again, ignoring extinction). Most impor tantly, se previous methods have all assumed that ancestral states can be assessed without accounting for effects character on speciation/extinction pro cesses. This is problematic, because reconstruction a particular ancestral state depends critically on how character affects speciation and extinction. For example, imagine a phylogeny in which one character state promotes speciation compared to a second state, resulting in a greater proportion extant species with first state. If we attempted to understand character change without taking into account character's effect on speciation, we would interpret state's abundance to reflect a higher rate change toward it, even if in 701

2 702_SYSTEMATIC BIOLOGY_VOL. 56 fact change were equal to and from state (Maddison, 2006). This in turn would bias ancestral state reconstruction and mislead us about process diversification. Our approach here is first to incor porate explicitly process character state change di rectly and simultaneously into likelihood assessment speciation and extinction. Here we describe how to calculate probability tree and observed states a given basic model with six parameters: instantaneous speciation and ex tinction when lineage is in state 0 (e.g., herbivory), when lineage is in state 1 (e.g., carnivory), and instantaneous character state change (0 to 1 and 1 to 0). We will occasionally abbreviate this model as BiSSE (binary-state speciation and extinction) model. Following a presentation model and likelihood cal culations, we apply method to some simulated data sets. Likelihood BiSSE Model We assume that an accurate rooted phylogenetic tree with branch lengths is known ( "inferred tree") and that character state is known for each termi nal taxa. (Alternatively, our methods could be applied to each fully resolved trees coming from a Bayesian MCMC analysis?yang and Rannala, 1997; Larget and Simon, 1999.) The tree is assumed complete: all extant species in group have been found and included. We consider only binary characters, but extending method to multistate characters would be straightfor ward. All terminal taxa are contemporaneous, and tree is ultrametric (i.e., total root-to-tip distance is same for all tips). Except where noted, we measure time backwards, with 0 being present. The parameters model are as follows. While a lineage has character state 0, instantaneous specia tion rate is Ao, extinction rate is?jlq, and rate transition to state 1 is?/oi- Similarly, while a lineage has character state 1, speciation, extinction, and transi tion are A4,?i\, and c\\?. These are as parameters sumed constant throughout tree, although it would be straightforward to extend model to explore hy poses about changes to se parameters. We assume that transitions happen instantaneously over time scales considered (i.e., we ignore periods time during which a species is polymorphic). We also assume that se events are independent one anor; in particu lar, we assume that character state does change not, in and itself, cause speciation (or vice versa). Probability Tree and Character States (D) Although reflect probabilities events mov ing forward in time, our calculations will move back ward in time, from tips tree to root. This is well-established "pruning" (Felsenstein, 1981) or "downpass" (Maddison and Maddison, 1992) approach and is used to handle more compactly all possibilities ancestral states at various parts tree (Felsenstein, 1981). This approach uses a simple principle: if we are N t -*r Assume: Dwo(r) and Dni(?) t+at%, Calculate: Dno(?+At) and Dw7(r+Ar) FIGURE 1. Calculation probabilities (D) observed tree and character states, a along branch tree. We assume that we know D's for time t on branch and attempt to calculate m for time t+at. able to use key probabilities at any point on a tree to de rive corresponding probabilities immediately ancestral (i.e., closer to root), n it must be possible to move step-by-step down tree toward root. When root is reached, calculated probabilities will apply to whole tree. Our calculations make use full tree un topology, like those Nee et al. (1994b), which use only timing branching events. We need to use full tree topology because we are considering simultaneously evolu tion character states. In Appendix 1 we describe how removing on dependence character states reduces our equations to those Nee et al. In BiSSE model, key probabilities, Dm)(?) or D]vi(0/ describe chance that a lineage beginning at time t with state 0 or 1 would evolve into a clade like that observed to have descended from node N, closer to present (Fig. 1). We track se probabilities back by a very short amount time, At, toward root, accounting for all possible events that could have hap pened along way. If we use a small enough time in terval At, we can ignore possibility that more than one event happens during time interval. Once we have derived equations for change in probability over At, we n shrink time interval and use definition a derivative to obtain differential equations describing change in se as we probabilities de scend toward root. By integrating se differential equations along branches, we are able to solve for overall probability data given BiSSE model. In we following, carry out se calculations, first along a branch and n across nodes in tree. Calculations within a branch.?let us assume that we have already calculated probability Dm)(0 that clade node N and its stem lineage would have evolved exactly as represented by inferred tree (including branch lengths and terminal character states) given that lineage started in state 0 at time t on branch be low node N (Fig. 1). We will describe calculations focused on state 0; calculations for state 1 are parallel and indeed need to be performed simultaneously.

3 2007 MADDISON ET AL.?A CHARACTER'S EFFECT ON DIVERSIFICATION 703 3) No state change, State b) change, No state c) change, No state (j) change, No speciation No speciation Speciation & Extinction Speciation & Extinction FIGURE 2. Alternative scenarios by which a lineage with state 0 at time i+ai on branch might yield clade decended from node N but no or living descendants. To move next step down tree (Fig. 1), we need to calculate corresponding probabilities for one small time step (Ai) furr down branch; i.e., DN0(t + Ai) and Dm(t + Ai). To calculate Dm)(? + Ai), we enumerate all ways that observed clade could have arisen by considering all possible events that could occur in Ai time interval. Specifically, we use law total probability to write D]yo(i + Ai) as sum all pos sible transitions forward in time from i + Ai to i times probability that clade would have evolved in manner observed from time i to present. There are four cases, each which requires that lineage in terest did not go extinct during Ai interval (Fig. 2). In first case (Fig. 2a), nothing happens in Ai interval. In second case (Fig. 2b), character changes. In third case (Fig. 2c), a speciation event occurs and left lineage gene node N; right lineage must thus go extinct sometime before present, which occurs with probability Eo(t), described furr in next section. The fourth case (Fig. 2d) is similar, except that left lineage goes extinct and right lineage gene node N. No or cases contribute to + probabilityd^o(t Ai) because assumptions that events are inde pendent and clade had to survive to present. The probabilities for se four cases can n be summed to yield D^o(t + Ai): ligibly small if Ai is small. Dropping Ai2, we get: all terms order DN0(t + Ai) = [1 - (A0 + /x0 + i?oi)ai]dno(i) + (q01 At) Similarly, Dm(t) + 2(X0At)E0(t)Dm(t) (2a) DN1(t + Ai) = [1 - (Ai + /X! +?io)ai]dni(i) + (q10at) DN0(t) + 2(AX A?)Ei(?)DM(?) (2b) Dividing [Dm(t + Ai - Dm(t)] and [DN1(t + Ai) - Dni(?)] by time interval, Ai, and taking limit as Ai goes to zero, we can derive two coupled differential equations: -?? = -(A0 + /xo + qoi)dno(t) + at q01dn1(t) +2X0Eo(t)DNO(t) (3a)? = -(Ai + mi + qio)dm(t) + qiodno(t) + 2AiEi(i)DM(i) (3b) DN0(i+Ai) = (1 -/x0a?)x [(l-?oiai)(l-a0a?)dno(i) + (<?oia?)(l-a0a?)dn1(?) + (l-q01at)(k0at)eo(t)dm(t) + (1-qoi At)(k0At)E0(t)DN0(t)] + (/x0a?) xo (in all cases no extinction in Ai) (see Fig. 2a: No state change, no speciation) (see Fig. 2b: State change, no speciation) (see Fig. 2c: No state ^ ' change, speciation, extinction) (see Fig. 2d: No state change, speciation, extinction) (if lineage went extinct, clade has zero probability being observed) For clarity in Equation (1), we have ignored several pos sible transitions that involve multiple events within time interval (e.g., speciation and extinction), because se occur with a probability ( order Ai2) that is neg We have not found analytical solutions for se equa tions. Neverless, given solutions for Eo(t) and E\{t) described below, y can be numerically integrated

4 704_SYSTEMATIC BIOLOGY_VOL. 56 along a branch. This permits us to derive D^o(t) and Dm(t) a* bottom branch (rootward) from probabilities at top branch. Initial conditions.?if N is a terminal node with state 0, n initial conditions are = DN0(0) 1 and D]vi(0) = 0. That is, state must be that observed at time 0. Likewise, if taxon has state 1, Dno(0) = 0 and = Dni(0) 1. Calculations at nodes.?when we arrive at bottom branch, i.e., at immediate ancestral node A, we need to combine probability from branch with probability from its sister branch before con tinuing journey down ancestral branch. For?s clade to have evolved as represented in inferred tree, speciation event must have occurred, and each daughter lineages must have survived to present day. The probability A clade evolving as in inferred tree equals probability that left daughter lineage evolved as in tree, times probability that right daughter lineage evolved as in tree, times proba bility speciation across a small interval time, Ai. This probability speciation, A Ai, thus introduces a Ai term at each node. These Ai terms refore contribute a factor ( At)n to probability over whole tree, where n is number internal nodes. Because this factor does not depend on parameter values, we can safely ignore it without changing proportionality likelihoods under different parameter values. Technically, ignoring it converts our probabilities to probability densities, but we will continue to speak probabilities for brevity. Thus, right before speciation event has occurred, probability that lineage is in state 0 or 1 and evolves into a clade like that observed to have descended from node A (including speciation event at node A) is given by: DA0(tA) DN0(tA)DM0(tA)X0 (4a) DA1(tA) DN1(tA)DM1(tA)X1 (4b) where M is sister node to N. Note that se equations include only case where character state is same for ancestor A and its two descendant lineages. That is, we treat speciation and character state changes as independent events, so that probability is zero for both changes to occur simultaneously. Now, having passed node A, we can continue toward root, using D^q(?a) and DAi(tA) as starting points for numerical integration down?s branch. At root.?when we get to root R we will have calculated Dro(?r) and D#i(i#), which describe probability observing phylogeny and extant character states, given that root was in state 0 or 1, respectively. To obtain a single likelihood to serve as basis for inference we need to account for proba bility that root was in state 0 or 1. We could, fol lowing Schl?ter et al. (1997) and Pagel (1999), add DR0 and Dri toger. This effectively assigns a probability 0.5 to root being in state 0 and to root being in state 1, even if transition probabilities BiSSE model would make it much more likely that charac ter is in one state or or. we Alternatively, could weight se root states by equilibrium frequencies 0 and 1 implicit in model (see Appendix 2), as done by default in Mesquite version 1.1 and later for similar likelihood calculations for ancestral states (Mad dison and Maddison, 2006). We take this latter approach here. Probability Extinction (E) In order to use differential Equations (3) to cal culate probability, Dm)(0 or D]vi(i), that a lineage evolves as represented in inferred tree, we must de termine probability that a lineage alive at time i goes extinct before present time. Here, we derive Eo(i) and E\(t) using a similar procedure, except that ex tinction probabilities do not depend on tree structure surviving lineages, on only time. a) Extinction in At b) No speciation, c) No state change Extinction since t No speciation, State change Extinction since t d) Speciation, No character change Extinction both since t f+af^o t t+af FIGURE 3. Alternative scenarios by which a lineage at time t with state 0 might go extinct.

5 2007_MADDISON ET AL.?A CHARACTER'S EFFECT ON DIVERSIFICATION_705 Assume that we have calculated E0(t), probability that a lineage starting at time i in state 0 leaves no descen dants at present day. Then Eo(i + Ai) can be obtained by considering four different possible events in Ai interval consistent with lineage (and all its de scendants) going extinct ultimately (Fig. 3). In first case (Fig. 3a), lineage goes extinct during Ai time interval. In second case (Fig. 3b), lineage neir goes extinct nor changes state nor speciates between i and i + Ai, but neverless it goes extinct eventually. In third case (Fig. 3c), lineage changes state dur ing Ai time interval and n goes extinct. Finally, in fourth case (Fig. 3d), lineage speciates during Ai time interval, but now both descendant lineages must go extinct before present day. We assume in fourth case that extinction events are independent one anor, thus contributing a term Eo(t)2 to prob ability. No or cases need be considered because assumption that events are independent and Ai is small. extremely The probabilities for se four cases can n be summed to yield probability extinction, Eo(i + Ai): Initial conditions.?at time i = 0, re is no time for extinction, and thus = = Eq(0) Ei(0) 0. Application to Estimation and Testing Maximizing BiSSE likelihood, calculated as de scribed above, can yield estimates for all six parameters. This means that method could be used not just to un derstand speciation and extinction, but entire process diversification including character state change. With such a flexible model, a variety hyposis tests are possible. For example, is speciation under state 0 more rapid than under state 1? Is net diversification rate (speci ation? extinction) higher under one state than or? What causes an excess species with one state (Maddi son, 2006): asymmetrical character change (rate 0 to 1 different from rate 1 to 0), asymmetrical extinction, or asymmetrical speciation? To answer se questions, one can perform likelihood ratio tests by comparing like lihoods given unconstrained versus appropriately con strained models. E0(i + Ai) = /x0ai + (1 - /x0ai)(l - qoiat)(l - AoAi)Eo(i) + (1 - fi0at)(q01at)(l - A0Ai)Ei(i) + (1 - MoAi)(l - qq1 Ai)(A0Ai)E0(i)2 (see Fig. 3a: Extinction in Ai) (see Fig. 3b: No state change, no speciation) (see Fig. 3c: State change, no speciation) (see Fig. 3d: No state change, speciation) Again, we have included some terms in Equation (5) that are negligibly small ( order Ai2). Dropping se terms, we get: E0(i + Ai) =?0At + [1 - (/x0 + coi + A0)Ai]E0(i) Similarly, + (qoi Ai)Ei(i) + (A0Ai)Eo(i)2 Ei(i + Ai) = tu At + [1 - (/xi + qio + Ai)Ai]Ei(i) (6a) + (??ioai)e0(i) + (AiAi)Ei(i)2 (6b) Dividing change in Eq and Ei by time interval, Ai and taking limit as Ai goes to zero yields coupled differential equations: deo di dei ~at = (Mo + qoi + ko)e0(t) +??oiei(0 + A.0E0(i)2 (7a) (/xi + q10 + Ai)Ei(i) + q10e0(t) + AiEi(i)2 (7b) As with equations for D(t), se equations for extinction probabilities can be solved using numerical integration. Mo - /xi - In we an following, provide initial exploration BiSSE using simulated phylogenetic trees and characters. First, we estimate rate parameters when true speciation, extinction, and character transition eir do or do not depend on character state. Second, we examine wher null hyposis equal can be rejected using likelihood-ratio tests. These examples do not attempt to a provide full ex ploration method. To understand method's capabilities and limitations, we would want to study power method to reject false null hyposes under a variety parameter values and assumptions, includ ing cases with multiple simultaneous asymmetries. We would also like to know size tree needed for adequate power, as well as accuracy and precision estimated parameter values. These explorations we leave for a subsequent paper. Implementation The BiSSE likelihood calculations described above have been programmed in Diverse package mod ules (Midford and Maddison, 2007) for Mesquite (Mad dison and Maddison, 2007). These calculations use a fourth-order Runge-Kutta method numerical inte gration (Raison and Rabinowitz, 1978) for proceeding along branches and Mesquite's implementation Brent's (1973) optimizer for seeking maximum likeli hood estimates. If choice is made to condition

6 706_SYSTEMATIC BIOLOGY_VOL. 56 on probabilities survival entire clade, module uses Nee at al.'s (1994b) convention (i.e., con dition used is that at least two descendants clade survive). In our examination method, maximum likeli hood optimization was first attempted using 10 random starting parameter values, with numerical integra tor using a coarse division each branch into segments (an average-length branch was assigned 100 segments; or branches were assigned up to a maximum 400 or a minimum 50 segments in proportion to ir length). Error in numerical integration is expected to decrease as segment length decreases, and refore most likely parameter set resulting from 10 at tempts was n used as a starting point for a final op timization using a finer division branches (average 1000 segments, maximum 4000, minimum 500). Details numerical integration method will be considered in a subsequent paper, but we note that reasonable estimates obtained in our results suggest that method is having some success. Likelihoods were conditional on clade survival. The prior used for character state at root (i.e., probability state 0 versus 1 at root) was set equilibrium frequencies implicit in proposed model parameter values. Simulated trees and terminal character states were generated in Mesquite via a prerelease version "BiSSE Trees & Characters" module Diverse pack age (Midford and Maddison, 2007). This module divided time into small steps, such that if an event's occurrence has an instantaneous rate r probability event in time slice is r /1000. In each time slice on each lin a eage, uniform random number was drawn to determine wher character changed, followed a by random number draw to determine if extinction occurred. If ex tinction did not occur on a lineage, n a random num ber was drawn to determine if speciation had occurred. If tree went extinct, simulation was restarted. A sim ulation continued until tree first had requested number species (e.g., 500). This is not ideal, because extinction and speciation could have caused tree to achieve requested number species several times; stopping simulation at first time biases re sult toward shorter terminal branches. The bias should, however, be very small for parameters investigated below, where speciation are high relative to extinction and character state change and where size tree is large. (For example, under scenario with highest extinction rate explored below, average height tree when 500 species was first reached was that height tree when 500 species was last reached, as determined by 100 simulations in which trees were allowed to grow to 600 species. See also Ap pendix 2 for a deterministic approximation to average times to reach a certain number species.) Parameter Estimation To explore parameter estimation, we simulated trees under each four parameter combinations. The first combination was entirely symmetrical: A0 = k\ = 0.1,?jlq =??i = 0.03, qoi = qio = The or three com binations introduced an asymmetry in each process (speciation, extinction, character change) by altering one parameter at a time in perfectly symmetrical model. Specifically, three alternative models involved raising speciation rate with state 1 = (Ai 0.2), raising ex tinction rate with state 0 = (/?o 0.06), or lowering rate character change to state 0 = (qio 0.005). Five hundred trees 500 extant species were simulated under each model. Maximum likelihood estimates parameters from resulting trees were obtained from Diverse package. The results are plotted in Figure 4, which focuses on estimate asymmetric parameter in each sim ulation. In general, estimates speciation An and Ai are fairly good, falling near correct param eter values. As well, estimates for symmetrical case = (An Ai = 0.1) are easily distinguished from es timates in asymmetrical case = (An 0.1, k\ = 0.2). For extinction and character state change, estimates do not so closely match simulated. Wher this reflects general difficulties in estimating ex tinction (Nee et al., 1994; Kubo and Iwasa, 1995; Paradis, 2005) or state change or instead depends on particu lar parameter values chosen remains to be explored in a future paper. We examined Hyposis Testing same simulated trees to explore what power method would have to reject null hypoth esis that a rate (e.g., speciation) was equal for two character states, when in fact it was different. In addition to unconstrained six-parameter likelihood estimates described above, we also calculated maximum likeli hood given constrained five-parameter models, holding eir speciation = equal (A0 Ai), extinction = equal (/?o Mi)/ or state change equal (q01 = q10). The log-likelihood difference between constrained five-parameter likelihood model and un constrained six-parameter likelihood model was used as a test statistic. If this difference was larger than a critical value (based on simulated trees with truly symmetri cal rate parameters), n we rejected null hyposis symmetrical for parameter interest. When investigating wher character affected speciation, 5% symmetric simulations yielded a 2 x log-likelihood difference 3.60 or more between constrained (An = Ai) and unconstrained (A.n. ^ Ai) models even though re was no difference in speci ation rate. Thus, we used 3.60 as our 5% cutf for signif icance. Of asymmetric simulations, 58% resulted in a greater log-likelihood difference than this cutf, leading to rejection null hyposis equal speciation = (Ao Ai). This shows at least some power to re ject null hyposis with trees 500 species when re is a twold difference in speciation. Rejec tion with asymmetries in extinction or character state change were, however, much lower. ex Exploring tinction rate differences, 21.2% asymmetric simula tions exceeded cutf 2 x log-likelihood difference 4.08 based on symmetric simulations. Exploring

7 2007 MADDISON ET AL.?A CHARACTER'S EFFECT ON DIVERSIFICATION 707 a) 0.4 Speciation b) 0.15 Extinction S 03 E 0.2 "O 0) "c? E ^OA *t estimated A estimated ji i State change 0.03 T3? "CO estimated q0i 0.04 FIGURE 4. Estimated parameter values from simulated trees and characters. Lines indicate true parameter values simulations (solid, equal for two states; dashed, unequal). Small symbols represent estimates; large symbols represent mean values estimates. Open circles represent parameter estimates from symmetrical simulations; i.e., equal parameter values for states 0 and 1 (speciation 0.1, extinction 0.03, character change 0.01). Closed triangles represent estimates from simulations with asymmetries in respectively estimated parameters. Asymmetries shown are (a) speciation rate = asymmetry (ki 0.1 versus 0.2), (b) extinction rate asymmetry (/x0 = 0.03 versus 0.06), and (c) character state change rate = asymmetry (q versus 0.005). In maximum likelihood analysis, all six parameters were free to vary, but each scatterplot focuses on only parameters interest in each case. character state change, only 16.4% asymmetric simulations exceeded cutf 2 x log-likelihood dif ference 4.52 based on symmetric simulations. As we noted above with respect to parameter estimation, se results do not necessarily indicate that extinction or character are changes generally harder to study, only that y are under parameter values studied. Given sufficient data, we would expect twice log likelihood difference from constrained versus uncon strained models to follow ax2 distribution with one de gree freedom, under null hyposis equal for two states. The 5% cutf for significance under this asymptotic distribution would be The 5% cut f values established by our equal-rate simulations are

8 708_SYSTEMATIC BIOLOGY_VOL. 56 close to this value, suggesting that x2 approximation may be a reasonable one for likelihood ratio tests involv ing trees this size. It remains to be studied how probability reject ing a false null hyposis (power) varies with number species and with degree difference in. From our initial exploration, however, we suspect that large phylogenetic trees will generally be needed to have suf ficient power. Rar than a being sign a weakness in method, we believe that this low power stems from fact that re are many similar ways to generate an observed phylogeny and observed character states (Maddison, 2006)?and only with much data can se be distinguished. Extensions In this paper, we have used se likelihood calcula tions to explore our ability to detect asymmetrical speciation, extinction, or character state change. The calculations could equally be applied to exploring how uncertainty in some se parameters (e.g., specia tion and extinction ) influences likelihood esti mates for or parameters (e.g., character state changes). They could also be used to infer ancestral states using likelihood (Schl?ter et al., 1997). Maddison (2006) has explained why methods are needed that consider a char acter 's effect on diversification when interpreting character's evolutionary history; our likelihood calcu lations can provide such methods. They could also be used in tree inference itself, permitting models in which character states affect diversification processes. The general approach describing differential equa tions for likelihood and integrating down branches should be applicable to a broad array questions. Us ing numerical integration methods frees us from constraint considering only those models that can be solved analytically. We could, for example, extend method to include multiple correlated characters or or models character evolution or or models speci ation (e.g., speciation that occurs simultaneously with character change). Anor useful extension would be to deal with continuous valued data, perhaps incorpo rating a model such as Slatkin's (1981). Paradis's (2005) method is already generalized for multiple characters different types, although as noted it does not integrate state change with speciation/extinction into a common model. A final caution about BiSSE method: even if it proves to have adequate ability to estimate parameters and test hyposes, it will suffer same limitation (Read and Nee, 1995; Maddison, 2000) shared by Pagel's (1994) and Maddison's (1990) tests character correla tion and Ree's (2005) test diversification. If we deter mine, for example, that Ai > Ao, we cannot on this basis alone say that character interest is controlling speci ation rate. Anor character, with parallel distribution character states, could in fact be responsible. This would be little concern if our character's states were scat tered with much homoplasy on tree, because n we could argue that any or character is unlikely to have a parallel distribution, unless it were causally linked to our examined character, and so indirect causality could still be argued. But, issue codistributed characters would be a concern if all species with state 1 (for example) occurred in a single clade. Our method, as well as ors cited, could give a significant result in such a case, even though any or synapomorphy clade could actually be responsible for effect, and such a synapomorphy might be unrelated to our character interest except by coincidence origin on same sin gle lineage (Read and Nee, 1995; Maddison, 2000). Thus, correct conclusion given a significant result using our method is that character examined or a codistributed character appear to be controlling diversification. Acknowledgments We thank Arne Mooers and Jeff Thorne for helpful discussion. Risa Sargent, Rick Ree, Brian Moore, and an anonymous reviewer gave use ful comments on a previous version this paper. This work was sup ported by U.S. National Science Foundation grant EF for CIPR?S project, by NSERC Discovery Grants to WPM and SPO, and by a sabbatical fellowship from National Evolutionary Synsis Center to SPO. REFERENCES Barraclough, T. G., P. H. Harvey, and S. Nee Rate rbcl gene sequence evolution and species diversification in flowering plants (angiosperms). Proc. R. Soc. Lond. B 263: Barraclough, T. G., and S. Nee Phylogenetics and speciation. Trends Ecol Evol. 16: Barraclough, T. G., A. P. Vogler, and P. H. Harvey Revealing factors that promote speciation. Phil. Trans. R. Soc. Lond. B 353: Brent, R. 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Rannala Bayesian phylogenetic inference using DNA sequences: A Markov chain Monte Carlo method. Mol. Biol. Evol. 14: First submitted 22 December 2006; reviews returned 23 May 2007; final acceptance 29 May 2007 Associate Editor: Todd Oakley Appendix 1 Character-independent model.?the likelihood model described in text can be reduced to a simple speciation-extinction (or birth-death) model if one assumes that speciation and extinction are not af fected a by character. As speciation-extinction model has been well studied (e.g., Nee et al., 1994b), we describe how our calculations re duce to irs in case that speciation and extinction are constant (k,?jl). Rederiving Equations (3) and (7) without reference to a character, we obtain differential equations for DN(t), probability that a lineage at time t gives rise to extant descendant lineages and no ors, and E (t), probability that a lineage and all its descendants go extinct before present day: =?f- -(k + ti)dn(t) + dr 2kE(t)DN(t) (8a) j p? = - fi (p + k)e(t) + ke{tf (8b) Using se two differential equations, tree can be traversed as described in text to a yield probability observing extant data. In this simple case, however, it is possible to derive an analytical solution for calculations probabilities a along single branch. We describe this briefly to show relationship between our method and that Nee et al. (1994b), and because character-independent model describes approximate behavior likelihood functions when character state has little effect on extinction and speciation. The solution for E(t) is:?(t) = 1-(x-Ae^) ***" (9a)?(f) w = -^? iik = p (9b) 1 + kt * v ; as derived by Kendall (1948). E(t) rises as we go furr back in time (increase t), approaching 1 if k < \x and?i/k orwise. Basically, furr back in time we go, more opportunity re is for entire lineage and its descendants to go extinct. The above solution for E(t) can be used to solve for DN(t): DN(0 = e-^""-'->^_g^?,^2dwfa) DN(t) = iu#m (10a) DN(fN) if X =? (10b) (1{1++Xt^y where tn is time depth node N (see Fig. 1). These equations permit us to jump from node N to its ancestor A without recourse to approximate numerical integration. To traverse node A, we need to account for speciation event, as we did in Equation (4). The above process a traversing branch and n accounting for node can be repeated until root is reached. Doing so, we get probability observing full phylogeny: DR(tR) DR(tR) = : n In,, (k-e-^-m-'ti) e-(k-p)(tj,b-ti,t) \_t_j_ (k - e-^-m-b p? (k)n iik^p (11a) In (l + U-,t)2 (k)n n i?k = fi (lib) (l + U-,&)2 where tirb is time at base z'th branch (rootward), tiit is time at terminus z'th branch length (nearest n present), is number nodes (including root node), and product is taken over all branches in tree. If we condition on existence a root node with two surviv ing lineages (dividing (11a) by k [1? E(tR)]2), n after some algebra, Equation (11a) can be shown to equal equation (21) Nee et al. (1994b) except for a factor, (N? 1)!, which does not depend on parameters. The (N? 1)! arises because Nee et al. consider probability all possible tree topologies with same branching times (regardless where branches occur), and re are? (N 1)! such tree topolo gies. Thus, our method gene same likelihood surface as that Nee et al. (1994b) when character does not influence rate speciation extended and extinction, but as we have shown, can be more readily to include more complex processes diversification. Appendix 2 Equilibrium frequencies.?the expected frequency state 0 and 1 within an assemblage species should, over evolutionary time, reach an equilibrium that can be calculated from model. Here, we consider passage time, T, in forward direction and track number lineages in state 0, n0, and number lineages in state 1, nx. Given that each lineage could speciate, go extinct, or switch state, expected number species each type should obey following ordinary differential equations:? = k0n0 - iiqn0 - qmn0 + q10m (12a) dnl -^, /iou\? = kxnx - /xini - qwni + q0lnq (12b)

10 710_SYSTEMATIC BIOLOGY_VOL. 56 From se equations, we can derive a differential equation for frequency lineage in state 0,x = n0/(n0 + ri\), using quotient rule: dtto d;ii dx ~?? Ul ~ ~?? n? df =(? +,? -gx<x-x)-x?n+v-x)<l? (13) where g = k0? po?ki + plx. The equilibrium frequency lineages in state 0, x, is thus single root quadratic equation, g x?? (1 x) x q0i + (1 - x) qw = 0, that lies between 0 and 1. When g = 0, this equi librium occurs at x = qio/(qoi +qw). Average number species over time.?we can also solve Equations (12) explicitly to determine average number species as we move forward in time from a single species to time T when a character affects diversification process. Averaging over those cases where root is in state 0 (with probability x) and in state 1 (with probability 1 - x), average number lineages, E(T), at time T grows exponentially according to: TTT)? g[*(*o-mo)+(i-*)(*i-mi)]t Q4\ If lineages in state 0 speciate more rapidly or go extinct less rapidly than lineages in state 1 (i.e., if > g 0), n more are species expected to exist after time T for groups that are initially in state 0. After some algebra, average number lineages, E0(T), at time T assuming that root state is 0 can be written Assuming E0(T) = E(T) 1 instead g(l-x)2 g(l-x)2-q0i_ Ei(T) = E(T) 1 gxz g(l-*)2 as: g(l - x)2 - q0 that root state is 1, we have: gx2 + qw + e (^o-mo-t^)t (15a) + eto-n-u)tf_g*_\ (15b) Vs*2 + w Equations (15a) and (15b) both reduce to E(T) given by Equation (14) when net diversification rate is same for two character states (i.e., g is zero). The above equations can be solved for T to estimate amount time required to generate a certain number species when charac ter state influences diversification. It should be emphasized, however, that se equations use a deterministic model (12) to approximate stochastic process diversification and character state change. Thus, Equations (14) and (15) provide only a heuristic guide to evolution ary process when character states influence speciation and extinction.