A Likelihood-ratio Test of Monophyly

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Syst. Biol. 45(4):546-558, 1996

A LIKELIHOOD-RATIO TEST OF MONOPHYLYJ O H N P. HUELSENBECK,1 DAVI D M. HILLIS,2 AND RASMUS NIELSEN1

department of Integrative Biobgy, University of California, Berkeley, California 94720-3140, USA;E-mail: [email protected] (J.P.H.), [email protected] (R.N.)2Department of Zoology, University of Texas, Austin, Texas 78712, USA; E-mail: [email protected] phylogenetic analyses are inspired by or depend upon the monophyly of a groupspecified a priori. Also, many evolutionary problems for which phytogenies are useful do notrequire every detail of the phylogeny to be estimated correctly but depend upon the monophyly(or lack thereof) of a particular group. We propose a likelihood-ratio test that compares whetherthe best trees estimated with and without the constraint of monophyly are significantly different.Simulation suggests that the test is conservative when the null hypothesis (a particular specifiedconstraint) is correct. We applied the likelihood-ratio test of monophyly to the question of therelationship of the presumed-extinct marsupial wolf (Thyladnus). Specifically, we examined thenull hypotheses that (1) the marsupial wolf does not form a monophyletic group with dasyuridsand (2) the marsupial wolf is a basal member of the Australian radiation of mammals. Bothhypotheses were rejected using the likelihood-ratio test of monophyly. [Likelihood-ratio test; max-imum likelihood; monophyly; phylogenetic methods; Thylacinus.]

Systematists are increasingly concernednot only with the accurate estimation ofphylogenetic relationships but also withthe testing of phylogenetic hypotheses.The method of maximum likelihood pro-vides a powerful framework to addressboth aspects of a phylogenetic analysis. Interms of the estimation of relationship,maximum likelihood appears to be bothefficient and robust (Kuhner and Felsen-stein, 1994; Tateno et al., 1994; Huelsen-beck, 1995a, 1995b). However, likelihoodalso provides a natural way of testing hy-potheses through the likelihood-ratio test(Edwards, 1972). The likelihood-ratio testhas been used to examine (1) the adequacyof models of DNA substitution (Goldman,1993), (2) the effect of additional parame-ters on the fit of a substitution model(Goldman, 1993; Yang et al., 1994; Yang,1996), (3) the existence of a molecular clock(Felsenstein, 1981), and (4) whether treesestimated from different data partitionsare heterogeneous (Huelsenbeck and Bull,1996).

Commonly, systematists are interestedin the monophyly of a group, and thisquestion forms the inspiration for phylo-genetic analysis. For example, Pettigrew's(1986, 1991) hypothesis that the "mega-bats7' (e.g., flying foxes) are more closelyrelated to primates than to "microbats"

spurred a flurry of DNA sequence analyseswith the primary question being, "Are batsmonophyletic?" (Bennet et al., 1988; Ad-kins and Honeycutt, 1991; Mindell et al.,1991; Ammerman and Hillis, 1992; Baileyet al., 1992; Stanhope et al., 1992). Similar-ly, Grauer et al.'s (1991, 1992) hypothesisthat the myomorph rodents (e.g., the rat-like rodents) are more closely related toprimates than to the caviomorph rodents(e.g., the guinea pig) motivated several pa-pers examining the monophyly of rodents(Hasegawa et al., 1992; Ma et al., 1993; Caoet al., 1994). In this paper, we propose alikelihood-ratio test to examine the mono-phyly of a group specified a priori. Thistest can be extended to more complicatednull hypotheses in which the constraint ontopology is not simply the monophyly ornonmonophyly of a group. We examinedthe statistical properties of this method us-ing simulation and provide an example ofthe method by testing several hypothesesof relationship for the marsupial wolf.

LIKELIHOOD-RATIO TEST OF MONOPHYLYThe method of maximum likelihood de-pends on the complete specification of a

stochastic model of evolution. We will as-sume aligned DNA sequences for the data.As an example, consider the following s =4 DNA sequences:546

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1996 HUELSENBECK ET AL.MONOPHYLY TEST 547Species 1Species 2Species 3Species 4

ACCAGTACCAGCAGCAGCAGCAGG

Each of the n = 6 site patterns representsa different datum in a maximum likeli-hood analysis. In this case, the observa-tions are xa = (A, A, A, A)T, x2 = (C, C, G,G)T, x3 = (C, C, Q C)T, x4 = (A, A, A, A)T,x5 = (G, G, G, G)T, and x6 = (T, C, C, G)T,where T is the transpose of the vector.Assuming independence among sites,the likelihood function is simply the jointprobability of observing the data

L(@|x1 / . . . ,x n) = flPr[x,|],where Pr[x,|] is the probability of observ-ing site pattern i. The vector denotes theparameters of the model of DNA substi-tution. We assume that the same model ofDNA substitution applies to all sites, al-though this constraint can be relaxed. Theprobability of observing a given site pat-tern depends on the topology of the tree,the lengths of the branches of the tree (interms of expected number of substitutionsper site), and the substitution model. Weassume a Markov chain model for the pro-cess of DNA substitution, as is commonlyused (Felsenstein, 1981).The unknown parameters of the phylo-genetic model, , are estimated by maxi-mizing the likelihood function to obtain. The topology of the tree is considereda parameter, so the best estimate of phy-logeny is simply that tree for which thelikelihood is maximized. In principle, thelikelihood is maximized over all possibletrees (i.e., every possible bifurcating tree isconsidered and the likelihood calculated),although in practice heuristic search strat-egies are used instead when the numberof sequences is large.The ratio of the likelihood of the samedata under two different hypotheses pro-vides a measure of the support of one hy-pothesis over the other. For example, ifj ^ . . . , xn)IL{H2\x v ..., x,) > 1, then

Hx is better supported than H2. The like-lihood ratio provides a framework for thestatistical testing of hypotheses.Our likelihood-ratio test of monophylycompares the likelihood calculated underthe search constraint that the group of in-terest is monophyletic with the likelihoodcalculated when this search constraint isrelaxed. The null hypothesis H o is that thegroup in question is monophyletic. Thelikelihood is maximized under the null hy-pothesis to obtain Lo. The alternative hy-pothesis Hj is more general in that itmakes no assumptions about the mono-phyly of the group in question. In otherwords, the best tree that fits the alternativehypothesis is the optimal tree. The likeli-hood is maximized under the uncon-strained hypothesis to obtain LT. The like-lihood-ratio test statistic is

8 = 2(ln Lx - In Lo).Usually, when the null hypothesis is asubset of the alternative hypothesis, thelikelihood ratio can be tested against a x2distribution with p q degrees of free-

dom, where p is the number of parametersunder the alternative hypothesis and q isthe number of parameters under the nullhypothesis (Rice, 1995). However, for thephylogeny problem, there are several po-tential problems with using the x2 approx-imation. First, the number of observationsfor many of the 4s possible site patterns isvery sm all or zero (Go ldman, 1993). Hence,the general rule of thumb that each cellshould contain five observations (or thatthe number of observations is four or fivetimes the number of site patterns) is vio-lated (McCullagh and Nelder, 1989). How-ever, the x2 approximation is known to berobust for comparing two parametricmodels in the case of sparse data (Haber-man, 1977; Agresti and Yang, 1987; Yang,1996). Another more serious problem isthat topology is not a standard statisticalparameter (Goldman, 1993). Hence, theusual results from statistics do not neces-sarily hold (e.g., the x2 approximation fornested hypotheses). For the present appli-cation, it is not clear how many parametersare represented by topology. In other



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SYSTEMATIC BIOLOGY

FIGURE 1. The space of all possible unrooted trees for five taxa. The vertices of the graph are representedby the trees. The trees are connected by lines (edges). Trees are adjacent if they are only one perturbation awayfrom each other. In this case, the perturbation is either a contraction or an expansion of an internal branch ofthe tree. Shaded portions of the tree space represent trees that contain an internal branch separating taxa 1 and3 from the remaining taxa.

words, the difference in the number of pa-rameters between the general and null hy-potheses is unknown. To avoid this prob-lem, we resort to simulation of the nulldistribution of 8. In the absence of suitableasymptotic results appropriate for all pos-sible values under the null hypothesis, themaximum likelihood values are insteadused in the simulations. In this case, themaximum likelihood topology, maximumlikelihood estimates of branch lengths, andother parameters (such as the transition:transversion rate ratio) under the null hy-pothesis are used to simulate replicate datasets of the same size as the original. This

simulation procedure for generating thenull distribution is widely used in statisticsand is known as parametric bootstrapping(Efron, 1985; Felsenstein, 1988; Goldman,1993; see also Gouy and Li, 1989; Bull eta l , 1993; Huelsenbeck et al, 1996). If thelikelihood-ratio statistic 8 calculated fromthe original data is greater than the 95%confidence interval determined throughsimulation, the null hypothesis that thegroup in question is monophyletic is re-jected.A representation of the space of all pos-sible trees gives a visual interpretation ofour test. Figure 1 shows the space of un-



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1996 HUELSENBECK ET AL.MONOPHYLY TEST 549rooted trees for five taxa. The trees are thevertices (nodes) of a graph and the linesconnecting the trees (edges) show the ad-jacencies of the trees. Trees that are adja-cent are only one perturbation away fromeach other (Charleston, 1995). In Figure 1,the perturbation used is the contraction orexpansion of internal branches (Robinsonand Foulds, 1981). The tree space wouldlook different for other perturbations(Charleston, 1995). Those trees that areshaded in Figure 1 represent the null hy-pothesis that taxa 1 + 3 and taxa 2 + 4 +5 form a bipartition (statements aboutmonophyly of either group require rootedtrees, but these shaded trees support themonophyly of taxa 1 + 3 as long as one ofthe other taxa is the outgroup). The likeli-hood is maximized over the shaded por-tion of the tree space to obtain Lo. Thewhole tree space represents a more generalhypothesis. The likelihood is also maxi-mized under the more general hypothesisto obtain Lv Obviously, the null hypo thesisis a subset of the general hypothesis.

This test also can be extended to exam-ine the nonmonophyly of a group. In thiscase, the null hypothesis, as describedabove, is reversed and becomes all thosetrees for which the group of interest is notmonophy letic. In term s of the tree space ofFigure 1, the null hypothesis could repre-sent all of those trees that are not shaded(i.e., those trees for which 1 + 3 are notmonophyletic, again assuming that theoutgroup is among taxa 2, 4, and 5). Thealternative hypothesis would again be theentire space of trees. The likelihood-ratiotest statistic would be calculated as de-scribed above, and the distribution of thetest statistic 5 would be simulated underthe null hypothesis.

SIMULATION ANALYSISWe performed a simulation analysis toexamine the behavior of the likelihood-ra-tio test of monophyly. Specifically, we ex-amined the probability of rejecting the null

hypothesis and the behavior of parametricbootstrapping for determining the nulldistribution of 8. The behavior of the testwas examined when the assumptions of

True Model ofDNA Substitution

FIGURE 2. The general strategy used to examinethe statistical properties of the likelihood-ratio test ofmonophyly. Data were generated using the "truemodel of DNA substitution" and then analyzed usingmaximum likelihood assuming the correct and (pos-sibly) incorrect model of DNA substitution to pro duce&p and 8|>. Parameters estimated using maximum like-lihood under the null hypothesis were used to gen-erate an additional simulated data set (S (Ifl) to obtainS*. JC69 = Jukes and Cantor (1969).

the test were completely satisfied and alsowh en one of the assum ptions was violated.Figure 2 shows the general strategy takento examine the behavior of the test. We as-sumed a model of DNA substitution (TrueModel of DNA Substitution, Fig. 2) and afour- or eight-taxon model tree with spec-ified branch lengths. The Jukes-Cantor(1969) or Kimura (1980) models of DNAsubstitution were assumed for the truemodel. Simulation was use d to construct nsimulated data sets, D*1), . . . , D


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550 SYSTEMATIC BIOLOGY VOL. 451-r

Q) 0.75

om 0.5-.8DC

100 Sites500 Sites1,000 Sites

0.2 0.4 0.6 0.8Ho True Ho False

FIGURE 3. The power of the likelihood-ratio test for 100, 500, and 1,000 simulated nucleotide sites. The rateof change over the entire tree for these simulations was constrained to be 2.0 substitutions per site. R is theratio of the length of the internal branch to that of one of the four external branches (all external branches areequal in length). As R increases, the internal branch becomes longer.

model, which may or may not representthe model of DNA substitution that wasused in generating the simulated data.This likelihood ratio is denoted 8|. Finally,the tree and branch lengths calculated as-suming the Jukes-Cantor model were usedto produce m simulated data sets [ S ( U ) , . . . ,S(!m)]. This part of the analysis mimics theprocedure of parametric bootstrappingused to determine if the null hypothesis ofmonophyly will be rejected for a particularrealization of D (0 . For each data set, S fe )simulated at this point in the analysis, wecalculated the likelihood-ratio test statisticand denote this value as 8* (I;).The values AT = 8^, . . . , 8^ representan approximation to the distribution of thelikelihood-ratio test statistic if the truemodel of DNA substitution is assumed inthe analysis. Similarly, the values AE =h% \ . . . , 8 |I) are an approximation of thedistribution of 8 under the Jukes-Cantormodel of DNA substitution. If the Jukes-Cantor model of DNA substitution hap-pens to match the process generating theda ta sets D, . . . , D should

approximate the distribution of AE (and ATas well, if the model of substitution is cor-rect). The match between AE and A* was ofprime interest in this study because thematch or mismatch of these distributionsprovides an indication of how well the testwill perform (i.e., whether the test will re-ject the null hypothesis more or less oftenthan it should).The likelihood-ratio test of monophylyappears to be powerful. Figure 3 showsthe results of a power analysis in whichone of two unrooted four-taxon trees wassimulated: tree 1 = ((A, B)C, D) or tree 2= ((A, C)B, D). Sequences of 100, 500, and1,000 sites were simulated. For both trees,the lengths of the external branches (thosebranches leading directly to a tip) wereequal, and the expected number of substi-tutions per site over the entire tree w as 2.0.The length of the internal branch, however,differed. Figure 3 shows the ratio of thelength of the internal branch to that of theexternal branch (R) along the x-axis andthe probability of rejecting the null hy-pothesis at the 5% level along the y-axis.When R = 0, the internal branch is zero in



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552 SYSTEMATIC BIOLOGY VOL. 45when the null hypothesis is false. In thesesimulations, all branches were equal inlength, and the expected number of sub-stitutions over the entire tree was 4.0. Inthe simulations of Figure 5, one would notexpect the distributions to match becausethe null hypothesis is false. In fact, notonly are the distributions different, but the5% critical values show that the null hy-pothesis should, on average, be rejected(A*005 is much smaller than A E005).The parametr ic boots t rap proceduredoes not perform as well for approximat-ing the distribution of AE when the modelof DNA substitution assumed in the anal-ysis does not match the processes gener-ating the sequences. Figure 6 shows thedistributions of AE and A* for two modeltrees. The conditions of the simulation ofFigure 6 were the same as for the simula-tions of Figure 4 except that transitionshad an instantaneous rate of change 10times that of transversions for the truemodel of DNA substitution (K = a/f i =10.0). Maximum likelihood assuming aJukes-Cantor model of DNA substitutionis consistent (will converge to the correctphylogeny given enough sequence data)for the tree of Figure 6a. However, maxi-mum likelihood assuming a Jukes-Cantormodel is inconsistent (converges to an in-correct phylogeny) for the tree of Figure6b . Hence, the simulation results depictedin Figure 6b repre sent an exceptionally dif-ficult problem for parametric bootstrap-ping. Not only are the distributions of AEand A* different for bo th sim ulations, butthe expected critical values show that thelikelihood-ratio test should reject the nullhypothesis more readi ly than desired(AEO.o5 is g reater than AE005). This resultsuggests that the l ikelihood-ratio testshould be performed with as realistic (pa-rameter rich) models of DNA substitutionas possible, otherwise one runs the risk ofa high level of Type I error.

Although parametric bootstrapping ap-pears to perform poorly when a wrongsubstitution model is used for data analy-sis, the simulations shown in Figure 6 rep-resent an exceptionally difficult problemfor the method because the rates of evolu-

(a)O


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1996 HUELSENBECK ET AL.MONOPHYLY TEST 553model is inaccurate. However, when ratesof evolution are low, the distribution A* ob-tained by parametric bootstrapping canclosely approxim ate AE even if the assump-tions of the method are violated. We alsoperformed simulations in which overallrates of evolution were low (1.0 substitu-tion per site over the entire tree). All otheraspects of the simulation were the same asthe simulations of Figure 6. In this case, thematch between AE and A* was close for thesimulations in which all branches wereequal in length (AE005 = 0.34 and A*005 =0.82). For the case in which two of thebranches were 10 times longer than the re-maining branches, however, the match be-tween AE and A* for the low-rate simu la-tions (1.0 substitutions per site) was betterthan that for the high-rate simulations (4.0substitutions per site, Fig. 4). However, thecritical values show that the null hypoth-esis would still be rejected more readilythan desired (AE005 = 6.37 and AE005 =2.05). Phylogenetic problems in whichsome of the branches are very long repre-sent a difficult estimation problem formost methods of phylogenetic estimationas well as for this test.

We also examined the behavior of thetest for a more complicated tree, one witheight taxa. In the case of four taxa, the cor-rect tree under the null hypothesis was al-ways used for simulating the distributionA* because only one tree satisfies the nullhypothesis (the constraint that taxa A andB form a grou p). However, in the eight-tax-on simulations, there are multiple treesthat satisfy the null hypothesis, only oneof which can be the tru e tree. It is possible,then, to simulate the null distribution withan incorrect tree that nonetheless satisfiesthe null hypothesis. How is parametricbootstrapping affected by inaccurate esti-mation of topology? The eight-taxon sim-ulations can address this question becauseoften an incorrect tree is used for simula-tion under the null hypothesis. In thesesimulations, 1,000 data sets were generatedassuming a Jukes-Cantor model of evolu-tion and the tree depicted in Figure 7. Allof the branch lengths on the model treewere assumed to be equal, but the total

FIGURE 7. The tree assumed in the eight-taxonsimulations. All internal and external branches are ofequal length. The null hypothesis tested in the simu-lations constitutes the con straint that the taxon bipar-tition between the gray and black region of the treeexists.

amount of change on the tree differed. Atotal of 100 sites were simulated. For eachsimulated data set, the test statistic 8E wascalculated under the null hypothesis thatthe taxa in the shaded portion of the treeconstitute a monophyletic group. The dis-tribution AE represents the distribution ofthe test statistic under the null hypothesis.For each of the 1,000 simulated trees, anadditional simulation was performed us-ing the most likely tree obtained under thenull hypothesis in the previous simulation.Again, 8 was calculated for the 1,000 newsimulated data sets. The distribution ofthese 8* values corresponds to the estimat-ed distribution AE as obtained by the p ara-metric bootstrap.

The results of the simulations are shownin Figure 8. For bo th the high-rate (5.0 sub -stitutions per site over the tree) and thelow-rate (3.0 substitutions per site over thetree) cases, the true distribution of the teststatistic is very similar or identical to theestimated distribution of the test statistic.For the low-rate simulations, A E005 = 0.0,whereas AE005 as determined using para-metric bootstrapping is 0.67. Similarly, forhigh-rate simulations, AE005 = 2.14, where-as AE005 = 2.72. The parametric bootstrapprocedure appears conservative with re-spect to rejection of the null hypothesis,which confirms that parametric bootstrap-ping in this case may be an appropriatemethod to obtain the null distribution of



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554 SYSTEMATIC BIOLOGY VOL. 4 5

(a)

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FIGURE 8. The distribu tions of AE and A for theeight-taxon simulations. The parametric bootstrapprocedu re for generating the null distribution of 8 per-forms well. The match between AE an d A* is very closefor both (a) the low-rate case in which there is a totalof 3.0 substitutions per site over the entire tree and(b) the high-rate case in which there is a total of 5.0substitutions per site.

8E, even for a moderately large number oftaxa, when the model of DNA substitutionis accurate.These simulations show that the likeli-hood-ratio test of monophyly appears tobe powerful and that the level of false sig-nificance (Type I error) is low when theassumptions of the analysis are satisfied.Like most tests, the likelihood-ratio testproposed here appears sensitive to viola-tion of its assumptions. However, whether

violation of assumptions will cause spuri-ous results should depend on the rate ofevolution, the severity of the violation, andthe difficulty of the phylogenetic problem.AN APPLICATION TO MARSUPIAL

RELATIONSHIPSThomas et al. (1989) examined the rela-tionship of the presumed-extinct marsu-pial wolf (Thylacinus cynocephalus) us ing12S ribosomal RNA (rRNA) data. The treeestimated using the parsimony criterionplaced the ma rsup ial wolf as the sister tax-on to the Dasyuridae (Sarcophilus harrisii[Tasmanian devil] and Dasyurus maculatus

[tiger cat]), supporting the Australian ori-gin for the thylacine (Thomas et al., 1989).Some morphological evidence, however,supports a close relationship of the mar-supial wolf with the South American bor-hyaenids (Archer, 1982). Faith (1991) useda randomization procedure to show thatthe degree of support for a Thylacinus +Dasyurid ae clade was significant (as deter-mined with the T-PTP test; Faith, 1991),even though he could not reject the nullhypothesis of random signal for the dataset as a whole (as determ ined with the PTPtest; Faith, 1990).

We reexamined the marsupial wolf re-lationship using the likelihood-ratio test ofmonophyly. The 12S rRNA and cyto-chrome b sequences of Thomas et al. (1989)were used in this analysis. We examinedtwo null hypotheses: H o ' = Thylacinus +Dasyuridae are not monophyletic, and H /= Thylacinus is a basal m ember of the Aus-tralian radiation of mammals. The first hy-pothesis (Ho ') is equivalent to the hypoth-esis tested by Faith (1991). However, thishypothesis cannot be used to examinewhether the marsupial wolf is basal to theother Australian and New Guinean mar-supials. To test the second hypothesis(HQ"), we calculated the likelihood underthe constraint that Bos, Philander (opos-sum), and Thylacinus form a group (whenthe tree estimated under this constraint isrooted using the cow sequence, the result-ing tree forces the marsupial wolf to a ba-sal position).

Figure 9 shows the tree estimated using



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1996 HUELSENBECK ET AL.MONOPHYLY TEST 555

65

48

63

Thylacinus

89

Sarcophilus

96

^ Dasyurus

Trichosurus

Phalanger

Echymipera

Philander

BosFIGURE 9. The maximum likelihood estimate ofphylogeny for the analysis of the 12S rRNA and cy-

tochrome b sequences of cow (Bos), opossum (Philan-der), bandicoot (Echymipera), phalangers (Tricosurus,Phalanger), the marsupial wolf (Thylacinus), and theDasyuridae (Sarcophilus, Dasyurus). The HKY85 modelof DNA substitution was assumed in the analysis withthe transition: transversion bias estimated using max-imum likelihood. The best estimate of phylogeny hada log likelihood score of -1005.937. The numbers atthe nodes represent nonparametric bootstrap values.

maximum likelihood implemented withthe HKY85 model of DNA substitution(Hasegawa et al., 1985) when the 12SrRNA and cytochrome b sequences wereanalyzed (a total of 212 sites with gapsomitted). The HKY85 model allows for adifferent rate for transitions and transver-sions and for unequal base frequencies.This tree places Thylacinus as the sister tax-on to the Dasyuridae, as did Thomas etal/s (1989) original parsimony analysis ofthe 12S rRNA data. The log likelihoods cal-culated for the hypotheses examined herewere -1005.937 for the unconstrained hy-pothesis, -1011.095 for the Thylacinus +Dasyuridae nonmonophyly hypothesis(HQ'), and -1012.519 for the basal marsu-

CDcrCD

Obs.

1.000.75

0.50

0.25

0.00

FIGURE 10. The simulated distribution of 8 underthe null hypothesis that the marsupial wolf is basal tothe Australian and New Guinean marsupials (Ho").The distribution is based on 100 replicates. The ob-served 8 is much greater than the 5% critical valuedetermined through parametric bootstrapping.

pial wolf hypothesis (Ho"). Likelihoodswere calculated using the program PAUP*4.0d52 (Swofford, 1996). Both null hypoth-eses are rejected using our likelihood-ratiotest (Ho': 8 = 10.315, P < 0.01; Ho": 8 =13.163, P < 0.01; see Fig. 10). The test isable to reject the null hypotheses despitethe small number of sites analyzed. Hence,useful hypotheses can be examined usingthe likelihood-ratio test even when somegroups on the maximum likelihood treehave little support (as evidenced by non-parametric bootstrapping). The T-PTP testalso rejects Ho' at P < 0.01 but could notreject Ho" (P = 0.54). However, Swofford etal. (1996) showed that the null hypothesisof random covariation among charactersused by the T-PTP test is inappropriate.We argue that the null hypothesis assumedby the likelihood-ratio test of monophyly,however, is appropriate. An appropriatenull hypothesis consists of a tree (or agroup of trees) as well as a model of DNAsubstitution.

DISCUSSIONThe likelihood-ratio test of monophylyproposed here can be used to test a wide

variety of hypotheses of relationship. Be-sides constructing hypotheses in whichgroups are constrained to be monophyleticor nonmonophyletic, more complex restric-



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556 SYSTEMATIC BIOLOGY VOL. 4 5tions of the space of all trees can be envi-sioned. At this time, however, no publiclyavailable phylogenetic program allowsconstraints m ore complex than monoph ylyor nonmonophyly of particular groups tobe imposed on the search. Such generalconstraints could be used to examine ques-tions such as the African origin of modernhumans; the null hypothesis would bethose trees that are consistent with an orig-ination of modern humans in Africa. Thishypothesis cannot be accommodated by asimple constraint of monophyly.Numerous other tests are currentlyavailable to test one tree against another.Some of these tests do not make strong as-sumptions about the probability densitydistribution underlying the data (e.g., non-parametric tests such as Templeton's test;Templeton, 1983). Typically, however, non-parametric tests are not as powerful asparametric tests (Sokal and Rohlf, 1981).Kishino and Hasegawa (1989) proposed analternative parametric test that comparesthe likelihood of two trees. Their test as-sumes that the variance of the likelihoodof the two trees is asymptotically normallydistributed. Faith (1991) proposed a per-mutation test that can be used with theparsimony criterion to examine the mono-phyly or nonmonophyly of a group. How-ever, the distribution of the test statisticused by Faith (1991) is not related to thenull hypothesis for the phylogeny problem(Swofford et al., 1996). A thorough studyexploring the behavior of the availabletests in terms of Type I and Type II errorsunder a variety of evolutionary scenariosmight establish which methods generallyhave acceptable statistical properties.

The likelihood-ratio test proposed hereshould be implemented with as realistic amodel of DNA substitution as possible,otherwise the null hypothesis may be re-jected too easily. Currently, the most pa-rameter-rich models of DNA substitutionallow different rates of change from onenucleotide to another, different nu mbers ofsubstitutions along each branch of the tree,and rate heterogeneity among sites (as de-termined by a gamma distribution or byallowing a certain proportion of the sites

to be invariant; Yang, 1993; Waddell andPenny, 1996). Also, the models of DNAsubstitution can be made to more closelyfit the observed sequence data by estimat-ing parameters separately for differentdata partitions (e.g., first, second, and thirdcodon positions; Yang, 1996). Using morerealistic models of DNA substitution mayalleviate the problem of an elevated TypeI error.Phylogenetic analysis cannot proceedwithou t m aking specific assumptions aboutthe process of evolution. These assumptionsoften evoke skepticism about a method. Forexample, maximum likelihood and distancemethods have often been criticized becausethey make explicit assumptions about theprocess of DNA substitution (Carpenter,1994). However, evolutionary modelsshould motivate an examination of theiradequacy in a hypothesis-testing frame-work and a search for more realistic mod-els rather than their complete dismissal asuseless. This approach has led to a demon-strable improvement in the models ofDNA substitution used in phylogeneticanalysis over the past 5 years (Yang et al.,1994). Furthermore, by making assump-tions explicit, our models can be improvedand other hypotheses (such as the molec-ular clock) can be tested. Here, we haveshown how a wide variety of hypothesesof phylogenetic relationship can be testedin a likelihood framework.

PROGR AM AVAILABILITYThe likelihood-ratio test of monophyly

can be implemented if PAUP* 4.0 is usedin conjunction with a program to simulatedata under the appropriate model of DNAsubstitution. A program written by J.P.H.for simulating DNA sequence data for theHKY85+r model of DNA substitution(Hasegawa et al., 1985; Yang, 1993) is avail-able from http://mw511.biol.berkeley.edu / homepage.html /.ACKNOWLEDGMENTS

We thank Monty Slatkin, Joanna Mountain, andZiheng Yang for critically evaluating this manuscript.This research was su pporte d by grants NIH GM40282awarded to Monty Slatkin, DEB-9221052 awarded to



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1996 HUELSENBECK ET AL.MONOPHYLY TEST 557D.M.H., a Danish Research Council Fellowship award-ed to R.N., and a Miller Postdoctoral Fellowshipawarded to J.P.H.

REFERENCESADKINS, R. M., AND R. L. HONEYCUTT. 1991. Molec-

ular phylogeny of the superorder Archonta. Proc.Natl. Acad. Sri. USA 88:10317-10321.

AGRESTI, A., AND M. C. YANG. 1987. An empiricalinvestigation of some effects of sparseness in con-tingency tables. Comput. Stat. Data Anal. 5:9-21.

AMMERMAN, L. K., AND D. M. HILLIS. 1992. A molec-ular test of bat relationships: Monophyly or diphy-ly? Syst. Biol. 41:222-232.

ARCHER, M. 1982. A review of Miocene thylacinids(Thylacinidae, Marsupialia), the phylogenetic posi-tion of the Thylacinidae and the problem of aprior-isms in character analysis. Pages 445^476 in Carniv-orous marsupials (M. Archer, ed.). Royal ZoologicalSociety of New South Wales, Sydney.

BAILEY, W. J., J. L. SLIGHTOM, AND M. GOODMAN. 1992.Rejection of the "flying primate" hypothesis byphylogenetic evidence from the e-globin gene. Sci-ence 256:86-89.

BENNET, S., L. J. ALEXANDER, R. H. CROZIER, AND A.G. MACKINLAY. 1988. Are megabats flying pri-mates? Contrary evidence from a mitochondrialDNA sequence. Aust. J. Biol. Sri. 41:327-332.

BULL, J. J., C. W. CUNNINGHAM, I. J. MOLINEUX, M. R.BADGETT, AND D. M. HILLIS. 1993. Experimentalmolecular evolution of bacteriophage T7. Evolution47:993-1007.

CAO, Y, J. ADACHI, T.-A. YANO, AND M. HASEGAWA.1994. Phylogenetic place of guinea pigs: No sup-port of the rodent-polyphyly hypothesis from max-imum-likelihood analyses of multiple protein se-quences. Mol. Biol. Evol. 11:593-604.

CARPENTER, J. 1994. Successive weighting, reliabilityand evidence. Cladistics 10:215-220.

CHARLESTON, M. 1995. Toward a characterisation oflandscapes of combinatorial optimisation problems,with special reference to the phylogeny problem.Comput. Biol. 2:439^50.

EDWARDS, A. W. F. 1972. Likelihood. Cambridge Univ.Press, Cambridge, England.EFRON, B. 1985. Bootstrap confidence intervals for aclass of parametric problems. Biometrika 72:45-58.FAITH, D. P. 1990. Chance marsupial relationships.Nature 345:393-394.

FAITH, D. P. 1991. Cladistic permutation tests formonophyly and nonmonophyly. Syst. Zool. 40:366-375.

FELSENSTEIN, J. 1981. Evolutionary trees from DNAsequences: A maximum likelihood approach. J. Mol.Evol. 17:368-376.

FELSENSTEIN, J. 1988. Phytogenies from molecular se-quences: Inference and reliability. Annu. Rev. Genet.22:521-565.

GOLDMAN, N. 1993. Statistical tests of models of DNAsubstitution. J. Mol. Evol. 36:182-198.

GOUY, M., AND W.-H. Li. 1989. Phylogenetic analysisbased on rRNA sequences supports the archaebac-

terial rather than the eocyte tree. Science 339:145-147.

GRAUER, D., W. A. HIDE, AND W.-H. LI. 1991. Is theguinea-pig a rodent? Nature 351:649-652.

GRAUER, D., W. A. HIDE, A. ZHARKIKH, AND W.-H. Li.1992. The biochemical phylogeny of guinea-pigsand gundis, and the paraphyly of the order Roden-tia. Comp. Biochem. Physiol. B 101:495-498.

HABERMAN, S. J. 1977. Log-linear models and fre-quency tables with small expected cell counts. Ann.Stat. 5:1148-1169.

HASEGAWA, M., Y. CAO, J. ADACHI, AND T. YANO.1992. Rodent polyphyly? Nature 355:595.

HASEGAWA, M., H. KISHINO, AND T. YANO. 1985. Dat-ing of the human-ape splitting by a molecular clockof mitochondrial DNA. J. Mol. Evol. 21:160-174.

HUELSENBECK, J. P. 1995a. Performance of phyloge-netic methods in simulation. Syst. Biol. 44:17-48.

HUELSENBECK, J. P. 1995b. The robustness of two phy-logenetic methods: Four-taxon simulations reveal aslight superiority of maximum likelihood overneighbor joining. Mol. Biol. Evol. 12:843-849.

HUELSENBECK, J. P., AND J. J. BULL. 1996. A likelihoodratio test to detect conflicting phylogenetic signal.Syst. Biol. 45:92-98.

HUELSENBECK, J. P., D. M . HILLIS, AND R. JONES. 1996.Parametric bootstrapping in molecular phylogenet-ics: Applications and performance. Pages 19^45 inMolecular zoology: Advances, strategies and pro-tocols (J. D. Ferraris and S. R. Palumbi, eds.). Wiley& Sons, New York.

JUKES, T. H., AND C. R. CANTOR. 1969. Evolution ofprotein molecules. Pages 21-132 in Mammalianprotein metabolism (H . Munro, ed.). AcademicPress, New York.

KIMURA, M. 1980. A simple method for estimatingevolutionary rate of base substitutions through com-parative studies of nucleotide sequences. J. Mol.Evol. 16:111-120.

KISHINO, H., AND M. HASEGAWA. 1989. Evaluation ofthe maximum likelihood estimate of the evolution-ary tree topologies from DNA sequence data, andthe branching order in Hominoidea. J. Mol. Evol. 29:170-179.

KUHNER, M. K., AND J. FELSENSTEIN. 1994. A simu-lation comparison of phylogeny algorithms underequal and unequal evolutionary rates. Mol. Biol.Evol. 11:459-468.

MA, D.-P, A. ZHARKIKH, D. GRAUER, J. L. VANDEBERG,AND W.-H. Li. 1993. Structure and evolution ofopossum, guinea pig, and porcupine cytochrome bgenes. J. Mol. Evol. 36:327-334.

MCCULLAGH, P., AND J. A. NELDER. 1989. Generalizedlinear models, 2nd edition. Chapman and Hall,London.

MINDELL, D. P., C. W. DICK, AND R. J. BAKER. 1991.Phylogenetic relationships among megabats, micro-bats and primates. Proc. Natl. Acad. Sri. USA 88:10322-10326.

PETTIGREW, J. D. 1986. Flying primates? Megabatshave the advanced pathway from eye to midbrain.Science 231:1304-1306.

PETTIGREW, J. D. 1991. Wings or brain? Convergent



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13/13

558 SYSTEMATIC BIOLOGY VOL. 45evolution in the origins of bats. Syst. Zool. 40:199-216.RICE, J. A. 1995. Mathematical statistics and d ata anal-ysis. Duxbury Press, Belmont, California.ROBINSON, D. F., AND L. R. FOULDS. 1981. Comparisonof phylogenetic trees. Math. Biosci. 53:131-147.SOKAL, R. R., AN D F. J. ROHLF. 1981. Biometry, 2ndedition. W. H. Freeman, New York.

STANHOPE, M. J., J. CZELUSNIAK, J.-S. SI, J. NICKERSON,AND M. GOODMAN. 1992. A molecular perspectiveon mammalian evolution from the gene encodinginterphotoreceptor retinoid binding protein, withconvincing evidence for bat monophyly. Mol. Phy-logenet. Evol. 1:148-160.SWOFFORD, D. L. 1996. PAUP*: Phylogenetic analysisusing parsimony (*and other methods), version 4.0.Sinauer, Sunderland, Massachusetts.SWOFFORD, D. L., J. L. THORNE, J. FELSENSTEIN, AN D B.M. WIEGMANN. 1996. The topology-dependent per-mutation test for monophyly does not test for mono-phyly. Syst. Biol. 45:575-579.TATENO, Y, N. TAKEZAKI, AND M. NEI. 1994. Relativeefficiencies of the maximum-likelihood, neighbor-joining, and maximum-parsimony methods whensubstitut ion rate varies w ith site. Mol. Biol. Evol. 11:261-277.

TEMPLETON, A. R. 1983. Phylogenetic inference fromrestriction site endonuclease cleavage site mapswith particular reference to the humans and apes.Evolution 37:221-244.THOMAS, R. H., W. SCHAFFNER, A. C. WILSON, AND S.

PAABO. 1989. DNA phylogeny of the extinct mar-supial wolf. Nature 340:465-467.WADDELL, P. J., AN D D. PENNY. 1996. Evolutionarytrees of apes and humans from DNA sequences.Pages 53-73 in Handbook of symbolic evolution (A.J. Lock and C. R. Peters, eds.). Oxford Univ. Press,Oxford, EnglandYANG, Z. 1993. Maximum likelihood estimation ofphylogeny from DNA sequences when substitutionrates differ over site s. Mol. Biol. Evol. 10:1396-1401.YANG, Z. 1996. Maximum likelihood models for com-bined analyses of multiple sequence data. J. Mol.Evol. 42:587-596.YANG, Z., N. GOLDMAN, AND A. FRIDAY. 1994. Com-parison of models for nucleotide substitution usedin maximum-likelihood phylogenetic estimation.Mol. Biol. Evol. 11:316-324.Received 28 December 1995; accepted 27 July 1996Associate Editor: Charles Marshall



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