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Bayesian Phylogenetic Analysis of Combined Data

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Bayesian Phylogenetic Analysis of Combined Data
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  Syst. Biol.  53(1):47–67, 2004Copyright c  Society of Systematic BiologistsISSN: 1063-5157 print / 1076-836X onlineDOI: 10.1080/10635150490264699 Bayesian Phylogenetic Analysis of Combined Data  J OHAN  A. A. N YLANDER , 1 F REDRIK  R ONQUIST , 1  J OHN  P. H UELSENBECK , 2 AND  J OS ´ E  L UIS  N IEVES -A LDREY 3 1 Department of Systematic Zoology, Evolutionary Biology Centre, Uppsala University, Norbyv¨ agen 18 D, SE-752 36 Uppsala, Sweden;E-mail: johan.nylander@ebc.uu.se (J.A.A.N) 2 Section of Ecology, Behavior and Evolution, Division of Biological Sciences, University of California–San Diego, La Jolla, California 92093-0116, USA 3 Departamento de Bioversidad y Biolog´ ıa Evolutiva, Museo Nacional de Ciencias Naturales, Jos´ e Guti´ errez Abascal 2, 28006 Madrid, Spain Abstract.—  TherecentdevelopmentofBayesianphylogeneticinferenceusingMarkovchainMonteCarlo(MCMC)techniqueshas facilitated the exploration of parameter-rich evolutionary models. At the same time, stochastic models have becomemorerealistic(andcomplex)andhavebeenextendedtonewtypesofdata,suchasmorphology.Basedonthisfoundation,wedevelopedaBayesianMCMCapproachtotheanalysisofcombineddatasetsandexploreditsutilityininferringrelationshipsamong gall wasps based on data from morphology and four genes (nuclear and mitochondrial, ribosomal and proteincoding). Examined models range in complexity from those recognizing only a morphological and a molecular partitionto those having complex substitution models with independent parameters for each gene. Bayesian MCMC analysis dealsefficiently with complex models: convergence occurs faster and more predictably for complex models, mixing is adequatefor all parameters even under very complex models, and the parameter update cycle is virtually unaffected by modelpartitioning across sites. Morphology contributed only 5% of the characters in the data set but nevertheless influencedthe combined-data tree, supporting the utility of morphological data in multigene analyses. We used Bayesian criteria(Bayes factors) to show that process heterogeneity across data partitions is a significant model component, although not asimportantasamong-siteratevariation.Morecomplexevolutionarymodelsareassociatedwithmoretopologicaluncertaintyandlessconflictbetweenmorphologyandmolecules.Bayesfactorssometimesfavorsimplermodelsoverconsiderablymoreparameter-rich models, but the best model overall is also the most complex and Bayes factors do not support exclusion of apparently weak parameters from this model. Thus, Bayes factors appear to be useful for selecting among complex models, but it is still unclear whether their use strikes a reasonable balance between model complexity and error in parameterestimates. [Bayes factors; Bayesian analysis; combined data; Cynipidae; gall wasps; MCMC; model heterogeneity; modelselection.] Increasingly, phylogenetic problems are being ad-dressed using data from several different sources:morphology and molecules, DNA and protein, mito-chondrial and nuclear genes, coding and noncodingsequences. Previously, it has been common to addresssuch mixed data sets using the parsimony method.Whereparametricmethodshavebeenapplied,theyhavetypically excluded some data (such as morphology) be-causeofalackofappropriatestochasticmodels,andtheyhave often ignored obvious heterogeneity across datapartitions because of the computational complexity of the maximum likelihood (ML) approach (for exceptions,see Yang, 1996b; DeBry, 1999; Pupko et al., 2002; Thorneand Kishino, 2002).The recent development of Bayesian inference of phy-logeny using Markov chain Monte Carlo (MCMC) esti-mation of posterior probability distributions has madeit easier to address complex, parameter-rich stochas-tic models within a statistical framework, opening upthe possibility for combined data analysis recognizingamong-partition heterogeneity in data source and inproperties of the evolutionary process. Recent stochas-tic models developed for new types of data, such asmorphology (Lewis, 2001a; Ronquist and Huelsenbeck,in prep.), now make it possible to include virtuallyany kind of character used today to infer phylogenyin such analyses, and the computational efficiency of the Bayesian MCMC approach allows each data par-tition to be treated using more realistic evolution-ary models. However, combined statistical analysis us-ing Bayesian MCMC techniques introduces a wholerange of questions that have not been addressed pre-viously, while providing a new perspective on oth-ers. Here, we describe a Bayesian MCMC approachto combined data analysis, using empirical resultsfrom one combined data set to address some of thesequestions. Bayesian MCMC Approach to Combined Data Bayesian phylogenetic inference based on heteroge-neousdataisastraightforwardextensionofthemethodsalready described for homogeneous data (see recent re-views by Huelsenbeck et al., 2001; Lewis, 2001b; Holderand Lewis, 2003). Assume that the data set  X   consistsof two distinct partitions  X  a  and  X  b  and allow the sub-stitution model parameters,  θ  a  and  θ  b , respectively, to becompletelydifferentforthetwopartitions.Inthemodelswe explored, we further assumed that the two data sub-sets evolve on the same topology,  τ  , with the same set of  branch lengths,  ν , but that the overall rate differs acrosspartitions according to a rate multiplier, denoted  m a  and m b  forthetwopartitions.Inotherwords,effectivebranchlengths are potentially different but proportional acrosspartitions, as in the ML model proposed by Yang (1996;note that Yang used  c  instead of   m  for the multiplier).Using Bayes’s rule (see for instance Huelsenbeck et al.,2001), the joint posterior probability distribution for thismodel becomes  f   ( τ  ,  ν ,  θ  a ,  θ  b , m a , m b  |  X  ) =  f   ( τ  ,  ν ,  θ  a ,  θ  b , m a , m b )  f   ( X  | τ  ,  ν ,  θ  a ,  θ  b , m a , m b )  f   ( X  )  , 47   b  y g u e  s  t   onF  e  b r  u a r  y1  8  ,2  0 1  6 h  t   t   p :  /   /   s  y s  b i   o . oxf   or  d  j   o ur n a l   s  . or  g /  D o wnl   o a  d  e  d f  r  om   48  SYSTEMATIC BIOLOGY VOL. 53where  f   ( τ  ,  ν ,  θ  a ,  θ  b , m a , m b )isthepriorprobabilityofthemodelparameters,  f   ( X  | τ  ,  ν ,  θ  a ,  θ  b , m a , m b )istheproba- bilityofthedatagivenmodelparameters(thelikelihoodfunction), and  f(X)  is the model likelihood (also calledthe integrated or predictive likelihood), which is a mul-tidimensional sum and integral of the probability of thedata over all parameter values.The posterior probability distribution, which is thecentral quantity in Bayesian inference, is typically es-timated using MCMC techniques instead of being de-rived analytically. The procedure is started with an arbi-trary set of parameter values. In each cycle (generation)of the Markov chain, one parameter or a block of param-eters is updated using a stochastic proposal mechanism.The most common mechanism used in Bayesian phy-logenetic inference, Metropolis sampling, involves theproposal of a new state based on an arbitrary proposaldistribution,  q , and then acceptance of this state with aprobabilitydeterminedbytheproductofthreeratios:theprior ratio, the likelihood ratio, and the proposal ratio.Assume, for instance, that we wish to update the sub-stitution model parameters for partition  a  from  θ  a  to  θ  ∗ a .The acceptance probability  r  would then become r  = min  1 , f   ( θ  ∗ a )  f   ( θ  a )  ×  f   ( X  a  | τ  ,  v , m a ,  θ  ∗ a )  f   ( X  a  | τ  ,  v , m a ,  θ  a )  × q ( θ  a  | θ  ∗ a ) q ( θ  ∗ a  | θ  a )  . When updating a homogeneous model or a param-eter shared across all partitions, the calculation of thelikelihood ratio (the second ratio in the product) alwaysinvolves the entire data set. However, updating a parti-tioned parameter only requires consideration of the af-fected data partition,  X  a  in this case. The calculation of the likelihood ratio is by far the most computationallycomplex operation in MCMC analysis, and the speed of the calculation is roughly proportional to the size of thedata set. Thus, the increase in the number of parametersin a partitioned model over that in a similar homoge-neousmodelislargelyoffsetbythespeedgainedineachcycle of the chain. The net result is that the time requiredfor updating all model parameters a given number of times will remain roughly constant regardless of modelpartitioning. However, more complex models will of course have more dimensions in their parameter spaces,which might cause difficulties for the MCMC samplingprocedure. Convergence and Mixing Theory predicts that a properly constructed Markovchain, if run long enough, will produce a valid sam-ple from the posterior probability distribution (Tierney,1994).However,thegreatestpracticalprobleminMCMCanalysis is to determine when the chain is sufficientlyclose to its target distribution (the posterior distributionof interest) for the samples to provide a good approx-imation of this distribution. One of the most powerfulapproaches used to address this question is compari-son of the results from independent runs started fromdifferent points in parameter space. In the phylogeneticcontext,weexpectintegrationovertopologytobepartic-ularly difficult; therefore, starting the independent runsfrom different, randomly chosen topologies should pro-vide a good test of whether the chains are providingvalidsamplesfromtheposteriorprobabilitydistribution(Huelsenbeck et al., 2002).Itisusefultodistinguishtwopotentialsourcesofprob-lems with MCMC estimation of a target distribution:convergence and mixing. The difference between themis best explained if we consider a posterior distributionwith two separate regions, each containing roughly half of the total probability. Typically, a MCMC run startssampling from a region with extremely low posteriorprobability because starting values are set arbitrarily orchosen randomly. When the chain has settled into thehigh-density regions of the distribution, it can be said tohave converged, and the overall likelihood will tend tovary less than during the initial burn-in period. How-ever, we still do not know how long it will take thechain to adequately sample both regions of high den-sity in the posterior distribution; this is determined bythe mixing behavior of the chain. The slower the mix-ing, the longer it will take the chain to move from one tothe other of the high-density regions. Whereas the gen-eration plot of the overall likelihood gives a preliminaryidea of whether convergence might have occurred, as-sessment of the mixing behavior requires examinationof the plots of all model parameters. This is particularlytruewhenMetropoliscouplingisused,becausethistech-niqueallowsthechaintojumpbetweendifferentregionsinparameterspacewithlittleeffectonoveralllikelihood(Huelsenbeck et al., 2001). Bayesian Model Selection Analyzing combined data using Bayesian MCMCmethods allows us to specify partition-specific substi-tution models. As more partitions are being considered,the complexity of the joint model increases as does thecomplexity of the issue of model selection. One strategyfor model selection for Bayesian MCMC analysis is to fita substitution model to each partition prior to the anal-ysis using, for example, a hierarchical likelihood-ratiotest(hLRT;HuelsenbeckandCrandall,1997;PosadaandCrandall, 2001), the Akaike information criterion (AIC;Akaike,1973),ortheBayesianinformationcriterion(BIC;Schwartz, 1978), all of which are based on ML estimates.The Markov chain is then run using a composite ‘super-model’ that consists of several submodels.It is not self-evident, however, that such an approachwill necessarily lead to an optimal composite model.Most importantly, the selection of an optimal model forone partition should not ignore information from otherpartitions. For example, the methods mentioned abovedepend on point estimates of the topology and other pa-rameters, and it is well known that different topologiesmightrankmodelsdifferently(SandersonandKim,2000;Posada and Crandall, 2001). Thus, selecting an optimalmodel for each partition separately, on the best tree im-plied by the data from that partition, might result in a   b  y g u e  s  t   onF  e  b r  u a r  y1  8  ,2  0 1  6 h  t   t   p :  /   /   s  y s  b i   o . oxf   or  d  j   o ur n a l   s  . or  g /  D o wnl   o a  d  e  d f  r  om   2004  NYLANDER ET AL.—BAYESIAN COMBINED ANALYSIS  49combination of models that could not be optimal on thesame topology. Furthermore, considering each partitionseparately may result in overparameterization, becausesuch an approach makes it difficult to discover when itis appropriate for two partitions to share parameters.Unfortunately, computational problems make it dif-ficult to apply directly the methods described aboveto parameter-rich partitioned models. Furthermore,Bayesian statisticians often object in general to modeltesting based on point estimates, because such methodsare not taking the uncertainty of the topology and otherparameters into account. The argument is that a modelwith substantial posterior probability for a large rangeof parameter values could have a higher marginal (total)likelihood than a model with a narrow peak in its likeli-hood, even though the latter model may have the high-est ML value. In such situations, Bayesian statisticianshaveargued,itwouldbeunwisetocomparemodelsonly based on the merits of a single point; instead, we shouldconsidertheentireparameterspaceandpreferthemodelwith the largest total likelihood (Bollback, 2002; HolderandLewis,2003).AnadditionalproblemwiththeMLap-proach is that it favors the more parameter-rich modelin comparisons of nested models unless the parameter-rich models are penalized as in AIC or BIC. That is, thefavored model might contain parameters that have littleornoexplanatoryvalue(BurnhamandAnderson,2002).The Bayesian approach does not always favor the moreparameter-rich of two nested models; on the contrary,there is some concern that Bayesian methods may, un-der some circumstances, put too much emphasis on thesimpler model. This phenomenon is known as Lindley’sparadox, and it can occur with large data sets when theestimate from the complex model is close to the simplemodel (Bartlett, 1957; Lindley, 1957).Becauseoftheproblemswiththelikelihoodapproach,weexploredBayesianmodelcomparisonbasedonBayesfactors. Assume that we wish to compare how well twomodels,  M 0  and  M 1 , describe the processes generating adata set  X  . The Bayes factor in favor of model 1 overmodel 0,  B 10 , is calculated as the ratio of the modellikelihoods  f   ( X  |  M i ): B 10  =  f   ( X  |  M 1 )  f   ( X  |  M 0 ) . The model likelihoods,  f   ( X  |  M i ), are the same as the  f   ( X  ) denominator of Bayes’s rule; the conditioning ona model is implicit in the latter.The Bayes factor can be interpreted as the posterioroddsofmodel1tomodel0inaBayesianinferenceprob-lem where we start with equal probability of the twomodels being true (Kass and Raftery, 1995; Wasserman,2000). Alternatively, the Bayes factor can be viewed sim-ply as a comparison of the predictive likelihoods of themodels (Gelfand and Dey, 1994; Kass and Raftery, 1995;Wasserman, 2000) or a comparison of the ability of themodelstoupdatethepriors(LavineandSchervish,1999;Wasserman,2000).Boththelattercomparisonswouldbe T ABLE  1. Interpretation of the Bayes factor ( B 10 ) (taken from Kassand Raftery, 1995). 2 log e ( B 10 )  B 10  Evidence against  M 0 0 to 2 1 to 3 not worth more thana bare mention2 to 6 3 to 20 positive6 to 10 20 to 150 strong > 10  > 150 very strong valid even, although strictly speaking none of the mod-els is likely to be an exact (true) description of the pro-cess under study. The Bayes factor comparison can beapplied to any set of models, regardless of whether theyare nested or not (as can AIC and BIC but not hLRT),and it is based on integration over the uncertainty in allparameter values rather than on ML point estimates (asopposed to AIC, BIC, and hLRT).The Bayes factor is not used in a normal statistical testof whether a hypothesis should be rejected or acceptedgiven some subjective cutoff value. Instead, the Bayesfactor evaluates the relative merits of competing mod-els, and the interpretation is left to the scientist. Jeffreys(1961) srcinally provided some guidelines for this in-terpretation, which have been modified by other work-ers. We use a version srcinally presented by Kass andRaftery (1995) (Table 1). Questions Regarding Combined Phylogenetic Analysis We applied combined Bayesian MCMC analysis to anempirical data set consisting of morphological and nu-cleotide data for 32 exemplar species of gall wasps (Hy-menoptera: Cynipidae) and outgroups. The exemplarsspan the entire diversity of the family and include phy-tophagous guests in galls (inquilines) and gall inducerson a variety of both herbaceous and woody host plants(Table 2; Ronquist, 1999).The morphological data consisted of 166 characters,which have previously been shown to partly resolvethe phylogeny with strong support values using parsi-mony methods (Liljeblad and Ronquist, 1998). The nu-cleotide data are almost entirely srcinal to this studyand consisted of a total of 3,080 aligned base pairs (bp)fromfourgenes:twonuclearprotein-codinggenes(elon-gation factor 1 α  F1 copy [EF1 α ] and long-wavelengthopsin [LWRh]), one mitochondrial protein-coding gene(cytochrome oxidase  c  subunit I [COI]), and nuclear 28Sribosomal DNA (rDNA). We analyzed the data using arange of models of varying complexity (dimensionality)and explored the following questions. What is the relationship between model complexity andcomputational complexity?—  It is difficult to predict howMCMC estimation of the posterior probability distribu-tion is affected by an increase in model complexity. Thechain can be updated faster in those generations wheremodel parameters affecting only some of the partitionsare changed; however, more parameters also means thateach parameter will be visited more rarely. More param-eters will also affect the complexity and the shape of theposterior distribution, which might slow convergence   b  y g u e  s  t   onF  e  b r  u a r  y1  8  ,2  0 1  6 h  t   t   p :  /   /   s  y s  b i   o . oxf   or  d  j   o ur n a l   s  . or  g /  D o wnl   o a  d  e  d f  r  om   50  SYSTEMATIC BIOLOGY VOL. 53 T ABLE  2. Taxa of gall wasps (Cynipidae) and outgroups (Figitidae, Liopteridae, Ibaliidae) used in the analysis. Brief biological data are givenfor each exemplar genus. GenBank accession numbers are given for all sequences; a dash indicates missing data. GenBank nos.Taxon Morphology a Host plant  b Biology c COI 28S EF1 α  LWRh CynipidaeSynergini Synergus crassicornis Quercus  (Fg) inquiline AY368909 AY368936 AY368962 AY371051 Ceroptres cerri C. clavicornis Quercus  (Fg) inquiline AY368910 AY368935 — AY371052 Periclistus brandtii Rosa  (Ro) inquiline AF395181 AF395152 AF395173 AF395189 Synophromorpha sylvestris S. rubi Rubus  (Ro) inquiline AY368911 AY368937 AY368961 —“Aylacini” Xestophanes potentillae Potentilla  (Ro) galler AY368912 AY368938 AY368963 — Diastrophus turgidus  Rosaceae galler AY368913 AY368939 AY368964 — Gonaspis potentillae Potentilla  (Ro) galler AY368914 AY368940 AY368965 — Liposthenes glechomae Glechoma  (La) galler AY368915 AY368941 AY368966 AY371053 Liposthenes kerneri Nepeta  (La) galler AY368916 AY368942 AY368967 AY371054  Antistrophus silphii A. pisum  Asteraceae galler AY368917 AY368943 AY368968 AY371055 Rhodus oriundus Salvia  (La) galler AY368918 AY368944 AY368969 AY371056  Hedickiana levantina Salvia  (La) galler AY368919 AY368945 AY368970 AY371057 Neaylax verbenaca N. salviae Salvia  (La) galler AY368920 AY368946 AY368971 AY371058  Isocolus rogenhoferi  Asteraceae galler AY368921 AY368947 AY368972 AY371059  Aulacidea tragopogonis  Asteraceae galler AY368922 AY368948 AY368973 AY371060 Panteliella bicolor P. fedtschenkoi Phlomis  (La) galler AF395180 AF395153 AF395172 AF395188 Barbotinia oraniensis Papaver  (Pa) galler AF395179 AF395150 AF395171 AF395187  Aylax papaveris Papaver  (Pa) galler AY368923 AY368949 AY368974 AY371061  Iraella luteipes Papaver  (Pa) galler AY368924 AY368950 AY368975 — Timaspis phoenixopodos  Asteraceae galler AY368925 AY368951 AY368976 AY371062 Phanacis hypochoeridis  Asteraceae galler AY368926 AY368952 AY368977 — Phanacis centaureae  Asteraceae galler AY368927 AY368953 AY368978 —Eschatocerini Eschatocerus acaciae Acacia  (Fb) galler AY368928 AY368954 AY368979 AY371063Diplolepidini Diplolepis rosae Rosa  (Ro) galler AF395174 AF395157 AF395166 AF395182Pediaspidini Pediaspis aceris Acer  (Sa) galler AY368929 AY368955 AY368980 AY371064Cynipini Plagiotrochus quercusilicis d Quercus  (Fg) galler AF395178 AF395154 AF395162 AF395186  Andricus kollari A. quercusradicis Quercus  (Fg) galler AF395176 AF395156 AF395168 AF395184 Neuroterus numismalis Quercus  (Fg) galler AY368930 AY368956 AY368981 — Biorhiza pallida Quercus  (Fg) galler AY368931 AY368957 AY368982 AY371065Figitidae Parnips nigripes  — parasitoid AY368932 AY368958 AY368983 AY371066Liopteridae Paramblynotus virginianus P. zonatus  — parasitoid AY368933 AY368959 AY368984 —Ibaliidae  Ibalia rufipes  — parasitoid AY368934 AY368960 AY368985 — a Species coded for morphology if different from the species sequenced.  b Genus or family of host plant attacked by the exemplar genus if phytophagous. A few rarely used host plants have been omitted; see Ronquist and Liljeblad(2001) for more information. If all members of the genus attack the same host-plant genus, then the family to which that genus belongs is indicated in brackets: Fb = Fabaceae; Fg = Fagaceae; La = Lamiaceae; Pa = Papaveraceae; Ro = Rosaceae; Sa = Sapindaceae. c Cynipidae are either inquilines (phytophagous guests) in galls or gall inducers. The outgroups are endoparasitoids attacking various insect larvae. d Species name recently designated a senior synonym of   P. fusifex . andmixing.However,morerealisticmodelsmayleadtoposterior distributions that are easier to traverse usingMCMC, despite the increase in the number of parame-ters.Weexaminedthecomputationalspeed,timetocon-vergence, and mixing over the entire range of models toexamine these questions empirically. Do morphological data influence multigene analyses?—  Morphological data are potentially important in phy-logenetic inference for many reasons. For instance,morphological characters are crucial in placing fossils inphylogenies and thus in dating branching events. How-ever, the ability to combine morphological and molecu-lar data in a single analysis is particularly important if itcan be shown that morphology has significant influenceon the phylogenetic estimate even when combined withmultigene data sets. This question has remained largelyunexplored with parametric methods, because only re-cently were stochastic models seriously considered formorphologicaldata(Lewis,2001a).WeusedanextendedversionofLewis’smodels(RonquistandHuelsenbeck,inprep.)inevaluatingwhetherthe166morphologicalchar-acters in our data set significantly affected the phyloge-netic estimate when combined with the 3,080 nucleotidecharacters from the four different genes.  Are composite models better?—  When it becomes pos-sible to analyze partitioned models easily, an obvious   b  y g u e  s  t   onF  e  b r  u a r  y1  8  ,2  0 1  6 h  t   t   p :  /   /   s  y s  b i   o . oxf   or  d  j   o ur n a l   s  . or  g /  D o wnl   o a  d  e  d f  r  om   2004  NYLANDER ET AL.—BAYESIAN COMBINED ANALYSIS  51question is how important it is to recognize across-partition heterogeneity in evolutionary processes. To ex-aminethisquestion,weusedBayesfactorcomparisonstolook at the increase in model likelihood associated withtheintroductionofdifferentmodelcomponentsaccount-ingforwithin-partitionoracross-partitionheterogeneityin the molecular portion of the data set.  Are complex models associated with increased variance of topologyestimates?—  Complexmodelsaregenerallyasso-ciated with more error variance in parameter estimates.If the error variance is excessive, it becomes a problemknownasoverparameterizationoroverfitting(Burnhamand Anderson, 2002). However, overly simple modelscan also be problematic. In particular, oversimplifiedevolutionary models might lead to dramatically low-ered topological variance and exaggerated clade proba- bility values in Bayesian phylogenetic inference (Suzukiet al., 2002). To examine the relationship between modelcomplexity and the precision of parameter estimates,we compared topology and tree-length estimates acrossmodels.Wealsolookedattheeffectofmodelcomplexityontheconflictbetweenthemorphologicalandmolecularpartitions.  Is the Bayesian MCMC approach sensitive to the inclu-sion of superfluous parameters in a complex model?—  It may be difficult to design complex models that adequatelyexplain a process under study without including oneor a few parameters that are superfluous in the sensethat (1) the data are not powerful enough to signifi-cantly alter their prior probability distribution or (2) theposterior probability distribution coincides with a lessparameter-rich submodel. Such “superfluous” parame-tersmightcauseproblemswithMCMCestimationoftheposterior distribution. We searched the posterior distri- butions of more complex models for such parametersto see whether they were present and, if so, whetherthere was any apparent effect on convergence or on theposterior distributions of other model parameters. If theBayesianMCMCapproachweresensitivetosuperfluousparameters, it might be difficult to design appropriatecomposite models that would result in successful com- bined analysis. Do Bayes factors strike a reasonable balance between modelcomplexity and error variance?—  The ability to allow het-erogeneity across data partitions in model parametersopens up a Pandora’s Box of model choice problems,which are difficult to address without good model se-lection criteria and procedures. Standard likelihood ra-tio tests have a tendency to prefer complex models(Gelfand and Day, 1994; Burnham and Anderson, 2002)and various procedures have been developed to punishparameter-richmodels(Akaike,1973;Schwartz,1978).Intheory, the Bayes factor comparison does not suffer fromthisproblem;asimplemodelcanbefavoredoveramoreparameter-rich model even if the models are nested. WelookedforinstancesofsimplemodelswinningovermorecomplexonesandcaseswheretheBayesfactorwouldfa-vormodelreductionbysupportingtheexclusionofweakparameters.M ATERIALS AND  M ETHODS Data WeassembledDNAandmorphologicaldatafor29gallwasp exemplars and three outgroup exemplars, the lat-ter representing the families Figitidae, Liopteridae, andIbaliidae (Table 2). Previous phylogenetic analyses in-dicate that Figitidae is the sister group to Cynipidaeand that the Liopteridae and Ibaliidae are successivelymore distant outgroups (Ronquist, 1999). The gall waspsample included representatives of all described tribesof the only extant subfamily. All major wasp genera of phytophagous guests in cynipid galls, also known asinquilines, were represented except for the genus  Sa-phonecrus , which is considered close to if not embeddedwithin  Synergus  (Ronquist, 1994, 1999; Nieves-Aldrey,2001; Ronquist and Liljeblad, 2001). A broad selectionof gall inducers attacking herbaceous and woody hostplants was also included. At least half the describedgenera were included for all tribes except the Cynipini,or the oak gallers. This tribe, comprising more than 40described genera, was represented by only four genera but is widely thought to be monophyletic (Kinsey, 1920;Askew, 1984; Ronquist, 1994, 1999; Liljeblad and Ron-quist,1998;Nieves-Aldrey,2001;RonquistandLiljeblad,2001; Stone et al., 2002).The morphological data were taken from Liljebladand Ronquist (1998) and consist of 166 parsimony-informativediscretecharacters:164externalmorpholog-ical characters and two ecological characters (alterationof sexual/asexual generations, and host-plant choice)(Liljeblad and Ronquist, 1998: appendix 1). Some mul-tistate characters were treated as ordered and othersas unordered, as specified by Liljeblad and Ronquist(1998).As far as possible, DNA data were collected from thesame species for which we had morphological data. Ina few cases, an exact match could not be obtained, butDNAsequenceswereobtained,fromacloserelativeandthese taxa were combined into a single terminal in the fi-nal analyses (Table 2). We sequenced parts of four genes:COI (1,078 bp), the nuclear protein-coding genes LWRh(481 bp) and EF1 α , (367 bp), and the nuclear 28S rDNA(1,154 bp) (GenBank accession numbers in Table 2). De-tails of the DNA amplification protocols and primerswere given by Rokas et al. (2002). The protein-codinggenes (COI, LWRh, and EF1 α ) were easily aligned byeye. The ribosomal sequences (28S) differed in length,and some of the more variable regions were difficultto align manually. We used ClustalW 1.81 (Thompsonet al., 1994) for this alignment. We applied a range of costs for the gap opening and gap extension penal-ties, and the individual alignments were subjected toparsimony bootstrap (Felsenstein, 1985) analyses usingPAUP ∗ (Swofford, 1998). Supported groups were largelycongruent among the resulting trees. The alignment re-sultingfromtheuseofthedefaultsettingsinClustalWisavailable from TreeBase (http://www.treebase.org, ac-cession S970).   b  y g u e  s  t   onF  e  b r  u a r  y1  8  ,2  0 1  6 h  t   t   p :  /   /   s  y s  b i   o . oxf   or  d  j   o ur n a l   s  . or  g /  D o wnl   o a  d  e  d f  r  om 
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