Biases with the Generalized Euclidean Distance measure in disparity analyses with high levels of missing data

The Generalized Euclidean Distance (GED) measure has been extensively used to conduct morphological disparity analyses based on palaeontological matrices of discrete characters. This is in part because some implementations allow the use of morphological matrices with high percentages of missing data without needing to prune taxa for a subsequent ordination of the data set. Previous studies have suggested that this way of using the GED may generate a bias in the resulting morphospace, but a detailed study of this possible effect has been lacking. Here, we test whether the percentage of missing data for a taxon artificially influences its position in the morphospace, and if missing data affects pre‐ and post‐ordination disparity measures. We find that this use of the GED creates a systematic bias, whereby taxa with higher percentages of missing data are placed closer to the centre of the morphospace than those with more complete scorings. This bias extends into pre‐ and post‐ordination calculations of disparity measures and can lead to erroneous interpretations of disparity patterns, especially if specimens present in a particular time interval or clade have distinct proportions of missing information. We suggest that this implementation of the GED should be used with caution, especially in cases with high percentages of missing data. Results recovered using an alternative distance measure, Maximum Observed Rescaled Distance (MORD), are more robust to missing data. As a consequence, we suggest that MORD is a more appropriate distance measure than GED when analysing data sets with high amounts of missing data.     

The complex effects of mass extinctions on morphological disparity

Studies of biodiversity through deep time have been a staple for biologists and paleontologists for over 60 years. Investigations of species richness (diversity) revealed that at least five mass extinctions punctuated the last half billion years, each seeing the rapid demise of a large proportion of contemporary taxa. In contrast to diversity, the response of morphological diversity (disparity) to mass extinctions is unclear. Generally, diversity and disparity are decoupled, such that diversity may decline as morphological disparity increases, and vice versa. Here, we develop simulations to model disparity changes across mass extinctions using continuous traits and birth-death trees. We find no simple null for disparity change following a mass extinction but do observe general patterns. The range of trait values decreases following either random or trait-selective mass extinctions, whereas variance and the density of morphospace occupation only decline following trait-selective events. General trends may differentiate random and trait-selective mass extinctions, but methods struggle to identify trait selectivity. Long-term effects of mass extinction trait selectivity change support for phylogenetic comparative methods away from the simulated Brownian motion toward Ornstein-Uhlenbeck and Early Burst models. We find that morphological change over mass extinction is best studied by quantifying multiple aspects of morphospace occupation.     

Journeys through discrete‐character morphospace: synthesizing phylogeny,tempo, and disparity

Palaeontologists have long employed discrete categorical data to capture morphological variation in fossil species, using the resulting character–taxon matrices to measure evolutionary tempo, infer phylogenies and capture morphological disparity. However, to date these have been seen as separate approaches despite a common goal of understanding morphological evolution over deep time. Here I argue that there are clear advantages to considering these three lines of enquiry in a single space: the phylomorphospace. Conceptually these high‐dimensional spaces capture how a phylogenetic tree explores morphospace and allow us to consider important process questions around evolutionary rates, constraints, convergence and directional trends. Currently the literature contains fundamentally different approaches used to generate such spaces, with no direct comparison between them, despite the differing evolutionary histories they imply. Here I directly compare five different phylomorphospace approaches, three with direct literature equivalents and two that are novel. I use a single empirical case study of coelurosaurian theropod dinosaurs (152 taxa, 853 characters) to show that under many analyses the literature‐derived approaches tend to reflect introduced phylogenetic (rather than the intended morphological) signal. The two novel approaches, which produce limited ancestral state estimates prior to ordination, are able to minimize this phylogenetic signal and thus exhibit more realistic amounts of phylogenetic signal, rate heterogeneity, and convergent evolution.     

Geometric morphometric analyses provide evidence for the adaptive character of the Tanganyikan cichlid fish radiations

The cichlids of East Africa are renowned as one of the most spectacular examples of adaptive radiation. They provide a unique opportunity to investigate the relationships between ecology, morphological diversity, and phylogeny in producing such remarkable diversity. Nevertheless, the parameters of the adaptive radiations of these fish have not been satisfactorily quantified yet. Lake Tanganyika possesses all of the major lineages of East African cichlid fish, so by using geometric morphometrics and comparative analyses of ecology and morphology, in an explicitly phylogenetic context, we quantify the role of ecology in driving adaptive speciation. We used geometric morphometric methods to describe the body shape of over 1000 specimens of East African cichlid fish, with a focus on the Lake Tanganyika species assemblage, which is composed of more than 200 endemic species. The main differences in shape concern the length of the whole body and the relative sizes of the head and caudal peduncle. We investigated the influence of phylogeny on similarity of shape using both distance-based and variance partitioning methods, finding that phylogenetic inertia exerts little influence on overall body shape. Therefore, we quantified the relative effect of major ecological traits on shape using phylogenetic generalized least squares and disparity analyses. These analyses conclude that body shape is most strongly predicted by feeding preferences (i.e., trophic niches) and the water depths at which species occur. Furthermore, the morphological disparity within tribes indicates that even though the morphological diversification associated with explosive speciation has happened in only a few tribes of the Tanganyikan assemblage, the potential to evolve diverse morphologies exists in all tribes. Quantitative data support the existence of extensive parallelism in several independent adaptive radiations in Lake Tanganyika. Notably, Tanganyikan mouthbrooders belonging to the C-lineage and the substrate spawning Lamprologini have evolved a multitude of different shapes from elongated and Lamprologus-like hypothetical ancestors. Together, these data demonstrate strong support for the adaptive character of East African cichlid radiations.     

Taxon incompleteness and discrete time bins affect character change rates in simulated data

Estimating how fast or slow morphology evolves through time (phenotypic change rate, PR) has become common in macroevolutionary studies and has been important for clarifying key evolutionary events. However, the inclusion of incompletely scored taxa (e.g. fossils) and variable lengths of discrete arbitrary time bins could affect PR estimates and potentially mask real PR patterns. Here, the impact of taxon incompleteness (unscored data) on PR estimates is assessed in simulated data. Three different time bin series were likewise evaluated: bins evenly spanning the tree length (i), a shorter middle bin and longer first and third bins (ii), and a longer middle bin and shorter first and third bins (iii). The results indicate that PR values decrease as taxon incompleteness increases. Statistically significant PR values, and the dispersion among PR values, depended on the time bins. These outcomes imply that taxon incompleteness can undermine our capacity to infer morphology evolutionary dynamics and that these estimates are also influenced by our choice of discrete time bins. More importantly, the present results stress the need for a better approach to deal with taxon incompleteness and arbitrary discrete time bins.     

Spatial patterns of disparity and diversity of the Recent cuttlefishes (Cephalopoda) across the Old World

Aim Diversity and disparity metrics of all Recent cuttlefishes are studied at the macroevolutionary scale (1) to establish the geographical biodiversity patterns of these cephalopods at the species level and (2) to explore the relationships between these two metrics. Location Sampling uses what is known about these tropical, subtropical and warm temperate cephalopods of the Old World based on a literature review and on measurements of museum specimens. Some 111 species spread across seventeen biogeographical areas serve as basic units for exploring diversity and disparity metrics in space. Methods Landmarks describe the shape of the cuttlebone (the inner shell of the sepiids) and differences between shapes are quantified using relative warp analyses. Relative warps are thus used as the morphological axis for constructing morphospaces whose characteristics are described by disparity indices: total variance, range, and minimum and maximum of relative warps. These are analysed and then compared with the diversity (species richness) metric. Results Results show no significant latitudinal or longitudinal gradients either for diversity or for disparity. Around the coast of southern Africa, disparity is high regardless of whether diversity (species richness) is high or low. In the ‘East Indies’ area disparity is low despite the high diversity. Main conclusions The relationship between diversity and disparity is clearly not linear and no simple adjustment models seem to fit. The number of species in a given area does not predict its disparity level. The particular pattern of southern Africa may be the result of paleogeographical changes since the Eocene, whereas that of the ‘East Indies’ may indicate that this area could act as a centre of origin. However, the lack of any clear phylogenetical hypothesis precludes the study from providing any explanation of the observed patterns.     

Discrete and morphometric traits reveal contrasting patterns and processes in the macroevolutionary history of a clade of scorpions

Many palaeontological studies have investigated the evolution of entire body plans, generally relying on discrete character‐taxon matrices. In contrast, macroevolutionary studies performed by neontologists have mostly focused on morphometric traits. Although these data types are very different, some studies have suggested that they capture common patterns. Nonetheless, the tests employed to support this claim have not explicitly incorporated a phylogenetic framework and may therefore be susceptible to confounding effects due to the presence of common phylogenetic structure. We address this question using the scorpion genus Brachistosternus Pocock 1893 as case study. We make use of a time‐calibrated multilocus molecular phylogeny, and compile discrete and traditional morphometric data sets, both capturing the overall morphology of the organisms. We find that morphospaces derived from these matrices are significantly different, and that the degree of discordance cannot be replicated by simulations of random character evolution. Moreover, we find strong support for contrasting modes of evolution, with discrete characters being congruent with an ‘early burst’ scenario whereas morphometric traits suggest species‐specific adaptations to have driven morphological evolution. The inferred macroevolutionary dynamics are therefore contingent on the choice of character type. Finally, we confirm that metrics of correlation fail to detect these profound differences given common phylogenetic structure in both data sets, and that methods incorporating a phylogenetic framework and accounting for expected covariance should be favoured.     

缺失数据和贝叶斯系统发育分析精确度之探索

The effect of missing data on phylogenetic methods is a potentially important issue in our attempts to reconstruct the Tree of Life. If missing data are truly problematic, then it may be unwise to include species in an analysis that lack data for some characters (incomplete taxa) or to include characters that lack data for some species. Given the difficulty of obtaining data from all characters for all taxa (e.g., fossils), missing data might seriously impede efforts to reconstruct a comprehensive phylogeny that includes all species. Fortunately, recent simulations and empirical analyses suggest that missing data cells are not themselves problematic, and that incomplete taxa can be accurately placed as long as the overall number of characters in the analysis is large. However, these studies have so far only been conducted on parsimony, likelihood, and neighbor-joining methods. Although Bayesian phylogenetic methods have become widely used in recent years, the effects of missing data on Bayesian analysis have not been adequately studied. Here, we conduct simulations to test whether Bayesian analyses can accurately place incomplete taxa despite extensive missing data. In agreement with previous studies of other methods, we find that Bayesian analyses can accurately reconstruct the position of highly incomplete taxa (i.e., 95% missing data), as long as the overall number of characters in the analysis is large. These results suggest that highly incomplete taxa can be safely included in many Bayesian phylogenetic analyses.     

Diversity trends and their ontogenetic basis: an exploration of allometric disparity in rodents

It has been hypothesized that most morphological evolution occurs by allometric differentiation. Because rodents encapsulate a phenomenal amount of taxonomic diversity and, among several clades, contrasting levels of morphological diversity, they represent an excellent subject to address the question: how variable are allometric patterns during evolution? We investigated the influence of phylogenetic relations and ecological factors on the results of the first quantification of allometric disparity among rodents by exploring allometric space, a multivariate morphospace here derived from, and encapsulating all, the ontogenetic trajectories of 34 rodent species from two parallel phylogenetic radiations. Disparity was quantified using angles between ontogenetic trajectories for different species and clades. We found an overlapping occupation of allometric space by muroid and hystricognath species, revealing both clades possess similar abilities to evolve in different directions of phenotypic space, and anatomical diversity does not act to constrain the labile nature of allometric patterning. Morphological features to enable efficient processing of food serve to group rodents in allometric space, reflecting the importance of convergent morphology, rather than shared evolutionary history, in the generation of allometric patterns. Our results indicate that the conserved level of morphological integration found among primates cannot simply be extended to all mammals.     

Patterns of cranial ontogeny in lacertid lizards: morphological and allometric disparity

We explored the ontogenetic dynamics of the morphological and allometric disparity in the cranium shapes of twelve lacertid lizard species. The analysed species (Darevskia praticola, Dinarolacerta mosorensis, Iberolacerta horvathi, Lacerta agilis, L. trilineata, L. viridis, Podarcis erhardii, P. melisellensis, P. muralis, P. sicula, P. taurica and Zootoca vivipara) can be classified into different ecomorphs: terrestrial lizards that inhabit vegetated habitats (habitats with lush or sparse vegetation), saxicolous and shrub‐climbing lizards. We observed that there was an overall increase in the morphological disparity (MD) during the ontogeny of the lacertid lizards. The ventral cranium, which is involved in the mechanics of jaw movement and feeding, showed higher levels of MD, an ontogenetic shift in the morphospace planes and more variable allometric patterns than more conserved dorsal crania. With respect to ecology, the allometric trajectories of the shrub‐climbing species tended to cluster together, whereas the allometric trajectories of the saxicolous species were highly dispersed. Our results indicate that the ontogenetic patterns of morphological and allometric disparity in the lacertid lizards are modified by ecology and functional constraints and that the identical mechanisms that lead to intraspecific morphological variation also produce morphological divergence at higher taxonomic levels.     

A two-dimensional morphospace for cyanobacteria and microalgae: Morphological diversity,evolutionary relatedness,and size constraints

1. Body metrics are considered as master traits that regulate physiological, behavioural and life history features of planktic cyanobacteria and microalgae. Although the distribution of their morphological traits reflects the various trade-offs and strategies needed for survival in pelagic habitats, previous methods for quantifying phytoplankton body shape do not adequately represent the intricate details of surface variation that are so important for their nutrient- and light-harvesting capabilities. Therefore, here we provide a new framework to quantify and illustrate the morphological diversity of cyanobacteria and microalgae.
2. Essential components of formulae used for surface area (A = Cs × d²) and volume (V = Cv × d³) calculations are provided by the shape-specific surface area and volume constants (Cs, Cv). Cs, the surface shape factor, characterises the coarseness of the object surface, and Cv, the volumetric shape factor, characterises the shape deviation from a sphere. Using these morphologically and biologically relevant variables, we defined a two-dimensional morphological space, in which all three-dimensional objects have well-defined positions.
3. By analysing morphologies of taxa representing various forms in major cyanobacterial and microalgal groups, we demonstrated that these groups show considerable differences in the area occupied within the morphospace and these differences are not affected by evolutionary relatedness. We showed that the ratio of surface and volume constants correlated with organism size, suggesting that the development of basic morphologies is constrained by their linear dimensions.
4. Using surface and volumetric shape factors, we created a two-dimensional Euclidean morphospace in which all three-dimensional objects have a specific position. Positioning uni- and multicellular cyanobacteria and microalgae of various shapes into this morphospace allows their geometrical and ecological limitations to be understood. Because of the close linkage between phytoplankton morphology and ecology, the proposed morphospace may serve as a proxy for an ecospace. Thus, in future the proposed morphospace can be used to visualise current ecological processes such as eutrophication or seasonal succession of phytoplankton.

     

Multiple imputation for discrete data: Evaluation of the joint latent normal model

Missing data are ubiquitous in clinical and social research, and multiple imputation (MI) is increasingly the methodology of choice for practitioners. Two principal strategies for imputation have been proposed in the literature: joint modelling multiple imputation (JM‐MI) and full conditional specification multiple imputation (FCS‐MI). While JM‐MI is arguably a preferable approach, because it involves specification of an explicit imputation model, FCS‐MI is pragmatically appealing, because of its flexibility in handling different types of variables. JM‐MI has developed from the multivariate normal model, and latent normal variables have been proposed as a natural way to extend this model to handle categorical variables. In this article, we evaluate the latent normal model through an extensive simulation study and an application on data from the German Breast Cancer Study Group, comparing the results with FCS‐MI. We divide our investigation in four sections, focusing on (i) binary, (ii) categorical, (iii) ordinal, and (iv) count data. Using data simulated from both the latent normal model and the general location model, we find that in all but one extreme general location model setting JM‐MI works very well, and sometimes outperforms FCS‐MI. We conclude the latent normal model, implemented in the R package jomo , can be used with confidence by researchers, both for single and multilevel multiple imputation.     

Relative benefits of amino‐acid,codon, degeneracy,DNA, and purine‐pyrimidine character coding for phylogenetic analyses of exons

Both traditional as well as 10 more recent methods of coding characters from exons of protein‐coding genes are reviewed. The more recent methods collectively blur the distinction between nucleotide and amino‐acid coding and enable investigators to carefully quantify the effects of different sources of phylogenetic signal as well as their potential biases. Codon models, which explicitly model silent and replacement substitutions, are a major advance and are expected to be broadly useful for simultaneously inferring recent and ancient divergences, unlike amino‐acid coding. Degeneracy coding, wherein ambiguity codes are used to eliminate silent substitutions at the individual‐nucleotide level, has clear advantages over scoring amino‐acid characters. Nucleotide, codon, and amino‐acid models are now directly comparable with easy‐to‐use programs, and widely used phylogenetics programs can analyze partitioned supermatrices that incorporate all three types of model. Therefore, it should become standard practice to test among these alternative model types before conducting parametric phylogenetic analyses. An earlier study of 78 protein‐coding genes from 360 green‐plant plastid genomes is used as an empirical example with which to quantify the relative performance of alternative character‐coding methods using five quantification measures. Codon models were selected as having the best fit to the data, yet were outperformed by nucleotide models for all five quantification measures. Third‐codon positions were found to be an important source of phylogenetic signal and even outperformed analyses of first and second positions for some measures. Degeneracy coding generally performed at least as well as amino‐acid coding and is an arguably more effective alternative.     

Integrating data‐deficient species in analyses of evolutionary history loss

There is an increasing interest in measuring loss of phylogenetic diversity and evolutionary distinctiveness which together depict the evolutionary history of conservation interest. Those losses are assessed through the evolutionary relationships between species and species threat status or extinction probabilities. Yet, available information is not always sufficient to quantify the threat status of species that are then classified as data deficient. Data‐deficient species are a crucial issue as they cause incomplete assessments of the loss of phylogenetic diversity and evolutionary distinctiveness. We aimed to explore the potential bias caused by data‐deficient species in estimating four widely used indices: HEDGE, EDGE, PDloss, and Expected PDloss. Second, we tested four different widely applicable and multitaxa imputation methods and their potential to minimize the bias for those four indices. Two methods are based on a best‐ vs. worst‐case extinction scenarios, one is based on the frequency distribution of threat status within a taxonomic group and one is based on correlates of extinction risks. We showed that data‐deficient species led to important bias in predictions of evolutionary history loss (especially high underestimation when they were removed). This issue was particularly important when data‐deficient species tended to be clustered in the tree of life. The imputation method based on correlates of extinction risks, especially geographic range size, had the best performance and enabled us to improve risk assessments. Solving threat status of DD species can fundamentally change our understanding of loss of phylogenetic diversity. We found that this loss could be substantially higher than previously found in amphibians, squamate reptiles, and carnivores. We also identified species that are of high priority for the conservation of evolutionary distinctiveness.     

Phenotypic disparity of the elbow joint in domestic dogs and wild carnivores*

In this article, I use geometric morphometrics in 2D from a sample of 366 elbow joints to quantify phenotypic disparity in domestic dog breeds, in wild canids, and across the order Carnivora. The elbow joint is a well‐established morphological indicator of forearm motion and, by extension, of functional adaptations toward locomotor or predatory behavior in living carnivores. The study of the elbow joint in domestic dogs allows the exploration of potential convergences between (i) pursuit predators and fast‐running dogs, and (ii) ambush predators and fighting breeds. The results indicate that elbow shape disparity among domestic dogs exceeds than in wolves; it is comparable to the disparity of wild Caninae, but is significantly lower than the one observed throughout Canidae and Carnivora. Moreover, fast‐running and fighting breeds are not convergent in elbow joint shape with extreme pursuit and ambush wild carnivores, respectively. The role of artificial selection and developmental constraints in shaping limb phenotypic disparity through the extremely fast evolution of the domestic dog is discussed in the light of this new evidence.     

Probabilistic methods surpass parsimony when assessing clade support in phylogenetic analyses of discrete morphological data

Fossil taxa are critical to inferences of historical diversity and the origins of modern biodiversity, but realizing their evolutionary significance is contingent on restoring fossil species to their correct position within the tree of life. For most fossil species, morphology is the only source of data for phylogenetic inference; this has traditionally been analysed using parsimony, the predominance of which is currently challenged by the development of probabilistic models that achieve greater phylogenetic accuracy. Here, based on simulated and empirical datasets, we explore the relative efficacy of competing phylogenetic methods in terms of clade support. We characterize clade support using bootstrapping for parsimony and Maximum Likelihood, and intrinsic Bayesian posterior probabilities, collapsing branches that exhibit less than 50% support. Ignoring node support, Bayesian inference is the most accurate method in estimating the tree used to simulate the data. After assessing clade support, Bayesian and Maximum Likelihood exhibit comparable levels of accuracy, and parsimony remains the least accurate method. However, Maximum Likelihood is less precise than Bayesian phylogeny estimation, and Bayesian inference recaptures more correct nodes with higher support compared to all other methods, including Maximum Likelihood. We assess the effects of these findings on empirical phylogenies. Our results indicate probabilistic methods should be favoured over parsimony.     

The use and misuse of regression models in landscape genetic analyses

The field of landscape genetics has been rapidly evolving, adopting and adapting analytical frameworks to address research questions. Current studies are increasingly using regression‐based frameworks to infer the individual contributions of landscape and habitat variables on genetic differentiation. This paper outlines appropriate and inappropriate uses of multiple regression for these purposes, and demonstrates through simulation the limitations of different analytical frameworks for making correct inference. Of particular concern are recent studies seeking to explain genetic differences by fitting regression models with effective distance variables calculated independently on separate landscape resistance surfaces. When moving across the landscape, organisms cannot respond independently and uniquely to habitat and landscape features. Analyses seeking to understand how landscape features affect gene flow should model a single conductance or resistance surface as a parameterized function of relevant spatial covariates, and estimate the values of these parameters by linking a single set of resistance distances to observed genetic dissimilarity via a loss function. While this loss function may involve a regression‐like step, the associated nuisance parameters are not interpretable in terms of organismal movement and should not be conflated with what is actually of interest: the mapping between spatial covariates and conductance/resistance. The growth and evolution of landscape genetics as a field has been rapid and exciting. It is the goal of this paper to highlight past missteps and demonstrate limitations of current approaches to ensure that future use of regression models will appropriately consider the process being modeled, which will provide clarity to model interpretation.     

Resolving issues of imprecise and habitat-biased locations in ecological analyses using GPS telemetry data

Global positioning system (GPS) technologies collect unprecedented volumes of animal location data, providing ever greater insight into animal behaviour. Despite a certain degree of inherent imprecision and bias in GPS locations, little synthesis regarding the predominant causes of these errors, their implications for ecological analysis or solutions exists. Terrestrial deployments report 37 per cent or less non-random data loss and location precision 30 m or less on average, with canopy closure having the predominant effect, and animal behaviour interacting with local habitat conditions to affect errors in unpredictable ways. Home-range estimates appear generally robust to contemporary levels of location imprecision and bias, whereas movement paths and inferences of habitat selection may readily become misleading. There is a critical need for greater understanding of the additive or compounding effects of location imprecision, fix-rate bias, and, in the case of resource selection, map error on ecological insights. Technological advances will help, but at present analysts have a suite of ad hoc statistical corrections and modelling approaches available—tools that vary greatly in analytical complexity and utility. The success of these solutions depends critically on understanding the error-inducing mechanisms, and the biggest gap in our current understanding involves species-specific behavioural effects on GPS performance.     

三种葱属植物花形态及花药解剖结构观察

蒙古韭(Allium mongolicum Regel)、细叶韭(Allium tenuissimum L.)、野韭(Allium ramosumL.)作为野生蔬菜近年来受到人们的青睐,一些学者对这三种葱属植物进行了初步研究,但对其解剖结构的研究未见报道.通过田间观察和石蜡切片法,对这三种葱属植物花的形态及花药解剖结构进行了观察,结果表明:三种葱属植物花的形态结构差异明显,花粉粒的形状及大小也有明显的差别:野韭花粉粒为长卵形,细叶韭为卵圆形,蒙古韭为近半圆形;从花粉粒大小来看,野韭最大,蒙古韭次之,细叶韭最小.通过以上研究揭示了三种葱属植物花药解剖学特征,从而为葱属植物分类标准、亲缘关系鉴定、生殖发育等方面的进一步研究提供理论依据.     

Detecting the historical signature of key innovations using stochastic models of character evolution and cladogenesis

Phylogenetic evidence for biological traits that increase the net diversification rate of lineages (key innovations) is most commonly drawn from comparisons of clade size. This can work well for ancient, unreversed traits and for correlating multiple trait origins with higher diversification rates, but it is less suitable for unique events, recently evolved innovations, and traits that exhibit homoplasy. Here I present a new method for detecting the phylogenetic signature of key innovations that tests whether the evolutionary history of the candidate trait is associated with shorter waiting times between cladogenesis events. The method employs stochastic models of character evolution and cladogenesis and integrates well into a Bayesian framework in which uncertainty in historical inferences (such as phylogenetic relationships) is allowed. Applied to a well-known example in plants, nectar spurs in columbines, the method gives much stronger support to the key innovation hypothesis than previous tests.