Polytomies and bayesian phylogenetic inference

Bayesian phylogenetic analyses are now very popular in systematics and molecular evolution because they allow the use of much more realistic models than currently possible with maximum likelihood methods. There are, however, a growing number of examples in which large Bayesian posterior clade probabilities are associated with very short edge lengths and low values for non-Bayesian measures of support such as nonparametric bootstrapping. For the four-taxon case when the true tree is the star phylogeny, Bayesian analyses become increasingly unpredictable in their preference for one of the three possible resolved tree topologies as data set size increases. This leads to the prediction that hard (or near-hard) polytomies in nature will cause unpredictable behavior in Bayesian analyses, with arbitrary resolutions of the polytomy receiving very high posterior probabilities in some cases. We present a simple solution to this problem involving a reversible-jump Markov chain Monte Carlo (MCMC) algorithm that allows exploration of all of tree space, including unresolved tree topologies with one or more polytomies. The reversible-jump MCMC approach allows prior distributions to place some weight on less-resolved tree topologies, which eliminates misleadingly high posteriors associated with arbitrary resolutions of hard polytomies. Fortunately, assigning some prior probability to polytomous tree topologies does not appear to come with a significant cost in terms of the ability to assess the level of support for edges that do exist in the true tree. Methods are discussed for applying arbitrary prior distributions to tree topologies of varying resolution, and an empirical example showing evidence of polytomies is analyzed and discussed.

Polytomies and bayesian phylogenetic inference

Bayesian phylogenetic analyses are now very popular in systematics and molecular evolution because they allow the use of much more realistic models than currently possible with maximum likelihood methods. There are, however, a growing number of examples in which large Bayesian posterior clade probabilities are associated with very short edge lengths and low values for non-Bayesian measures of support such as nonparametric bootstrapping. For the four-taxon case when the true tree is the star phylogeny, Bayesian analyses become increasingly unpredictable in their preference for one of the three possible resolved tree topologies as data set size increases. This leads to the prediction that hard (or near-hard) polytomies in nature will cause unpredictable behavior in Bayesian analyses, with arbitrary resolutions of the polytomy receiving very high posterior probabilities in some cases. We present a simple solution to this problem involving a reversible-jump Markov chain Monte Carlo (MCMC) algorithm that allows exploration of all of tree space, including unresolved tree topologies with one or more polytomies. The reversible-jump MCMC approach allows prior distributions to place some weight on less-resolved tree topologies, which eliminates misleadingly high posteriors associated with arbitrary resolutions of hard polytomies. Fortunately, assigning some prior probability to polytomous tree topologies does not appear to come with a significant cost in terms of the ability to assess the level of support for edges that do exist in the true tree. Methods are discussed for applying arbitrary prior distributions to tree topologies of varying resolution, and an empirical example showing evidence of polytomies is analyzed and discussed.

An Extreme Case of Plant-Insect Codiversification: Figs and Fig-Pollinating Wasps

It is thought that speciation in phytophagous insects is often due to colonization of novel host plants, because radiations of plant and insect lineages are typically asynchronous. Recent phylogenetic comparisons have supported this model of diversification for both insect herbivores and specialized pollinators. An exceptional case where contemporaneous plant-insect diversification might be expected is the obligate mutualism between fig trees (Ficus species, Moraceae) and their pollinating wasps (Agaonidae, Hymenoptera). The ubiquity and ecological significance of this mutualism in tropical and subtropical ecosystems has long intrigued biologists, but the systematic challenge posed by >750 interacting species pairs has hindered progress toward understanding its evolutionary history. In particular, taxon sampling and analytical tools have been insufficient for large-scale cophylogenetic analyses. Here, we sampled nearly 200 interacting pairs of fig and wasp species from across the globe. Two supermatrices were assembled: on an average, wasps had sequences from 77% of 6 genes (5.6 kb), figs had sequences from 60% of 5 genes (5.5 kb), and overall 850 new DNA sequences were generated for this study. We also developed a new analytical tool, Jane 2, for event-based phylogenetic reconciliation analysis of very large data sets. Separate Bayesian phylogenetic analyses for figs and fig wasps under relaxed molecular clock assumptions indicate Cretaceous diversification of crown groups and contemporaneous divergence for nearly half of all fig and pollinator lineages. Event-based cophylogenetic analyses further support the codiversification hypothesis. Biogeographic analyses indicate that the present-day distribution of fig and pollinator lineages is consistent with a Eurasian origin and subsequent dispersal, rather than with Gondwanan vicariance. Overall, our findings indicate that the fig-pollinator mutualism represents an extreme case among plant-insect interactions of coordinated dispersal and long-term codiversification.

Caractérisation et délimitation des sous-espèces de Gesneria viridiflora (Gesneriaceae) dans les Antilles

Une taxonomie révisée et une connaissance des limites d’espèces demeurent toujours importantes dans les points chauds en biodiversité comme les Antilles où de nombreuses espèces endémiques sont retrouvées. Des limites d’espèces divergentes impliquent un différent nombre d’espèces retrouvées dans un écosystème, ce qui peut exercer une influence sur les décisions prises face aux enjeux de conservation. Les genres Gesneria et Rhytidophyllum qui forment les principaux représentants de la famille des Gesneriaceae dans les Antilles comprennent plusieurs taxons aux limites d’espèces ambigües et quelques espèces qui ont des sous-espèces reconnues. C’est le cas de Gesneria viridiflora (Decne.) Kuntze qui comprend quatre sous-espèces géographiquement isolées et qui présentent des caractères végétatifs et reproducteurs similaires et variables. Une délimitation d’espèces approfondie de ce complexe d’espèce est effectuée ici à partir d’une approche de taxonomie intégrative considérant des données morphologiques, génétiques et bioclimatiques. Les données morphologiques quantitatives et qualitatives obtenues à partir de spécimens d’herbier sont utilisées pour délimiter des groupes morphologiques à l’aide d’une analyse en coordonnées principales. Ces groupes sont ensuite testés à l’aide de séquences d’ADN de quatre régions nucléaires en utilisant une méthode bayesienne basée sur la théorie de la coalescence. Finalement, les occurrences et les valeurs de variables de température et de précipitation qui y prévalent sont utilisées dans une analyse en composantes principales bioclimatique pour comparer les groupes délimités morphologiquement et génétiquement. Les résultats de l’analyse morphologique multivariée supportent la distinction entre les groupes formés par les sous-espèces actuellement reconnues de G. viridiflora. Les résultats, incluant des données génétiques, suggèrent une distinction jusqu’ici insoupçonnée des populations du Massif de la Hotte au sud-ouest d’Haïti qui sont génétiquement plus rapprochées des populations de Cuba que de celles d’Hispaniola. Bioclimatiquement, les groupes délimités par les analyses morphologiques et génétiques sont distincts. L’approche de taxonomie intégrative a permis de distinguer cinq espèces distinctes plutôt que les quatre sous-espèces acceptées jusqu’à aujourd’hui. Ces espèces sont : G. acrochordonanthe, G. quisqueyana, G. sintenisii, G. sylvicola et G. viridiflora. Une carte de distribution géographique, un tableau de la nouvelle taxonomie applicable et une clé d’identification des espèces sont présentés. La nouvelle taxonomie déterminée dans cette étude démontre un endémisme insoupçonné dans plusieurs régions du point chaud en biodiversité des Antilles et souligne l’importance d’investiguer les limites d’espèces dans les groupes diversifiés comprenant des taxons aux limites d’espèces incomprises.

Caractérisation et délimitation des sous-espèces de Gesneria viridiflora (Gesneriaceae) dans les Antilles

Une taxonomie révisée et une connaissance des limites d’espèces demeurent toujours importantes dans les points chauds en biodiversité comme les Antilles où de nombreuses espèces endémiques sont retrouvées. Des limites d’espèces divergentes impliquent un différent nombre d’espèces retrouvées dans un écosystème, ce qui peut exercer une influence sur les décisions prises face aux enjeux de conservation. Les genres Gesneria et Rhytidophyllum qui forment les principaux représentants de la famille des Gesneriaceae dans les Antilles comprennent plusieurs taxons aux limites d’espèces ambigües et quelques espèces qui ont des sous-espèces reconnues. C’est le cas de Gesneria viridiflora (Decne.) Kuntze qui comprend quatre sous-espèces géographiquement isolées et qui présentent des caractères végétatifs et reproducteurs similaires et variables. Une délimitation d’espèces approfondie de ce complexe d’espèce est effectuée ici à partir d’une approche de taxonomie intégrative considérant des données morphologiques, génétiques et bioclimatiques. Les données morphologiques quantitatives et qualitatives obtenues à partir de spécimens d’herbier sont utilisées pour délimiter des groupes morphologiques à l’aide d’une analyse en coordonnées principales. Ces groupes sont ensuite testés à l’aide de séquences d’ADN de quatre régions nucléaires en utilisant une méthode bayesienne basée sur la théorie de la coalescence. Finalement, les occurrences et les valeurs de variables de température et de précipitation qui y prévalent sont utilisées dans une analyse en composantes principales bioclimatique pour comparer les groupes délimités morphologiquement et génétiquement. Les résultats de l’analyse morphologique multivariée supportent la distinction entre les groupes formés par les sous-espèces actuellement reconnues de G. viridiflora. Les résultats, incluant des données génétiques, suggèrent une distinction jusqu’ici insoupçonnée des populations du Massif de la Hotte au sud-ouest d’Haïti qui sont génétiquement plus rapprochées des populations de Cuba que de celles d’Hispaniola. Bioclimatiquement, les groupes délimités par les analyses morphologiques et génétiques sont distincts. L’approche de taxonomie intégrative a permis de distinguer cinq espèces distinctes plutôt que les quatre sous-espèces acceptées jusqu’à aujourd’hui. Ces espèces sont : G. acrochordonanthe, G. quisqueyana, G. sintenisii, G. sylvicola et G. viridiflora. Une carte de distribution géographique, un tableau de la nouvelle taxonomie applicable et une clé d’identification des espèces sont présentés. La nouvelle taxonomie déterminée dans cette étude démontre un endémisme insoupçonné dans plusieurs régions du point chaud en biodiversité des Antilles et souligne l’importance d’investiguer les limites d’espèces dans les groupes diversifiés comprenant des taxons aux limites d’espèces incomprises.

Phylogenetics, Ancestral State Reconstruction, And A New Infrafamilial Classification Of The Pantropical Ochnaceae (medusagynaceae, Ochnaceae S.str., Quiinaceae) Based On Five Dna Regions.

Ochnaceae s.str. (Malpighiales) are a pantropical family of about 500 species and 27 genera of almost exclusively woody plants. Infrafamilial classification and relationships have been controversial partially due to the lack of a robust phylogenetic framework. Including all genera except Indosinia and Perissocarpa and DNA sequence data for five DNA regions (ITS, matK, ndhF, rbcL, trnL-F), we provide for the first time a nearly complete molecular phylogenetic analysis of Ochnaceae s.l. resolving most of the phylogenetic backbone of the family. Based on this, we present a new classification of Ochnaceae s.l., with Medusagynoideae and Quiinoideae included as subfamilies and the former subfamilies Ochnoideae and Sauvagesioideae recognized at the rank of tribe. Our data support a monophyletic Ochneae, but Sauvagesieae in the traditional circumscription is paraphyletic because Testulea emerges as sister to the rest of Ochnoideae, and the next clade shows Luxemburgia+Philacra as sister group to the remaining Ochnoideae. To avoid paraphyly, we classify Luxemburgieae and Testuleeae as new tribes. The African genus Lophira, which has switched between subfamilies (here tribes) in past classifications, emerges as sister to all other Ochneae. Thus, endosperm-free seeds and ovules with partly to completely united integuments (resulting in an apparently single integument) are characters that unite all members of that tribe. The relationships within its largest clade, Ochnineae (former Ochneae), are poorly resolved, but former Ochninae (Brackenridgea, Ochna) are polyphyletic. Within Sauvagesieae, the genus Sauvagesia in its broad circumscription is polyphyletic as Sauvagesia serrata is sister to a clade of Adenarake, Sauvagesia spp., and three other genera. Within Quiinoideae, in contrast to former phylogenetic hypotheses, Lacunaria and Touroulia form a clade that is sister to Quiina. Bayesian ancestral state reconstructions showed that zygomorphic flowers with adaptations to buzz-pollination (poricidal anthers), a syncarpous gynoecium (a near-apocarpous gynoecium evolved independently in Quiinoideae and Ochninae), numerous ovules, septicidal capsules, and winged seeds with endosperm are the ancestral condition in Ochnoideae. Although in some lineages poricidal anthers were lost secondarily, the evolution of poricidal superstructures secured the maintenance of buzz-pollination in some of these genera, indicating a strong selective pressure on keeping that specialized pollination system.

Molecular phylogenetics of Chaetodon and the Chaetodontidae (Teleostei : Perciformes) with reference to morphology

Butterflyfish are colourful, pan-tropical coastal fish that are important and distinctive members of coral reef communities. A successful systematic scheme and a robust phylogeny is considered essential in understanding further their biogeography and ecology, although recent cladistic treatments of butterflyfish phylogeny, based on soft tissue and bone morphology and coded at the generic and subgeneric levels, differ in character coding and subsequently tree topology. This study provides an independent test of the morphologically based hypotheses, using molecular systematic data from two partial mitochondrial gene fragments, cytochrome b (cytb) and small subunit rRNA (rrnS), for 52 ingroup chaetodontids and seven pomacanthids used to root the molecular trees. Individual gene trees were largely compatible and a combined molecular phylogeny, inferred from Bayesian analysis, was used to test alternative hypotheses suggested by morphological analyses. The tree was also used to map the latest morphological matrix in order to evaluate potential synapomorphies for various nodes defining butterflyfish interrelationships. A clade comprised of Chelmon and Coradion was sister group to other chaetodontids. Heniochus and Hemitaurichthys were each resolved as monophyletic groups, and as sister taxa Of the taxa sampled, Prognothodes was resolved as the sister genus to Chaeotodon. Of the ten Chaetodon subgenera sampled, all were monophyletic but their interrelationships differed significantly from that inferred from morphological characters. Lepidochaetodon was the most basal subgenus followed by Exornator and the remaining subgenera. Molecular data support the sister group relationship between Corallochaetodon and Citharoedus suggested by morphology, but major differences occur among the remaining more derived taxa. Chaetodon trifascialis and C. oligacanthus were resolved as sister taxa adding weight to the inclusion of the latter in C. Megaprotodon. Of those pairs of taxa known to hybridize and sampled with molecular data, all were closely related phylogenetically, except those hybrids known to occur in the Rabdophorus subgenus. Two base changes separated C. pelewensis from C. paucifasciatus which have been regarded previously as a single species. Cytb provided greater resolution than rrnS and will likely provide additional resolution with greater taxon sampling.

Phylogenetics of Olea (Oleaceae) based on plastid and nuclear ribosomal DNA sequences: tertiary climatic shifts and lineage differentiation times.

BACKGROUND AND AIMS: The genus Olea (Oleaceae) includes approx. 40 taxa of evergreen shrubs and trees classified in three subgenera, Olea, Paniculatae and Tetrapilus, the first of which has two sections (Olea and Ligustroides). Olive trees (the O. europaea complex) have been the subject of intensive research, whereas little is known about the phylogenetic relationships among the other species. To clarify the biogeographical history of this group, a molecular analysis of Olea and related genera of Oleaceae is thus necessary. METHODS: A phylogeny was built of Olea and related genera based on sequences of the nuclear ribosomal internal transcribed spacer-1 and four plastid regions. Lineage divergence and the evolution of abaxial peltate scales, the latter character linked to drought adaptation, were dated using a Bayesian method. KEY RESULTS: Olea is polyphyletic, with O. ambrensis and subgenus Tetrapilus not sharing a most recent common ancestor with the main Olea clade. Partial incongruence between nuclear and plastid phylogenetic reconstructions suggests a reticulation process in the evolution of subgenus Olea. Estimates of divergence times for major groups of Olea during the Tertiary were obtained. CONCLUSIONS: This study indicates the necessity of revising current taxonomic boundaries in Olea. The results also suggest that main lines of evolution were promoted by major Tertiary climatic shifts: (1) the split between subgenera Olea and Paniculatae appears to have taken place at the Miocene-Oligocene boundary; (2) the separation of sections Ligustroides and Olea may have occurred during the Early Miocene following the Mi-1 glaciation; and (3) the diversification within these sections (and the origin of dense abaxial indumentum in section Olea) was concomitant with the aridification of Africa in the Late Miocene.

Friends or relatives? Phylogenetics and species delimitation in the controversial European orchid Genus ophrys

Background and Aims Highly variable, yet possibly convergent, morphology and lack of sequence variation have severely hindered production of a robust phylogenetic framework for the genus Ophrys. The aim of this study is to produce this framework as a basis for more rigorous species delimitation and conservation recommendations. Methods Nuclear and plastid DNA sequencing and amplified fragment length polymorphism (AFLP) were performed on 85 accessions of Ophrys, spanning the full range of species aggregates currently recognized. Data were analysed using a combination of parsimony and Bayesian tree-building techniques and by principal coordinates analysis. Key Results Complementary phylogenetic analyses and ordinations using nuclear, plastid and AFLP datasets identify ten genetically distinct groups (six robust) within the genus that may in turn be grouped into three sections (treated as subgenera by some authors). Additionally, genetic evidence is provided for a close relationship between the O. tenthredinifera, O. bombyliflora and O. speculum groups. The combination of these analytical techniques provides new insights into Ophrys systematics, notably recognition of the novel O. umbilicata group. Conclusions Heterogeneous copies of the nuclear ITS region show that some putative Ophrys species arose through hybridization rather than divergent speciation. The supposedly highly specific pseudocopulatory pollination syndrome of Ophrys is demonstrably 'leaky', suggesting that the genus has been substantially over-divided at the species level.

Phyloclimatic modeling: Combining phylogenetics and bioclimatic modeling

We investigate the impact of past climates on plant diversification by tracking the "footprint" of climate change on a phylogenetic tree. Diversity within the cosmopolitan carnivorous plant genus Drosera (Droseraceae) is focused within Mediterranean climate regions. We explore whether this diversity is temporally linked to Mediterranean-type climatic shifts of the mid-Miocene and whether climate preferences are conservative over phylogenetic timescales. Phyloclimatic modeling combines environmental niche (bioclimatic) modeling with phylogenetics in order to study evolutionary patterns in relation to climate change. We present the largest and most complete such example to date using Drosera. The bioclimatic models of extant species demonstrate clear phylogenetic patterns; this is particularly evident for the tuberous sundews from southwestern Australia (subgenus Ergaleium). We employ a method for establishing confidence intervals of node ages on a phylogeny using replicates from a Bayesian phylogenetic analysis. This chronogram shows that many clades, including subgenus Ergaleium and section Bryastrum, diversified during the establishment of the Mediterranean-type climate. Ancestral reconstructions of bioclimatic models demonstrate a pattern of preference for this climate type within these groups. Ancestral bioclimatic models are projected into palaeo-climate reconstructions for the time periods indicated by the chronogram. We present two such examples that each generate plausible estimates of ancestral lineage distribution, which are similar to their current distributions. This is the first study to attempt bioclimatic projections on evolutionary time scales. The sundews appear to have diversified in response to local climate development. Some groups are specialized for Mediterranean climates, others show wide-ranging generalism. This demonstrates that Phyloclimatic modeling could be repeated for other plant groups and is fundamental to the understanding of evolutionary responses to climate change.

Mitochondrial sequence data and Dipsacales phylogeny: Mixed models, partitioned Bayesian analyses, and model selection

Phylogenetic analyses of chloroplast DNA sequences, morphology, and combined data have provided consistent support for many of the major branches within the angiosperm, clade Dipsacales. Here we use sequences from three mitochondrial loci to test the existing broad scale phylogeny and in an attempt to resolve several relationships that have remained uncertain. Parsimony, maximum likelihood, and Bayesian analyses of a combined mitochondrial data set recover trees broadly consistent with previous studies, although resolution and support are lower than in the largest chloroplast analyses. Combining chloroplast and mitochondrial data results in a generally well-resolved and very strongly supported topology but the previously recognized problem areas remain. To investigate why these relationships have been difficult to resolve we conducted a series of experiments using different data partitions and heterogeneous substitution models. Usually more complex modeling schemes are favored regardless of the partitions recognized but model choice had little effect on topology or support values. In contrast there are consistent but weakly supported differences in the topologies recovered from coding and non-coding matrices. These conflicts directly correspond to relationships that were poorly resolved in analyses of the full combined chloroplast-mitochondrial data set. We suggest incongruent signal has contributed to our inability to confidently resolve these problem areas. (c) 2007 Elsevier Inc. All rights reserved.

Evolutionary history of Ramphastos toucans: Molecular phylogenetics, temporal diversification, and biogeography

The toucan genus Ramphastos (Piciformes: Ramphastidae) has been a model in the formulation of Neotropical paleobiogeographic hypotheses. Weckstein (2005) reported on the phylogenetic history of this genus based on three mitochondrial genes, but some relationships were weakly supported and one of the subspecies of R. vitellinus (citreolaemus) was unsampled. This study expands on Weckstein (2005) by adding more DNA sequence data (including a nuclear marker) and more samples, including R v. citreolaemus. Maximum parsimony, maximum likelihood, and Bayesian methods recovered similar trees, with nodes showing high support. A monophyletic R. vitellinus complex was strongly supported as the sister-group to R. brevis. The results also confirmed that the southeastern and northern populations of R. vitellinus ariel are paraphyletic. X v. citreolaemus is sister to the Amazonian subspecies of the vitellinus complex. Using three protein-coding genes (COI, cytochrome-b and ND2) and interval-calibrated nodes under a Bayesian relaxed-clock framework, we infer that ramphastid genera originated in the middle Miocene to early Pliocene, Ramphastos species originated between late Miocene and early Pleistocene, and intra-specific divergences took place throughout the Pleistocene. Parsimony-based reconstruction of ancestral areas indicated that evolution of the four trans-Andean Ramphastos taxa (R. v. citreolaemus, R. a. swainsonii, R. brevis and R. sulfuratus) was associated with four independent dispersals from the cis-Andean region. The last pulse of Andean uplift may have been important for the evolution of R. sulfuratus, whereas the origin of the other trans-Andean Ramphastos taxa is consistent with vicariance due to drying events in the lowland forests north of the Andes. Estimated rates of molecular evolution were higher than the ""standard"" bird rate of 2% substitutions/site/million years for two of the three genes analyzed (cytochrome-b and ND2). (C) 2009 Elsevier Inc. All rights reserved.

Phylogenetics, conservation, and historical biogeography of the West Indian members of the Adelieae (Euphorbiaceae)

The Caribbean Island Biodiversity Hotspot is the largest insular system of the New World and a priority for biodiversity conservation worldwide. The tribe Adeliae (Euphorbiaceae) has over 35 species endemic to this hotspot, representing one of the most extraordinary cases of speciation in the West Indies, involving taxa from Cuba, Hispaniola, Jamaica, and the Bahamas. These species form a monophyletic group and traditionally have been accommodated in two endemic genera: Lasiocroton and Leucocroton. A study based on: (1) scanning electron microscopy of pollen and trichomes, (2) macromorphology, and (3) molecular data, was conducted to reveal generic relationships within this group. Phylogenies were based on parsimony and Bayesian analyses of nucleotide sequences of the ITS regions of the nuclear ribosomal DNA and the non-coding chloroplast DNA spacers psbM-trnD and ycf6-pcbM. One species, Lasiocroton trelawniensis, was transferred from the tribe into the genus Bernardia. Of the remaining species, three major monophyletic assemblages were revealed, one was restricted to limestone ares of Hispaniola and was sister to a clade with two monophyletic genera, Lasiocroton and Leucocroton. Morphological, biogeographical, and ecological data provided additional support for each of these three monophyletic assemblages. The Hispaniolan taxa were accommodated in a new genus with four species: Garciadelia. Leucocroton includes the nickel hyperaccumulating species from serpentine soils of Cuba, while the rest of the species were placed in Lasiocroton, a genus restricted to limestone areas. The geographic history of the islands as well as the phylogenetic placement of the Leucocroton-alliance, allows the research to include the historical biogeography of the alliance across the islands of the Caribbean based on a dispersal-vicariance analysis. The alliance arose on Eastern Cuba and Hispaniola, with Lasiocroton and Leucocroton diverging on Eastern Cuba according to soil type. Within Leucocroton, the analysis shows two migrations across the serpentine soils of Cuba. Additional morphological, ecological, and phylogenetic analyses support four new species in Cuba (Lasiocroton gutierrezii) and Hispaniola ( Garciadelia abbottii, G. castilloae, and G. mejiae). ^

The full Bayesian significance test for mixture models: results in gene expression clustering

Gene clustering is a useful exploratory technique to group together genes with similar expression levels under distinct cell cycle phases or distinct conditions. It helps the biologist to identify potentially meaningful relationships between genes. In this study, we propose a clustering method based on multivariate normal mixture models, where the number of clusters is predicted via sequential hypothesis tests: at each step, the method considers a mixture model of m components (m = 2 in the first step) and tests if in fact it should be m - 1. If the hypothesis is rejected, m is increased and a new test is carried out. The method continues (increasing m) until the hypothesis is accepted. The theoretical core of the method is the full Bayesian significance test, an intuitive Bayesian approach, which needs no model complexity penalization nor positive probabilities for sharp hypotheses. Numerical experiments were based on a cDNA microarray dataset consisting of expression levels of 205 genes belonging to four functional categories, for 10 distinct strains of Saccharomyces cerevisiae. To analyze the method's sensitivity to data dimension, we performed principal components analysis on the original dataset and predicted the number of classes using 2 to 10 principal components. Compared to Mclust (model-based clustering), our method shows more consistent results.

Disequilibrium Coefficient: A Bayesian Perspective

Hardy-Weinberg Equilibrium (HWE) is an important genetic property that populations should have whenever they are not observing adverse situations as complete lack of panmixia, excess of mutations, excess of selection pressure, etc. HWE for decades has been evaluated; both frequentist and Bayesian methods are in use today. While historically the HWE formula was developed to examine the transmission of alleles in a population from one generation to the next, use of HWE concepts has expanded in human diseases studies to detect genotyping error and disease susceptibility (association); Ryckman and Williams (2008). Most analyses focus on trying to answer the question of whether a population is in HWE. They do not try to quantify how far from the equilibrium the population is. In this paper, we propose the use of a simple disequilibrium coefficient to a locus with two alleles. Based on the posterior density of this disequilibrium coefficient, we show how one can conduct a Bayesian analysis to verify how far from HWE a population is. There are other coefficients introduced in the literature and the advantage of the one introduced in this paper is the fact that, just like the standard correlation coefficients, its range is bounded and it is symmetric around zero (equilibrium) when comparing the positive and the negative values. To test the hypothesis of equilibrium, we use a simple Bayesian significance test, the Full Bayesian Significance Test (FBST); see Pereira, Stern andWechsler (2008) for a complete review. The disequilibrium coefficient proposed provides an easy and efficient way to make the analyses, especially if one uses Bayesian statistics. A routine in R programs (R Development Core Team, 2009) that implements the calculations is provided for the readers.