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.

Bayesian generalized linear models for meta-analysis of diagnostic tests

With the recognition of the importance of evidence-based medicine, there is an emerging need for methods to systematically synthesize available data. Specifically, methods to provide accurate estimates of test characteristics for diagnostic tests are needed to help physicians make better clinical decisions. To provide more flexible approaches for meta-analysis of diagnostic tests, we developed three Bayesian generalized linear models. Two of these models, a bivariate normal and a binomial model, analyzed pairs of sensitivity and specificity values while incorporating the correlation between these two outcome variables. Noninformative independent uniform priors were used for the variance of sensitivity, specificity and correlation. We also applied an inverse Wishart prior to check the sensitivity of the results. The third model was a multinomial model where the test results were modeled as multinomial random variables. All three models can include specific imaging techniques as covariates in order to compare performance. Vague normal priors were assigned to the coefficients of the covariates. The computations were carried out using the 'Bayesian inference using Gibbs sampling' implementation of Markov chain Monte Carlo techniques. We investigated the properties of the three proposed models through extensive simulation studies. We also applied these models to a previously published meta-analysis dataset on cervical cancer as well as to an unpublished melanoma dataset. In general, our findings show that the point estimates of sensitivity and specificity were consistent among Bayesian and frequentist bivariate normal and binomial models. However, in the simulation studies, the estimates of the correlation coefficient from Bayesian bivariate models are not as good as those obtained from frequentist estimation regardless of which prior distribution was used for the covariance matrix. The Bayesian multinomial model consistently underestimated the sensitivity and specificity regardless of the sample size and correlation coefficient. In conclusion, the Bayesian bivariate binomial model provides the most flexible framework for future applications because of its following strengths: (1) it facilitates direct comparison between different tests; (2) it captures the variability in both sensitivity and specificity simultaneously as well as the intercorrelation between the two; and (3) it can be directly applied to sparse data without ad hoc correction. ^

A multivariate frailty model for disease recurrences and survival

A multivariate frailty hazard model is developed for joint-modeling of three correlated time-to-event outcomes: (1) local recurrence, (2) distant recurrence, and (3) overall survival. The term frailty is introduced to model population heterogeneity. The dependence is modeled by conditioning on a shared frailty that is included in the three hazard functions. Independent variables can be included in the model as covariates. The Markov chain Monte Carlo methods are used to estimate the posterior distributions of model parameters. The algorithm used in present application is the hybrid Metropolis-Hastings algorithm, which simultaneously updates all parameters with evaluations of gradient of log posterior density. The performance of this approach is examined based on simulation studies using Exponential and Weibull distributions. We apply the proposed methods to a study of patients with soft tissue sarcoma, which motivated this research. Our results indicate that patients with chemotherapy had better overall survival with hazard ratio of 0.242 (95% CI: 0.094 - 0.564) and lower risk of distant recurrence with hazard ratio of 0.636 (95% CI: 0.487 - 0.860), but not significantly better in local recurrence with hazard ratio of 0.799 (95% CI: 0.575 - 1.054). The advantages and limitations of the proposed models, and future research directions are discussed. ^

A Bayesian analysis for spatial processes with application to mapping

In geographical epidemiology, maps of disease rates and disease risk provide a spatial perspective for researching disease etiology. For rare diseases or when the population base is small, the rate and risk estimates may be unstable. Empirical Bayesian (EB) methods have been used to spatially smooth the estimates by permitting an area estimate to "borrow strength" from its neighbors. Such EB methods include the use of a Gamma model, of a James-Stein estimator, and of a conditional autoregressive (CAR) process. A fully Bayesian analysis of the CAR process is proposed. One advantage of this fully Bayesian analysis is that it can be implemented simply by using repeated sampling from the posterior densities. Use of a Markov chain Monte Carlo technique such as Gibbs sampler was not necessary. Direct resampling from the posterior densities provides exact small sample inferences instead of the approximate asymptotic analyses of maximum likelihood methods (Clayton & Kaldor, 1987). Further, the proposed CAR model provides for covariates to be included in the model. A simulation demonstrates the effect of sample size on the fully Bayesian analysis of the CAR process. The methods are applied to lip cancer data from Scotland, and the results are compared. ^

Bayesian and survival model for tumor relapse status and disease-specific survival, with applications to breast cancer

Breast cancer is the most common non-skin cancer and the second leading cause of cancer-related death in women in the United States. Studies on ipsilateral breast tumor relapse (IBTR) status and disease-specific survival will help guide clinic treatment and predict patient prognosis.^ After breast conservation therapy, patients with breast cancer may experience breast tumor relapse. This relapse is classified into two distinct types: true local recurrence (TR) and new ipsilateral primary tumor (NP). However, the methods used to classify the relapse types are imperfect and are prone to misclassification. In addition, some observed survival data (e.g., time to relapse and time from relapse to death)are strongly correlated with relapse types. The first part of this dissertation presents a Bayesian approach to (1) modeling the potentially misclassified relapse status and the correlated survival information, (2) estimating the sensitivity and specificity of the diagnostic methods, and (3) quantify the covariate effects on event probabilities. A shared frailty was used to account for the within-subject correlation between survival times. The inference was conducted using a Bayesian framework via Markov Chain Monte Carlo simulation implemented in softwareWinBUGS. Simulation was used to validate the Bayesian method and assess its frequentist properties. The new model has two important innovations: (1) it utilizes the additional survival times correlated with the relapse status to improve the parameter estimation, and (2) it provides tools to address the correlation between the two diagnostic methods conditional to the true relapse types.^ Prediction of patients at highest risk for IBTR after local excision of ductal carcinoma in situ (DCIS) remains a clinical concern. The goals of the second part of this dissertation were to evaluate a published nomogram from Memorial Sloan-Kettering Cancer Center, to determine the risk of IBTR in patients with DCIS treated with local excision, and to determine whether there is a subset of patients at low risk of IBTR. Patients who had undergone local excision from 1990 through 2007 at MD Anderson Cancer Center with a final diagnosis of DCIS (n=794) were included in this part. Clinicopathologic factors and the performance of the Memorial Sloan-Kettering Cancer Center nomogram for prediction of IBTR were assessed for 734 patients with complete data. Nomogram for prediction of 5- and 10-year IBTR probabilities were found to demonstrate imperfect calibration and discrimination, with an area under the receiver operating characteristic curve of .63 and a concordance index of .63. In conclusion, predictive models for IBTR in DCIS patients treated with local excision are imperfect. Our current ability to accurately predict recurrence based on clinical parameters is limited.^ The American Joint Committee on Cancer (AJCC) staging of breast cancer is widely used to determine prognosis, yet survival within each AJCC stage shows wide variation and remains unpredictable. For the third part of this dissertation, biologic markers were hypothesized to be responsible for some of this variation, and the addition of biologic markers to current AJCC staging were examined for possibly provide improved prognostication. The initial cohort included patients treated with surgery as first intervention at MDACC from 1997 to 2006. Cox proportional hazards models were used to create prognostic scoring systems. AJCC pathologic staging parameters and biologic tumor markers were investigated to devise the scoring systems. Surveillance Epidemiology and End Results (SEER) data was used as the external cohort to validate the scoring systems. Binary indicators for pathologic stage (PS), estrogen receptor status (E), and tumor grade (G) were summed to create PS+EG scoring systems devised to predict 5-year patient outcomes. These scoring systems facilitated separation of the study population into more refined subgroups than the current AJCC staging system. The ability of the PS+EG score to stratify outcomes was confirmed in both internal and external validation cohorts. The current study proposes and validates a new staging system by incorporating tumor grade and ER status into current AJCC staging. We recommend that biologic markers be incorporating into revised versions of the AJCC staging system for patients receiving surgery as the first intervention.^ Chapter 1 focuses on developing a Bayesian method to solve misclassified relapse status and application to breast cancer data. Chapter 2 focuses on evaluation of a breast cancer nomogram for predicting risk of IBTR in patients with DCIS after local excision gives the statement of the problem in the clinical research. Chapter 3 focuses on validation of a novel staging system for disease-specific survival in patients with breast cancer treated with surgery as the first intervention. ^

Fully adaptive gaussian mixture metropolis-hastings algorithm

Markov Chain Monte Carlo methods are widely used in signal processing and communications for statistical inference and stochastic optimization. In this work, we introduce an efficient adaptive Metropolis-Hastings algorithm to draw samples from generic multimodal and multidimensional target distributions. The proposal density is a mixture of Gaussian densities with all parameters (weights, mean vectors and covariance matrices) updated using all the previously generated samples applying simple recursive rules. Numerical results for the one and two-dimensional cases are provided.

Orthogonal MCMC algorithms

Monte Carlo (MC) methods are widely used in signal processing, machine learning and stochastic optimization. A well-known class of MC methods are Markov Chain Monte Carlo (MCMC) algorithms. In this work, we introduce a novel parallel interacting MCMC scheme, where the parallel chains share information using another MCMC technique working on the entire population of current states. These parallel ?vertical? chains are led by random-walk proposals, whereas the ?horizontal? MCMC uses a independent proposal, which can be easily adapted by making use of all the generated samples. Numerical results show the advantages of the proposed sampling scheme in terms of mean absolute error, as well as robustness w.r.t. to initial values and parameter choice.

Maximum likelihood estimation of a migration matrix and effective population sizes in n subpopulations by using a coalescent approach

A maximum likelihood estimator based on the coalescent for unequal migration rates and different subpopulation sizes is developed. The method uses a Markov chain Monte Carlo approach to investigate possible genealogies with branch lengths and with migration events. Properties of the new method are shown by using simulated data from a four-population n-island model and a source–sink population model. Our estimation method as coded in migrate is tested against genetree; both programs deliver a very similar likelihood surface. The algorithm converges to the estimates fairly quickly, even when the Markov chain is started from unfavorable parameters. The method was used to estimate gene flow in the Nile valley by using mtDNA data from three human populations.

Magnetic activity and differential rotation in the young Sun-like stars KIC 7985370 and KIC 7765135

Aims. We present a detailed study of the two Sun-like stars KIC 7985370 and KIC 7765135, to determine their activity level, spot distribution, and differential rotation. Both stars were previously discovered by us to be young stars and were observed by the NASA Kepler mission. Methods. The fundamental stellar parameters (vsini, spectral type, T_eff, log g, and [Fe/H]) were derived from optical spectroscopy by comparison with both standard-star and synthetic spectra. The spectra of the targets allowed us to study the chromospheric activity based on the emission in the core of hydrogen Hα and Ca ii infrared triplet (IRT) lines, which was revealed by the subtraction of inactive templates. The high-precision Kepler photometric data spanning over 229 days were then fitted with a robust spot model. Model selection and parameter estimation were performed in a Bayesian manner, using a Markov chain Monte Carlo method. Results. We find that both stars are Sun-like (of G1.5 V spectral type) and have an age of about 100–200 Myr, based on their lithium content and kinematics. Their youth is confirmed by their high level of chromospheric activity, which is comparable to that displayed by the early G-type stars in the Pleiades cluster. The Balmer decrement and flux ratio of their Ca ii-IRT lines suggest that the formation of the core of these lines occurs mainly in optically thick regions that are analogous to solar plages. The spot model applied to the Kepler photometry requires at least seven persistent spots in the case of KIC 7985370 and nine spots in the case of KIC 7765135 to provide a satisfactory fit to the data. The assumption of the longevity of the star spots, whose area is allowed to evolve with time, is at the heart of our spot-modelling approach. On both stars, the surface differential rotation is Sun-like, with the high-latitude spots rotating slower than the low-latitude ones. We found, for both stars, a rather high value of the equator-to-pole differential rotation (dΩ ≈ 0.18 rad d^-1), which disagrees with the predictions of some mean-field models of differential rotation for rapidly rotating stars. Our results agree instead with previous works on solar-type stars and other models that predict a higher latitudinal shear, increasing with equatorial angular velocity, that can vary during the magnetic cycle.

Modelling of the surface emission of the low magnetic field magnetar SGR 0418+5729

We perform a detailed modelling of the post-outburst surface emission of the low magnetic field magnetar SGR 0418+5729. The dipolar magnetic field of this source, B=6×1012G estimated from its spin-down rate, is in the observed range of magnetic fields for normal pulsars. The source is further characterized by a high pulse fraction and a single-peak profile. Using synthetic temperature distribution profiles, and fully accounting for the general-relativistic effects of light deflection and gravitational redshift, we generate synthetic X-ray spectra and pulse profiles that we fit to the observations. We find that asymmetric and symmetric surface temperature distributions can reproduce equally well the observed pulse profiles and spectra of SGR 0418. None the less, the modelling allows us to place constraints on the system geometry (i.e. the angles ψ and ξ that the rotation axis makes with the line of sight and the dipolar axis, respectively), as well as on the spot size and temperature contrast on the neutron star surface. After performing an analysis iterating between the pulse profile and spectra, as done in similar previous works, we further employed, for the first time in this context, a Markov-Chain Monte Carlo approach to extract constraints on the model parameters from the pulse profiles and spectra, simultaneously. We find that, to reproduce the observed spectrum and flux modulation: (a) the angles must be restricted to 65° ≲ ψ + ξ ≲ 125° or 235° ≲ ψ + ξ ≲ 295°; (b) the temperature contrast between the poles and the equator must be at least a factor of ∼6, and (c) the size of the hottest region ranges between 0.2 and 0.7 km (including uncertainties on the source distance). Lastly, we interpret our findings within the context of internal and external heating models.

Analysis of Melanoma Onset: Assessing Familial Aggregation by Using Estimating Equations and Fitting Variance Components via Bayesian Random Effects Models

We investigate whether relative contributions of genetic and shared environmental factors are associated with an increased risk in melanoma. Data from the Queensland Familial Melanoma Project comprising 15,907 subjects arising from 1912 families were analyzed to estimate the additive genetic, common and unique environmental contributions to variation in the age at onset of melanoma. Two complementary approaches for analyzing correlated time-to-onset family data were considered: the generalized estimating equations (GEE) method in which one can estimate relationship-specific dependence simultaneously with regression coefficients that describe the average population response to changing covariates; and a subject-specific Bayesian mixed model in which heterogeneity in regression parameters is explicitly modeled and the different components of variation may be estimated directly. The proportional hazards and Weibull models were utilized, as both produce natural frameworks for estimating relative risks while adjusting for simultaneous effects of other covariates. A simple Markov Chain Monte Carlo method for covariate imputation of missing data was used and the actual implementation of the Bayesian model was based on Gibbs sampling using the free ware package BUGS. In addition, we also used a Bayesian model to investigate the relative contribution of genetic and environmental effects on the expression of naevi and freckles, which are known risk factors for melanoma.

Phylogeny and biogeography of the freshwater crayfish Euastacus (Decapoda : Parastacidae) based on nuclear and mitochondrial DNA

Euastacus crayfish are endemic to freshwater ecosystems of the eastern coast of Australia. While recent evolutionary studies have focused on a few of these species, here we provide a comprehensive phylogenetic estimate of relationships among the species within the genus. We sequenced three mitochondrial gene regions (COI, 16S, and 12S) and one nuclear region (28S) from 40 species of the genus Euastacus, as well as one undescribed species. Using these data, we estimated the phylogenetic relationships within the genus using maximum-likelihood, parsimony, and Bayesian Markov Chain Monte Carlo analyses. Using Bayes factors to test different model hypotheses, we found that the best phylogeny supports monophyletic groupings of all but two recognized species and suggests a widespread ancestor that diverged by vicariance. We also show that Eitastacus and Astacopsis are most likely monophyletic sister genera. We use the resulting phylogeny as a framework to test biogeographic hypotheses relating to the diversification of the genus. (c) 2005 Elsevier Inc. All rights reserved.

Assessing the impacts of grazing levels on bird density in woodland habitat: a Bayesian approach using expert opinion

Many studies on birds focus on the collection of data through an experimental design, suitable for investigation in a classical analysis of variance (ANOVA) framework. Although many findings are confirmed by one or more experts, expert information is rarely used in conjunction with the survey data to enhance the explanatory and predictive power of the model. We explore this neglected aspect of ecological modelling through a study on Australian woodland birds, focusing on the potential impact of different intensities of commercial cattle grazing on bird density in woodland habitat. We examine a number of Bayesian hierarchical random effects models, which cater for overdispersion and a high frequency of zeros in the data using WinBUGS and explore the variation between and within different grazing regimes and species. The impact and value of expert information is investigated through the inclusion of priors that reflect the experience of 20 experts in the field of bird responses to disturbance. Results indicate that expert information moderates the survey data, especially in situations where there are little or no data. When experts agreed, credible intervals for predictions were tightened considerably. When experts failed to agree, results were similar to those evaluated in the absence of expert information. Overall, we found that without expert opinion our knowledge was quite weak. The fact that the survey data is quite consistent, in general, with expert opinion shows that we do know something about birds and grazing and we could learn a lot faster if we used this approach more in ecology, where data are scarce. Copyright (c) 2005 John Wiley & Sons, Ltd.

A Bayesian approach to imposing curvature on distance functions

The estimated parameters of output distance functions frequently violate the monotonicity, quasi-convexity and convexity constraints implied by economic theory, leading to estimated elasticities and shadow prices that are incorrectly signed, and ultimately to perverse conclusions concerning the effects of input and output changes on productivity growth and relative efficiency levels. We show how a Bayesian approach can be used to impose these constraints on the parameters of a translog output distance function. Implementing the approach involves the use of a Gibbs sampler with data augmentation. A Metropolis-Hastings algorithm is also used within the Gibbs to simulate observations from truncated pdfs. Our methods are developed for the case where panel data is available and technical inefficiency effects are assumed to be time-invariant. Two models-a fixed effects model and a random effects model-are developed and applied to panel data on 17 European railways. We observe significant changes in estimated elasticities and shadow price ratios when regularity restrictions are imposed. (c) 2004 Elsevier B.V. All rights reserved.

Segmenting eukaryotic genomes with the generalized Gibbs sampler

Eukaryotic genomes display segmental patterns of variation in various properties, including GC content and degree of evolutionary conservation. DNA segmentation algorithms are aimed at identifying statistically significant boundaries between such segments. Such algorithms may provide a means of discovering new classes of functional elements in eukaryotic genomes. This paper presents a model and an algorithm for Bayesian DNA segmentation and considers the feasibility of using it to segment whole eukaryotic genomes. The algorithm is tested on a range of simulated and real DNA sequences, and the following conclusions are drawn. Firstly, the algorithm correctly identifies non-segmented sequence, and can thus be used to reject the null hypothesis of uniformity in the property of interest. Secondly, estimates of the number and locations of change-points produced by the algorithm are robust to variations in algorithm parameters and initial starting conditions and correspond to real features in the data. Thirdly, the algorithm is successfully used to segment human chromosome 1 according to GC content, thus demonstrating the feasibility of Bayesian segmentation of eukaryotic genomes. The software described in this paper is available from the author's website (www.uq.edu.au/similar to uqjkeith/) or upon request to the author.