Markov chain Monte Carlo segregation and linkage analysis for oligogenic models.

A new method for segregation and linkage analysis, with pedigree data, is described. Reversible jump Markov chain Monte Carlo methods are used to implement a sampling scheme in which the Markov chain can jump between parameter subspaces corresponding to models with different numbers of quantitative-trait loci (QTL's). Joint estimation of QTL number, position, and effects is possible, avoiding the problems that can arise from misspecification of the number of QTL's in a linkage analysis. The method is illustrated by use of a data set simulated for the 9th Genetic Analysis Workshop; this data set had several oligogenic traits, generated by use of a 1,497-member pedigree. The mixing characteristics of the method appear to be good, and the method correctly recovers the simulated model from the test data set. The approach appears to have great potential both for robust linkage analysis and for the answering of more general questions regarding the genetic control of complex traits.     

Bayesian phylogenetic inference via Markov chain Monte Carlo methods

We derive a Markov chain to sample from the posterior distribution for a phylogenetic tree given sequence information from the corresponding set of organisms, a stochastic model for these data, and a prior distribution on the space of trees. A transformation of the tree into a canonical cophenetic matrix form suggests a simple and effective proposal distribution for selecting candidate trees close to the current tree in the chain. We illustrate the algorithm with restriction site data on 9 plant species, then extend to DNA sequences from 32 species of fish. The algorithm mixes well in both examples from random starting trees, generating reproducible estimates and credible sets for the path of evolution.     

A Monte Carlo method for combined segregation and linkage analysis.

We introduce a Monte Carlo approach to combined segregation and linkage analysis of a quantitative trait observed in an extended pedigree. In conjunction with the Monte Carlo method of likelihood-ratio evaluation proposed by Thompson and Guo, the method provides for estimation and hypothesis testing. The greatest attraction of this approach is its ability to handle complex genetic models and large pedigrees. Two examples illustrate the practicality of the method. One is of simulated data on a large pedigree; the other is a reanalysis of published data previously analyzed by other methods.     

Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models

Inference for Dirichlet process hierarchical models is typicallyperformed using Markov chain Monte Carlo methods, which canbe roughly categorized into marginal and conditional methods.The former integrate out analytically the infinite-dimensionalcomponent of the hierarchical model and sample from the marginaldistribution of the remaining variables using the Gibbs sampler.Conditional methods impute the Dirichlet process and updateit as a component of the Gibbs sampler. Since this requiresimputation of an infinite-dimensional process, implementationof the conditional method has relied on finite approximations.In this paper, we show how to avoid such approximations by designingtwo novel Markov chain Monte Carlo algorithms which sample fromthe exact posterior distribution of quantities of interest.The approximations are avoided by the new technique of retrospectivesampling. We also show how the algorithms can obtain samplesfrom functionals of the Dirichlet process. The marginal andthe conditional methods are compared and a careful simulationstudy is included, which involves a non-conjugate model, differentdatasets and prior specifications.     

Bayesian phylogenetic model selection using reversible jump Markov chain Monte Carlo

A common problem in molecular phylogenetics is choosing a model of DNA substitution that does a good job of explaining the DNA sequence alignment without introducing superfluous parameters. A number of methods have been used to choose among a small set of candidate substitution models, such as the likelihood ratio test, the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), and Bayes factors. Current implementations of any of these criteria suffer from the limitation that only a small set of models are examined, or that the test does not allow easy comparison of non-nested models. In this article, we expand the pool of candidate substitution models to include all possible time-reversible models. This set includes seven models that have already been described. We show how Bayes factors can be calculated for these models using reversible jump Markov chain Monte Carlo, and apply the method to 16 DNA sequence alignments. For each data set, we compare the model with the best Bayes factor to the best models chosen using AIC and BIC. We find that the best model under any of these criteria is not necessarily the most complicated one; models with an intermediate number of substitution types typically do best. Moreover, almost all of the models that are chosen as best do not constrain a transition rate to be the same as a transversion rate, suggesting that it is the transition/transversion rate bias that plays the largest role in determining which models are selected. Importantly, the reversible jump Markov chain Monte Carlo algorithm described here allows estimation of phylogeny (and other phylogenetic model parameters) to be performed while accounting for uncertainty in the model of DNA substitution.     

Bayesian adaptive Markov chain Monte Carlo estimation of genetic parameters

Accurate and fast estimation of genetic parameters that underlie quantitative traits using mixed linear models with additive and dominance effects is of great importance in both natural and breeding populations. Here, we propose a new fast adaptive Markov chain Monte Carlo (MCMC) sampling algorithm for the estimation of genetic parameters in the linear mixed model with several random effects. In the learning phase of our algorithm, we use the hybrid Gibbs sampler to learn the covariance structure of the variance components. In the second phase of the algorithm, we use this covariance structure to formulate an effective proposal distribution for a Metropolis-Hastings algorithm, which uses a likelihood function in which the random effects have been integrated out. Compared with the hybrid Gibbs sampler, the new algorithm had better mixing properties and was approximately twice as fast to run. Our new algorithm was able to detect different modes in the posterior distribution. In addition, the posterior mode estimates from the adaptive MCMC method were close to the REML (residual maximum likelihood) estimates. Moreover, our exponential prior for inverse variance components was vague and enabled the estimated mode of the posterior variance to be practically zero, which was in agreement with the support from the likelihood (in the case of no dominance). The method performance is illustrated using simulated data sets with replicates and field data in barley.     

A tutorial introduction to Bayesian inference for stochastic epidemic models using Markov chain Monte Carlo methods

Recent Bayesian methods for the analysis of infectious disease outbreak data using stochastic epidemic models are reviewed. These methods rely on Markov chain Monte Carlo methods. Both temporal and non-temporal data are considered. The methods are illustrated with a number of examples featuring different models and datasets.     

Reversible jump Markov chain Monte Carlo computation and Bayesian model determination

Markov chain Monte Carlo methods for Bayesian computation haveuntil recently been restricted to problems where the joint distributionof all variables has a density with respect to some fixed standardunderlying measure. They have therefore not been available forapplication to Bayesian model determination, where the dimensionalityof the parameter vector is typically not fixed. This paper proposesa new framework for the construction of reversible Markov chainsamplers that jump between parameter subspaces of differingdimensionality, which is flexible and entirely constructive.It should therefore have wide applicability in model determinationproblems. The methodology is illustrated with applications tomultiple change-point analysis in one and two dimensions, andto a Bayesian comparison of binomial experiments.     

Study on mapping Quantitative Trait Loci for animal complex binary traits using Bayesian-Markov chain Monte Carlo approach

It is a challenging issue to map Quantitative Trait Loci (QTL) underlying complex discrete traits, which usually show discontinuous distribution and less information, using conventional statistical methods. Bayesian-Markov chain Monte Carlo (Bayesian-MCMC) approach is the key procedure in mapping QTL for complex binary traits, which provides a complete posterior distribution for QTL parameters using all prior information. As a consequence, Bayesian estimates of all interested variables can be obtained straightforwardly basing on their posterior samples simulated by the MCMC algorithm. In our study, utilities of Bayesian-MCMC are demonstrated using simulated several animal outbred full-sib families with different family structures for a complex binary trait underlied by both a QTL and polygene. Under the Identity-by-Descent-Based variance component random model, three samplers basing on MCMC, including Gibbs sampling, Metropolis algorithm and reversible jump MCMC, were implemented to generate the joint posterior distribution of all unknowns so that the QTL parameters were obtained by Bayesian statistical inferring. The results showed that Bayesian-MCMC approach could work well and robust under different family structures and QTL effects. As family size increases and the number of family decreases, the accuracy of the parameter estimates will be improved. When the true QTL has a small effect, using outbred population experiment design with large family size is the optimal mapping strategy.     

Study on mapping Quantitative Trait Loci for animal complex binary traits using Bayesian-Markov chain Monte Carlo approach

It is a challenging issue to map Quantitative Trait Loci (QTL) underlying complex discrete traits,which usually show discontinuous distribution and less information,using conventional statisti-cal methods. Bayesian-Markov chain Monte Carlo (Bayesian-MCMC) approach is the key procedure in mapping QTL for complex binary traits,which provides a complete posterior distribution for QTL parameters using all prior information. As a consequence,Bayesian estimates of all interested vari-ables can be obtained straightforwardly basing on their posterior samples simulated by the MCMC algorithm. In our study,utilities of Bayesian-MCMC are demonstrated using simulated several ani-mal outbred full-sib families with different family structures for a complex binary trait underlied by both a QTL and polygene. Under the Identity-by-Descent-Based variance component random model,three samplers basing on MCMC,including Gibbs sampling,Metropolis algorithm and reversible jump MCMC,were implemented to generate the joint posterior distribution of all unknowns so that the QTL parameters were obtained by Bayesian statistical inferring. The results showed that Bayesian-MCMC approach could work well and robust under different family structures and QTL effects. As family size increases and the number of family decreases,the accuracy of the parameter estimates will be im-proved. When the true QTL has a small effect,using outbred population experiment design with large family size is the optimal mapping strategy.     

Comparison of marker types and map assumptions using Markov chain Monte Carlo-based linkage analysis of COGA data

We performed multipoint linkage analysis of the electrophysiological trait ECB21 on chromosome 4 in the full pedigrees provided by the Collaborative Study on the Genetics of Alcoholism (COGA). Three Markov chain Monte Carlo (MCMC)-based approaches were applied to the provided and re-estimated genetic maps and to five different marker panels consisting of microsatellite (STRP) and/or SNP markers at various densities. We found evidence of linkage near the GABRB1 STRP using all methods, maps, and marker panels. Difficulties encountered with SNP panels included convergence problems and demanding computations.     

Mapping multiple QTL using linkage disequilibrium and linkage analysis information and multitrait data

A multi-locus QTL mapping method is presented, which combines linkage and linkage disequilibrium (LD) information and uses multitrait data. The method assumed a putative QTL at the midpoint of each marker bracket. Whether the putative QTL had an effect or not was sampled using Markov chain Monte Carlo (MCMC) methods. The method was tested in dairy cattle data on chromosome 14 where the DGAT1 gene was known to be segregating. The DGAT1 gene was mapped to a region of 0.04 cM, and the effects of the gene were accurately estimated. The fitting of multiple QTL gave a much sharper indication of the QTL position than a single QTL model using multitrait data, probably because the multi-locus QTL mapping reduced the carry over effect of the large DGAT1 gene to adjacent putative QTL positions. This suggests that the method could detect secondary QTL that would, in single point analyses, remain hidden under the broad peak of the dominant QTL. However, no indications for a second QTL affecting dairy traits were found on chromosome 14.     

Data cloning: easy maximum likelihood estimation for complex ecological models using Bayesian Markov chain Monte Carlo methods

We introduce a new statistical computing method, called data cloning, to calculate maximum likelihood estimates and their standard errors for complex ecological models. Although the method uses the Bayesian framework and exploits the computational simplicity of the Markov chain Monte Carlo (MCMC) algorithms, it provides valid frequentist inferences such as the maximum likelihood estimates and their standard errors. The inferences are completely invariant to the choice of the prior distributions and therefore avoid the inherent subjectivity of the Bayesian approach. The data cloning method is easily implemented using standard MCMC software. Data cloning is particularly useful for analysing ecological situations in which hierarchical statistical models, such as state-space models and mixed effects models, are appropriate. We illustrate the method by fitting two nonlinear population dynamics models to data in the presence of process and observation noise.     

A study of deleterious gene structure in plants using Markov chain Monte Carlo

The characteristics of deleterious genes have been of great interest in both theory and practice in genetics. Because of the complex genetic mechanism of these deleterious genes, most current studies try to estimate the overall magnitude of mortality effects on a population, which is characterized classically by the number of lethal equivalents. This number is a combination of several parameters, each of which has a distinct biological effect on genetic mortality. In conservation and breeding programs, it is important to be able to distinguish among different combinations of these parameters that lead to the same number of lethal equivalents, such as a large number of mildly deleterious genes or a few lethal genes, The ability to distinguish such parameter combinations requires more than one generation of mating. We propose a model for survival data from a two-generation mating experiment on the plant species Brassica rapa, and we enable inference with Markov chain Monte Carlo. This computational strategy is effective because a vast amount of missing genotype information must be accounted for. In addition to the lethal equivalents, the two-generation data provide separate information on the average intensity of mortality and the average number of deleterious genes carried by an individual. In our Markov chain Monte Carlo algorithm, we use a vector proposal distribution to overcome inefficiency of a single-site Gibbs sampler. Information about environmental effects is obtained from an outcrossing experiment conducted in parallel with the two-generation mating experiments.     

Bayesian models and Markov chain Monte Carlo methods for protein motifs with the secondary characteristics.

Statistical methods have been developed for finding local patterns, also called motifs, in multiple protein sequences. The aligned segments may imply functional or structural core regions. However, the existing methods often have difficulties in aligning multiple proteins when sequence residue identities are low (e.g., less than 25%). In this article, we develop a Bayesian model and Markov chain Monte Carlo (MCMC) methods for identifying subtle motifs in protein sequences. Specifically, a motif is defined not only in terms of specific sites characterized by amino acid frequency vectors, but also as a combination of secondary characteristics such as hydrophobicity, polarity, etc. Markov chain Monte Carlo methods are proposed to search for a motif pattern with high posterior probability under the new model. A special MCMC algorithm is developed, involving transitions between state spaces of different dimensions. The proposed methods were supported by a simulated study. It was then tested by two real datasets, including a group of helix-turn-helix proteins, and one set from the CATH Protein Structure Classification Database. Statistical comparisons showed that the new approach worked better than a typical Gibbs sampling approach which is based only on an amino acid model.