Three-point appraisal of genetic linkage maps

This paper develops a simple diagnostic for the investigation of uncertainty within genetic linkage maps using a Bayesian procedure. The method requires only the genotyping data and the proposed genetic map, and calculates the posterior probability for the possible orders of any set of three markers, accounting for the presence of genotyping error (mistyping) and for missing genotype data. The method uses a Bayesian approach to give insight into conflicts between the order in the proposed map and the genotype scores. The method can also be used to assess the accuracy of a genetic map at different genomic scales and to assess alternative potential marker orders. Simulation and two case studies were used to illustrate the method. In the first case study, the diagnostic revealed conflicts in map ordering for short inter-marker distances that were resolved at a distance of 8–12?cM, except for a set of markers at the end of the linkage group. In the second case study, the ordering did not resolve as distances increase, which could be attributed to regions of the map where many individuals were untyped.     

Bayesian meta-analysis models for microarray data: a comparative study

Background

With the growing abundance of microarray data, statistical methods are increasingly needed to integrate results across studies. Two common approaches for meta-analysis of microarrays include either combining gene expression measures across studies or combining summaries such as p-values, probabilities or ranks. Here, we compare two Bayesian meta-analysis models that are analogous to these methods.

Results

Two Bayesian meta-analysis models for microarray data have recently been introduced. The first model combines standardized gene expression measures across studies into an overall mean, accounting for inter-study variability, while the second combines probabilities of differential expression without combining expression values. Both models produce the gene-specific posterior probability of differential expression, which is the basis for inference. Since the standardized expression integration model includes inter-study variability, it may improve accuracy of results versus the probability integration model. However, due to the small number of studies typical in microarray meta-analyses, the variability between studies is challenging to estimate. The probability integration model eliminates the need to model variability between studies, and thus its implementation is more straightforward. We found in simulations of two and five studies that combining probabilities outperformed combining standardized gene expression measures for three comparison values: the percent of true discovered genes in meta-analysis versus individual studies; the percent of true genes omitted in meta-analysis versus separate studies, and the number of true discovered genes for fixed levels of Bayesian false discovery. We identified similar results when pooling two independent studies of Bacillus subtilis. We assumed that each study was produced from the same microarray platform with only two conditions: a treatment and control, and that the data sets were pre-scaled.

Conclusion

The Bayesian meta-analysis model that combines probabilities across studies does not aggregate gene expression measures, thus an inter-study variability parameter is not included in the model. This results in a simpler modeling approach than aggregating expression measures, which accounts for variability across studies. The probability integration model identified more true discovered genes and fewer true omitted genes than combining expression measures, for our data sets.     

Data partitions and complex models in Bayesian analysis: the phylogeny of Gymnophthalmid lizards

Phylogenetic studies incorporating multiple loci, and multiple genomes, are becoming increasingly common. Coincident with this trend in genetic sampling, model-based likelihood techniques including Bayesian phylogenetic methods continue to gain popularity. Few studies, however, have examined model fit and sensitivity to such potentially heterogeneous data partitions within combined data analyses using empirical data. Here we investigate the relative model fit and sensitivity of Bayesian phylogenetic methods when alternative site-specific partitions of among-site rate variation (with and without autocorrelated rates) are considered. Our primary goal in choosing a best-fit model was to employ the simplest model that was a good fit to the data while optimizing topology and/or Bayesian posterior probabilities. Thus, we were not interested in complex models that did not practically affect our interpretation of the topology under study. We applied these alternative models to a four-gene data set including one protein-coding nuclear gene (c-mos), one protein-coding mitochondrial gene (ND4), and two mitochondrial rRNA genes (12S and 16S) for the diverse yet poorly known lizard family Gymnophthalmidae. Our results suggest that the best-fit model partitioned among-site rate variation separately among the c-mos, ND4, and 12S + 16S gene regions. We found this model yielded identical topologies to those from analyses based on the GTR+I+G model, but significantly changed posterior probability estimates of clade support. This partitioned model also produced more precise (less variable) estimates of posterior probabilities across generations of long Bayesian runs, compared to runs employing a GTR+I+G model estimated for the combined data. We use this three-way gamma partitioning in Bayesian analyses to reconstruct a robust phylogenetic hypothesis for the relationships of genera within the lizard family Gymnophthalmidae. We then reevaluate the higher-level taxonomic arrangement of the Gymnophthalmidae. Based on our findings, we discuss the utility of nontraditional parameters for modeling among-site rate variation and the implications and future directions for complex model building and testing.     

Branch-length prior influences Bayesian posterior probability of phylogeny

The Bayesian method for estimating species phylogenies from molecular sequence data provides an attractive alternative to maximum likelihood with nonparametric bootstrap due to the easy interpretation of posterior probabilities for trees and to availability of efficient computational algorithms. However, for many data sets it produces extremely high posterior probabilities, sometimes for apparently incorrect clades. Here we use both computer simulation and empirical data analysis to examine the effect of the prior model for internal branch lengths. We found that posterior probabilities for trees and clades are sensitive to the prior for internal branch lengths, and priors assuming long internal branches cause high posterior probabilities for trees. In particular, uniform priors with high upper bounds bias Bayesian clade probabilities in favor of extreme values. We discuss possible remedies to the problem, including empirical and full Bayesian methods and subjective procedures suggested in Bayesian hypothesis testing. Our results also suggest that the bootstrap proportion and Bayesian posterior probability are different measures of accuracy, and that the bootstrap proportion, if interpreted as the probability that the clade is true, can be either too liberal or too conservative.     

Frequentist properties of Bayesian posterior probabilities of phylogenetic trees under simple and complex substitution models

What does the posterior probability of a phylogenetic tree mean?This simulation study shows that Bayesian posterior probabilities have the meaning that is typically ascribed to them; the posterior probability of a tree is the probability that the tree is correct, assuming that the model is correct. At the same time, the Bayesian method can be sensitive to model misspecification, and the sensitivity of the Bayesian method appears to be greater than the sensitivity of the nonparametric bootstrap method (using maximum likelihood to estimate trees). Although the estimates of phylogeny obtained by use of the method of maximum likelihood or the Bayesian method are likely to be similar, the assessment of the uncertainty of inferred trees via either bootstrapping (for maximum likelihood estimates) or posterior probabilities (for Bayesian estimates) is not likely to be the same. We suggest that the Bayesian method be implemented with the most complex models of those currently available, as this should reduce the chance that the method will concentrate too much probability on too few trees.     

应用贝叶斯理论推断DNA分子标记基因型

引入贝叶斯理论用以从DNA分子标记的表现型（电泳谱带）推断其基因型（DNA来源）。结果表明,根据标记座位独立贫富而确定的遗传信息不完全标记的基因型概率,与根据邻近的遗传信息完全标记的基因型和有关重组率算得的相应贝叶斯概率,通常都有很大的差异,所以在进行数量性状基因定位和标记辅助选择等工作前前,应当计算每一个体基因组上所有遗传信息不完全座位的有关基因型的贝叶斯概率,文中列出计算未知基因型的贝叶斯概率的详细过程,也讨论了贝叶斯概率的若干推广应用。     

Comparing bootstrap and posterior probability values in the four-taxon case

Assessment of the reliability of a given phylogenetic hypothesis is an important step in phylogenetic analysis. Historically, the nonparametric bootstrap procedure has been the most frequently used method for assessing the support for specific phylogenetic relationships. The recent employment of Bayesian methods for phylogenetic inference problems has resulted in clade support being expressed in terms of posterior probabilities. We used simulated data and the four-taxon case to explore the relationship between nonparametric bootstrap values (as inferred by maximum likelihood) and posterior probabilities (as inferred by Bayesian analysis). The results suggest a complex association between the two measures. Three general regions of tree space can be identified: (1) the neutral zone, where differences between mean bootstrap and mean posterior probability values are not significant, (2) near the two-branch corner, and (3) deep in the two-branch corner. In the last two regions, significant differences occur between mean bootstrap and mean posterior probability values. Whether bootstrap or posterior probability values are higher depends on the data in support of alternative topologies. Examination of star topologies revealed that both bootstrap and posterior probability values differ significantly from theoretical expectations; in particular, there are more posterior probability values in the range 0.85-1 than expected by theory. Therefore, our results corroborate the findings of others that posterior probability values are excessively high. Our results also suggest that extrapolations from single topology branch-length studies are unlikely to provide any general conclusions regarding the relationship between bootstrap and posterior probability values.     

Recombination, gene conversion, and identity-by-descent at three loci

We investigate the probabilities of identity-by-descent at three loci in order to find a signature which differentiates between the two types of crossing over events: recombination and gene conversion. We use a Markov chain to model coalescence, recombination, gene conversion and mutation in a sample of size two. Using numerical analysis, we calculate the total probability of identity-by-descent at the three loci, and partition these probabilities based on a partial ordering of coalescent events at the three loci. We use these results to compute the probabilities of four different patterns of conditional identity and non-identity at the three loci under recombination and gene conversion. Although recombination and gene conversion do make different predictions, the differences are not likely to be useful in distinguishing between them using three locus patterns between pairs of DNA sequences. This implies that measures of genetic identity in larger samples will be needed to distinguish between gene conversion and recombination.     

Measures of clade confidence do not correlate with accuracy of phylogenetic trees

Metrics of phylogenetic tree reliability, such as parametric bootstrap percentages or Bayesian posterior probabilities, represent internal measures of the topological reproducibility of a phylogenetic tree, while the recently introduced aLRT (approximate likelihood ratio test) assesses the likelihood that a branch exists on a maximum-likelihood tree. Although those values are often equated with phylogenetic tree accuracy, they do not necessarily estimate how well a reconstructed phylogeny represents cladistic relationships that actually exist in nature. The authors have therefore attempted to quantify how well bootstrap percentages, posterior probabilities, and aLRT measures reflect the probability that a deduced phylogenetic clade is present in a known phylogeny. The authors simulated the evolution of bacterial genes of varying lengths under biologically realistic conditions, and reconstructed those known phylogenies using both maximum likelihood and Bayesian methods. Then, they measured how frequently clades in the reconstructed trees exhibiting particular bootstrap percentages, aLRT values, or posterior probabilities were found in the true trees. The authors have observed that none of these values correlate with the probability that a given clade is present in the known phylogeny. The major conclusion is that none of the measures provide any information about the likelihood that an individual clade actually exists. It is also found that the mean of all clade support values on a tree closely reflects the average proportion of all clades that have been assigned correctly, and is thus a good representation of the overall accuracy of a phylogenetic tree.     

Reanalysis of Murphy et al.'s data gives various mammalian phylogenies and suggests overcredibility of Bayesian trees

Murphy and colleagues reported that the mammalian phylogeny was resolved by Bayesian phylogenetics. However, the DNA sequences they used had many alignment gaps and undetermined nucleotide sites. We therefore reanalyzed their data by minimizing unshared nucleotide sites and retaining as many species as possible (13 species). In constructing phylogenetic trees, we used the Bayesian, maximum likelihood (ML), maximum parsimony (MP), and neighbor-joining (NJ) methods with different substitution models. These trees were constructed by using both protein and DNA sequences. The results showed that the posterior probabilities for Bayesian trees were generally much higher than the bootstrap values for ML, MP, and NJ trees. Two different Bayesian topologies for the same set of species were sometimes supported by high posterior probabilities, implying that two different topologies can be judged to be correct by Bayesian phylogenetics. This suggests that the posterior probability in Bayesian analysis can be excessively high as an indication of statistical confidence and therefore Murphy et al.'s tree, which largely depends on Bayesian posterior probability, may not be correct.     

Genomic Targets of Positive Selection in Giant Mice from Gough Island

A key challenge in understanding how organisms adapt to their environments is to identify the mutations and genes that make it possible. By comparing patterns of sequence variation to neutral predictions across genomes, the targets of positive selection can be located. We applied this logic to house mice that invaded Gough Island (GI), an unusual population that shows phenotypic and ecological hallmarks of selection. We used massively parallel short-read sequencing to survey the genomes of 14 GI mice. We computed a set of summary statistics to capture diverse aspects of variation across these genome sequences, used approximate Bayesian computation to reconstruct a null demographic model, and then applied machine learning to estimate the posterior probability of positive selection in each region of the genome. Using a conservative threshold, 1,463 5-kb windows show strong evidence for positive selection in GI mice but not in a mainland reference population of German mice. Disproportionate shares of these selection windows contain genes that harbor derived nonsynonymous mutations with large frequency differences. Over-represented gene ontologies in selection windows emphasize neurological themes. Inspection of genomic regions harboring many selection windows with high posterior probabilities pointed to genes with known effects on exploratory behavior and body size as potential targets. Some genes in these regions contain candidate adaptive variants, including missense mutations and/or putative regulatory mutations. Our results provide a genomic portrait of adaptation to island conditions and position GI mice as a powerful system for understanding the genetic component of natural selection.     

Hardy-Weinberg equilibrium diagnostics

We propose two diagnostics for the statistical assessment of Hardy-Weinberg equilibrium. One diagnostic is the posterior probability of the complement of the smallest highest posterior density credible region that includes points in the parameter space consistent with the hypothesis of equilibrium. The null hypothesis of equilibrium is to be rejected if this probability is less than a pre-selected critical level. The second diagnostic is the proportion of the parameter space occupied by the highest posterior density credible region associated with the critical level. These Bayesian diagnostics can be interpreted as analogues of the classical types I and II error probabilities. They are broadly applicable: they can be computed for any hypothesis test, using samples of any size generated according to any distribution.     

Effects of branch length uncertainty on Bayesian posterior probabilities for phylogenetic hypotheses

In Bayesian phylogenetics, confidence in evolutionary relationships is expressed as posterior probability--the probability that a tree or clade is true given the data, evolutionary model, and prior assumptions about model parameters. Model parameters, such as branch lengths, are never known in advance; Bayesian methods incorporate this uncertainty by integrating over a range of plausible values given an assumed prior probability distribution for each parameter. Little is known about the effects of integrating over branch length uncertainty on posterior probabilities when different priors are assumed. Here, we show that integrating over uncertainty using a wide range of typical prior assumptions strongly affects posterior probabilities, causing them to deviate from those that would be inferred if branch lengths were known in advance; only when there is no uncertainty to integrate over does the average posterior probability of a group of trees accurately predict the proportion of correct trees in the group. The pattern of branch lengths on the true tree determines whether integrating over uncertainty pushes posterior probabilities upward or downward. The magnitude of the effect depends on the specific prior distributions used and the length of the sequences analyzed. Under realistic conditions, however, even extraordinarily long sequences are not enough to prevent frequent inference of incorrect clades with strong support. We found that across a range of conditions, diffuse priors--either flat or exponential distributions with moderate to large means--provide more reliable inferences than small-mean exponential priors. An empirical Bayes approach that fixes branch lengths at their maximum likelihood estimates yields posterior probabilities that more closely match those that would be inferred if the true branch lengths were known in advance and reduces the rate of strongly supported false inferences compared with fully Bayesian integration.     

Bayesian Adaptive Randomization and Trial Monitoring with Predictive Probability for Time-to-Event Endpoint

There has been much development in Bayesian adaptive designs in clinical trials. In the Bayesian paradigm, the posterior predictive distribution characterizes the future possible outcomes given the currently observed data. Based on the interim time-to-event data, we develop a new phase II trial design by combining the strength of both Bayesian adaptive randomization and the predictive probability. By comparing the mean survival times between patients assigned to two treatment arms, more patients are assigned to the better treatment on the basis of adaptive randomization. We continuously monitor the trial using the predictive probability for early termination in the case of superiority or futility. We conduct extensive simulation studies to examine the operating characteristics of four designs: the proposed predictive probability adaptive randomization design, the predictive probability equal randomization design, the posterior probability adaptive randomization design, and the group sequential design. Adaptive randomization designs using predictive probability and posterior probability yield a longer overall median survival time than the group sequential design, but at the cost of a slightly larger sample size. The average sample size using the predictive probability method is generally smaller than that of the posterior probability design.     

Bayesian estimation of concordance among gene trees

Multigene sequence data have great potential for elucidating important and interesting evolutionary processes, but statistical methods for extracting information from such data remain limited. Although various biological processes may cause different genes to have different genealogical histories (and hence different tree topologies), we also may expect that the number of distinct topologies among a set of genes is relatively small compared with the number of possible topologies. Therefore evidence about the tree topology for one gene should influence our inferences of the tree topology on a different gene, but to what extent? In this paper, we present a new approach for modeling and estimating concordance among a set of gene trees given aligned molecular sequence data. Our approach introduces a one-parameter probability distribution to describe the prior distribution of concordance among gene trees. We describe a novel 2-stage Markov chain Monte Carlo (MCMC) method that first obtains independent Bayesian posterior probability distributions for individual genes using standard methods. These posterior distributions are then used as input for a second MCMC procedure that estimates a posterior distribution of gene-to-tree maps (GTMs). The posterior distribution of GTMs can then be summarized to provide revised posterior probability distributions for each gene (taking account of concordance) and to allow estimation of the proportion of the sampled genes for which any given clade is true (the sample-wide concordance factor). Further, under the assumption that the sampled genes are drawn randomly from a genome of known size, we show how one can obtain an estimate, with credibility intervals, on the proportion of the entire genome for which a clade is true (the genome-wide concordance factor). We demonstrate the method on a set of 106 genes from 8 yeast species.     

Utility of the dentin matrix protein 1 (DMP1) gene for resolving mammalian intraordinal phylogenetic relationships

We sequenced exon 6 of the nuclear dentin matrix protein 1 (DMP1) gene from 19 species of bats (order Chiroptera) to assess the utility of this gene for higher-level phylogenetic studies. Bayesian analysis revealed high support (posterior probabilities >/=0.95) for monophyly of Noctilionoidea (Phyllostomidae, Noctilionidae, and Mormoopidae), all genera and most families examined. Comparison of the phylogenetic information present in DMP1 with mitochondrial rDNA and nuclear RAG2 genes indicated no significant heterogeneity. Thus, we concatenated these three data sets into a single "total evidence" phylogenetic analysis. Combined analysis was congruent with study of RAG2 and combined RAG2 and mtrDNA sequences, but improved support (Bayesian posterior probabilities) for many nodes. Our results indicate that exon 6 of DMP1 is rapidly evolving, able to tolerate non-frame shifting insertion and deletion events, is more variable than RAG2, and provides phylogenetic resolution from the interfamilial to infraclass levels in mammals.     

Stopping rules in Bayesian adaptive threshold estimation

Threshold estimation with sequential procedures is justifiable on the surmise that the index used in the so-called dynamic stopping rule has diagnostic value for identifying when an accurate estimate has been obtained. The performance of five types of Bayesian sequential procedure was compared here to that of an analogous fixed-length procedure. Indices for use in sequential procedures were: (1) the width of the Bayesian probability interval, (2) the posterior standard deviation, (3) the absolute change, (4) the average change, and (5) the number of sign fluctuations. A simulation study was carried out to evaluate which index renders estimates with less bias and smaller standard error at lower cost (i.e. lower average number of trials to completion), in both yes-no and two-alternative forced-choice (2AFC) tasks. We also considered the effect of the form and parameters of the psychometric function and its similarity with the model function assumed in the procedure. Our results show that sequential procedures do not outperform fixed-length procedures in yes-no tasks. However, in 2AFC tasks, sequential procedures not based on sign fluctuations all yield minimally better estimates than fixed-length procedures, although most of the improvement occurs with short runs that render undependable estimates and the differences vanish when the procedures run for a number of trials (around 70) that ensures dependability. Thus, none of the indices considered here (some of which are widespread) has the diagnostic value that would justify its use. In addition, difficulties of implementation make sequential procedures unfit as alternatives to fixed-length procedures.     

The temporal order of genetic and pathway alterations in tumorigenesis

Cancer evolves through the accumulation of mutations, but the order in which mutations occur is poorly understood. Inference of a temporal ordering on the level of genes is challenging because clinically and histologically identical tumors often have few mutated genes in common. This heterogeneity may at least in part be due to mutations in different genes having similar phenotypic effects by acting in the same functional pathway. We estimate the constraints on the order in which alterations accumulate during cancer progression from cross-sectional mutation data using a probabilistic graphical model termed Hidden Conjunctive Bayesian Network (H-CBN). The possible orders are analyzed on the level of genes and, after mapping genes to functional pathways, also on the pathway level. We find stronger evidence for pathway order constraints than for gene order constraints, indicating that temporal ordering results from selective pressure acting at the pathway level. The accumulation of changes in core pathways differs among cancer types, yet a common feature is that progression appears to begin with mutations in genes that regulate apoptosis pathways and to conclude with mutations in genes involved in invasion pathways. H-CBN models provide a quantitative and intuitive model of tumorigenesis showing that the genetic events can be linked to the phenotypic progression on the level of pathways.     

Missense mutations in disease genes: a Bayesian approach to evaluate causality.

The problem of interpreting missense mutations of disease-causing genes is an increasingly important one. Because these point mutations result in alteration of only a single amino acid of the protein product, it is often unclear whether this change alone is sufficient to cause disease. We propose a Bayesian approach that utilizes genetic information on affected relatives in families ascertained through known missense-mutation carriers. This method is useful in evaluating known disease genes for common disease phenotypes, such as breast cancer or colorectal cancer. The posterior probability that a missense mutation is disease causing is conditioned on the relationship of the relatives to the proband, the population frequency of the mutation, and the phenocopy rate of the disease. The approach is demonstrated in two cancer data sets: BRCA1 R841W and APC I1307K. In both examples, this method helps establish that these mutations are likely to be disease causing, with Bayes factors in favor of causality of 5.09 and 66.97, respectively, and posterior probabilities of .836 and .985. We also develop a simple approximation for rare alleles and consider the case of unknown penetrance and allele frequency.     

Bayesian model averaging: development of an improved multi-class, gene selection and classification tool for microarray data

MOTIVATION: Selecting a small number of relevant genes for accurate classification of samples is essential for the development of diagnostic tests. We present the Bayesian model averaging (BMA) method for gene selection and classification of microarray data. Typical gene selection and classification procedures ignore model uncertainty and use a single set of relevant genes (model) to predict the class. BMA accounts for the uncertainty about the best set to choose by averaging over multiple models (sets of potentially overlapping relevant genes). RESULTS: We have shown that BMA selects smaller numbers of relevant genes (compared with other methods) and achieves a high prediction accuracy on three microarray datasets. Our BMA algorithm is applicable to microarray datasets with any number of classes, and outputs posterior probabilities for the selected genes and models. Our selected models typically consist of only a few genes. The combination of high accuracy, small numbers of genes and posterior probabilities for the predictions should make BMA a powerful tool for developing diagnostics from expression data. AVAILABILITY: The source codes and datasets used are available from our Supplementary website.