Bayesian mapping of quantitative trait loci for complex binary traits

A complex binary trait is a character that has a dichotomous expression but with a polygenic genetic background. Mapping quantitative trait loci (QTL) for such traits is difficult because of the discrete nature and the reduced variation in the phenotypic distribution. Bayesian statistics are proved to be a powerful tool for solving complicated genetic problems, such as multiple QTL with nonadditive effects, and have been successfully applied to QTL mapping for continuous traits. In this study, we show that Bayesian statistics are particularly useful for mapping QTL for complex binary traits. We model the binary trait under the classical threshold model of quantitative genetics. The Bayesian mapping statistics are developed on the basis of the idea of data augmentation. This treatment allows an easy way to generate the value of a hypothetical underlying variable (called the liability) and a threshold, which in turn allow the use of existing Bayesian statistics. The reversible jump Markov chain Monte Carlo algorithm is used to simulate the posterior samples of all unknowns, including the number of QTL, the locations and effects of identified QTL, genotypes of each individual at both the QTL and markers, and eventually the liability of each individual. The Bayesian mapping ends with an estimation of the joint posterior distribution of the number of QTL and the locations and effects of the identified QTL. Utilities of the method are demonstrated using a simulated outbred full-sib family. A computer program written in FORTRAN language is freely available on request.     

贝叶斯方法估计阈性状育种值的原理和计算技术

本文介绍了估计阈性状育种值的贝叶斯方法的原理,演示了描述阈性状观察值、建立后验概率密度函数、以及导出非线性方程组的方法．并就这一估计方法的计算技术进行了讨论,针对动物遗传育种中方程组系数矩阵往往很大,超出计算机内存的情况,提出了不需要建立方程组,在数据上迭代求解的计算方法．本文还综述了这一非线性方法与线性方法在阈性状育种值估计上的比较．     

Building food networks from molecular data: Bayesian or fixed-number thresholds for including links

DNA metabarcoding of faeces or gut contents has greatly increased our ability to construct networks of predators and prey (food webs) by reducing the need to observe predation events directly. The possibility of both false positives and false negatives in DNA sequences, however, means that constructing food networks using DNA requires researchers to make many choices as to which DNA sequences indicate true prey for a particular predator. To date, DNA-based food networks are usually constructed by including any DNA sequence with more than a threshold number of reads. The logic used to select this threshold is often not explained, leading to somewhat arbitrary-seeming networks. As an alternative strategy, we demonstrate how to construct food networks using a simple Bayesian model to suggest which sequences correspond to true prey. The networks obtained using a well-chosen fixed cutoff and our Bayesian approach are very similar, especially when links are resolved to prey families rather than species. We therefore recommend that researchers reconstruct diet data using a Bayesian approach with well-specified assumptions rather than continuing with arbitrary fixed cutoffs. Explicitly stating assumptions within a Bayesian framework will lead to better-informed comparisons between networks constructed by different groups and facilitate drawing together individual case studies into more coherent ecological theory. Note that our approach can easily be extended to other types of ecological networks constructed by DNA metabarcoding of pollen loads, identification of parasite DNA in faeces, etc.     

ANCESTRAL CHARACTER ESTIMATION UNDER THE THRESHOLD MODEL FROM QUANTITATIVE GENETICS

Evolutionary biology is a study of life's history on Earth. In researching this history, biologists are often interested in attempting to reconstruct phenotypes for the long extinct ancestors of living species. Various methods have been developed to do this on a phylogeny from the data for extant taxa. In the present article, I introduce a new approach for ancestral character estimation for discretely valued traits. This approach is based on the threshold model from evolutionary quantitative genetics. Under the threshold model, the value exhibited by an individual or species for a discrete character is determined by an underlying, unobserved continuous trait called “liability.” In this new method for ancestral state reconstruction, I use Bayesian Markov chain Monte Carlo (MCMC) to sample the liabilities of ancestral and tip species, and the relative positions of two or more thresholds, from their joint posterior probability distribution. Using data simulated under the model, I find that the method has very good performance in ancestral character estimation. Use of the threshold model for ancestral state reconstruction relies on a priori specification of the order of the discrete character states along the liability axis. I test the use of a Bayesian MCMC information theoretic criterion based approach to choose among different hypothesized orderings for the discrete character. Finally, I apply the method to the evolution of feeding mode in centrarchid fishes.     

Modular analysis of the probabilistic genetic interaction network

MOTIVATION: Epistatic Miniarray Profiles (EMAP) has enabled the mapping of large-scale genetic interaction networks; however, the quantitative information gained from EMAP cannot be fully exploited since the data are usually interpreted as a discrete network based on an arbitrary hard threshold. To address such limitations, we adopted a mixture modeling procedure to construct a probabilistic genetic interaction network and then implemented a Bayesian approach to identify densely interacting modules in the probabilistic network. RESULTS: Mixture modeling has been demonstrated as an effective soft-threshold technique of EMAP measures. The Bayesian approach was applied to an EMAP dataset studying the early secretory pathway in Saccharomyces cerevisiae. Twenty-seven modules were identified, and 14 of those were enriched by gold standard functional gene sets. We also conducted a detailed comparison with state-of-the-art algorithms, hierarchical cluster and Markov clustering. The experimental results show that the Bayesian approach outperforms others in efficiently recovering biologically significant modules.     

Analyzing marginal cases in differential shotgun proteomics

SUMMARY: We present an approach to statistically pinpoint differentially expressed proteins that have quantitation values near the quantitation threshold and are not identified in all replicates (marginal cases). Our method uses a Bayesian strategy to combine parametric statistics with an empirical distribution built from the reproducibility quality of the technical replicates. AVAILABILITY: The software is freely available for academic use at http://pcarvalho.com/patternlab.     

Bayesian multiple quantitative trait loci mapping for complex traits using markers of the entire genome

A Bayesian methodology has been developed for multiple quantitative trait loci (QTL) mapping of complex binary traits that follow liability threshold models. Unlike most QTL mapping methods where only one or a few markers are used at a time, the proposed method utilizes all markers across the genome simultaneously. The outperformance of our Bayesian method over the traditional single-marker analysis and interval mapping has been illustrated via simulations and real data analysis to identify candidate loci associated with colorectal cancer.     

Genetic parameters for litter size in sheep: natural versus hormone-induced oestrus

The litter size in Suffolk and Texel-sheep was analysed using REML and Bayesian methods. Litters born after hormonal induced oestrus and after natural oestrus were treated as different traits in order to estimate the genetic correlation between the traits. Explanatory variables were the age of the ewe at lambing, period of lambing, a year*flock-effect, a permanent environmental effect associated with the ewe, and the additive genetic effect. The heritability estimates for litter size ranged from 0.06 to 0.13 using REML in bi-variate linear models. Transformation of the estimates to the underlying scale resulted in heritability estimates from 0.12 to 0.17. Posterior means of the heritability of litter size in the Bayesian approach with bi-variate threshold models varied from 0.05 to 0.18. REML estimates of the genetic correlations between the two types of litter size ranged from 0.57 to 0.64 in the Suffolk and from 0.75 to 0.81 in the Texel. The posterior means of the genetic correlation (Bayesian analysis) were 0.40 and 0.44 for the Suffolk and 0.56 and 0.75 for the Texel in the sire and animal model respectively. A bivariate threshold model seems appropriate for the genetic evaluation of prolificacy in the breeds concerned.     

Accurate and rapid estimation of phosphene thresholds (REPT)

To calibrate the intensity of transcranial magnetic stimulation (TMS) at the occipital pole, the phosphene threshold is used as a measure of cortical excitability. The phosphene threshold (PT) refers to the intensity of magnetic stimulation that induces illusory flashes of light (phosphenes) on a proportion of trials. The existing PT estimation procedures lack the accuracy and mathematical rigour of modern threshold estimation methods. We present an improved and automatic procedure for estimating the PT which is based on the well-established Ψ Bayesian adaptive staircase approach. To validate the new procedure, we compared it with another commonly used procedure for estimating the PT. We found that our procedure is more accurate, reliable, and rapid when compared with an existing PT measurement procedure. The new procedure is implemented in Matlab and works automatically with the Magstim Rapid(2) stimulator using a convenient graphical user interface. The Matlab program is freely available for download.     

Probabilistic approach to age estimation of children by dental maturation

Two probabilist methods of age prediction in children are proposed: they are both based on the radiological presence of erupted teeth or germs. Using an apprenticeship sample of known age and sex, we established several discriminant models (+/- 13, +/- 16, +/- 18 years old). We also evaluated a Bayesian model with the following age groups: < 13, [13-16[, [16-18[, > or = 18 years old, or [X and Y] years old. When applied on a known test sample, Fisher's linear functions presented a success rate greater than 90%, above 13 years threshold, and below 16 and 18 years thresholds, and Bayesian approach, greater than 85%. Therefore, these methods provide an interesting alternative for children age determination that can be applied in biological and forensic anthropology, too.     

Sample size calculations for surveys to substantiate freedom of populations from infectious agents

We develop a Bayesian approach to sample size computations for surveys designed to provide evidence of freedom from a disease or from an infectious agent. A population is considered "disease-free" when the prevalence or probability of disease is less than some threshold value. Prior distributions are specified for diagnostic test sensitivity and specificity and we test the null hypothesis that the prevalence is below the threshold. Sample size computations are developed using hypergeometric sampling for finite populations and binomial sampling for infinite populations. A normal approximation is also developed. Our procedures are compared with the frequentist methods of Cameron and Baldock (1998a, Preventive Veterinary Medicine34, 1-17.) using an example of foot-and-mouth disease. User-friendly programs for sample size calculation and analysis of survey data are available at http://www.epi.ucdavis.edu/diagnostictests/.     

Modeling multi-robot task allocation with limited information as global game

Continuous response threshold functions to coordinate collaborative tasks in multi-agent systems are commonly employed models in a number of fields including ethology, economics, and swarm robotics. Although empirical evidence exists for the response threshold model in predicting and matching swarm behavior for social insects, there has been no formal argument as to why natural swarms use this approach and why it should be used for engineering artificial ones. In this paper, we show, by formulating task allocation as a global game, that continuous response threshold functions used for communication-free task assignment result in system level Bayesian Nash equilibria. Building up on these results, we show that individual agents not only do not need to communicate with each other, but also do not need to model each other’s behavior, which makes this coordination mechanism accessible to very simple agents, suggesting a reason for their prevalence in nature and motivating their use in an engineering context.     

Model for fitting longitudinal traits subject to threshold response applied to genetic evaluation for heat tolerance

A semi-parametric non-linear longitudinal hierarchical model is presented. The model assumes that individual variation exists both in the degree of the linear change of performance (slope) beyond a particular threshold of the independent variable scale and in the magnitude of the threshold itself; these individual variations are attributed to genetic and environmental components. During implementation via a Bayesian MCMC approach, threshold levels were sampled using a Metropolis step because their fully conditional posterior distributions do not have a closed form. The model was tested by simulation following designs similar to previous studies on genetics of heat stress. Posterior means of parameters of interest, under all simulation scenarios, were close to their true values with the latter always being included in the uncertain regions, indicating an absence of bias. The proposed models provide flexible tools for studying genotype by environmental interaction as well as for fitting other longitudinal traits subject to abrupt changes in the performance at particular points on the independent variable scale.     

Using the quantitative genetic threshold model for inferences between and within species

Sewall Wright's threshold model has been used in modelling discrete traits that may have a continuous trait underlying them, but it has proven difficult to make efficient statistical inferences with it. The availability of Markov chain Monte Carlo (MCMC) methods makes possible likelihood and Bayesian inference using this model. This paper discusses prospects for the use of the threshold model in morphological systematics to model the evolution of discrete all-or-none traits. There the threshold model has the advantage over 0/1 Markov process models in that it not only accommodates polymorphism within species, but can also allow for correlated evolution of traits with far fewer parameters that need to be inferred. The MCMC importance sampling methods needed to evaluate likelihood ratios for the threshold model are introduced and described in some detail.     

Bayesian methods for estimating GEBVs of threshold traits

Estimation of genomic breeding values is the key step in genomic selection (GS). Many methods have been proposed for continuous traits, but methods for threshold traits are still scarce. Here we introduced threshold model to the framework of GS, and specifically, we extended the three Bayesian methods BayesA, BayesB and BayesCπ on the basis of threshold model for estimating genomic breeding values of threshold traits, and the extended methods are correspondingly termed BayesTA, BayesTB and BayesTCπ. Computing procedures of the three BayesT methods using Markov Chain Monte Carlo algorithm were derived. A simulation study was performed to investigate the benefit of the presented methods in accuracy with the genomic estimated breeding values (GEBVs) for threshold traits. Factors affecting the performance of the three BayesT methods were addressed. As expected, the three BayesT methods generally performed better than the corresponding normal Bayesian methods, in particular when the number of phenotypic categories was small. In the standard scenario (number of categories=2, incidence=30%, number of quantitative trait loci=50, h²=0.3), the accuracies were improved by 30.4%, 2.4%, and 5.7% points, respectively. In most scenarios, BayesTB and BayesTCπ generated similar accuracies and both performed better than BayesTA. In conclusion, our work proved that threshold model fits well for predicting GEBVs of threshold traits, and BayesTCπ is supposed to be the method of choice for GS of threshold traits.     

A critical thermal transition driving spring phenology of Northern Hemisphere conifers

Despite growing interest in predicting plant phenological shifts, advanced spring phenology by global climate change remains debated. Evidence documenting either small or large advancement of spring phenology to rising temperature over the spatio-temporal scales implies a potential existence of a thermal threshold in the responses of forests to global warming. We collected a unique data set of xylem cell-wall-thickening onset dates in 20 coniferous species covering a broad mean annual temperature (MAT) gradient (−3.05 to 22.9°C) across the Northern Hemisphere (latitudes 23°–66° N). Along the MAT gradient, we identified a threshold temperature (using segmented regression) of 4.9 ± 1.1°C, above which the response of xylem phenology to rising temperatures significantly decline. This threshold separates the Northern Hemisphere conifers into cold and warm thermal niches, with MAT and spring forcing being the primary drivers for the onset dates (estimated by linear and Bayesian mixed-effect models), respectively. The identified thermal threshold should be integrated into the Earth-System-Models for a better understanding of spring phenology in response to global warming and an improved prediction of global climate-carbon feedbacks.     

Bayesian inference in genetic parameter estimation of visual scores in Nellore beef-cattle

The aim of this study was to estimate the components of variance and genetic parameters for the visual scores which constitute the Morphological Evaluation System (MES), such as body structure (S), precocity (P) and musculature (M) in Nellore beef-cattle at the weaning and yearling stages, by using threshold Bayesian models. The information used for this was gleaned from visual scores of 5,407 animals evaluated at the weaning and 2,649 at the yearling stages. The genetic parameters for visual score traits were estimated through two-trait analysis, using the threshold animal model, with Bayesian statistics methodology and MTGSAM (Multiple Trait Gibbs Sampler for Animal Models) threshold software. Heritability estimates for S, P and M were 0.68, 0.65 and 0.62 (at weaning) and 0.44, 0.38 and 0.32 (at the yearling stage), respectively. Heritability estimates for S, P and M were found to be high, and so it is expected that these traits should respond favorably to direct selection. The visual scores evaluated at the weaning and yearling stages might be used in the composition of new selection indexes, as they presented sufficient genetic variability to promote genetic progress in such morphological traits.     

On Indifference Zone Selection with a Preference Threshold

This paper discusses a selection criterion that generalizes the well-known concept of indifference zone selection through a preference threshold. A population is preferred to another population if the difference in the sums of observed values exceeds a given nonnegative threshold value. We present an argument for this selection rule by modelling preference by imprecise previsions. We aim at guidelines to design a selection experiment, which is characterized by two numbers: the number of necessary observations per population, and the preference threshold. Next to the probability of correct selection we also need a second specification. In this paper we consider a probability of false selection that is strongly related to the minimum probability of correct selection. Based on this model the outcome of an experiment may be ‘no selection’, at least not based on strong preference of a single population. The ideas are presented through a simple selection problem for normal populations with common known variance. Although the theory has a frequentist nature, the derivation and justification of the selection rule through imprecise previsions relies on Bayesian foundations, and via this route we gain more insight into the selection criterion.     

Source population of dispersing rock-wallabies (Petrogale lateralis) identified by assignment tests on multilocus genotypic data

The ability to confidently identify or exclude a population as the source of an individual has numerous powerful applications in molecular ecology. Several alternative assignment methods have recently been developed and are yet to be fully evaluated with empirical data. In this study we tested the efficacy of different assignment methods by using a translocated rock-wallaby (Petrogale lateralis) population, of known provenance. Specimens from the translocated population (n = 43), its known source population (n = 30) and four other nearby populations (n = 19-32) were genotyped for 11 polymorphic microsatellite loci. The results identified Bayesian clustering, frequency and Bayesian methods as the most consistent and accurate, correctly assigning 93-100% of individuals up to a significance threshold of P = 0.01. Performance was variable among the distance-based methods, with the Cavalli-Sforza and Edwards chord distance performing best, whereas Goldstein et al.'s (deltamu)2 consistently performed poorly. Using Bayesian clustering, frequency and Bayesian methods we then attempted to determine the source of rock-wallabies which have recently recolonized an outcrop (Gardners) 8 km from the nearest rock-wallaby population. Results indicate that the population at Gardners originated via a recent dispersal event from the eastern end of Mt. Caroline. This is only the second published record of dispersal by rock-wallabies between habitat patches and is the longest movement recorded to date. Molecular techniques and methods of analysis are now available to allow detailed studies of dispersal in rock-wallabies and should also be possible for many other taxa.