A posterior probability of linkage-based re-analysis of schizophrenia data yields evidence of linkage to chromosomes 1 and 17

OBJECTIVE: Linkage analysis using 22 Canadian pedigrees identified a promising schizophrenia candidate region on 1q23 with a maximum 2-point HLOD under a recessive model of 5.8 [Brzustowicz et al. 2000]. In the current study, we revisited this data set using a Bayesian linkage analysis technique, namely the posterior probability of linkage (PPL). METHODS: The PPL has been developed as an alternative to traditional linkage analysis. It differs from both LOD scores and 'non-parametric' methods in that it directly measures the probability of linkage given the data, and incorporates prior genomic information. RESULTS: As expected, PPL results for 1q23 supported the previously observed linkage, with an estimated multipoint PPL of 99.7%. However, the PPL supported two further results: a second peak on chromosome 1 at 1p13 with a multipoint with PPL of 70% and a chromosome 17 marker (D17S784 at 17q25) with a multipoint PPL of 44%. CONCLUSIONS: The PPL-based analysis presented has the advantage over other likelihood-based linkage methods in that it avoids maximization and produces a less complex view of the strength of evidence for linkage.     

Cytological Studies of Human Meiosis: Sex-Specific Differences in Recombination Originate at,or Prior to,Establishment of Double-Strand Breaks

Meiotic recombination is sexually dimorphic in most mammalian species, including humans, but the basis for the male:female differences remains unclear. In the present study, we used cytological methodology to directly compare recombination levels between human males and females, and to examine possible sex-specific differences in upstream events of double-strand break (DSB) formation and synaptic initiation. Specifically, we utilized the DNA mismatch repair protein MLH1 as a marker of recombination events, the RecA homologue RAD51 as a surrogate for DSBs, and the synaptonemal complex proteins SYCP3 and/or SYCP1 to examine synapsis between homologs. Consistent with linkage studies, genome-wide recombination levels were higher in females than in males, and the placement of exchanges varied between the sexes. Subsequent analyses of DSBs and synaptic initiation sites indicated similar male:female differences, providing strong evidence that sex-specific differences in recombination rates are established at or before the formation of meiotic DSBs. We then asked whether these differences might be linked to variation in the organization of the meiotic axis and/or axis-associated DNA and, indeed, we observed striking male:female differences in synaptonemal complex (SC) length and DNA loop size. Taken together, our observations suggest that sex specific differences in recombination in humans may derive from chromatin differences established prior to the onset of the recombination pathway.     

A new method for computing the multipoint posterior probability of linkage

The posterior probability of linkage (PPL) is a Bayesian statistic which directly measures the probability of linkage between a trait locus and a marker (in the 2-point case) or a genomic region (in the multipoint case). It has several benefits, including ease of interpretation, the ability to incorporate prior genomic information, and a mathematically rigorous and robust procedure for accumulating linkage information across multiple heterogeneous datasets. To date, the majority of work on the PPL has focused on the development of the 2-point statistic, with only preliminary attempts at the development of an equivalent multipoint version. In this paper we present a new way of computing of the multipoint PPL. This new version imputes to each genomic point an estimate of the 2-point PPL we would have obtained from a fully informative marker giving similar evidence for linkage. This version, which we call the imputed PPL, is shown to be superior to previously developed versions.     

The appropriate threshold for declaring linkage when allowing sex-specific recombination rates.

In human genetics, two loci are declared to be linked when the lod score at the maximum likelihood recombination fraction theta exceeds the threshold of 3.0. Since recombination rates differ between the sexes, one can alternatively detect linkage by estimating separate recombination rates, theta m and theta f, for male and female meiosis and examining the corresponding sex-specific lod scores. The question arises: In order to maintain the same chance of falsely declaring linkage, what is the correct threshold for declaring linkage when sex-specific lod scores are used? We show here that the appropriate threshold is about 3.5. If the restriction that theta f greater than theta m is added, the appropriate threshold falls to about 3.25. We also discuss the relative efficiency of detecting linkage by using sex-specific and sex-averaged lod scores.     

First evidence of higher female recombination in a species with temperature-dependent sex determination: the saltwater crocodile

The first evidence of genetic linkage and sex-specific recombination in the order Crocodylia is reported. This study was conducted using a resource pedigree of saltwater crocodiles consisting of 16 known-breeding pairs (32 adults) and 101 juveniles. A total of 21 microsatellite loci were available for analysis. Ten of the 21 loci showed linkage with 4 linkage groups: 3 pairwise (Cj131/Cj127, CUD68/Cj101, and Cj107/Cp10) and 1 four-locus (Cj122, CUD78, Cj16, and Cj104) being found. Linkage analysis on the 21 loci revealed evidence of sex-specific differences in recombination rates. All 5 nonzero interlocus intervals were longer in females than in males, with the 4-loci linkage group 3-fold longer in females than in males (41.63 cM and 14.1 cM, respectively). This is the first report of sex-specific recombination rates in a species that exhibits temperature-dependent sex determination.     

Evolution of the genomic recombination rate in murid rodents

Although very closely related species can differ in their fine-scale patterns of recombination hotspots, variation in the average genomic recombination rate among recently diverged taxa has rarely been surveyed. We measured recombination rates in eight species that collectively represent several temporal scales of divergence within a single rodent family, Muridae. We used a cytological approach that enables in situ visualization of crossovers at meiosis to quantify recombination rates in multiple males from each rodent group. We uncovered large differences in genomic recombination rate between rodent species, which were independent of karyotypic variation. The divergence in genomic recombination rate that we document is not proportional to DNA sequence divergence, suggesting that recombination has evolved at variable rates along the murid phylogeny. Additionally, we document significant variation in genomic recombination rate both within and between subspecies of house mice. Recombination rates estimated in F(1) hybrids reveal evidence for sex-linked loci contributing to the evolution of recombination in house mice. Our results provide one of the first detailed portraits of genomic-scale recombination rate variation within a single mammalian family and demonstrate that the low recombination rates in laboratory mice and rats reflect a more general reduction in recombination rate across murid rodents.     

Learning Priors for Bayesian Computations in the Nervous System

Our nervous system continuously combines new information from our senses with information it has acquired throughout life. Numerous studies have found that human subjects manage this by integrating their observations with their previous experience (priors) in a way that is close to the statistical optimum. However, little is known about the way the nervous system acquires or learns priors. Here we present results from experiments where the underlying distribution of target locations in an estimation task was switched, manipulating the prior subjects should use. Our experimental design allowed us to measure a subject''s evolving prior while they learned. We confirm that through extensive practice subjects learn the correct prior for the task. We found that subjects can rapidly learn the mean of a new prior while the variance is learned more slowly and with a variable learning rate. In addition, we found that a Bayesian inference model could predict the time course of the observed learning while offering an intuitive explanation for the findings. The evidence suggests the nervous system continuously updates its priors to enable efficient behavior.     

Sex-specific demographic behaviours that shape human genomic variation

In the human species, the two uniparental genetic systems (mitochondrial DNA and Y chromosome) exhibit contrasting diversity patterns. It has been proposed that sex-specific behaviours, and in particular differences in migration rate between men and women, may explain these differences. The availability of high-density genomic data and the comparison of genetic patterns on autosomal and sex chromosomes at global and local scales allow a reassessment of the extent to which sex-specific behaviours shape our genome. In this article, we first review studies comparing the genetic patterns at uniparental and biparental genetic systems and assess the extent to which sex-specific migration processes explain the differences between these genetic systems. We show that differences between male and female migration rates matter, but that they are certainly not the only contributing factor. In particular, differences in effective population size between men and women are also likely to account for these differences. Then, we present and discuss three anthropological processes that may explain sex-specific differences in effective population size and thus human genomic variation: (i) variance in reproductive success arising from, for example, polygyny; (ii) descent rules; and (iii) transmission of reproductive success.     

Sex: differences in mutation, recombination, selection, gene flow, and genetic drift

In many instances, there are large sex differences in mutation rates, recombination rates, selection, rates of gene flow, and genetic drift. Mutation rates are often higher in males, a difference that has been estimated both directly and indirectly. The higher male mutation rate appears related to the larger number of cell divisions in male lineages but mutation rates also appear gene- and organism-specific. When there is recombination in only one sex, it is always the homogametic sex. When there is recombination in both sexes, females often have higher recombination but there are many exceptions. There are a number of hypotheses to explain the sex differences in recombination. Sex-specific differences in selection may result in stable polymorphisms or for sex chromosomes, faster evolutionary change. In addition, sex-dependent selection may result in antagonistic pleiotropy or sexually antagonistic genes. There are many examples of sex-specific differences in gene flow (dispersal) and a number of adaptive explanations for these differences. The overall effective population size (genetic drift) is dominated by the lower sex-specific effective population size. The mean of the mutation, recombination, and gene flow rates over the two sexes can be used in a population genetics context unless there are sex-specific differences in selection or genetic drift. Sex-specific differences in these evolutionary factors appear to be unrelated to each other. The evolutionary explanations for sex-specific differences for each factor are multifaceted and, in addition, explanations may include chance, nonadaptive differences, or mechanistic, nonevolutionary factors.     

Model parameterization, prior distributions, and the general time-reversible model in Bayesian phylogenetics

Bayesian phylogenetic methods require the selection of prior probability distributions for all parameters of the model of evolution. These distributions allow one to incorporate prior information into a Bayesian analysis, but even in the absence of meaningful prior information, a prior distribution must be chosen. In such situations, researchers typically seek to choose a prior that will have little effect on the posterior estimates produced by an analysis, allowing the data to dominate. Sometimes a prior that is uniform (assigning equal prior probability density to all points within some range) is chosen for this purpose. In reality, the appropriate prior depends on the parameterization chosen for the model of evolution, a choice that is largely arbitrary. There is an extensive Bayesian literature on appropriate prior choice, and it has long been appreciated that there are parameterizations for which uniform priors can have a strong influence on posterior estimates. We here discuss the relationship between model parameterization and prior specification, using the general time-reversible model of nucleotide evolution as an example. We present Bayesian analyses of 10 simulated data sets obtained using a variety of prior distributions and parameterizations of the general time-reversible model. Uniform priors can produce biased parameter estimates under realistic conditions, and a variety of alternative priors avoid this bias.     

A major susceptibility locus for specific language impairment is located on 13q21

Children who fail to develop language normally-in the absence of explanatory factors such as neurological disorders, hearing impairment, or lack of adequate opportunity-are clinically described as having specific language impairment (SLI). SLI has a prevalence of approximately 7% in children entering school and is associated with later difficulties in learning to read. Research indicates that genetic factors are important in the etiology of SLI. Studies have consistently demonstrated that SLI aggregates in families. Increased monozygotic versus dizygotic twin concordance rates indicate that heredity, not just shared environment, is the cause of the familial clustering. We have collected five pedigrees of Celtic ancestry that segregate SLI, and we have conducted genomewide categorical linkage analysis, using model-based LOD score techniques. Analysis was conducted under both dominant and recessive models by use of three phenotypic classifications: clinical diagnosis, language impairment (spoken language quotient <85) and reading discrepancy (nonverbal IQ minus non-word reading >15). Chromosome 13 yielded a maximum multipoint LOD score of 3.92 under the recessive reading discrepancy model. Simulation to correct for multiple models and multiple phenotypes indicated that the genomewide empirical P value is <.01. As an alternative measure, we also computed the posterior probability of linkage (PPL), obtaining a PPL of 53% in the same region. One other genomic region yielded suggestive results on chromosome 2 (multipoint LOD score 2.86, genomic P value <.06 under the recessive language impairment model). Our findings underscore the utility of traditional LOD-score-based methods in finding genes for complex diseases, specifically, SLI.     

A comparative analysis of the rainbow trout genome with 2 other species of fish (Arctic charr and Atlantic salmon) within the tetraploid derivative Salmonidae family (subfamily: Salmoninae).

We updated the genetic map of rainbow trout (Oncorhynchus mykiss) for 2 outcrossed mapping panels, and used this map to assess the putative chromosome structure and recombination rate differences among linkage groups. We then used the rainbow trout sex-specific maps to make comparisons with 2 other ancestrally polyploid species of salmonid fishes, Arctic charr (Salvelinus alpinus) and Atlantic salmon (Salmo salar) to identify homeologous chromosome affinities within each species and ascertain homologous chromosome relationships among the species. Salmonid fishes exhibit a wide range of sex-specific differences in recombination rate, with some species having the largest differences for any vertebrate species studied to date. Our current estimate of female:male recombination rates in rainbow trout is 4.31:1. Chromosome structure and (or) size is associated with recombination rate differences between the sexes in rainbow trout. Linkage groups derived from presumptive acrocentric type chromosomes were observed to have much lower sex-specific differences in recombination rate than metacentric type linkage groups. Arctic charr is karyotypically the least derived species (i.e., possessing a high number of acrocentric chromosomes) and Atlantic salmon is the most derived (i.e., possessing a number of whole-arm fusions). Atlantic salmon have the largest female:male recombination ratio difference (i.e., 16.81:1) compared with rainbow trout, and Arctic charr (1.69:1). Comparisons of recombination rates between homologous segments of linkage groups among species indicated that when significant experiment-wise differences were detected (7/24 tests), recombination rates were generally higher in the species with a less-derived chromosome structure (6/7 significant comparisons). Greater similarity in linkage group syntenies were observed between Atlantic salmon and rainbow trout, suggesting their closer phylogenetic affinities, and most interspecific linkage group comparisons support a model that suggests whole chromosome arm translocations have occurred in the evolution of this group. However, some possible exceptions were detected and these findings are discussed in relation to their influence on segregation distortion patterns. We also report unusual meiotic segregation patterns in a female parent involving the duplicated (homeologous) linkage group pair 12/16 and discuss several models that may account for these patterns.     

Evolution of the genomic rate of recombination in mammals

Rates of recombination vary considerably between species. Despite the significance of this observation for evolutionary biology and genetics, the evolutionary mechanisms that contribute to these interspecific differences are unclear. On fine physical scales, recombination rates appear to evolve rapidly between closely related species, but the mode and tempo of recombination rate evolution on the broader scale is poorly understood. Here, we use phylogenetic comparative methods to begin to characterize the evolutionary processes underlying average genomic recombination rates in mammals. We document a strong phylogenetic effect in recombination rates, indicating that more closely related species tend to have more similar average rates of recombination. We demonstrate that this phylogenetic signal is not an artifact of errors in recombination rate estimation and show that it is robust to uncertainty in the mammalian phylogeny. Neutral evolutionary models present good fits to the data and we find no evidence for heterogeneity in the rate of evolution in recombination across the mammalian tree. These results suggest that observed interspecific variation in average genomic rates of recombination is largely attributable to the steady accumulation of neutral mutations over evolutionary time. Although single recombination hotspots may live and die on short evolutionary time scales, the strong phylogenetic signal in genomic recombination rates indicates that the pace of evolution on this scale may be considerably slower.     

Linkage mapping reveals sex-dimorphic map distances in a passerine bird

Linkage maps are lacking for many highly influential model organisms in evolutionary research, including all passerine birds. Consequently, their full potential as research models is severely hampered. Here, we provide a partial linkage map and give novel estimates of sex-specific recombination rates in a passerine bird, the great reed warbler (Acrocephalus arundinaceus). Linkage analysis of genotypic data at 51 autosomal microsatellites and seven markers on the Z-chromosome (one of the sex chromosomes) from an extended pedigree resulted in 12 linkage groups with 2-8 loci. A striking feature of the map was the pronounced sex-dimorphism: males had a substantially lower recombination rate than females, which resulted in a suppressed autosomal map in males (sum of linkage groups: 110.2 cM) compared to females (237.2 cM; female/male map ratio: 2.15). The sex-specific recombination rates will facilitate the building of a denser linkage map and cast light on hypotheses about sex-specific recombination rates.     

Moving beyond noninformative priors: why and how to choose weakly informative priors in Bayesian analyses

Throughout the last two decades, Bayesian statistical methods have proliferated throughout ecology and evolution. Numerous previous references established both philosophical and computational guidelines for implementing Bayesian methods. However, protocols for incorporating prior information, the defining characteristic of Bayesian philosophy, are nearly nonexistent in the ecological literature. Here, I hope to encourage the use of weakly informative priors in ecology and evolution by providing a ‘consumer's guide’ to weakly informative priors. The first section outlines three reasons why ecologists should abandon noninformative priors: 1) common flat priors are not always noninformative, 2) noninformative priors provide the same result as simpler frequentist methods, and 3) noninformative priors suffer from the same high type I and type M error rates as frequentist methods. The second section provides a guide for implementing informative priors, wherein I detail convenient ‘reference’ prior distributions for common statistical models (i.e. regression, ANOVA, hierarchical models). I then use simulations to visually demonstrate how informative priors influence posterior parameter estimates. With the guidelines provided here, I hope to encourage the use of weakly informative priors for Bayesian analyses in ecology. Ecologists can and should debate the appropriate form of prior information, but should consider weakly informative priors as the new ‘default’ prior for any Bayesian model.     

Robust modeling of additive and nonadditive variation with intuitive inclusion of expert knowledge

We propose a novel Bayesian approach that robustifies genomic modeling by leveraging expert knowledge (EK) through prior distributions. The central component is the hierarchical decomposition of phenotypic variation into additive and nonadditive genetic variation, which leads to an intuitive model parameterization that can be visualized as a tree. The edges of the tree represent ratios of variances, for example broad-sense heritability, which are quantities for which EK is natural to exist. Penalized complexity priors are defined for all edges of the tree in a bottom-up procedure that respects the model structure and incorporates EK through all levels. We investigate models with different sources of variation and compare the performance of different priors implementing varying amounts of EK in the context of plant breeding. A simulation study shows that the proposed priors implementing EK improve the robustness of genomic modeling and the selection of the genetically best individuals in a breeding program. We observe this improvement in both variety selection on genetic values and parent selection on additive values; the variety selection benefited the most. In a real case study, EK increases phenotype prediction accuracy for cases in which the standard maximum likelihood approach did not find optimal estimates for the variance components. Finally, we discuss the importance of EK priors for genomic modeling and breeding, and point to future research areas of easy-to-use and parsimonious priors in genomic modeling.     

Impact of prior specifications in ashrinkage-inducing Bayesian model for quantitative trait mapping and genomic prediction

Background

In quantitative trait mapping and genomic prediction, Bayesian variable selection methods have gained popularity in conjunction with the increase in marker data and computational resources. Whereas shrinkage-inducing methods are common tools in genomic prediction, rigorous decision making in mapping studies using such models is not well established and the robustness of posterior results is subject to misspecified assumptions because of weak biological prior evidence.

Methods

Here, we evaluate the impact of prior specifications in a shrinkage-based Bayesian variable selection method which is based on a mixture of uniform priors applied to genetic marker effects that we presented in a previous study. Unlike most other shrinkage approaches, the use of a mixture of uniform priors provides a coherent framework for inference based on Bayes factors. To evaluate the robustness of genetic association under varying prior specifications, Bayes factors are compared as signals of positive marker association, whereas genomic estimated breeding values are considered for genomic selection. The impact of specific prior specifications is reduced by calculation of combined estimates from multiple specifications. A Gibbs sampler is used to perform Markov chain Monte Carlo estimation (MCMC) and a generalized expectation-maximization algorithm as a faster alternative for maximum a posteriori point estimation. The performance of the method is evaluated by using two publicly available data examples: the simulated QTLMAS XII data set and a real data set from a population of pigs.

Results

Combined estimates of Bayes factors were very successful in identifying quantitative trait loci, and the ranking of Bayes factors was fairly stable among markers with positive signals of association under varying prior assumptions, but their magnitudes varied considerably. Genomic estimated breeding values using the mixture of uniform priors compared well to other approaches for both data sets and loss of accuracy with the generalized expectation-maximization algorithm was small as compared to that with MCMC.

Conclusions

Since no error-free method to specify priors is available for complex biological phenomena, exploring a wide variety of prior specifications and combining results provides some solution to this problem. For this purpose, the mixture of uniform priors approach is especially suitable, because it comprises a wide and flexible family of distributions and computationally intensive estimation can be carried out in a reasonable amount of time.     

Integrating biological knowledge into variable selection: an empirical Bayes approach with an application in cancer biology

ABSTRACT: BACKGROUND: An important question in the analysis of biochemical data is that of identifying subsets of molecular variables that may jointly influence a biological response. Statistical variable selection methods have been widely used for this purpose. In many settings, it may be important to incorporate ancillary biological information concerning the variables of interest. Pathway and network maps are one example of a source of such information. However, although ancillary information is increasingly available, it is not always clear how it should be used nor how it should be weighted in relation to primary data. RESULTS: We put forward an approach in which biological knowledge is incorporated using informative prior distributions over variable subsets, with prior information selected and weighted in an automated, objective manner using an empirical Bayes formulation. We employ continuous, linear models with interaction terms and exploit biochemically-motivated sparsity constraints to permit exact inference. We show an example of priors for pathway- and network-based information and illustrate our proposed method on both synthetic response data and by an application to cancer drug response data. Comparisons are also made to alternative Bayesian and frequentist penalised-likelihood methods for incorporating network-based information. CONCLUSIONS: The empirical Bayes method proposed here can aid prior elicitation for Bayesian variable selection studies and help to guard against mis-specification of priors. Empirical Bayes, together with the proposed pathway-based priors, results in an approach with a competitive variable selection performance. In addition, the overall procedure is fast, deterministic, and has very few user-set parameters, yet is capable of capturing interplay between molecular players. The approach presented is general and readily applicable in any setting with multiple sources of biological prior knowledge.     

Quantification of prior impact in terms of effective current sample size

Bayesian methods allow borrowing of historical information through prior distributions. The concept of prior effective sample size (prior ESS) facilitates quantification and communication of such prior information by equating it to a sample size. Prior information can arise from historical observations; thus, the traditional approach identifies the ESS with such a historical sample size. However, this measure is independent of newly observed data, and thus would not capture an actual “loss of information” induced by the prior in case of prior-data conflict. We build on a recent work to relate prior impact to the number of (virtual) samples from the current data model and introduce the effective current sample size (ECSS) of a prior, tailored to the application in Bayesian clinical trial designs. Special emphasis is put on robust mixture, power, and commensurate priors. We apply the approach to an adaptive design in which the number of recruited patients is adjusted depending on the effective sample size at an interim analysis. We argue that the ECSS is the appropriate measure in this case, as the aim is to save current (as opposed to historical) patients from recruitment. Furthermore, the ECSS can help overcome lack of consensus in the ESS assessment of mixture priors and can, more broadly, provide further insights into the impact of priors. An R package accompanies the paper.     

The neglected tool in the Bayesian ecologist's shed: a case study testing informative priors' effect on model accuracy

Despite benefits for precision, ecologists rarely use informative priors. One reason that ecologists may prefer vague priors is the perception that informative priors reduce accuracy. To date, no ecological study has empirically evaluated data‐derived informative priors' effects on precision and accuracy. To determine the impacts of priors, we evaluated mortality models for tree species using data from a forest dynamics plot in Thailand. Half the models used vague priors, and the remaining half had informative priors. We found precision was greater when using informative priors, but effects on accuracy were more variable. In some cases, prior information improved accuracy, while in others, it was reduced. On average, models with informative priors were no more or less accurate than models without. Our analyses provide a detailed case study on the simultaneous effect of prior information on precision and accuracy and demonstrate that when priors are specified appropriately, they lead to greater precision without systematically reducing model accuracy.