The remaining markers are specified to be random draws from a Student t distribution with key hyperparameters being degrees of freedom v and scale s2.The reliability of whole-genome prediction models (WGP) based on using high-density single nucleotide polymorphism (SNP) panels critically depends on proper specification of key hyperparameters. A currently popular WGP model labeled BayesB specifies a hyperparameter s2 is drawn by a Gibbs step and v is drawn by a MH step. The second strategy, termed UNIMH, substitutes MH for Gibbs when drawing s2 and further collapses or marginalizes the full conditional density of v. The third strategy, termed BIVMH, is based on jointly drawing the two hyperparameters in a bivariate MH step. We also tested the effect of misspecification of s2 for its effect on accuracy of genomic estimated breeding values (GEBV), yet allowing for inference on the other hyperparameters.We consider three alternative Markov chain Monte Carlo (MCMC) approaches based on the use of Metropolis-Hastings (MH) to estimate these key hyperparameters. The first approach, termed DFMH, is based on a previously published strategy for which P < 0.05) computational efficiencies for estimating v and s2 than DFMH in BayesA (π = 1) and BayesB implementations. We drew similar conclusions based on an analysis of the public domain heterogeneous stock mice data. We also determined significant drops (P < 0.01) in accuracies of GEBV under BayesA by overspecifying s2, whereas BayesB was more robust to such misspecifications. However, understating s2 was compensated by counterbalancing inferences on v in BayesA and BayesB, and on π in BayesB.The UNIMH and BIVMH strategies had significantly greater  contains supplementary material, which is available to authorized users. Note that Meuwissen et al. [π in the opposite manner, whereas we prefer the definition of π that is used most commonly in the variable selection literature [π is typically believed to be the proportion of SNPs that are associated or in linkage disequilibrium (LD) with causal variants, although this interpretation is somewhat complicated by varying degrees of LD across the genome. These hyperparameters  are relevant in that they partly characterize the genetic architecture of traits, but more importantly depend upon the density or characteristics of the SNPs used in the analyses [Genomic predictions based on the use of high-density single nucleotide polymorphisms (SNPs) distributed across the genome have been increasingly adopted for animal and plant breeding programs. Parametric Bayesian methods have become particularly popular, most notably BayesA and BayesB, which were first introduced by Meuwissen et al. . BayesB analyses .et al. [v and s2 at 4.2 and 0.04, respectively, in the BayesB model of their simulation study. In various simulation and/or empirical data analysis scenarios, it seems necessary to carefully tune key hyperparameters, since their optimal specifications should depend upon several key factors. For example, one could loosely interpret s2 as a “typical” SNP effect variance component in a BayesA model , such that its value should be inversely related to the number of SNPs used in an analysis. We have demonstrated this previously [Key hyperparameters in BayesA or BayesB whole-genome prediction (WGP) models have been arbitrarily specified in a wide selection of WGP studies. Meuwissen et al.  used argeviously .v and s2 across MCMC cycles is generally so large that these two hyperparameters are nearly non-identifiable from each other [s2, since the conditional conjugate is a Gamma prior, whereas a Metropolis Hastings (MH) update was used on sampling from the FCD of v, since it is not recognizable [v using MH), and it is the control or reference strategy in this paper.Inference in BayesA- or BayesB-like models is conducted using either Markov chain Monte Carlo (MCMC) methods for fully Bayesian inference or using faster, albeit approximate, methods based on the use of the expectation maximization (EM) algorithm or its various derivatives . Unfortuch other ,8. A MCMch other  who applgnizable . We labeComputational efficiency in MCMC schemes is related to the degree of mixing or autocorrelation between subsequent samples of the same parameter. The most popular metric for determining the degree of mixing or autocorrelation for a fixed number of MCMC cycles on a particular parameter is the effective sample size (ESS), which can be readily computed using software packages like CODA . The ESSv and s2 are both log-transformed, the resulting scatterplot of MCMC samples of the transformed variables against each other often demonstrates a more linear relationship, although we have also observed counterexamples, where a linear relationship is maintained on the original scale. At any rate, a change of variable might facilitate potentially more efficient MCMC sampling strategies based on multivariate Gaussian proposal densities, for example, in Metropolis sampling schemes [We surmised that, compared to the use of DFMH, there may be a number of MCMC strategies that could improve the computational efficiency of inferring upon key hyperparameters in a BayesA or BayesB WGP model. Furthermore, the efficiency of any strategy could markedly depend on the choice of an appropriate scale for the parameters being sampled. For example, a highly nonlinear relationship between two variables can often be rendered linear by transforming either one or both of the corresponding parameters. When  schemes .s2 on the accuracy of genomic estimated breeding values (GEBV) or even whether s2 could be reasonably extrapolated based on analyses derived from other marker densities.There were two primary objectives in this study. First, we wanted to explore alternative strategies to improve the computational efficiency of estimating hyperparameters in BayesA and BayesB WGP models. Second, given the prevalent practice of specifying rather than estimating these hyperparameters, we wanted to assess the impact of misspecifying The WGP model used for comparison of the various computational strategies and/or hyperparameter specifications can be denoted as follows:yi is the phenotype for animal i , β is a vector of fixed effects such that yi to β, zij is the genotype covariate for SNP j on animal i, coded as either 0, 1, or 2 copies of a reference allele for SNP j on animal i, gj is the random effect for SNP j, and ei is the residual. We assume where zij for each SNP genotype j as a deviation from that SNP average; i.e. using gj}; however, we assumed that this recoding could only help to facilitate mixing on all other parameters whose FCD are functions of {gj}. Hence, we applied this strategy for all computational efficiency comparisons considered in this paper.Stranden and Christensen  recentlyistensen  further π, and a spike at 0 with probability (1 − π). Here, π loosely represents the proportion of SNPs having associated genetic effects on the phenotype. Recall that BayesA is a special case of BayesB when π = 1. Following Yang and Tempelman [ν ~ p(ν) ∝ (ν + 1)− 2 in both BayesA and BayesB, the Gelman prior [s2 ~ χ− 2 in BayesA, a proper but diffuse conjugate prior s2 ~ Gamma such that αs = 0.1 and rate parameter βs = 0.1 in BayesB; more informative alternative specifications could be provided for s2 in either model as desired. Finally, we specify π ~ p = Beta with απ = 1 and βπ = 8 in BayesB. A Beta prior is equivalent to specifying a prior mean of απ/(απ + βπ) based on a prior sample size of απ + βπ. For all three computational strategies that we describe subsequently, we adapt the same commonly used MCMC strategies for sampling from all parameters or random variables as outlined by Meuwissen et al. [v, s2, and π that Meuwissen et al. [π in Bayes B was deemed to be Beta, where m1 denotes the number of non-zero gj sampled for that MCMC cycle [The prior distribution of empelman , we specan prior  s2 ~ χ− mentclass2pt{minimmentclass2pt{minimMC cycle . We now v does not have a recognizable form; hence sampling from this FCD requires a strategy other than a Gibbs step. Here, we used the MH algorithm to sample from the FCD of v, drawing from our experiences in various other applications [ξ = log(ν), ensuring that the FCD of ξ takes into account the Jacobian of the transformation from v to ξ [See Additional file ξ can conceptually be defined anywhere on the continuous real line, we believe that this transformation better justifies the use of a Gaussian proposal density centered on the value of ξ from the previous MCMC cycle; i.e., a random walk MH step [t proposal density could be used as well. During the first half of burn-in, we adaptively tune the variance of this proposal density such that the MH acceptance ratios are intermediate , adapting the strategy described by Muller [s2 is Gamma, provided that a conditionally conjugate Gamma prior is used. Hence, for DFMH, we sampled v, using the described MH update, and s2 with a Gibbs update. To facilitate even better the mixing of v through the joint posterior density, given that the MH acceptance rates are potentially less than 50%, we drew 10 MH samples for v per MCMC cycle conditional on all other unknowns. Further details on the DFMH strategy are provided in [See Additional file This strategy, which we designate as DFMH, closely follows Yi and Xu . The FCDications ,16-18. M MH step ; alternay Muller  and in ay Muller ,22. The y Muller  demonstrv but also use MH to sample from s2. However, we also embed an additional strategy, whereby the FCD for both v and s2 is based on integrating out t distribution, rather than a scaled inverted chi-square distribution mixture of normals. This strategy leads to a “collapsed” sampler, which is also known to facilitate faster MCMC mixing [v and s2.Metropolis Hastings sampling, if properly tuned with good proposal densities and intermediate acceptance rates, can often lead to faster mixing and hence greater MCMC efficiency relative to Gibbs sampling. This is because MH sampling typically proposes bigger jumps throughout the posterior density compared to the use of Gibbs sampling. Hence, we propose a second strategy, UNIMH, whereby we again use MH to sample from C mixing  since unv in DFMH, we sample s2 by invoking a change of variable to its logarithm ), allowing for the Jacobian of the transformation from s2 to ψ, and use a random walk MH algorithm based on a Gaussian proposal density for ψ. Similar to what was done for v, the variance of this proposal density was only tuned for intermediate acceptance rates during the first half of burn-in, to ensure a properly convergent MCMC chain. For reasons similar to those provided for DFMH, 10 MH samples per MCMC cycle were specified for sampling v and s2 in UNIMH. Further details on this strategy are provided in [See Additional file As with v and s2 can be high; hence, it might be advantageous to jointly sample these two parameters with a bivariate random walk MH sampler, as demonstrated with a different application by Ntzoufras [v) and log(s2) from their respective FCD using the UNIMH strategy previously described, fine-tuning the variances of the two separate Gaussian proposal densities to ensure that MH acceptance rates fall between 25 and 75%. In Stage 2, we sampled log (v) and log (s2) using UNIMH, fixing the variances of their respective proposal densities to those values tuned at the end of Stage 1, while computing the empirical correlation between the samples of log (v) and log (s2) drawn from within the same cycle. In Stage 3, log (v) and log (s2) were jointly sampled using a bivariate Gaussian proposal density with variances based on those tuned at the end of Stage 1 and a covariance based on the correlation computed from Stage 2. During Stage 3, we further fine-tuned the proposal variances to ensure intermediate acceptance rates for joint samples of log (v) and log (s2), with the proposal covariance based on the same correlation derived in Stage 3. In Stage 4, we drew samples using the same joint MH random walk from the newly tuned bivariate Gaussian proposal density in Stage 3 but without further tuning, in order to ensure a properly convergent MCMC chain before the end of the burn-in period. At the end of Stage 4, and hence burn-in, we saved samples for the hyperparameters of v and s2  for MCMC-based fully Bayesian inference. As with UNIMH, 10 MH samples were drawn for each of v and s2 at each stage per MCMC cycle. Details on this strategy are provided in [See Additional file As previously noted, the posterior correlation between tzoufras . Hence, th locus on this chromosome was heterozygous , whereas the remaining 80 000 loci were completely monomorphic, similar to the simulation in Coster et al. [−5 per locus per generation.In order to test the computational efficiency of the three sampling strategies, DFMH, UNIMH and BIVMH, under the BayesA and BayesB modeling specifications, we simulated 15 replicated datasets using the HaploSim package in R . The simr et al. . Individet al. [In Generation 6000, random matings were used to increase the population size to 1000 individuals in Generation 6001. In Generation 6001, we deleted loci with a minor allele frequency (MAF) less than 0.05 and randomly selected 30 from the remaining loci to be quantitative trait loci (QTL). Following Meuwissen th and 10th SNP for inclusion in analyses in order to consider the effect of different marker densities; i.e., high (average of 2394 SNPs), medium (average of 598 SNPs), and low (average of 239 SNPs) across the 15 replicates; these marker densities subsequently influence the average pairwise LD measures between adjacent markers. We compared the computational efficiencies between the three strategies for inferring on key hyperparameters  under these three marker densities. We ignored additionally fitting polygenic effects [After discarding SNPs with a MAF less than 0.05, we then selected every single, 4 effects  for all We compared the computational efficiencies of the three MCMC strategies for each replicated dataset, considering each marker density. Since computational efficiency pertains to a particular hyperparameter, it was defined as the effective sample size (ESS) for the post-burn-in MCMC cycles divided by total CPU time, i.e., ESS/CPU recorded in number per second. That is, the greater ESS/CPU, the greater the computational efficiency for inferring the posterior density of that particular hyperparameter.th marker). Each scenario was based on setting s2 as an arbitrary multiplicative constant of the average posterior mean at the medium marker density (v and π where applicable) were inferred upon. Values for s2 for these five scenarios were set to 1) s2 = s2 = 0.1 s2 = 0.01 s2 = 10 s2 = 100 v and π where applicable) were still inferred upon within each of these scenarios in order to remove confounding effects of their misspecification.Recognizing that many researchers do not infer some or even all hyperparameters in WGP models, because of the perceived inferential challenges, we thought it was important to assess the impact of misspecification of these parameters on the accuracy of genomic prediction. Using the same simulated data as described previously, we focused on five scenarios, all at the medium marker density  were inferred upon as well.We also investigated if it was possible to specify a roughly good working value for reported . For exai was defined as gj and TBV is defined as before. Hypothesis testing between accuracies of different methods were based on blocking on replicated datasets, using a Wilcoxon signed rank test. Furthermore, every single MCMC analysis was based on 120 000 cycles during burn-in, followed by 400 000 cycles after burn-in, saving every 10 for a total of 40 000 samples.In all cases, accuracy of prediction was defined as the correlation between GEBV and TBV, where GEBV for animal r2 = 0.6 [th sample of 800 000 MCMC cycles, after a burn-in period of 120 000 samples.In the real dataset, 2296 mice were genotyped for 12 147 SNPs with a high pairwise LD of r2 = 0.6 . After fr2 = 0.6 , there wr2 = 0.6 , we alsor2 = 0.6  for the th and 10th SNP for inclusion, the average LD measure (r2) between adjacent SNPs, was 0.32, 0.24 and 0.17 for the three marker densities over the 15 replicated datasets. Pairwise scatterplots used to assess quality control on our procedures are in [See Additional file s2 using each of the three sampling strategies DFMH, UNIMH and BIVMH were compared under both BayesA [See Additional file s2 decreased as the marker density increased and that estimates derived from BayesB were generally one order of magnitude greater than those in BayesA. Furthermore, posterior medians of π generally increased [See Additional file v generally decreased as marker density increased, for both BayesA [See Additional file By selecting every single, 4measure r between measure r between measure r between measure r between measure r between measure r between s2, π and v under the three computational strategies for each of the three marker densities indicated very good agreement, as expected, since the joint posterior density should not differ regardless of whether one uses DFMH, UNIMH or BIVMH, provided that the prior specifications are identical. Better agreement between the three algorithms was generally found for BayesA than for BayesB inference on s2 and v. This was as anticipated, since BayesB requires inference on one more hyperparameter (π) and hence potentially greater Monte Carlo error for the same number of cycles; furthermore, inference on s2 and v is essentially based on information from only non-zero SNP effects, which is appreciably less than the effective total number of SNPs used for estimating s2 and v in BayesA [Overall, pairwise scatterplots of the posterior medians of n BayesA . Moreovev and s2, respectively, under the BayesA model. In all cases, ESS/CPU increased with lower marker densities, since that leads to an inferential situation of more data information per marker  or a greater level of determinedness [P < 0.05). For v at the high marker density (r2 = 0.32), UNIMH had higher ESS/CPU than BIVMH (P < 0.05), whereas for s2 at the medium marker density (r2 = 0.24), median ESS/CPU was higher for BIVMH than UNIMH (P < 0.05). Interestingly, the ratios of computing efficiencies for both hyperparameters between UNIMH and BIVMH with DFMH hovered between 3 and 6 across marker densities. Efficiencies for the three alternative sampling strategies were also compared under the BayesB model for v, s2, and π (Table P < 0.05), with values several times greater for v and s2, whereas the computing efficiency advantage was somewhat less pronounced for π.Table inedness . Furthers2. We assessed the impact on accuracy of GEBV based on setting s2 equal to a wide range of multiples (0.01× to 100×) of the average posterior mean  across the 15 replicates under the medium marker density. For BayesA . For BayesB . For the high-density panel that included four times as many markers, we specified s2 = 0.25 s2 = 2.5 s2 at the higher marker density using BayesA (P = 0.04).We also wondered if estimates of s2, in [See Additional file v, and for π in [See Additional file v between DFMH versus UNIMH and BIVMH under a BayesB model. Using the 950-SNP panel, the ESS/CPU for s2 was up to 3 times greater using UNIMH and BIVMH compared to DFMH, whereas the ESS/CPU for v was up to 50 times greater in UNIMH and BIVMH compared to DFMH. In other words, UNIMH and BIVMH had greater computing efficiency. For the 1900-SNP panel  during the MCMC chain; in order words, the MCMC chain mixed very poorly for DFMH.In all cases, it should be noted that the posterior medians were generally in better agreement between algorithms than were the posterior means, especially for rs Table . That isPairwise scatterplots of posterior means of SNP effects and GEBV between the three computing strategies are also provided in Additional file v, s2 and π) that partly characterize the genetic architecture and/or marker LD or densities in BayesA or BayesB WGP models. This default action may be in part due to the high posterior correlation that exists between some of these hyperparameters, in particular, v and s2 [Most researchers have not inferred key hyperparameters  to infer either v or s2 but not necessarily both, given this compensation effect, in conjunction with the high degree of posterior correlation between the two hyperparameters. Note, for example, that the BayesD-π model strategy of Habier et al. [v is fixed while inferring upon s2 and π, might not be greatly different in its effect from fixing s2 and inferring upon v and π, as we did in our misspecification study. Our results are consistent with our previous work [s2 are expected to be larger with BayesB than with BayesA models since genetic variability is distributed over fewer SNPs in BayesB if π < < 1; for similar reasons, one would also expect smaller estimates of s2 in higher density SNP panels.Gianola  and Leheet al.  have recet al.  provided [et al.  demonstrr et al.  arbitrarous work , in whics2 with a MH on a collapsed FCD, or joint sampling of s2 and v with a bivariate MH step (BIVMH), lead to substantial improvements in ESS/CPU compared to the more common strategy based on a Gibbs step for s2 (UNIMH). We concede that our investigation is not exhaustive with respect to assessing all possible strategies to improve computing efficiency in these models; in fact, a hybrid approach may exist that combines some or all of the three presented sampling strategies that might be even computationally more efficient. Deviations of MH sampling such as Langevin-Hastings could also have been explored and assessed here, although its advantage relative to MH sampling has not been too convincing in other animal breeding models [v) and log (s2). However, we determined that this matrix is positive definite generally only when v < 50, thereby negating its use in this way. Recently, non-MCMC (i.e. expectation-maximization) schemes have been increasingly popular. However, it is often not straightforward to estimate key hyperparameters in these implementations [We considered two alternative sampling strategies to DFMH, each involving the use of MH, in an attempt to improve the computational efficiency of WGP analyses, as measured by the ratio ESS/CPU. Using simulation studies and empirical data analyses, we demonstrated that strategies that are more heavily based on MH and/or collapsed sampling had better computing efficiencies than the DFMH procedure first advocated by Yi and Xu . Simple g models ,12. In wntations .We also examined the components of computing efficiency; i.e., ESS and CPU/cycle in seconds separately for each parameter in both models and under all three strategies (results not reported). As anticipated, DFMH was computationally less expensive in terms of CPU/cycle compared to the proposed strategies UNIMH and BIVMH; however, the ESS were such that the values for UNIMH and BIVMH generally far exceeded those for DFMH for the same number of MCMC cycles. What was particularly worrying was how quickly the ESS/CPU measures degraded with increasing marker densities, which suggests that higher density panels, without concomitant increases in sample size, lead to analyses that require not only greater CPU/cycle but also a greater number of MCMC cycles to ensure that ESS values are sufficiently large to ensure reliable posterior inference on these hyperparameters.gj , as well as on fixed effects . In the simulation study and in the mouse data analysis, we saved all MCMC samples from three randomly chosen gj , three randomly chosen gj and μ were substantially larger compared to analyses for which these allele coding strategies were not implemented (results not shown).We investigated the impact of these different computing strategies on GEBV and SNP effects mentclass2pt{minimv, partly because MCMC draws of v often got stuck in high values, yet rather low probability areas of the posterior density. This situation thereby rendered inferences on v that were less reliable than with UNIMH and BIVMH, which facilitated good mixing throughout the joint posterior density. Since UNIMH and BIVMH are not consistently better or worse than each other in terms of computing efficiency, we recommend the use of UNIMH for its greater simplicity of implementation.The joint posterior density was the same for all three algorithms; this was readily established in the simulation study but was less clear in our analysis of mice weights, where the data structure and statistical model were considerably more complex. In that situation, we noted that DFMH was particularly unstable for MCMC inference on s2 appeared to have particularly deleterious effects on accuracy of genomic prediction in BayesA models, although no such effect was observed in BayesB models, which is likely due to the counteracting influence of π. This may be a key reason why we did not observe any differences in accuracies of GEBV in BayesB between various specifications of s2 in Figure s2 is understated, the estimate of v should decrease accordingly to compensate, so that there is a good deal of flexibility in maintaining a narrow range in s2 is overstated, then there is very little flexibility to accordingly bring down v , since s2. We hypothesize that this is the reason why understating the value of s2 is less serious than overstating it, at least for BayesA, as further indicated by our results, provided that the other hyperparameter v is inferred upon. Furthermore, specifying a very large value for s2 could create overfitting problems.Over-specifying s2) is strongly linked to the other hyperparameters (v and/or π) being inferred upon. However, all three hyperparameters are often arbitrarily specified in most BayesA or BayesB analyses. We also determined that it may be reasonable to consider specifying values for s2 for one marker density based on a previous estimate from another marker density by taking into account the inverse relationship between s2 and marker density. This might be particularly appealing, given that MCMC mixing tends to be substantially faster and to better behave for low-density panels. Hence, a reasonable strategy might be to use such analyses to extrapolate s2 specifications for higher densities, where the MCMC mixing behavior of s2 might tend to be more problematic. Or, at the very least, one could base an informative prior density for s2 based on an extrapolation from the use of a lower density panel. Perez and de los Campos [Our work on misspecification of one hyperparameter , 17 950 (598×30) or 71 820 (2394×30) SNPs for low, medium, and high densities on a 30 M genome. Although the ESS should also be roughly the same between these two scenarios, ESS/CPU would be expected to be substantially smaller in the larger scale analyses because of larger m and n; however, this needs to be tested more extensively.Our simulation study and empirical study admittedly seem rather small compared to more typical analyses based on tens or even hundreds of thousands of SNPs. For example, our simulation study was based on the use of an average of  Goddard  demonstrv progressively decreased with increasing marker densities, whether in the simulation study or within the data application. While we did not specifically investigate the effect of misspecification of v per se, we recognize that work has been indirectly extensively addressed in the context of BayesA (finite v) versus GBLUP (v → ∞ ) comparisons [et al. [v close to 1 can lead to a greater GEBV accuracy compared to default specifications (4 ≤ v ≤ 5) that are typically applied in BayesA or BayesB analyses [v as opposed to setting v to an arbitrary value would be context-specific, as shown with s2; nevertheless, we also anticipate that misspecification of v would not have such dramatic consequences given the smaller differences in performance often seen between BayesA and GBLUP models.We noted that estimates of parisons -38. In f [et al.  demonstranalyses ,8,32. Wes2 in BayesA should not be much different in magnitude from the variance component for SNP effects in a GBLUP [At any rate, we provide evidence that these hyperparameters should not be arbitrarily specified in BayesA or BayesB models. We anticipate that these issues are also relevant to determine tuning parameters for various nonparametric approaches as well. However, we realize that the computational challenges may be huge for panels with marker densities that far exceed those that we considered in this paper. At the very least, some hyperparameters should be specified based on simple methods of moments-like (i.e. heritability-based) determinations ,7 or oth a GBLUP  analysisv and scale parameter s2 and using collapsed representations of FCD can improve MCMC efficiency for inference on all hyperparameters. Even separate univariate Metropolis-Hastings draws on v and s2 are substantially more efficient than using Gibbs sampling of s2. Overspecification rather than underspecification of the key hyperparameter s2 can adversely affect accuracy of GEBV under a BayesA model, even when v is estimated. Conversely, the BayesB model is more robust to misspecification of s2 provided that inference on π, the probability of association, and v are also inferred upon.In hierarchical Bayesian WGP models such as BayesA and BayesB, jointly drawing the degrees of freedom


Rare variants may, in part, explain some of the hereditability missing in current genome-wide association studies. Many gene-based rare-variant analysis approaches proposed in recent years are aimed at population-based samples, although analysis strategies for family-based samples are clearly warranted since the family-based design has the potential to enhance our ability to enrich for rare causal variants. We have recently developed the generalized least squares, sequence kernel association test, or GLS-SKAT, approach for the rare-variant analyses in family samples, in which the kinship matrix that was computed from the high dimension genetic data was used to decorrelate the family structure. We then applied the SKAT-O approach for gene-/region-based inference in the decorrelated data. In this study, we applied this GLS-SKAT method to the systolic blood pressure data in the simulated family sample distributed by the Genetic Analysis Workshop 18. We compared the GLS-SKAT approach to the rare-variant analysis approach implemented in family-based association test-v1 and demonstrated that the GLS-SKAT approach provides superior power and good control of type I error rate. Rare variants may, in part, explain some of the missing heritability in current genome-wide association studies . Many raIndependent individuals in the GAW18 data were extracted based on the list of unrelated individuals provided by the GAW18 organizers. For each simulation, a linear regression model was built based on the independent individuals using covariates including gender, age, age square, antihypertension medicine usage, and smoking. The linear regression model built from the unrelated subjects was then projected to the correlated subjects and the residuals were calculated for all individuals in GAW18.Because the GLS-SKAT approach  can implicitly control for population substructure, neither ethnicity nor principal components were included as covariates in the residual calculation.The software EMMAX  was usedGLS transformation was performed using R code developed in-house. The GLS transformation has been described in detail elsewhere (manuscript in preparation). Briefly, a transformation matrix was calculated as the inverse of the decomposition of the kinship matrix. Then, this transformation matrix was used to decorrelate the family data by multiplying it with both the phenotype and genotypes matrices. The residual of SBP after controlling for the covariates was used as the phenotype. Given the time limitation, only simulated chromosome 3 genotype data was used as recommended by the GAW organizer. All variants, both common and rare, were transformed in this step.For gene-based inference, the SKAT-O approach  was applThis GLS-SKAT procedure was applied to replicates 1 to 100 in the GAW18 simulated data. Power and type I error rate were calculated based on the 100 simulated data sets. To achieve a family-wise error rate of 0.05 after applying the Bonferroni correction, an alpha level of 4.0E-5 was utilized in the power calculation to account for the 1247 gene regions on chromosome 3.As a comparison we analyzed the same 100 data sets using the family-based association test (FBAT) test for rare variants (described in the updated FBAT version 2.04). There are 2 versions of the FBAT test for rare variants: FBAT-v0, which weights all the variants equally, and FBAT-v1, which weights the variants inversely to their MAF. Here we used the FBAT-v1 in the analysis. For consistency, both rare and common variants also were analyzed via the FBAT-v1 test.We were blind to the GAW18 answer sheet when we carried out the analyses. We compared the results to the answer sheet and carried out power calculations only after the GAW18 meeting.Detailed description of the GAW data can be found in the work of Almasy et al . In the Figure To calculate the type I error rate, we pooled the statistics across all of the 100 simulated data sets after excluding genes located within 1 megabit (Mb) to any of the causal SNPs on chromosome 3. The type I error rate was calculated for both the GLS-SKAT and FBAT-v1 analyses , we have followed the SKAT-O approach  for geneApplication of the GLS-SKAT approach to GAW18 simulated family data showed that the GLS-SKAT approach does not have inflated type I error rates with complex family structures, indicating that the GLS-based framework can properly decorrelate family data. Moreover, in all analyses when using GLS-SKAT, this was observed without incorporating any covariates related to ethnicity or population substructure, indicating that, by accounting for the pairwise relatedness, the kinship matrix can be used to control for both major scale population structure  and the much finer scale substructures . This is consistent with what has been observed in previous investigations ,14.By incorporating all family members in the analysis, the GLS-SKAT approach makes full use of all phenotype and genotype data. This is an advantage compared to the FBAT-based test, in which only trios were included. Moreover, because the FBAT-based analysis is based on transmission disequilibrium, the phenotypes of parents do not contribute to the test statistics. This leads to additional loss of information and consequently lower power, as observed in this study.Another advantage of the GLS-SKAT approach is that it does not rely on the known family structure because the kinship matrix is estimated using the high dimensional genetics data, making it robust to the potential errors of unidentified cryptic family structure. This is particularly important for samples with complex large pedigrees, in which the cleaning and correction of the self-reported family structure using genetic data are cumbersome.The GLS approach is similar in spirit to the previously proposed EMMAX and GRAMMAR approach ,14, as wIn the application of the GLS approach to GAW18 data, the transformation was performed using the kinship matrix after controlling for all covariates that contribute to SBP in the simulation model. However, in practice there could be residual correlation as a result of the structure of unknown covariates within the family, as one reviewer pointed out. This potential problem can be solved by using a covariance matrix instead of the projection matrix in the GLS transformation, in which the covariance matrix is a weighted average of the kinship matrix and an identical matrix with the weight being the calculated hereditability. This is similar to the approach used in GRAMMAR-GAMMA .In summary, we applied the GLS-SKAT approach to GAW18 simulated SBP data. The results demonstrated that the GLS-SKAT approach can properly control for both population and family structure using the kinship matrix estimated from high dimensional genetics data. By using all the available phenotype and genotype information, we demonstrated that the GLS-SKAT approach has a clear advantage in terms of power compared to the FBAT-based test.The authors declare that they have no competing interests.DL conceived and performed the analysis and drafted the manuscript. JIR and XG supervised the whole study and helped to prepare the manuscript. All authors read and approved the final manuscript.


MutS Homolog 1 (MSH1) encodes a plant-specific protein that functions in mitochondria and chloroplasts. We showed previously that disruption or suppression of the MSH1 gene results in a process of developmental reprogramming that is heritable and non-genetic in subsequent generations. In Arabidopsis, this developmental reprogramming process is accompanied by striking changes in gene expression of organellar and stress response genes. This developmentally reprogrammed state, when used in crossing, results in a range of variation for plant growth potential. Here we investigate the implications of MSH1 modulation in a crop species. We found that MSH1-mediated phenotypic variation in Sorghum bicolor is heritable and potentially valuable for crop breeding. We observed phenotypic variation for grain yield, plant height, flowering time, panicle architecture, and above-ground biomass. Focusing on grain yield and plant height, we found some lines that appeared to respond to selection. Based on amenability of this system to implementation in a range of crops, and the scope of phenotypic variation that is derived, our results suggest that MSH1 suppression provides a novel approach for breeding in crops. One increasingly problematic threat to plant improvement is the depletion of natural stores of genetic diversity for most of our major crop species For years, breeders have speculated that non-genetic variation plays a key role in conventional crop improvement strategies MSH1 is a plant-specific gene that encodes a mitochondrial and plastid-localized protein MSH1 appears to be influenced by environmental stress msh1 mutants are characterized by variable phenotypes including dwarfing, variegation, delay in maturity transition and flowering, altered branching, and woody growth with aerial rosettes at short day length growth MSH1 in crop plants, including tomato, soybean, tobacco, millet and sorghum produces a similar MSH1-dr phenotypic range in each that is subsequently inherited independent of the RNAi transgene Here we investigate the consequences of incorporating the MSH1-dr condition to plant selection, using sorghum as a model. We show that crossing with a transgene-null MSH1-dr line produces an unexpected range of phenotypic variation that is both heritable and responsive to selection. This variation appears to be stable over at least four generations. We also show evidence of line×environment interactions. Finally, we demonstrate gains in grain yield over two generations of selection, suggesting that this non-genetic variation may prove valuable for agricultural production as a potential crop breeding strategy.3 individuals displaying the MSH1-dr phenotype but null for the MSH1-RNAi transgene were used as females in crosses to wild type inbred Tx430 to derive F1 seed. Another three T3 individuals were used as males in the reciprocal crosses to Tx430. The number of F1 plants derived from each cross ranged from 5 to 19 individuals. Parents and F1 progeny were grown under greenhouse conditions on a 14 hr/10 hr day-night cycle with 28°C/22°C day-night temperatures. Self-pollinated seed of F1 plants was harvested individually to generate corresponding F2 families.Sorghum MSH1-dr plants used in these experiments were derived as described in 2 and fertilized according to standard growing practices. The 2010 field experiment was used to propagate F2 lines, and contained F2 and wild type Tx430. The 2011 field experiment contained F2, F3, and F4 lines randomized across seven blocks with 28 rows per block  and two field replicates. Replicates were augmented with wild type Tx430 .In all field plots, plants were thinned to a final density of 15 plants/m2 based on final plant density, with 2–3 such measurements taken per row. For comparison of panicle yield distributions in F2 versus in wild type Tx430, individual panicle grain yield  was used. For flowering time, plant height, and rachis length, measurements were taken on individual plants. For each dry biomass measurement, three fully dried plants were pooled together then converted to grams/plant. Plants showing the DR phenotype were not included in phenotypic variation analysis.For estimating grain yield, threshed panicles from three plants were pooled and converted to grams/m2.The 2012 multi-location experiment included Lincoln, NE  and Mead, NE  sites, which received 178 mm and 158 mm of precipitation over the growing season, respectively. Within each location, lines were grown in two-row plots arranged in a randomized complete block design with two replicates. For this experiment, grain yield was estimated by taking threshed panicles from a meter-length area of each row and converting to grams/mijky = μ+iα+kr+(b/r)jk+ijkε where ijky is the trait response, μ is the population mean, iα is the effect of line i, kr is the effect of replicate k, (b/r)jk is the effect of block j nested within replicate k, and ijkε is the residual error. For evaluations over multiple environments, mean phenotypic values and confidence intervals for each line were estimated using the linear mixed model ijkmy = μ+iα+me+(r/e)km+(b/r/e)jkm+(imαe)+ijkmε where ijkmy is the trait response, μ is the population mean, iα is the effect of line i, me is the effect of environment m, (r/e)km is the effect of replicate k nested within environment m, (b/r/e)jkm is the effect of block j nested within replicate k of environment m, (imαe) is the interaction between line i and environment m, and ijkmε is the residual. Line, environment, and line×environment effects were treated as fixed while block and replicate effects were treated as random. Models were fit by restricted maximum likelihood using the R package “nlme” 4 models for plant height and biomass excluded lines exhibiting mixed heights to avoid heteroscedasticity.For evaluations in a single environment, mean phenotypic values and confidence intervals for each line were estimated using the linear mixed model 5′-GTGTACT CATCTGGATCTGTATTG-3′ and RNAi-R 5′-GGTTGAGGAGCCTGAATCTCTGAAC-3′. Positive and negative controls were included from a confirmed transgenic line and wild type Tx430, respectively.PCR assay for MSH1-RNAi transgene presence in sorghum materials used primers RNAi-F SSR marker analysis used SSR primers that were developed and mapped previously 4 lines and wild type Tx430 was carried out at the Monsanto Applied Genotyping Labs . The platform used was an exclusive custom-designed Sorghum bicolor Infinium high-density beadchip containing 1,885 internally validated SNP markers.Leaf tissue sample was collected from plants grown under controlled greenhouse conditions three weeks after germination. Genomic DNA was extracted from freeze-dried leaf tissue and processed following manufacturer's recommendations prior to Infinium beadchip hybridization . The genotyping of five FFor the six samples, 107 of the 1,885 SNP markers, ca 5.68%, provided invalid data due to one of the following: low marker signal intensity, marker failed data QC, or unscorable allele calls. The remaining 1,778 SNP markers were used for the analysis. These 1,778 SNP markers are distributed across all 10 sorghum chromosomes with genome coverage approximating 90%. The number of heterozygotes (# Het) and percentage of heterozygotes (% Het) were calculated based on the 1778 SNP markers.MSH1 +/+ individuals that retained the characteristic msh1 phenotype despite having normal MSH1 transcript levels Previously, we described MSH1-RNAi lines displaying numerous physiological changes, a condition of developmental reprogramming that was termed MSH1-dr 1 progeny no longer showed the dwarfed, tillering, and late flowering phenotype; instead, many of the plants grew taller and produced more seed than the wild type. This was repeatedly observed in F1 populations derived from nine separate crosses, three of which used an MSH1-dr plant as the pollen donor To investigate the mechanism of inheritance, we performed reciprocal crosses in sorghum of MSH1-dr individuals to their wild type counterpart. 1 generation from either direct or reciprocal crosses argues against the observed phenotypes in this sorghum material being inherited via cytoplasmic organellar genomes. Analogously generated crosses in Arabidopsis with msh1 point or T-DNA insertion mutations also display enhanced vigor; in other species, including tomato, soybean and tobacco, heritable MSH1-dr phenotypes also persist despite restored MSH1 expression following RNAi silencing, and crosses in those species to their respective wild type counterparts similarly produce progeny with enhanced growth phenotypes 1 observations involve a conserved, programmed pathway.Lack of the MSH1-dr phenotype in the F1 plants produced an F2 population variable in plant phenotype . These data, together with reproducibility of the phenomenon, argue against the developmental reprogramming phenotype as a consequence of genome hypermutability.To ensure that the observed variation was not the consequence of inadvertent seed contamination or outcrossing, 50 SSR markers were used to test a number of derived lines, which produced no evidence of polymorphism . This anariation . In Arab2 families, individuals were self-pollinated and selected for grain yield and plant height to the F3 and F4 generations. F4 lines, along with F3 and F2 lines from remnant seed, were evaluated together in a 2011 field experiment. Despite weak selection intensity , derived F3 and F4 lines showed differences in grain yield and plant height, as well as differences in dry biomass and panicle length .Although the Fneration . Further2 to F4 showed high response to selection for plant height but variable response for grain yield , they showed large differences at the second site, which was more drought-stressed, demonstrating a line×environment effect  have major impacts on phenotype. The causes underlying G×E effects can potentially come from multiple sources, both genetic and non-genetic t effect . ResultsMSH1 suppression in a previous generation. The MSH1-dr lines used in this study were maintained as transgene-nulls seven generations following segregation of the transgene, suggesting that the non-genetic properties of the MSH1-dr line are stable through multiple rounds of self-pollination dw3, in line Tx430 A substantial range of sorghum phenotypic variation observed in this study appears to be primarily non-genetic, and is induced by crossing to a MSH1-dr line, altered through 3 and the F4 generations showed highly significant increases in above-ground biomass and grain yield over Tx430 wild type. One interpretation of these increases would be that dw3 reversion could cause pleiotropic changes in plant architecture. However, the greater range of plant height, panicle architecture and yield variation observed in this study appears to exclude that possibility The range of phenotypic variation observed is surprising. While we were not able to take measurements of all parameters for this initial study, the FMSH1 system may be useful in understanding this type of environmental influence and selecting for enhanced stability of crop performance.The observation of line×environment interaction in test plots suggests that at least some portion of the genotype×environment interaction that is commonly observed in varietal studies may be non-genetic, which is supported by other recent studies 1 populations, suggesting that the effects we observe in this system may be distinct from heterosis. It is possible that self- or open-pollination breeding will prove more effective at capturing maximal growth gain derived from MSH1 manipulation. The transgene-null MSH1-dr line crossed to its wild type counterpart produces maximum variation in the F2 population, at which point selection appears to be most effective. Large-scale seed increase in F3 and F4 generations permits rapid capture of the growth enhancement as variation tapers off. In our experience with this system, variation observed in the F2 population tends to produce above wild type performance more often than below Click here for additional data file.Figure S2F4 lines show trait differences compared to wild type. Lines from the MSH1 F4 generation show differences between each other and compared to wild type Tx430. Points represent line means; error bars represent 95% confidence intervals, which are shorter in wild type Tx430 due to higher sample size.(TIFF)Click here for additional data file.Figure S3Sample SSR marker analysis. Sorghum genomic DNAs were prepared from wild type Tx430, Tx430 MSH1-DR line , one F2 and seven F4 lines selected for phenotypic diversity. The sweet sorghum variety Wray was included as a control. The SSR marker shown is generated with SAM16073 primers. Arrow shows detected DNA polymorphism. M designates marker lane, with fragment sizes (bp) shown at left. The 1500 and 35 bp fragments are internal markers used to calibrate each lane.(TIFF)Click here for additional data file.Figure S4Sorghum genetic map with markers displaying heterozygous genotype.(TIFF)Click here for additional data file.Figure S5No correlation was found between plant height and grain yield. Each point represents a line mean. Spearman's rank correlation coefficient = 0.02 .(TIFF)Click here for additional data file.Figure S6Shifts in phenotypic means by individual lineages. Several lineages from F2 to F4 were (re)planted within the same field experiment and measured for (A) yield per panicle and (B) plant height, with varying results in terms of response to selection. Representative lineages from independent crosses are shown.(TIFF)Click here for additional data file.Figure S7Line×environment effects were detected from a multiple location experiment. Although three F4 lines (bulked from indicated F3 parent) were similar to wild type Tx430 in grain yield when grown in one location (Havelock), significant differences emerged when grown another location with a more challenging environment (Mead). Data were collected from a field experiment in 2012.(TIFF)Click here for additional data file.Table S1MSH1-dr×Tx430 derived populations show more variation compared to wild type Tx430. Data were acquired from 2011 field experiment. Brown-Forsythe tests for homogeneous variances were performed between all individuals of indicated generation versus wild type .(DOCX)Click here for additional data file.Table S2Frequency of MSH1-dr phenotype (8.4%) in F3 families derived from sorghum MSH1-dr×Tx430. Data were acquired from plants grown in greenhouse conditions.(DOCX)Click here for additional data file.Table S3Msh1-dr phenotype shows a partially heritable or metastable component. From each of ten lines, a single individual that did not display the MSH1-dr phenotype was grown along with its parental generation. Parental and progeny generation frequencies were then counted with N≥105 in each generation.(DOCX)Click here for additional data file.Table S4SSR marker polymorphism data for 43 markers. Markers were scored as + or − relative the pattern of Tx430 wild type. SSR markers were selected based on their polymorphic behavior in comparisons of Tx430 and a sweet sorghum variety, Wray. Assays included a transgene-null Tx430 line displaying the developmental reprogramming phenotype (DR), one F2, two F3, and seven F4 lines.(DOCX)Click here for additional data file.Table S5SNP marker analysis. (A) Summary of Het %. (B) A list of all the markers with heterozygous genotype. Markers with heterozygous genotypes are ordered by chromosome and genetic distance. The remainder (not shown) had homozygous genotypes. Marker genotypes of the six lines are similar but for the two markers highlighted in yellow. Markers showing a heterozygous genotype represent the true heterozygous genotype, not heterogeneity at the markers since only a single plant was sampled for DNA.(DOCX)Click here for additional data file.Table S6Line effects on trait values are significant. Data for each trait listed below were fit to a linear mixed model, with results indicating differences between lines. Line was treated as a fixed effect while block and replicate were treated as random effects. Separately analyzing lines by generation or general height class, or adding a model term for generation and height class, did not affect conclusions. The models were used to estimate trait means and confidence intervals Click here for additional data file.Table S7Analysis for significant effects using a mixed model indicates that line, location, and line×location are all significant. See (DOCX)Click here for additional data file.


Genome-wide association studies are becoming computationally more demanding with the growing amounts of data. Combinatorial traits can increase the data dimensions beyond the computational capabilities of the current tools. We addressed this issue by creating an application for quick association analysis that is ten to hundreds of times faster than the leading fast methods. Our tool (RegScan) is designed for performing basic linear regression analysis with continuous traits maximally fast on large data sets. RegScan specifically targets association analysis of combinatorial traits in metabolomics. It can both generate and analyze the combinatorial traits efficiently. RegScan is capable of analyzing any number of traits together without the need to specify each trait individually. The main goal of the article is to show that RegScan can be the preferred analytical tool when large amounts of data need to be analyzed quickly using the allele frequency test.Availability: Precompiled RegScan , source code, user guide and examples are freely available at www.biobank.ee/regscan.Requirements: Qt 4.4.3 or newer for dynamic compilations. P-value of association is the main metric for initial filtering [Genome-Wide Association Studies (GWAS) have successfully identified common variants of the human genome associated with the common diseases and traits. International research consortia have uncovered the effects of thousands of genetic makers in complex traits and diseases . Traditiiltering . It is biltering . These aWe created a tool (RegScan) that considerably accelerates association studies. RegScan performs linear regression analysis  and identifies the statistically significant associations between markers and traits maximally fast. Its computational speed benefit is best used in studying large numbers of combinatorial traits.P-value calculation. Additional methods of accelerating the analysis include setting various restrictive filters before the analysis to bypass the calculations that are not of interest to the user. These optional filters are fully controlled by the user and are explained in the RegScan User Manual.RegScan is a command line application written primarily in C++/Qt . It usesP-value, standard error (SE) of slope and minor allele count can be used to filter the results during runtime if needed. The output can be additionally filtered and analyzed with RegScan to detect true positives or create subsets. RegScan is a collection of analytical scripts that provide functions additional to the association analysis by linear regression: (i) counting and extracting markers associated with combinatorial traits or traits associated with markers, (ii) applying unlimited number of filters to identify associations of interest or extract any subset from the results, (iii) evaluating the statistical significance of the combinatorial trait–marker associations (elaborated below). If needed, RegScan can be efficiently used to prepare condensed data sets for other tools for the types of analyses not supported by RegScan, as it is capable of quickly assessing the associations.RegScan uses allele dosage to perform linear regression analysis. It works with the Oxford GEN/SAMPLE file format  and can P-value, effect size (β) and SE. We conducted tests to estimate (i) the analysis speed gain of RegScan over the reference tools, (ii) computational speed as a function of data size and various settings. The results are presented in Supplementary Data. Briefly, compared with the other software packages, RegScan was always the fastest, and QuickTest the second fastest. QuickTest was therefore chosen for computational speed comparison tests. A typical RegScan analysis was 10× faster than QuickTest on a 2.3 GHz (Scientific Linux 6.3) with one trait and various numbers of individuals . Further significant speed gain results from setting various restrictive filters such as a higher minor allele count level, lower SE level, etc. Combining all methods of computational speed reduction can result in analysis speed that is several orders of magnitude faster than with the other common GWAS tools.Computational speed can be of greatest importance in certain situations such as creating large databases of associations by brute force or studying combinatorial traits. For example, a data set with only 112 metabolite concentrations will create 6160 pair-wise combinations. When analyzed individually with other tools and the TDS (see above) the time spent will be >26 processor years. Analyzing these traits individually with RegScan would take about 2.6 processor years. However, when fed into the RegScan analysis all at once, these traits will require only about one processor month, or even less when applying additional optional RegScan filters.P-value, effect size (β) and SE agreed well between all tools . The RS indicates how much stronger the association between the combinatorial trait and a given marker is compared with the corresponding single traits. The RS is calculated as RS = Psmaller_single/Pcombinatorial, where Psmaller_single is the P-value of association for the single trait that yielded the lower P-value of the two single traits; Pcombinatorial is the P-value of association for the combinatorial trait. As an example, if P(A) = 10−6, P(B) = 10−2, P(A/B) = 10−10, then RS(A/B) = 10−6/10−10 = 104. We suggest that this simple score is effective in identifying the associations that are likely to be biologically more relevant. RegScan has functions to compute the RS and filter the results based on its value.We conducted proof-of-principle tests with 1000 Genomes-imputed 38.02 million markers , 873 indSupplementary Data).To validate the use of the RS in studying the combinatorial traits we conducted theoretical simulations that represent the theoretical ‘real-life situations’. We generated data for the scenario where the genetic marker affects a phenotype ratio as well as the scenario where the marker has linear effects on one or both phenotypes, but not on their ratio. The study indicated that the RS is able to identify the correct model in >95% of cases  the statistical parameter selected by the user, or (b) the maximal effect size.Post-run data analysis for combinatorial traits. Several functions allow studying the association analysis results. We introduced a simple, yet useful, method (RS) for identifying associations with combinatorial traits based on the statistical parameters of the corresponding single traits.The main advantages of RegScan include the following:The main disadvantage of RegScan is the absence of higher-level analytical functions in addition to the allele frequency association analysis. RegScan also relies on external R scripts for data adjustments.RegScan’s main focus is to find marker–trait associations in metabolomics in the context of combinatorial traits. Another predicted use is marker associations with gene expression. RegScan addresses the main obstacle in these studies—the heavy computational burden to find the main associations. RegScan is currently lacking several common analytical options. Our intent is to develop RegScan into a full-capability GWAS tool based on the user feedback.Supplementary data are available online at http://bib.oxfordjournals.org/.Depending on the data size, RegScan performs association analysis between markers and continuous traits ten to several hundred times faster than the other GWAS tools. Analyses that used to take weeks or months now take days.RegScan can automatically generate and analyze combinatorial traits; it can analyze any number of traits in one go.RegScan provides functions for filtering and additional analysis of the association analysis results; it introduces the concept of RS to study the combinatorial traits.RegScan is designed for metabolomics GWAS but is not limited to that.


Motivation: Analysing the joint association between a large set of responses and predictors is a fundamental statistical task in integrative genomics, exemplified by numerous expression Quantitative Trait Loci (eQTL) studies. Of particular interest are the so-called ‘hotspots’, important genetic variants that regulate the expression of many genes. Recently, attention has focussed on whether eQTLs are common to several tissues, cell-types or, more generally, conditions or whether they are specific to a particular condition.Results: We have implemented MT-HESS, a Bayesian hierarchical model that analyses the association between a large set of predictors, e.g. SNPs, and many responses, e.g. gene expression, in multiple tissues, cells or conditions. Our Bayesian sparse regression algorithm goes beyond ‘one-at-a-time’ association tests between SNPs and responses and uses a fully multivariate model search across all linear combinations of SNPs, coupled with a model of the correlation between condition/tissue-specific responses. In addition, we use a hierarchical structure to leverage shared information across different genes, thus improving the detection of hotspots. We show the increase of power resulting from our new approach in an extensive simulation study. Our analysis of two case studies highlights new hotspots that would remain undetected by standard approaches and shows how greater prediction power can be achieved when several tissues are jointly considered.Availability and implementation: Chttp://www.mrc-bsu.cam.ac.uk/software/.Contact:sylvia.richardson@mrc-bsu.cam.ac.uk or lb664@cam.ac.ukSupplementary information:Supplementary data are available at Bioinformatics online. Integrating different layers of genomic information is essential to improve our understanding of the genetic basis of complex diseases. The development of integrative analysis strategies has become an important part of experimental design in the era of next-generation genomics (One prime example is expression quantitative trait loci (eQTL) analysis, which links DNA polymorphisms to gene expression levels on a genome-wide basis. Typically, such analyses would involve testing for association of all transcript-SNP pairs. When eQTL studies involve large numbers of variants and genes, performing millions of such tests can become slow and it has been necessary to develop fast implementations . To go bsimultaneously the expression of multiple genes under multiple conditions in a multivariate way.There is currently much interest in dissecting tissue, cell type or condition specificity of eQTLs  between multiple genes, thus increasing the power to detect hotspots. We build on our previous work  to move away from single feature at-a-time analysis and account for the correlated nature of the predictors by implementing a fully multivariate model search over the space of predictors (ii) to allow the analysis of multi-dimensional responses, and (iii) to exploit the Even though our case studies relate to eQTL analyses, the model and algorithm that we present are generic and can be used for a large range of integrative genomics analyses that can be formulated within a parallel regression framework, such as finding targets for miRNAs by regressing gene expression levels on miRNA levels , and many predictors, here the SNPs. We present a Bayesian variable selection method that acts simultaneously on three levels:p, small n’ framework in terms of separation between signal and background noise, and in terms of genetic resolution as it can handle the highly correlated predictors that result from linkage disequilibrium (LD).The first level is the key driver of the method. This single response variable selection is accurately described in joint control is interesting to quantify. Note that once a predictor is selected, tissue-specific regression coefficients can be estimated from the posterior distribution.The second level is built on top of this variable selection, in the sense that evidence from all the tissues influences the probability for a predictor to be selected. This strong assumption is reasonable only when considering a small number of correlated traits for which evidence of The third level pools information across all responses in order to enhance the detection of hotspots. It also has the great benefit of eliminating many false positives. The performance of this choice of prior was explored in Our model is an extension of HESS algorithm . The sparsity prior is given by a priori independent. To enforce that kjω is a probability, we impose the constraint kjω and the sparsity priors assigned to its components. Finally, the prior for the cross-tissue error covariance matrix Σk is an Inverse Wishart distribution, For the binary matrix Γ, we introduce a kB, the intercepts kA and the residual cross conditions covariances Σk. For each response k, g, a popular choice is the Zellner–Siow prior q related (multiple conditions/tissues) regressions with an exchangeable shrinkage prior on g. In our set-up we have found empirically that when q is large the level of shrinkage is too low. We adapted the Zellner–Siow prior to related multiple responses by taking g as in q = 1, and at the same time increasing the precision of the prior proportionally to the number of responses. In our experiments, this improved the performance.The priors we have chosen allow to integrate out the regression coefficients j, in order to centre jρ on average around 1. This choice of parameters for the Gamma distribution ensures a finite mode while allowing a large coefficient of variation, thus providing necessary shrinkage for most jρs towards 1 but also supporting some large jρs to enhance hot-spot detection. Alternative parameterizations cannot achieve these two competing goals simultaneously. See Supplementary Material, Section S.4, for a sensitivity analysis on a simulated and real dataset. We let the parameters k, a priori expected number of predictors and its variance for each response k) are specified.We fix k, we choose the convenient value d = 3, which brings kh is more complicated, as it should be comparable in size with the likely error variance. We use the same empirical Bayes approach as in k, we run a stepwise regression for each condition kh as the median of the condition-specific estimates of the error variance.Finally, for the covariance matrices ΣThe inference on this high-dimensional model is performed using a specifically designed MCMC algorithm. This algorithm has already been presented in great detail in q × p matrix g. The challenging part is the sampling of the matrices Γ, and this problem is solved by relying on an Evolutionary Monte Carlo (EMC) structure as introduced in The variables sampled are those that appear in the posterior density , that isT sweeps and C chains, its output is given by g has been kept in a single chain because this setting allowed for a faster convergence of our algorithm , we focus on functions of those parameters in order to summarize the evidence of association between predictors and responses.The posterior distribution of any function of any subset of model parameters can be calculated straightforwardly in the MCMC estimation of the model. Since the model includes explicit variable selection parameters Pairwise association between response jx is associated with the k.Y It is straightforwardly estimated by the average number of times kjπ is greater than some threshold value c. The threshold is set according to the user’s required balance between false positives and false negatives. An alternative approach is to calculate the so-called Bayesian FDR .The signal-to-noise ratio in each tissue is tuned thorough the tissue-specific noise standard deviation μ. For the plots presented here we chose We have conducted an extensive investigation into the effect of the signal-to-noise ratio imbalance across tissues and here we show the output of the simulations for independent residuals  and 10 cis markers. Half the cis-associations are completely isolated, for the other half the response is also associated with other (trans) markers. The pattern is shown in cis-associations to be stronger than those in the hotspots, as trans-associations are expected to be weaker, thus we now have two parameters for signal strength cis- and trans-association pattern, due to the lower proportion of associations contained in hotspots.We now investigate the ability of the model to detect different types of association patterns. Since the model is designed to borrow information across responses, it is of interest to see whether the model can also detect predictors associated with a single response, for instance, if we can still detect cis- and trans-results, we classify the response–predictor pairs as ‘true negative’, ‘true cis isolated’, ‘true cis other’ and ‘true trans’ (part of hotspot). cis-associations very well, the multiple tissue model is able to pick up more of the associations that are contained in hotspots, thus showing the benefit of combining the tissues in a fully Bayesian way. The MANOVA misses all hotspot associations in this example. Looking directly at the hotspots, To compare in detail IRF7-driven inflammatory network (IDIN) found to be associated with immune response and increased risk of type 1 diabetes in humans (The rat dataset consists of multiple tissue expression and genome-wide genotype data (∼1400 SNPs) in a panel of 29 RI rat strains  focusing on 146 probe sets (corresponding to 143 protein coding genes), which are both robustly expressed and varying in expression across the three tissues. See a MANOVA  analysiskw and jρ and g show no erratic behaviour. Convergence is reached after few iterations due to the combined effect of the automatic tuning of the proposal density in the Metropolis–Hastings algorithms and the parallel chains implementation which allow good mixing. Focusing on probe set-marker associations significant when using a Bayes FDR (bFDR) of 5% (which corresponds to a MPPI > 0.8), MT-HSSS identified three loci controlling more than 30% of the probe sets under analysis (We ran HESS on a 2.90-GHz Intel(R) Xeon(R) CPU computer with 130 GB of memory for 15 000 sweeps of which 5000 as burn-in and the same hyper-parameters as in the simulation studies. Computational time was 70 and 114 min for ST-HESS (average across tissues) and MT-HESS, respectively. Visual inspection of the trace of the parameters analysis . In addiP-values for declaring associations, the MANOVA analysis does not return any SNPs associated with five or more genes. In order to see if a MANOVA analysis shows some of the evidence for the hotspots discovered by MT-HESS, we also computed for each locus the sum over all transcripts of the negative log P-values for the MANOVA test . This plot is very noisy, but it is clear that the P-values contain evidence for the hotspots on chromosomes 10 and 15.Note that with the same 5% FDR threshold on Given the different output produced by MT-HESS and ST-HESS, we have also investigated their model adequacy (IRF7 (adjusted PIRF2 (adjusted PNFKB (adjusted PIRF1 or IRF10 (adjusted PWe comment in more details on the regulatory hotspot located in rat chromosome 10 (rat marker J343641), which showed association with 53 probe sets (MPPI > 0.8), representing 53 protein coding distinct genes. Enrichment analysis of transcription factor binding sites (TFBSs) in the putative promoter of the corresponding human orthologous genes using PASTAA  and IRF4 (Foxk2 as a candidate master regulator of the subset of the IDIN network genes whose expression is associated with the regulatory loci located in rat chromosome 10.Examination of the genes in the LD block around the regulatory locus on rat chromosome 10 (Supplementary Methods for full details of the cohort). Note that CD4 and CD8 T cells are both T lymphocyte subsets, whereas monocytes are from the myeloid lineage. We ran HESS separately on each cell type, and simultaneously on the joint dataset. The analysis was restricted to 21 788 SNPs on chromosome 5 and 3248 probesets selected by highest variance  for computational feasibility. HESS was run using a burn-in of 4000 sweeps and values recorded for 8000 sweeps. MT-HESS took 110 h on a 2.80-GHz Intel(R) Xeon(R) computer.The human dataset comprised genotype and corresponding expression data from three purified leukocyte subsets  isolated from the peripheral blood of 59 patients with inflammatory bowel disease , and this may account for the lack of hotspot detection in this cell type.In the single cell type analysis, hotspots were most frequently detected in CD4 T cells, followed by CD8 T cells, whereas in monocytes we found a paucity of hotspot SNPs . Using aSupplementary Fig. S.18). The signal in the joint analysis appears to be driven predominantly by signal in either the CD4 T cells or the CD8 T cells .In the joint MT-HESS analysis 28 SNPs were associated with expression of five or more genes at a 5% bFDR significance threshold, with four of these SNPs associated with 20 or more genes . The SNP lies in an intergenic region, ∼143 kB downstream from the end of the nearest gene, PIK3R1 ). Histone marks from ENCODE . Interestingly, this hotspot SNP lies 10.7 kb from the start site of a long intergenic non-coding (linc) RNA (Ensembl transcript ENST00000507733). Emerging evidence suggests that long non-coding RNAs play an important role in orchestrating transcriptional programmes through a variety of mechanisms, commonly involving ribonucleic acid–protein interactions . P-values from simple univariate regression of the expression of these genes on rs11745891 were skewed towards zero . However, after multiple-testing of The 78 genes whose expression in CD4 T cells was associated with rs11745891 genotype are located on 21 chromosomes, most frequently chromosome 1 (nine genes) . The resulting residuals were substantially less correlated than the residuals from regression on just the hotspot SNP alone  at a time, where the vector of responses correspond to the expression in multiple tissues or conditions. We see that MT-HESS outperforms MANOVA, demonstrating that our hierarchical modelling approach is more powerful than pairwise methods.When faced with an eQTL analysis task which involves multiple tissues, a natural alternative to MANOVA test is to compare results from the intersection of single-tissue model (iST-HESS) with those from the multiple-tissue one (MT-HSSS). Our extensive simulation study show that when there is a similar pattern of association but varying levels of signal-to-noise ratio across tissues, MT-HESS clearly outperforms iST-HESS. This is well in keeping with the construction of the model underlying MT-HESS, which performs a statistically sound evidence synthesis for detecting hotspots between the set of conditions or tissues. When signal-to-noise ratio is constant across conditions/tissues MT-HESS results become closer to those of iST-HESS pointing to the benefit of adjusting for external sources of correlated noise prior to MT-HESS analysis, for example by using a latent factor approach as in In our simulation study we show a substantial gain of power of MT-HESS when the association in each tissue were too weak to be detected by ST-HESS, a situation also illustrated in our first case study where two new hotspots were highlighted by MT-HESS, whereas they were not detected in the single tissue analyses. Here the ST-HESS signal is relatively noisy and the ability of MT-HESS to borrow and synthesize information between different layers of the responses results in higher power for MT-HESS.trans master regulator situated near a long intergenic non-coding (linc) RNA (Ensembl transcript ENST00000507733) (see Section 4.2) is a good example of the power of our approach for generating hypotheses which can be validated by follow-up experiments. The clear separation between noise and signal offered by our sparse regression approach is paramount for successfully prioritizing such further validation.Our new C++ implementation of the complex hierarchical structure in MT-HESS  permits efficient analysis of very large datasets. It made feasible an MT-HESS analysis of our human dataset, where a multivariate analysis and variable selection between 3000 responses in three cell types and nearly 22 000 SNPs was carried out. The putative novel In conclusion, our new algorithm MT-HESS with its efficient C++ architecture is tailored to jointly analyse realistic case studies that comprise a large number of responses collected over dependent multiple tissues or conditions and a large set of potentially collinear predictors.


ALMS1/LAMB2/TNRC6A/TACR3/PRKAG1) are associated with age at menarche . In addition, we identify common X-chromosome loci at IGSF1  and FAAH2 . Highlighted genes implicate cellular energy homeostasis, post-transcriptional gene silencing and fatty-acid amide signalling. A frequently reported mutation in TACR3 for idiopathic hypogonatrophic hypogonadism (p.W275X) is associated with 1.25-year-later menarche (P=2.8 × 10−11), illustrating the utility of population studies to estimate the penetrance of reportedly pathogenic mutations. Collectively, these novel variants explain ∼0.5% variance, indicating that these overlooked sources of variation do not substantially explain the ‘missing heritability' of this complex trait.More than 100 loci have been identified for age at menarche by genome-wide association studies; however, collectively these explain only ∼3% of the trait variance. Here we test two overlooked sources of variation in 192,974 European ancestry women: low-frequency protein-coding variants and X-chromosome variants. Five missense/nonsense variants (in  Previous studies have linked over 100 genomic loci to age-at-menarche but that work was restricted to common autosomal variation. Here, Lunetta et al. identify associations with rare protein-coding and X-linked variants, implicating new mechanisms that regulate puberty timing. Age at menarche, the onset of first menstruation in females, indicates the start of reproductive maturity and is a commonly reported marker of pubertal timing. One hundred and six genomic loci for this highly heritable trait have been mapped by genome-wide association studies (GWAS), implicating many previously unsuspected mechanismsHere we report a dual strategy for assessing genetic variation overlooked by those prior efforts: low-frequency protein-coding variants genotyped by large-scale exome-focussed arrays and high-density X-chromosome single-nucleotide polymorphism (SNP) genotyping and imputation. We identify several new associations between rare protein-coding and X-linked variants with age at menarche in women of European ancestry. The findings implicate new mechanisms that regulate puberty timing, but collectively these novel variants explained only ∼0.5% of the variance, indicating that these often overlooked sources of variation that do not substantially explain the ‘missing heritability' of this complex trait.P values obtained in this study. Meta-analysis of individual variant associations with questionnaire-reported variation in age at menarche (restricted to the ages of 9–17 years) in this discovery phase identified one signal at genome-wide statistical significance (P<5 × 10−8); this was a rare missense variant in the Alström's syndrome gene . For follow-up testing in up to 116,317 independent women of European ancestry from the deCODE  and 23andMe studies, we selected rs45501594 and 23 other variants that met the following criteria: protein coding, present in over half of the exome array studies, and with association P<5 × 10−4. In the follow-up samples, 7 of the 20 variants that passed QC showed directionally concordant confirmatory associations with P<0.05, of which five reached genome-wide significance in a combined meta-analysis of discovery phase and follow-up data  <5%) variants passed quality-control (QC) criteria in a combined sample of up to 76,657 women of European ancestry from 19 studies . Gene-ba-up data . No sign-up data .ALMS1 . In the follow-up samples, each rare allele was associated with 0.23-year-later age at menarche, an effect size more than double that of any genetic variant previously reported for puberty timing in the general population. This strong signal was not detected by the previous HapMap2-based GWAS as it is poorly tagged by common SNPs in that reference panel . Deleterious mutations in this gene cause Alström's syndrome (OMIM no. 203800), a rare, autosomal-recessive disorder characterized by cone–rod dystrophy, sensorineural hearing loss, dilated cardiomyopathy, childhood obesity, insulin resistance, diabetes mellitus, hypogonadotropic hypogonadism in males, menstrual irregularities and early puberty in females, and short stature in adulthoodALMS1 gene-trapped mouse modelThe rare missense variant in TACR3; rs144292455, MAF=0.08%, combined P=2.8 × 10−11, TACR3 locus were previously associated with age at menarcheTACR3 SNP in a sensitivity analysis within a participating study , TACR3 mutation in the rare reproductive disorder idiopathic hypogonadotropic hypogonadism (idiopathic hypogonadotropic hypogonadism (IHH), OMIM no. 614840)The variant with largest effect was a rare stop-gain mutation in the tachykinin receptor 3 gene , and both signals remained significant when jointly tested in a follow-up sample of 76,831 women from the 23andMe study . LAMB2 encodes one of 15 subunits of Laminin, an extracellular matrix glycoprotein with a key role in the attachment, migration and organization of cells into tissues during embryonic development. Rare recessive mutations in LAMB2 cause Pierson's syndrome (OMIM no. 609049), a disorder characterized by congenital nephrotic syndrome and ocular anomalies, typically with microcoriaA low-frequency missense variant in the TNRC6A gene was associated with later age at menarche , which reportedly showed subgenome-wide significant association with age at menarche PRKAG1 missense variant is in the same locus as, but not correlated to, a reportedr2=0.02). PRKAG1 encodes the gamma-1 regulatory subunit of AMP-activated protein kinase, which senses and maintains cellular energy homeostasis by promoting fatty-acid oxidation and inhibiting fatty-acid synthesis; PRKAG1 is overexpressed in ovarian carcinomasA low-frequency missense variant in IGSF1 and FAAH2, reached genome-wide significance for association with age at menarche and both associations were confirmed in 39,486 independent women of European ancestry from the deCODE study.Our second genotyping approach considered X-chromosome GWAS SNPs in up to 76,831 women of European ancestry from the 23andMe studyIGSF1 were robustly associated with age at menarche 18Common variants in and near FAAH2, MAF 24%, β=0.05 years per allele, P=1.9 × 10−10; FAAH2 encodes fatty-acid amide hydrolase 2. This enzyme catalyses the hydrolysis and degradation of bioactive fatty-acid amides, a large class of endogenous signalling lipids including the endocannabinoids, which modulate several physiological processes, including feeding, inflammation, pain, sleep and various reproductive processes, including hypothalamic gonadotropin-releasing hormone secretion21The second X-chromosome locus, in Xp11.21 (lead SNP rs5914101 is intronic in P>0.05), indicating that their effects on puberty timing are unlikely to be mediated by body mass index.We sought to further functionally characterize the seven genes implicated by these analyses using expression data on 53 tissue types from the Genotype-Tissue Expression consortiumTACR3 variants in patients with rare disorders. In the deCODE study these novel variants collectively explained only 0.5% of the variance in age at menarche, suggesting that these often overlooked sources of genetic variation do not contribute disproportionately to the missing heritability of this complex trait. While variants with MAF below 1% are likely not well represented here, our findings indicate that, similar to other complex traitsIn summary, by large-scale analysis of genetic variation not captured by previous GWAS for age at menarche, we identified several low-frequency exonic variants of relatively large effect and two common X-chromosome signals. The implicated genes provide new insights into the mechanisms that link energy homeostasis to puberty timing, indicate possible roles of RNA-mediated gene silencing and fatty-acid amide signalling, and link genes behind rare autosomal, X-linked and syndromic disorders of puberty to normal variation in reproductive timing. Our findings using dense exome arrays in large unselected populations are informative for the clinical interpretation of heterozygous N∼9,700) used the cluster file made available by CHARGE to call genotypes. Other studies followed standard calling and QC protocols for the Exome array (http://www.chargeconsortium.com/main/exomechip/) and variants with MAF>5% in the meta-analysis were excluded. We performed gene-based testing (within seqMeta) for low-frequency variants using fixed effect burden tests, which assume that all rare variants have the same effect direction and size , and SKAT tests, which assume that rare variant effects are random and can contain a mixture of null, protective and risk rare alleles. These tests were run using three variant filters, all of which included only variants with MAF<1%: (1) all non-synonymous; (2) non-synonymous annotated as ‘damaging' ; and (3) only loss of function. The multiple testing adjustment included two tests × three filters × number of genes, requiring study-wise significance threshold P<1.14 × 10−6. For individual variants, a fixed-effects inverse variance-weighted meta-analysis was performed across all studies using METAL , with associations considered significant at a conservative genome-wide significance threshold of P<5 × 10−8.Exome array genotype data were generated across 19 studies in up to 76,657 women of genetically determined European ancestry with questionnaire-reported age at menarche between ages 9 and 17 years . Exome ame array . Each coWe performed follow-up testing of selected exome array variants in the 23andMe study (as described below) and also in 39,486 independent women of European ancestry from the deCODE study, Iceland, who had genotypes on over 34 million variants by imputation of whole-genome sequencing-identified SNPs and indels on Illumina SNP chip data 26. VariaP<10−20, call rate <95% or with large allele frequency discrepancies compared with European 1000 Genomes reference data. Frequency discrepancies were identified by computing a 2 × 2 table of allele counts for European 1000 Genomes samples and 2,000 randomly sampled 23andMe customers with European ancestry, and identifying SNPs with a χ2P<10−15. Genotype data were imputed against the March 2012 ‘v3' release of 1000 Genomes reference haplotypes. Age at menarche was assessed by questionnaire and recorded in 2-year-age bins, which were rescaled to 1-year effect estimates post analysis. The validity of this approach was confirmed by the lack of significant heterogeneity between rescaled 23andMe menarche estimates for the 123 previously identified signals and their reported effectsP values were computed from likelihood ratio tests.X-chromosome SNP data were generated in up to 76,831 women of European ancestry from the 23andMe study17Identified X-chromosome variants were replicated in 39,486 women from the deCODE study, as described above.How to cite this article: Lunetta, K. L. et al. Rare coding variants and X-linked loci associated with age at menarche. Nat. Commun. 6:7756 doi: 10.1038/ncomms8756 (2015).Supplementary Figures 1-5 and Supplementary Tables 1-3


Pinus contorta var. latifolia) and jack pine (Pinus banksiana) hybridize in western Canada, an area of recent mountain pine beetle range expansion. Given the heterogeneity of the environment, and indications of local adaptation, there are many unknowns regarding the response of these forests to future outbreaks. To better understand this we aim to identify genetic regions that have adaptive potential.Lodgepole pine (arlequin and bayescan) and correlation-based (matsam and bayenv) methods.We used data collected on 472 single nucleotide polymorphism (SNP) loci from 576 tree samples collected across 13 lodgepole pine-dominated sites and four jack pine-dominated sites. We looked at the relationship of genetic diversity with the environment, and we identified candidate loci using both frequency-based , we did not find any loci in common between lodgepole and jack pine. Many of the outlier loci identified were correlated with environmental variation.Using rigorous criteria we have been able to identify potential outlier SNPs. We have also found evidence of contrasting environmental adaptations between lodgepole and jack pine which could have implications for beetle spread risk. A fundamental goal of evolutionary biology is to identify adaptive variation within a genome. While in the past this may have been hampered by a paucity of appropriate methods and limited access to relevant genomic data, this area is rapidly expanding. With the advent of single nucleotide polymorphism (SNP)-chip genotyping and next-generation sequencing , and an Forest trees present an interesting system in which to study adaptation as many are sessile and long-lived, with large out-breeding populations. This combination of characteristics could potentially result in maladapted forests if there is insufficient time for them to respond at the genome level to rapid environmental change (Pinus banksiana) is a keystone species of the boreal forest and has recently been identified as a new host for mountain pine beetle (Pinus contorta var. latifolia) host populations, a sister species to jack pine , jack pine  and their interspecific hybrids that have been previously characterized at SNP loci described in To identify outlier loci within and between species, we used samples of lodgepole  gene catalogue  terms using blast2go  calculated using a 30-yr average (1931–1960) from weather stations interpolated for Canada using ANNUSPLIN . We transformed the allele frequencies per site using the Hellinger transformation  using thR2 values for each component. Following that we performed RDA for the models followed by ANOVA to test for significance using permutation. For each data set we reduced the environmental variables to include only those that were not correlated above 0.80. We performed variance partitioning for all three data sets.The correlation between genetic diversity and the environment can be the result of a causative process, or the result of shared spatial processes  using muarlequin which identifies outliers by comparing the levels of genetic diversity and differentiation between populations. This method is based on the calculation of arlequin include more flexibility for the mutation model, calculation of heterozygosity to account for different mutation rates, and an option to estimate outliers with either an infinite island or hierarchical demographic model. We estimated the number of putative outliers for all three subsets using both demographic models. For the hierarchical model, we split the pure species data into lodgepole and jack pine groups, and the lodgepole pine data were split into north and south and jack pine into east and west based on population structure analysis . For each data set we calculated outliers three times to ensure robustness, each time using a burn-in of 10 000 iterations, a thinning interval of 50, and a sample size of 10 000. We ran different values of the prior odds of the neutral model to quantify how this affected our results  data, but can be used for SNP data  with 50 000 MCMC steps using a batch file to estimate the BayesFactor. We used Jeffreys' scale of evidence  using gmap  and exhibited extreme values for one of the two species, but not both (Supporting Information Table S1). There was no evidence of null alleles for these loci based on the proportion of missing data. We identified 24 linkage groups in lodgepole pine and 20 groups in jack pine; nine of these were shared between the species. The number of loci in the linkage groups ranged from two to four (Table S1). Thirteen of the linkage groups were locus pairs that occurred on the same contig. Interestingly, there were an additional 11 SNP locus pairs that were not linked but were characterized from the same contig. While the inclusion of linked loci may create a small bias for some of the outlier detection methods we were using, we decided to include all of them in the outlier analysis. Linkage indicates correlated allele frequencies regardless of physical linkage; therefore the linked loci may be linked statistically but not physically if they are under the same evolutionary pressures , many ofWe successfully annotated most contigs through sequence similarity with informatively annotated sequences from other species. Fifty contigs showed sequence similarity to either Arabidopsis or white spruce sequence annotated as having unknown function, and six of the contigs showed insufficient similarity to sequences in the databases that were queried (Table S1). Annotation for the remaining contigs included a wide range of proteins involved in an assortment of molecular processes with > 90 contigs falling under cellular processes, plastid proteins, biosynthetic pathways, membrane function and response to stress categories.The amount of genetic variation that partitioned the sites estimated using RDA varied among the three data sets. The first two axes of the RDA for lodgepole pine explained 40.17% of the variation among the sites Table . Of the R2 = 0.1604; P < 0.01). For jack pine, the environmental variables were highly correlated with each other, and with geography. Only elevation was retained in the final model, and the variation in genetic data explained by this was almost entirely shared with the variation explained by geography (adjusted R2 = 0.3637). For the combined subset, environmental variables were reduced to: elevation, average PET, minimum temperature – coldest period, precipitation – wettest period, precipitation – driest period, and annual temperature range. We found that both the environment and spatial heterogeneity independently contributed significantly to explaining allele count variation among sites , and a large proportion of the variation in genetic diversity was explained by both processes (adjusted R2 = 0.4407).The shared and separate contributions of environment and spatial heterogeneity in explaining the variation in genetic diversity for each data set were examined Table . For lodarlequin, respectively , precipitation – driest period , longitude , maximum temperature – warmest period , and average PET  following Bonferroni correction (α = 0.004).We identified 31 loci in lodgepole pine with environmental correlations using matsam; in this case, one locus was detected across the three other methods, eight loci were detected across two methods, and 34 loci were detected using a single method . The genomic cline patterns were complex and diverse, but included evidence of lodgepole introgression into jack pine  and vice versa, as well as evidence of heterozygote advantage , positive selection , and negative selection . Of the significant loci, six were consensus outliers: two were identified in the lodgepole pine subset and of these one was identified in the combined subset. An additional locus was identified in just the combined subset, and one in jack pine (Table Based on the genomic clines estimated in ne Table .Through the application of multiple contrasting methods, we have identified candidate genes representing putative adaptive variation both within and between lodgepole and jack pine. As expected, loci for which we detected signatures of selection within the species subsets were not shared, potentially as a result of contrasting environmental adaptations. A high number of synonymous substitutions suggest that, despite a stringent approach to identifying candidates, a proportion may still represent false positives. The roles of regulatory, noncoding regions as a source of adaptive variation are implicated given that a large number of our candidate loci were found in the 3′ and 5′ untranslated regions.The performance of outlier detection methods has been reviewed using simulated data (arelquin and matsam, for the combined subset, resulted in over-estimation of outliers (Fig. arlequin, resulting in a more conservative estimate of putative outliers, which is likely to reflect a more reasonable approximation (matsam, there is no means to account for population demographics, and the processes of drift and gene flow can result in genetic clines resulting in a high proportion of false positives (bayenv and matsam was quite large, prompting us to discard the results from the combined subset analysis using matsam.Our analyses demonstrate the impact that population demography can have on the detection of outliers. Using ers Fig. . Both Nebayescan represents how much more likely this model is over the selection model; for example, a prior odds of 1 assumes that they are equally likely. The manual suggests setting the prior odds to 10 for data with a few hundred loci, and therefore we ran the model with the suggested value of 10 then, following The prior odds for the neutral model in bayescan is rarely reported in the literature (e.g. The contrasting results between the combined subset and the pure species analyses relates to their underlying population structure. The combined subset has a very strong hierarchy while there is limited population structure within lodgepole and jack pine , which wAmong our candidate outliers, five of them contributed to the variation explained by the first axis for the combined subset of our RDA analysis. Pinus genome (FST outliers can result from other processes aside from local adaptation, including stochastic effects, background selection, and pre- and post-zygotic isolation (see Among our three data sets, the proportion of outlier loci detected ranges from 1% to 4%; if one considers the large size of the s genome , 1983, as genome , evidenction see . This sution see . Among oHybridization between species is considered an important source of novel genetic variation for adaptation to act on . For exaThe number of consensus outliers that we detected for each data set is below the range found in other studies 5–10%; . The hig