Arabidopsis thaliana were re-analyzed by four methods . As a result, the new method identified most previously reported genes. Therefore, the new method is a good alternative for multi-locus GWAS.Genome-wide association study (GWAS) entails examining a large number of single nucleotide polymorphisms (SNPs) in a limited sample with hundreds of individuals, implying a variable selection problem in the high dimensional dataset. Although many single-locus GWAS approaches under polygenic background and population structure controls have been widely used, some significant loci fail to be detected. In this study, we used an iterative modified-sure independence screening (ISIS) approach in reducing the number of SNPs to a moderate size. Expectation-Maximization (EM)-Bayesian least absolute shrinkage and selection operator (BLASSO) was used to estimate all the selected SNP effects for true quantitative trait nucleotide (QTN) detection. This method is referred to as ISIS EM-BLASSO algorithm. Monte Carlo simulation studies validated the new method, which has the highest empirical power in QTN detection and the highest accuracy in QTN effect estimation, and it is the fastest, as compared with efficient mixed-model association (EMMA), smoothly clipped absolute deviation (SCAD), fixed and random model circulating probability unification (FarmCPU), and multi-locus random-SNP-effect mixed linear model (mrMLM). To further demonstrate the new method, six flowering time traits in  Genome-wide association study is concerned with the associations between markers and traits of interest so as to identify all the significantly associated markers. In genome-wide association studies, hundreds of thousands of markers are genotyped for several hundreds of individuals. Usually, only a very minor subset of these markers is associated with the trait. Most penalization methods fail when the number of markers is much larger than the sample size. Based on this fact, we have developed an algorithm that proceeds in two stages. In the first stage (screening), we reduced the number of markers via correlation learning to a moderate size. We then used a moderate-scale variable selection method to select variables in the reduced model. Conditional on the selected variables, we repeated the screening procedure and chose another set of variables. In the second stage (estimation), all the above-selected variables are accurately estimated in a multi-locus model. Our approach is simple, accurate in estimation, fast and shows high statistical power of detecting relevant markers on simulated data. We have also used this method to identify relevant genes in real data analysis. We recommend our approach for conducting a multi-locus genome-wide association study. In Gp) is several times larger than the number of individuals (n) [In genetics, only a small subset of SNPs is associated with the phenotype of a trait. This is an example of a variable selection problem for high-dimensional data, where the number of SNPs  . To solvuals (n) , nonnegauals (n) , least auals (n) , smoothluals (n) , elasticuals (n) , fused Luals (n) , adaptivuals (n)  and miniuals (n) . Among tuals (n) , penalizuals (n) , sure inuals (n) , adaptivuals (n) , elasticuals (n) , LASSO [uals (n) , empiricuals (n)  and empiuals (n)  have beeuals (n) . Despiteuals (n)  and Bayeuals (n) . Howeveruals (n) , multi-luals (n) , and fixuals (n) .p, via correlation learning (i.e. Iterative modified-Sure Independence Screening) to a moderate number. A moderate-scale variable selection method, SCAD, was then employed to select variables in the reduced model. We chose SCAD because of its nice oracle property. Conditional on the selected variables, we repeated the screening procedure and chose another set of variables. All the effects of the above-selected variables were estimated by Expectation-Maximization (EM)-Bayesian LASSO algorithm [In this study, we developed an approach that reduced the number of markers, lgorithm  and testlgorithm  and Farmlgorithm , multi-llgorithm  and mrMLlgorithm . Severallgorithm  has beenlgorithm  reduces lgorithm  is used lgorithm  integratThree Monte-Carlo simulation experiments were carried out to measure the effectiveness of our new method. We used statistical power to evaluate the effectiveness of ISIS EM-BLASSO method alongside the three methods for comparison purposes. For each QTN, we defined its statistical power as the fraction of the samples in which the QTN was detected see . Indeed,Mean squared error (MSE) was used to measure the accuracy of each estimated QTN effect for all the methods. We evaluated the accuracies of all the six simulated QTNs effects in the three simulation experiments. The results of all methods considered are presented in Type 1 errors for all the methods in the three simulation experiments were calculated . The valReceiver operating characteristic (ROC) curve is used to compare different methods for their efficiencies in the detection of significant effects. ROC curve is a plot of the statistical power against the controlled Type 1 error. A method with the highest ROC curve is deemed the best. We simulated various 67 probability levels of significance between 1e-8 to 1e-2, with these values we calculated the corresponding powers in the first simulation experiment. Comparing the computing times  Tables oArabidopsis flowering time traits in Atwell et al. [Six l et al.  have beeThe numbers of known genes in the proximity of SNPs for the above six traits were in total 67 genes from ISIS EM-BLASSO, 22 genes from mrMLM, 15 genes from FarmCPU, and 13 genes from EMMA. ISIS EM-BLASSO detected more known genes than the other methods . Among tDOGI was discovered to be associated with LDV, SD, 0W, and 4W, gene SVP was found to be related to LD and LDV, gene ETC3 was found to be associated with LD, 0W, and 2W, and gene ABCB19 was discovered to be related to SD, 2W, and 4W. These results are consistent with previous studies related to these traits as outlined in the references presented in It is interesting to note that for the trait SD, EMMA was not able to determine any significant gene whereas the new approach identified 21 genes. A similar trend is also observed when we considered trait 0W where FarmCPU did not detect any gene. Based on these results, we observe that the new approach can capture the genes associated with the trait under study. We also noted that some genes tested were found significant in nearly all the traits considered. For example, gene Arabidopsis thaliana GWAS results of this study are reliable.Based on the results obtained from this study, we observe that correlation learning can be used as a screening tool to reduce the number of markers in GWAS study. As already noted, most methods used in variable selection in linear regression fail in high and ultra-high dimensional settings. This study has presented a simple yet powerful tool to solve this problem. Therefore, the et al. [p, is several times larger than the sample size (n).The single locus tests in standard GWAS methods have been used successfully in identifying thousands of genetic variants associated with hundreds of various traits. As noted by Segura et al. , when weIn this article, we developed an algorithm, ISIS EM-BLASSO that screens and significantly reduces the number of SNPs to a moderate number. A moderate-scale variable selection method was used to select variables in the reduced set. We chose SCAD since it is asymptotically oracle efficient. Parameter estimation and significance testing were done in the last stage by applying EM-Bayesian LASSO  and likep, is moderately greater than or equal to the number of observations, n. Therefore the choice of a level of significance of 0.01 without any multiple test correction is justified. Note that if several iterations are performed, then many unimportant variables are selected. Even so, this algorithm is still valid because the extension of this method with EM-Bayesian Lasso [et al. [In the screening stage of ISIS EM-BLASSO, iterative sure independence screening can be performed several times. In the article, we only performed a single iteration since we chose a high level of significance, 0.01, for identifying predictors that are significantly correlated with the response. At this level of significance, we expect to have moderately more variables in the screening stage so that SCAD can be applied to shrink some of these variables to zero. A lower level of significance and/or multiple test correction in the screening stage might be too stringent and may result in missing significant SNPs at this juncture. Most shrinkage methods will still perform effectively even when the number of variables, an Lasso  in the lan Lasso  have bee [et al.  is inadeISIS EM-BLASSO considers the joint effect of all SNPs that passes the screening criterion. Unlike EMMA and FarmCPU, we do not apply the Bonferroni correction for multiple testing hence ISIS EM-BLASSO performs better than these methods regarding statistical power. Bonferroni correction is indeed too stringent hence it removes some significant SNPs in the final results when EMMA and FarmCPU are used. Although we still used a slightly stringent criterion of LOD value 3 in our final stage, ISIS EM-BLASSO still has high statistical power and low false positive rate, indicating a better performance of the new algorithm over EMMA and FarmCPU. mrMLM compares almost equally with ISIS EM-BLASSO regarding power and MSE  Tables. http://www.arabidopsis.org/) the work of Atwell et al. [m criterion. Therefore, the Arabidopsis thaliana GWAS results presented by our algorithm in this study are reliable.ISIS EM-BLASSO was used to analyze six flowering time traits in this study. As a result, 67 known genes were detected. Among these known genes, 50 were identified only by ISIS EM-BLASSO . Many gel et al.  and othel et al.  listed mWe considered an iterative modified sure independence screening (ISIS) and extended it by applying EM Bayesian LASSO algorithm herein referred to as ISIS EM-BLASSO. This approach was used to identify relevant genes in GWAS study in real data. The new approach is simple, fast and shows high statistical power of detecting relevant SNPs on simulated data. Mean squared errors of the estimated effects are also minimal. We recommend this approach as an accurate and fast alternative for carrying out multi-locus GWAS study especially in high dimensional settings. ISIS EM-BLASSO reduces the search to a moderate number of SNPs that are significantly correlated with the trait of interest. As a result, we reduce the computing time for GWAS, and also ensure that GWAS can be carried out on a small computer. Our extension by EM-Bayesian LASSO ensures that the parameter estimates are reliable.y being a n×1 vector of phenotypic quantitative trait, μ is the overall average, Qj is the jth fixed effect which must be included in the model, for example, the population structure, Xi is a n×1 vector of the ith SNP values and ε∼MVNn is the residual error.In this study, we considered the regression model,y for the fixed effects Qj, j = 1,2,…,q before applying the screening procedure. The effects Given the model in Eq  we correβ of regression coefficients in Eq ) and a penalty functionρλ,γ(⋅) is a function of the vector of coefficients β = T and the tuning  parameter λ > 0 controls the trade-off between minimizing the loss and the penalty term. γ>0 is a shrinkage parameter that determines the order of the penalty function. Notice that we avoided μ in Eq (ρ is symmetric about the origin (ρ(0) = 0) and its non-decreasing in the interval  [All regularized regression methods aim at estimating the vector ts in Eq  by minimfunction}where ρλfunction}where ρλal  .λ ≥ 0, λ > 2 and 1x∈A}{ being an indicator function. This penalty gradually reduces the penalization rate to zero as |β| gets larger. The regularization parameter γ controls the degree of concavity, with smaller γ corresponding to more concave penalty when γ → ∞, SCAD converges to the L1 penalty. [γ = 3.7 for SCAD. SCAD is asymptotically oracle-efficient [Of great importance to us in this study is the SCAD penalty which is defined aspenalty.  Suggesteβj is the jth non-QTN effect, Xj is the corresponding design matrix, γk is a vector of SNP effects for locus k and Zk is the corresponding incidence matrix determined by the genotypes of the locus k. The dimensions of γk and Zk depend on the number, mk of genotypes for the locus k. The residual error vector ε is assumed to be distributed as MVNn. In our case X will include the overall mean and the population structure matrix we will use the original before Y correction. Using a normal prior for γk, i.e., E(γk) = E. The EM steps of Xu [γk be the missing values and by using the following expectations for maximization step.Xu  developefor σ^k2 . As for for σ^k2 ,β^= in Fan and Lv . Herein,ScreeningStageKT = {1 ≤ i ≤ p: βi ≠ 0} be the exact sparse model of size k = |KT|. The other p−k variables can also be correlated with the response variable via linkage to the predictors that are contained in the model.We correct the phenotypes using fixed effects that must be included in the model such as the population structure. We center each input variables so that the observed mean is 0, and scale each predictor so that the sample standard deviation is 1. Let r = {r1,r2,⋯,rp} be a p-vector that is obtained by component-wise regression, i.e.,X first being standardized column-wise. r is a vector of marginal correlations of predictors with the response variable, rescaled by the standard deviation of the response.We let KT has a size less than n hence any moderate shrinkage method will remove unimportant predictors. SCAD is a moderate-scale variable selection method since it selects variables from moderately high dimensional linear model shrinking the unimportant ones to zero. Hence, we applied penalty in Eq  method is applied. After the modified SIS-SCAD (step (iii) above) has been implemented, we obtain the selected variables of say size EstimationstageT1∪T2 of size τ = τ1 + τ2. Using the model,γkfor k = 1,2,…,τ with the EM steps being updated using Eqs (Var(γk) is the prediction error for γk. Notice that at this stage, we use the original Y before correction, so that Zk denotes the kth SNP values and X includes overall mean and population structure.We apply EM-Bayesian LASSO algorithm  to filtesing Eqs  and 7)  T1∪T2 ofhttps://cran.r-project.org/web/packages/mrMLM/index.html. For comparison purposes, we have compared ISIS EM-BLASSO with four methods, which are EMMA (http://mouse.cs.ucla.edu/emma/) [http://cran.r-project.org/web/packages/ncvreg/) [http://cran.r-project.org/web/packages/mrMLM/index.html) [http://www.ZZLab.net/FarmCPU) [A diagrammatic representation of our overall algorithm is presented in u/emma/) , SCAD (hncvreg/) , mrMLM (ex.html) , and FarFarmCPU) .γk which must be tested for their significance to the phenotype under study. We propose that all k = 1,2,⋯,p could be viewed to be associated with the trait under study. With the model (10) above at hand, and all q, we consider the model’s likelihood functions. Let ISIS-EM-BLASSO gives the marker effects estimates, H0:γk)( = 0 that there is no QTN linked to the marker k. We use log of odds (LOD) score as the test statistic. The original likelihood functions  are In essence, we test the null hypothesis, LOD ≥ 3 and regard them as significant. Note that LOD value 3 is a slightly stringent criterion and is equivalent to p-value: Pr(χ2 > 3×4.6052) = 0.0002. If the null hypothesis is true, LOD × 4.6052 follows a Chi-square distribution with one degree of freedom. A similar procedure is used to test the significance of the estimates obtained from SCAD and mrMLM.The LOD score is easy to interpret because of base 10. We select all markers with a score m, where m is the number of markers.It is important to point out that, for the EMMA and FarmCPU methods we select significant markers based on Bonferroni correction for multiple tests by setting a threshold for P-value at 0.05/http://www.arabidopsis.org/) and all the phenotypes values for quantitative traits were simulated. The dataset of Atwell et al. [In our simulation studies, all the genetic values for SNP markers were randomly sampled from Atwell et al.  , where m is the number of markers. Each QTN within 1kb of the simulated QTN was considered a true QTN. A fraction of the number of such samples over the total number of run (1000) represented the empirical power of this QTN. The false positive rate (FPR) was calculated as a fraction of the number of false positive effects over the total number of zero effects considered in the full model. We calculated the mean squared error (MSE) for each QTN to measure the bias of QTN effect estimate. We also performed a paired t-test for the differences (A–B) of statistical power or MSE between ISIS EM-BLASSO (A) and other methods (B).In our first simulation study, the dataset consisted of 10000 SNP genotyped from 199 inbred lines. We set six QTN and placed them on the SNPs with allelic frequencies of 0.30; the heritabilities were set as 0.10, 0.05, 0.05, 0.15, 0.05 and 0.05, respectively. The positions and effects of these QTN are listed in K kinship coefficient matrix between a pair of individuals. Here h2), average, residual variance, and other values were the same as those in the first simulation study.In our second simulation study, we investigated the effect of the polygenic  background on ISIS EM-BLASSO. The polygenic effect was simulated by multivariate normal distribution, h2), average, residual variance, and other values were the same as those in the first simulation study.In our third simulation study, we studied the effect of epistatic background on ISIS EM-BLASSO. Three epistatic QTN each with g et al. . The QTNAll the above simulated datasets are saved in http://www.arabidopsis.usc.edu/. In the real data analyses, the significantly associated SNPs were determined by the critical threshold of LOD ≥ 3 for the new method, SCAD, and mrMLM, and with the P-value less than 0.05/m for EMMA and FarmCPU. Candidate genes for the trait under study were mined within 30 kb of the significantly associated SNP.Six flowering time-related traits in Arabidopsis  were anaS1 Table(DOC)Click here for additional data file.S2 Table(DOC)Click here for additional data file.S3 Table(DOC)Click here for additional data file.S4 Table(DOC)Click here for additional data file.S5 Table(DOC)Click here for additional data file.S1 Datasethttp://www.arabidopsis.org/). In the “Phenotypes_*” sheet, all the phenotypic values with 1000 replicates were simulated based on genotypic values in “Genotypes” sheet and simulated parameters. The ith sample consists of the (199(i−1)+1)th to 199ith rows in the first column . “*” indicates the sequence number  of simulated experiments.In the “Genotypes” sheet, there are 199 diverse inbred lines, each with 10000 SNP genotypes, obtained by sampling 2000 SNPs randomly from each chromosome in Atwell et al.  (http://(XLSX)Click here for additional data file.


Genomic selection was recently introduced in plant breeding. The objective of this study was to develop genomic prediction for important seed quality parameters in spring barley. The aim was to predict breeding values without expensive phenotyping of large sets of lines. A total number of 309 advanced spring barley lines tested at two locations each with three replicates were phenotyped and each line was genotyped by Illumina iSelect 9Kbarley chip. The population originated from two different breeding sets, which were phenotyped in two different years. Phenotypic measurements considered were: seed size, protein content, protein yield, test weight and ergosterol content. A leave-one-out cross-validation strategy revealed high prediction accuracies ranging between 0.40 and 0.83. Prediction across breeding sets resulted in reduced accuracies compared to the leave-one-out strategy. Furthermore, predicting across full and half-sib-families resulted in reduced prediction accuracies. Additionally, predictions were performed using reduced marker sets and reduced training population sets. In conclusion, using less than 200 lines in the training set can result in low prediction accuracy, and the accuracy will then be highly dependent on the family structure of the selected training set. However, the results also indicate that relatively small training sets (200 lines) are sufficient for genomic prediction in commercial barley breeding. In addition, our results indicate a minimum marker set of 1,000 to decrease the risk of low prediction accuracy for some traits or some families. Hordeum vulgare) is one of the major cereals and it is used for multiple purposes such as feed, food and malting [Barley . Genomi malting , and is  malting  were som malting . Since t malting –13. Howe malting ,15)Genomic prediction has the benefit of using information based on the total marker information without need of significant single markers associations. The marker information is used in models to calculate genomic estimated breeding values (GEBVs) which can assist the breeders in their selection decisions . The GEB2), and the effective number of QTL (Me).One study  highlighAccording to this theory, the accuracy increases with higher heritability. Studies also showed that most traits with high heritability are predicted well . The stuAnother important parameter influencing the accuracy is the number of individulals included in the training population as indicated by Genetic gains are obtained by selection in the breeding material. Therfore, it is of great interest to predict a new breeding set from previous ones. Consequently, it is relevant to enable prediction in new progeny sets across years and predict across less related breeding sets. However, this issue is a major challenge in genomic prediction in plants, since prediction accuracy between genetically unrelated populations (or families) is often low ,21. InveThe traits: seed size fractions, test weight, protein content and protein yield are all important and routinely measured in commercial breeding programs. Seed size and shape is known to be associated with malting performance . One stu2014 and S2015. However, the two sets of crossing parents were overlapping; hence, 16 of the parents were used in both years. All lines were grown in two locations in Denmark: Dyngby (Jutland) and Holeby (Lolland). Dyngby is characterized as fine sandy loam (JB6) soil, while Holeby is characterized as loam soil (JB7). The fields were divided in trials consisting of smaller plots, and each trial was a randomized block-design with three replicates of each line. In few cases, phenotypic measurements were only obtained from two of the three replicates. Moreover, two check lines were sown in each of the replications in each trial; thus, there were more replications of these two check lines. Each plot measured 5.5 × 1.5 m (8.25 m2) and each plot was harvested separately and a random sample of the seeds harvested were retained for further analysis. The yield was obtained from each plot by total weight of the harvested seeds.The Danish breeding company, Nordic Seed A/S, delivered genetic material and conducted all field trials. The germplasm consisted of 309 advanced spring malting barley lines from Nordic Seed. A total of 119 of the 309 lines were harvested in year 2014, while the remaining (190) were harvested in 2015. The lines in the two harvest years originated from two different sets of parents, and are referred to as SPhenotyping was done on the seeds obtained from each plot. The phenotypic measurements included seed size fractions, protein content, protein yield, test weight and ergosterol content. The seed size fractions were obtained using a SORTIMAT (Baumann Saatzuchtbedarf) instrument to separate the seed samples of 100 g into four seed size classes of >2.8, 2.8–2.5, 2.50–2.2 and < 2.2 mm. Seed fractions in each sample were weighted. The smallest fraction (<2.2 mm) was considered as dust and not used in any of the analysis. Test weight, ergosterol content and protein content were all measured using an instrument based on NIT (Near Infrared Transmission)-technology ). Ergosterol content is based on a test calibration, and is not a verified method. An overview of the phenotypes is given in The statistical models assume traits to follow a normal distribution. Since the three seed size fraction weights were not following a normal distribution, a standardized seed measure was created. A normally distributed seed size measure (SSW) was calculated based on the inverse cumulative normal distribution function. This was done for each fraction and the two measures of seed size were averaged. Since the predictions were similar between all seed size fractions only the results from SSW are shown.Genomic DNA was extracted from each of the 309 lines. Three, two-weeks old seedlings of each line, were bulked and DNA was extracted by a modified CTAB procedure (Cetyl Trimetheyl Ammonium Bromide ). All liM is the matrix that specifies the marker alleles, with the elements of −1, 0, and 1 for the homozygote, heterozygote and opposite homozygote, respectively. P  is calculated by using the frequency of the second allele at locus i (pi). I.e. The i’th columns is computed as: Pi = j2(pi-0.5), where j is a vector of ones. Z is calculated as M-P, and mean value of the alleles effects are 0. The final scaled G-matrix was calculated as:The genomic relationship matrix used in the genomic prediction model was calculated using principles described by , as follG-matrix using the build in R-packages prcomp and a heat map was constructed using the R-package hclust [A PCA was performed based on the e hclust .Linear mixed models with different fixed and random effects were tested by evaluation of maximum likelihood and variance components. The final model used for all phenotypes, except protein yield, was:y is the vector of phenotypic observations, t is the vector of fixed effects. For SSW, protein and PY the fixed effect was year×location, while for TW and ergosterol it was year×location×trial. Z1 is the design matrices for random factors, genotypes and lines, g is the vector of breeding values G is the G-matrix; l is the vector of lines I is the identity-matrix; Z2 is the design matrix for the G×E, c is the vector of G×E interactions, e is the vector of random residuals. The residuals were assumed independent normally distributed values described as For protein yield the model was:Variance components due to random effects were estimated by REML. These variances were used to calculate the heritability for the traits. All analysis of mixed linear models were conducted using the DMU Package for Analyzing Multivariate Mixed Model developed by Aarhus University, Center for Quantitative Genetics and Genomics ,31.Heritability estimates for the two models (5 and 6) were calculated as:GP) were calculated as the correlation between the phenotypic values corrected for the fixed effects and the GEBVs when own line performance were not included in the model. The bias was calculated as the regression between the phenotypic values corrected for the fixed effects and GEBVs. A deviation from the expected regression of 1.0 shows if the model yields GEBVs with too large or too small variance.The standard error of heritability estimates was calculated from the inverse of the information matrix related to the model using a Taylor series ,33. MaxiThe general ability of predicting the phenotypes of future lines based on their genotypes was tested. For this purpose, a leave-one-out (LOO) cross-validation strategy was used. In the LOO strategy, one individual line is excluded from the training population, and the GEBV of same line is predicted from a model trained on all other lines. This was done for all 309 lines, and the accuracy of the 309 predictions was calculated as the correlation between GEBVs and average phenotype corrected for fixed effects in the model.random) strategy using random sets of same sizes as the breeding sets, S2014 and S2015. In the LSOrandom, the standard error of accuracy was calculated as the standard deviation between the ten repeated cross validations.The first aim was to test the ability to predict across breeding sets, and was evaluated by using leave-set-out (LSO). In the LSO strategy, individuals from one year were excluded and the remaining individuals from the other year were used as training population. Genomic estimated breeding values were calculated for the individuals in the excluded year. The number of individuals differed between years, with 119 in year 2014 and 190 in year 2015. Results were compared to ten repeated predictions calculated using a leave-set-out  was excluded from the training set and predicted. One family was defined as lines having at least one parent in common. A number of 104 parents were present in the set. In total, prediction accuracies and biases could be obtained from 104 families of different sizes. The accuracy and bias is given as the correlation between the GEBVs and the corrected phenotype of all 309 lines.Thirdly, the effect of both training population size and the size of the marker set on prediction accuracy was studied. For this purpose, a LOO cross-validation strategy was used. The minimum size of the training population was found by performing predictions using randomly selected reduced populations. The starting population was the full set of 309 lines that was then reduced nine times by 10% each time. For each reduction step the prediction ability was obtained 100 times using 100 different randomly selected lines. The minimum marker number was investigated using the same principle. The starting set included 3,540 markers, and predictions were made by using reduced marker sets. The size of the marker set was reduced by 10% for each reduction step, and the prediction accuracy was obtained 100 times for each reduction step, using 100 different randomly selected marker sets. The markers were selected such that they were equally distributed on all seven chromosomes, and so that there was at least one cM between selected markers. For the larger marker sets, the minimum distance was reduced, when required. Since variance estimates could not be obtained by using small datasets, variance components and heritability were estimated using the full marker set of 3,540 markers, and predictions were performed by using reduced marker sets and training population.To gain information about the genetic relationship among lines, a PCA was made based on the 3,540 markers . No cleaThe phenotypes, seed size fractions, test weight, ergosterol content, protein content and protein yield were obtained from 309 lines over two years. In each year, three replicates were harvested in two different locations. A summary of the phenotypic values is given in The seed size fractions represent the weight of a given fraction, and as seen the largest fraction (f2.8) represents the main part of the seeds with a mean weight of 82.2 g. The largest coefficient of variation (CV) was found in the smallest fraction (f2.2) with a CV on 68% (results not shown). In addition, all seed size fraction weights were highly correlated to each other (results not shown). An overview of the seed size fraction weights in two years (2014 and 2015) and in the two locations (Dyngby and Holeby) is given in An overview of the phenotypes i.e. protein content, PY, TW and ergosterol is given in Heritability and variance component estimates are seen from The potential to use genomic prediction of quality traits in barley was investigated. First, the ability to predict quality traits of a variety when close relatives are in the training population was investigated. For this purpose, the cross-validation strategy was LOO. Accuracies were calculated for all traits as the correlation between the 309 GEBVs and the corrected phenotypes averaged over replicates for each line .The accuracies ranged from 0.40 for protein to 0.83 for ergosterol. To evaluate the level of the accuracy, the accuracies were compared to the maximum prediction accuracy being the square root of the line mean heritability. For all traits, the accuracy was close to the maximum possible prediction accuracy which indicates very high accuracies of the GEBVs.After testing the general ability to predict the performance in the population LOO in , the abi1 and R2) being the same size as the progeny sets (S2014 and S2015)  . Using SSecondly, the ability to predict across less related families was investigated by using a LFO cross-validation strategy. The prediction accuracy from most distant relative is seen from To investigate the effect of the size of the training population, a similar experiment was performed using a leave-one-out cross-validation strategy. Thus, predictions were done using reduced training sets. The outcome for both TW and SSW is seen in The effects of the marker number included in the prediction were evaluated by reducing marker set used for prediction . The meaGenomic selection can be a valuable tool to accelerate genetic gains in plant breeding programs, and has been described as a valuable tool in several plant species . GenPrediction ability is affected by several factors such as training population size, relatedness of training population with validation population and the genetic architecture of the traits studied. In this study, we found heritabilities of line means ranging from 0.21 for protein content up to 0.61 for ergosterol content. The results confirm that the most heritable traits display the highest prediction accuracies. The study of  detected2014 lines to predict S2015 and vice versa. In general, the predictions were similar between the LSO and the LSOrandom. Compared to the LOO method, the accuracies decreased. Since the PCA plot indicated (2014 and S2015) which may indicate that the traits are influenced by genotype by environment interactions.Genetic gains in breeding are obtained by selection of lines which are performing well in all environments in the coming years. Thus, the ability to predict the performance of lines to be tested next year is important. Therefore, we investigated the possibility of using Sndicated  that no Lines in a breeding program are not necessarily closely related, and the ability to predict performance across less related families can be relevant in practical breeding. Therefore, the LFO cross-validation strategy was performed. The prediction accuracy of the LFO prediction were slightly higher than prediction across years, and lower than the LOO strategy accuracies. The training population was not extremely decreased when one family was left out compared to the leave breeding set out strategy, and the decrease in the accuracy was caused by a reduction in the genetic relationship between lines. Likewise, the low accuracy when predicting across breeding sets may be explained by a poorer family relationship across breeding sets. Several studies have noted that the accuracy of genomic predictions is highly influenced by the population used to calibrate the model ,40.Prediction accuracy is affected by training population size ,16,41. OResults further showed that the prediction accuracy decreased, when the number of markers used for prediction was lowered. The decrease in accuracy could be explained by a reduction of markers in LD with the QTL. Hence, the number and distribution of markers must be enough to describe the relationships between lines. Results showed, that the prediction accuracy varied, not only due to the size of the marker set but depended on which markers were included. This highlights the fact that the number of traced QTL influence prediction accuracy, but also that the accuracy is influenced by the markers ability to capture family relationships between lines. Our results suggest that at least 1,000 markers are needed for accurate genomic predictions in barley. A minimum marker number of 1,000 markers may not be valid for all traits, since the two traits tested in this study show differences in their sensitivity to a decrease in marker density. Different traits may be more sensitive to the numbers of markers. When fewer markers were included low prediction accuracies were observed for some marker sets. If fewer markers are to be used, several factors should be considered. Firstly, the markers should be distributed all over the seven chromosomes. Secondly, the markers must be chosen for the specific trait, since the optimum marker set may not be the same for the different traits.To our knowledge, no barley breeders in Northern Europe currently use genomic prediction routinely in their breeding programs. Our results showed high prediction accuracies for several traits using a training set of 308 lines. However, using less than 200 lines in the training set can result in low prediction accuracy, and the accuracy will be highly dependent on the genetic relationships between lines in training population and lines to be predicted. These results confirmed the results of ,25, who In conclusion, since the effective population size of elite barley lines is low, small training sets of around 200 lines is enough to obtain high prediction accuracies. However, the genetic relationship between the training population and the predicted genotypes is highly relevant. It is therefore recommended to use training sets of 200 lines or larger, if information about the training and validation sets is limited. The number of well-distributed markers is recommended to be at least 1,000 markers.The prediction accuracies achieved using genomic selection were comparable to the maximum accuracies of predicting future performance. This corresponds to the accuracy of GEBVs that are close to unity. Very considerable gains can therefore be expected from the application of genomic selection in barley breeding since it would allow selection on large numbers of lines that only need genotyping and thus avoiding costly phenotyping for many traits. In addition, genomic selection can be applied earlier than phenotypic selection and thus reducing time from initial crosses to marketing. Implementation of genomic selectin in an early stage in a breeding program can increase the selection intensity. This is especially relevant concerning quality related traits, since these are often costly to measure. In general, our results confirm that high prediction accuracies can be obtained in barley breeding programs by implementing general methods for genomic selection. Therefore, genomic prediction has the potential to improve genetic gains in commercial barley breeding programs.S1 Table(XLSX)Click here for additional data file.S2 TableThe 309 lines were genotyped by Illumina iSelect 9K barley chip. The genotyping was outsourced to TraitGenetics .(CSV)Click here for additional data file.


The number of model parameters increases with the number of SNPs, so the multicollinearity between covariates can affect the results obtained using whole genome regression methods. In this study, dependencies between SNPs due to linkage and linkage disequilibrium among the chromosome segments were explicitly considered in methods used to estimate the effects of SNPs. The population structure affects the extent of such dependencies, so the covariance among SNP genotypes was derived for half-sib families, which are typical in livestock populations. Conditional on the SNP haplotypes of the common parent (sire), the theoretical covariance was determined using the haplotype frequencies of the population from which the individual parent (dam) was derived. The resulting covariance matrix was included in a statistical model for a trait of interest, and this covariance matrix was then used to specify prior assumptions for SNP effects in a Bayesian framework. The approach was applied to one family in simulated scenarios (few and many quantitative trait loci) and using semireal data obtained from dairy cattle to identify genome segments that affect performance traits, as well as to investigate the impact on predictive ability. Compared with a method that does not explicitly consider any of the relationship among predictor variables, the accuracy of genetic value prediction was improved by 10–22%. The results show that the inclusion of dependence is particularly important for genomic inference based on small sample sizes.In livestock, current statistical approaches utilize extensive molecular data,  Moreover, linkage and linkage disequilibrium (LD) between loci adds a second source of dependency among the predictors. The number of genomic markers such as single nucleotide polymorphisms (SNPs) is still growing, e.g., e.g., In whole-genome regression analyses, it is often the case that the number of genomic markers, le e.g., , but thee.g., e.g., e.g., For “approach , but thee.g., dairy cattle). This required the haplotypes and recombination rates of the common parent (sire) as well as the LD of gametes in the population from which the individual parent (dam) was derived. The obtained covariance structure was then integrated into a statistical model for genomic evaluation. Genome segments with significant impacts on a quantitative trait were inferred, and the precision of the parameter estimates as well as the accuracy of genomic prediction were evaluated. This report ends with a discussion of partial successes, drawbacks, and further options for considering dependencies.The objective of the present study was to extend the Bayesian approach by using an autoregressive prior for marker effects, and to determine the autocorrelation parameter explicitly according to genetic theory. The population structure influences the extent of associations. Family stratification leads to different levels of LD among families, and may result in a biased population-wide measure of LD. Thus, dependencies caused by LD were theoretically derived for a single half-sib family that is part of a typical livestock population . Furthermore, μ denotes the overall mean, and 1 is a vector of n ones. The residuals are assumed to be independent and normally distributed, To study the genetic effects captured by SNPs, each with two alleles, A and B, the following whole genome regression model is fitted to a trait s) and maternal (d) contributions because the alleles are inherited independently. In extension to j and k is determined according to the sire’s diplotype In this section, model (1) is extended to consider the dependencies among predictor variables in To determine the paternal contribution to the covariance in Double homozygous sire (haplotypes AA and AA): all half-sibs inherit the same paternal haplotype AA, and the paternal covariance is zero.The sire is heterozygous at one locus (haplotypes AA and AB): two haplotypes AA and AB can be observed among half-sibs, where each is equally frequent. The paternal covariance is also zero.Double heterozygous sire (haplotypes AA and BB or AB and BA): all possible haplotypes appear among daughters with a probability that depends on the recombination rate Using the allele frequency, In summary, the covariance between SNP genotypes among half-sibs can be split into a linkage part contributed by the sire and an LD part added by the mother. The covariance matrix ν degrees of freedom and scaling parameter S. Let To estimate the unknown parameters of model (1), a Bayesian shrinkage approach is employed. The matrix e.g., i.e., (P1) Uncorrelated prior i.e., a posteriori as (P2) Correlated prior γ, a single Metropolis iteration is used. A log-normal density is applied to the γs as the proposal density t, the γs are individually drawn from the proposal distribution:(P3) Correlated and adaptive prior possible . For eacThe tuning parameter ε is set to one. (The choice of ε is discussed below.) Let jth component is replaced by the current proposed value, and R is obtained as (see Appendix):α; otherwise, γ move slowly away from the initial estimate toward zero if there is evidence of a null effect, or they increase for nonzero SNP effects. Finally, after The SNP effects can be estimated as the mean of their posterior distribution, which can be obtained by Gibbs sampling from the conditional distribution e.g., m|y,Ψ,μ,σAppendix).(P4) Correlated and adaptive prior i.e., e.g., BayesA-type of model (γs which are sampled during Markov chain Monte Carlo (MCMC) simulations. Shrinkage based on P4 is related to differences in the effects of adjacent markers because the γs influence the entries of the lower triangular matrix If covariances between SNPs are not considered, rporated . For P3,a posteriori asFurthermore, the overall mean and residual variance component are distributed lgorithm .r). A range of r values between 0.05 and 1 was considered for evaluation. It is expected that the different methods would yield different rankings for the EGV. In general, the maximum mean TGV is obtained when only a few individuals are selected as parents of future offspring, and it approaches zero if more individuals are selected. Thus, the performance of a method can be determined by comparing the relationship between the mean TGV and r.The accuracy of genetic value prediction was verified by cross-validation. As a measure of accuracy, the mean correlation was calculated between the estimated and simulated genetic values in test data sets, lth training block, respectively, The precision of the estimates of the residual and genetic variance component was determined as the mean squared error (MSE) based on repeated simulations:lth sampling round. These effects resemble haplotype effects. Second, the posterior probability of being positive was obtained from the Gibbs samples:N the number of Gibbs samples after the burn-in period. If P-value, in a similar manner to that described by An appropriate measure of significance was required to evaluate the suitability of the suggested priors for understanding the genetic architecture, particularly relevant genomic regions. First, because the SNP effects were correlated, segment effects Two sets of data were explored. First, simulated data were used to evaluate the performance of the Bayesian approach depending on the different prior choices. Second, real genotype data were used to study the pattern of covariance between SNPs in a real half-sib family and for a particular chromosome. A real phenotype was not analyzed because the present study focused on the impact of the covariance matrix on the outcomes of genomic evaluations. Thus, challenges related to real observations (other nuisance effects or uncertainty about genetic effects on the selected chromosome) were eliminated completely. Hence, a phenotype was simulated based on the real genotypes to investigate the feasibility of the method in a general manner.File S1. In total, 500 marker genotypes for 10,000 progeny were simulated on a chromosome segment with a length of 12 cM, but only the loci at which the sire was heterozygous were considered in further analyses . It was found that 407 of the 903 eigenvalues of The data set comprised a single half-sib family of Holstein-Friesian cows .The data are provided as www.netlib.org/lapack) and module random 1.13 (jblevins.org/mirror/amiller). The program ran on a 2.93 GHz multi-user system. A single chain comprising 50,000 sampling rounds was executed, and 20,000 iterations were omitted as the burn-in. The R package coda was used for MCMC diagnostics. In particular, the effective sample size (ESS) and Heidelberger and Welch’s test, which tests the null hypothesis that the samples were drawn from a stationary distribution, were employed to determine the convergence of a Markov chain.(Metropolis-within-) Gibbs sampling algorithms were implemented in Fortran 90 embedding LAPACK 3.5.0 .The authors state that all data necessary for confirming the conclusions presented in the article are given in the Supplemental Material  to the covariance between SNPs, as discussed later.The diagonal elements of the theoretical covariance matrix i.e., > 10,309, and > 30,000 for a few exceptions; these values hardly differed between SNPs being the QTL or not. Heidelberger and Welch’s test was generally passed. The test failed only at the key SNP with the smallest simulated effect in the five-QTL scenario when the prior P4 was selected. Furthermore, for the hybrid strategy with prior P3, the acceptance rate α ranged from 0.54 to 0.79 (from 0.55 to 0.80) in the five-QTL (50-QTL) scenario.MCMC diagnostics indicated the convergence of the MCMC algorithm for all of the proposed priors. ESS at key SNP effects was rather high, e.g., see For all of the selected priors, and with five and 50 simulated QTL, the estimated SNP effects agreed well with the simulated effects when the complete data set  of selected candidates gave mixed results  in contrast to the antedependence model proposed by γs are expected, whereas large γ-values are related to QTL regions. This was also observed in our data analyses (see File S1). Therefore, such a covariance structure may be classified as nonstationary in a similar manner to the antedependence model because the covariances depend on the distances between markers and on the marker positions, i.e., their proximity to QTL.Previous studies have suggested that some covariance or correlation structure for marker effects should be included in regression models to estimate their effects. Candidates selected from covariance structures have proved useful in other fields, such as time series analysis, an decay . All of e.g., e.g., full-sib families often occur in chickens. Gametes of full-sibs are derived from a single sire and dam, and therefore the covariance between SNPs is restricted to the linkage part.  Then, the haplotypes of both parents should be considered when setting up the covariance matrix, as shown by Unlike haplotype-based approaches (θ and D could be estimated by the expectation-maximization (EM) algorithm  to the covariance matrix was more important than the maternal part , where the interquartile range of D was zero in the simulations , although it is implemented as a variable selection approach. In the present study, variable selection was not executed because it was assumed that each marker contributes to the genetic variation, at least indirectly through LD. However, this strategy led to poor mixing in the MCMC algorithm compared with the adaptive prior P4 using semireal data (results not shown). The trace plots contained repetitions of effect samples with low variability, followed by samples with very high variability, which was not observed to the same extent for the simulated data with small sample sizes. The correlated prior P2 was the most robust prior choice for both the simulations and semireal data because it could detect significant chromosome segments and estimate the variance components with the smallest MSE. To facilitate realistic genome-wide evaluations, we recommend specifying chromosome-wise priors, i.e., The prior covariance matrix used in P3 is similar to that described by e.g., by α was obtained, where α was higher when ε was smaller. However, the impact on a single-effect estimate was negligible for both the simulated and semireal data, whereas the estimated variance components α roughly agreed with the rule of thumb given by α (25–50%) in a similar manner to Bayesian approaches require the specification of prior distributions but the use of hyper-parameters was minimized in the present study. The sensitivity to the selected priors has been evaluated and discussed previously, e.g., in File S1). The variation around the two modes was due to variation in the maternally inherited gametes. Thus, the individuals selected with the highest EGV comprised a mixture of high/medium/low-TGV individuals, and the curve obtained for the mean TGVs decayed slowly. The curve approached the population mean rapidly above File S1). This pattern was not detected with a medium sample size (The sample size had a major effect on the outcome. For small sample sizes (In this study, the dependence between SNP genotypes was derived theoretically from the genetic parameters for a half-sib family. Integrating this information into genomic evaluations improved the estimates of the variance components and the genetic value predictions when the sample size was small (


Mastitis is the most prevalent disease in dairy sheep with major economic, hygienic and welfare implications. The disease persists in all dairy sheep production systems despite the implementation of improved management practises. Selective breeding for enhanced mastitis resistance may provide the means to further control the disease. In the present study, we investigated the genetic architecture of four mastitis traits in dairy sheep. Individual animal records for clinical mastitis occurrence and three mastitis indicator traits  were collected monthly throughout lactation for 609 ewes of the Greek Chios breed. All animals were genotyped with a custom-made 960-single nucleotide polymorphism (SNP) DNA array based on markers located in quantitative trait loci (QTL) regions for mastitis resistance previously detected in three other distinct dairy sheep populations.SOCS2, CTLA4, C6, C7, C9, PTGER4, DAB2, CARD6, OSMR, PLXNC1, IDH1, ICOS, FYB, and LYFR.Heritable variation and strong positive genetic correlations were estimated for clinical mastitis occurrence and the three mastitis indicator traits. SNP markers significantly associated with these mastitis traits were confirmed on chromosomes 2, 3, 5, 16 and 19. We identified pathways, molecular interaction networks and functional gene clusters for mastitis resistance. Candidate genes within the detected regions were identified based upon analysis of an ovine transcriptional atlas and transcriptome data derived from milk somatic cells. Relevant candidate genes implicated in innate immunity included The results confirmed the presence of animal genetic variability in mastitis resistance and identified genomic regions associated with specific mastitis traits in the Chios sheep. The conserved genetic architecture of mastitis resistance between distinct dairy sheep breeds suggests that across-breed selection programmes would be feasible.The online version of this article (doi:10.1186/s12864-017-3982-1) contains supplementary material, which is available to authorized users. Mastitis is an inflammation of the mammary gland usually caused by pathogens, mainly bacteria, developed in the gland cistern after penetration through the teat canal. It is the most prevalent and costly disease in the dairy industry due to reduced and discarded milk, early involuntary culling, veterinary service and labour costs , 2. Ther, Escherichia coli (E. coli) and Staphylococcus aureus (S. aureus), activate the mammary immune system in different ways, influencing disease severity and chronicity [The susceptibility of dairy animals to udder infection is heritable , 7–9 butronicity . The recronicity . Breedinronicity  as well ronicity , 14. Marhttp://cordis.europa.eu/result/rcn/163471_en.html). For the design of the custom-made array, SNPs were selected from both the 50 K and the 800 K SNP DNA arrays, as well as the available re-sequencing data; the array had an average density of 1 SNP every 23 Kb. The present study builds on results from previous studies on Lacaune, Churra and Sarda sheep [Quantitative trait loci (QTLs) and SNPs associated with SCC have been previously identified based on linkage and linkage disequilibrium analyses of three distinct dairy sheep breeds . Sevda sheep –19 in orDst) of 0.33, confirming its independent population status for the purposes of the present study.French Lacaune, Spanish Churra, Italian Sarda and Greek Chios were among the sheep breeds analysed within the framework of the sheep HapMap project . PrincipCoagulase negative Staphylococci, 22% S. aureus and 7.7% Streptococcus spp) bacteria were mainly responsible for CM. Nevertheless, Gram negative bacteria (9.3% Pasteurella spp. and 3% Proteus) were also isolated from the milk samples. Smoothed curves illustrating progression for each trait throughout lactation, after adjusting for all relevant sources of systematic variation, are shown in Fig. Descriptive statistics for SCC, California mastitis test (CMT) and total viable bacterial count in milk (TVC) are listed in Additional file Estimates of heritability and repeatability of the studied traits Table  based onGenomic association studies were conducted separately for early, mid, late and overall lactation for each trait. Multidimensional scaling analysis (MDS) revealed no substructure in the Chios population. In general, similar genomic associations were detected for CM and the three mastitis indicator traits but distinct associations were observed in the different lactation stages. SNPs significantly associated with the studied mastitis traits are shown in Table r2 = 0.20–0.97), implying that they likely correspond to the same causative variation  and to a non-coding lincRNA variant, respectively. Annotations of all significant SNP markers for CM, SCC, CMT and TVC, are presented in Additional file A relatively small number of genes (53 protein-coding and 26 microRNAs) were identified in the candidate regions for mastitis resistance .Based upon the significant heritability estimates and the large amount of genetic variance accounted for by the identified SNPs, we reasoned that the corresponding QTL regions would contain genes contributing to a common pathway associated with mastitis resistance. We therefore identified the sets of annotated genes lying within QTL intervals and sought evidence of gene set enrichment. These genes were enriched for pathways involved in inflammatory and immune responses, both innate and adaptive Fig. . MoreoveGenes that contribute to mastitis resistance are likely to be expressed in milk somatic cells, in other immune cells, and/or in the mammary gland, and much of the genetic variation is likely to affect their transcriptional regulation. In humans, more than 80% of genes that are induced in stimulated monocytes have shown evidence of heritable variation in their level of expression (eQTL) . To asseDAB2, IDH1, NDUFS1, MRPL42, FYB, SOCS2, OSMR, RGMB, PPP4R2, EGFLAM and NUDT4) were highly expressed in both mammary gland and various immune cell lines as shown in Additional file CTLA4, CCNYL1, CEP83, CHD1, TTC33, METTL21A, PLXNC1, CARD6, FASTKD2, PRKAA1 and PTGER4) were also inducible by LPS in sheep BMDMs  were differentially expressed in the milk somatic cells of sheep with divergent levels of SCC  were also differentially expressed in BMDMs after LPS treatment and/or in sheep with differing SCC. In general, no systematic differences were observed in TFBS profiles between dairy and meat sheep except for the following: (1) Mat1-Mc and AP-1 binding sites upstream of the OSMR and PDZRN3 genes, respectively, only in dairy sheep; (2) an HNF-1 binding site upstream of DAB2, only in meat sheep; and (3) HSF and c-Rel sites upstream of C9, only in meat and dairy sheep, respectively.Selective breeding in dairy sheep has led to the development of selective sweeps associated with milk production traits that are common to different breeds . The appBased on all above results, a total of 14 genes were selected as the most promising candidate genes for mastitis resistance , suppressor of cytokine signalling 2 (SOCS2), oncostatin M receptor (OSMR), FYN oncogene related to SRC (FYN), complement factor B (CFB) and isocitrate dehydrogenase 2 (NADP+), soluble (IDH2) – with mastitis susceptibility [CTLA4, SOCS2, OSMR), along with some from the same family , were also found to be LPS-inducible in BFxT sheep BMDMs and/or differentially expressed in the milk somatic cells of Churra and Assaf, indicating they are likely to be involved in protective immunity across sheep populations. The SNP marker located within the IDH1 gene corresponds to a missense mutation but further work is needed to verify if this is the actual causative mutation. For the OSMR and C9 genes, differences in the TFBS in the 1 kb upstream regions were also identified among dairy and meat sheep. The proto-oncogene c-Rel – whose binding site was identified in the sequence upstream of C9 in dairy sheep – is a member of the nuclear factor kappa b (NFkb) family of transcription factors, whose activity has been previously related with mastitis resistance in dairy cattle [CTLA4 [IDH1, OSMR, C6, C7 and C9, mitogen-responsive phosphoprotein, homolog 2 (DAB2), caspase recruitment domain family, member 6 (CARD6) [PTGER4) [In Lacaune sheep, a previous transcriptomic study of SCC associated 7 genes – cytotoxic T lymphocyte-associated protein 4  , 42, and[PTGER4) , 44. ThiICOS), leukaemia inhibitory factor receptor A (LIFR), plexin C1 (PLXNC1) and chromodomain-helicase-DNA-binding protein 1 (CHD1). The proteins encoded by these genes have been previously associated with multiple diseases in humans [cis- and trans-acting eQTLs that are relevant to mastitis, similar to a previous study of human monocytes that highlighted loci associated with inflammatory disease susceptibility [Based upon their transcriptional profiles and association with SCC, novel functional candidates for mastitis resistance were identified in the present study. These include inducible T-cell co-stimulator  traits such as SCC, CMT and TVC but a combination of both clinical mastitis and indicator traits would be preferable. A possible implementation scenario might entail marker-assisted selection based on validated selectable markers and the candidate genes identified in the present study. Another option would be the implementation of genomic prediction and selection schemes across different dairy sheep breeds sharing common QTLs for mastitis resistance, leading to enhanced reference population size and greater accuracy.The study was approved by the Ethics and Research Committee of the Faculty of Veterinary Medicine, Aristotle University of Thessaloniki, Greece. Permits for access and use of the commercial farms were granted by the individual farm owners, who were members of the Chios Sheep Breeders’ Cooperative “Macedonia”. During sampling, animals were handled by qualified veterinarians. Permission to qualified veterinarians to perform milk and blood sampling was granted by the National (Greek) Legislature for the Veterinary Profession, No. 344/29–12-2000. For the ovine transcriptional atlas study approval was obtained from The Roslin Institute’s and the University of Edinburgh’s Protocols and Ethics Committees. All animal work was carried out under the regulations of the Animals (Scientific Procedures) Act 1986.A total of 23 Chios, 103 Lacaune, 120 Churra and 20 Sarda dairy sheep were analysed within the sheep HapMap project . Genotyp2EDTA Vacutainer blood collection tubes (BD diagnostics) by jugular venepuncture for genomic DNA extraction.A total of 609 purebred Chios dairy ewes in the first or second lactation, raised in four commercial farms in Greece, were used. Presence or absence (0/1) of CM was recorded by an experienced veterinarian in monthly visits during the first five months of lactation. On the day of visit, two 50 ml milk samples were collected in the milking parlour under aseptic conditions for the measurement of three traits indirectly related with mastitis: CMT, SCC and TVC. CMT was scored on a scale from 0 to 4 , with hiA total of 638 samples with a CMT score greater than or equal to 2 and/or at least 500,000 somatic cells/ml milk , were fThe distributions of SCC and TVC records were skewed; therefore, records were naturally log-transformed in order to achieve normality of the respective distributions. Monthly animal records were analysed with the following mixed model to derive variance components and genetic parameters for the overall lactation; each trait was analysed separately:o in week of lactation (i.e. week post-lambing) mWhere: Y = record of ewe μ = overall meanF = fixed effect of flock iYS = fixed effect of year-season of lambing jα1 = linear regression on age at lambing (age)L = fixed effect of lactation number kW = fixed effect of week of lactation mbn = fixed regression coefficient on week of lactation m (order n = 2)Pn = orthogonal polynomial of week of lactation m (order n = 2)g = random additive genetic effect of ewe o, including pedigree genetic relationships among animalspe = random permanent environment effect of ewe oe = random residual effectA Logit function was fitted to the CM analysis to account for the binary nature (0/1) of the trait. For all traits, the fixed regression on week of lactation yielded a smoothed curve illustrating the trait progression throughout lactation. Heritability and repeatability estimates were derived from the variance components of the random effects for each trait. Residuals from these analyses were kept to be used as dependent variables in the ensuing genomic association studies aiming to assess the effect of SNPs on the respective traits expressed across the entire lactation. Subsequently, bivariate analyses were conducted with the above model to estimate genetic and phenotypic correlations among traits.In separate analyses of SCC, CMT and TVC, second-order polynomial functions of week of lactation were fitted to the additive genetic and permanent environment effects of model (1), resulting in separate variance component and genetic parameter estimates by week of lactation. The latter were then combined to derive average heritability and repeatability estimates for early (weeks of lactation 1–7), mid (weeks 8–17) and late (weeks 18–23) lactation. The corresponding residuals were also kept as input variables to the genomic association studies aiming to assess the impact of SNPs on traits expressed in different stages of lactation. These analyses were not possible with the Logit function fitted to the analysis of binary CM. Therefore, a linear model was fitted to this trait in order to derive residuals by lactation stage for the genomic association analysis. All mixed model analyses were conducted with ASReml v4.0 .Genomic DNA for all samples was extracted from blood buffy coat using a modified protocol (Modified Blood) as described by Psifidi et al. .P < 10−6. After quality control, 710 SNP markers remained for further analysis. SNP locations were obtained from the Oar v3.1 assembly using the Ensembl genome browser (www.ensembl.org). To account for possible population stratification, a genomic relationship matrix was generated including all individual animals. This genomic relationship matrix was converted to a distance matrix that was used to carry out classical MDS analysis with the R package GenABEL [P ≤ 0.05 significance threshold was set at P ≤ 7.04 × 10−5 and a suggestive significance threshold  was set at P ≤ 1.41 × 10−3.All animals were genotyped with a customized 960-SNP DNA array containing SNPs located in seven previously identified QTL regions for mastitis resistance on chromosomes 2, 3, 5, 12, 16 and 19. Details of the SNPs comprising the DNA array are presented in Additional file  GenABEL . Individ GenABEL  was usedr2 value using PLINK v1.9 [LD among significant SNPs was calculated as an INK v1.9  in orderINK v1.9 A) and total phenotypic variance (PVP) due to each SNP locus were calculated as follows:Individual markers found to be significant in the previous step were further examined with a mixed model that included the fixed effects of model (1), the fixed effect of the corresponding SNP locus genotype and the random effect of the animal. Additive (a) and dominance (d) effects, and the proportion of additive genetic variance (PVeml v4.0 .http://www.ensembl.org/Tools/VEP) and BioMart data mining tools (http://www.ensembl.org/biomart/martview/) were used to map and annotate the significant SNP markers derived with the genomic association analysis on the reference genome, and to locate genes in the corresponding candidate regions for mastitis resistance.For sheep assembly Oar v3.1, the Ensembl Variant Effect Predictor (VEP) (www.ingenuity.com) in order to identify canonical pathways and gene networks constructed by the products of these genes. IPA constructs multiple possible upstream regulators, pathways and networks serving as hypotheses for the biological mechanism underlying the trait. This analysis uses data from a large-scale causal network derived from the Ingenuity Knowledge Base. IPA then infers the most suitable pathways and networks based on their statistical significance, thereby setting a threshold above which the pathways are significant.Gene lists in the candidate regions for mastitis resistance were analysed using the Ingenuity Pathway Analysis (IPA) programme  [P-value, with increasing ES (>1) reflecting over-representation of that functional category.Gene lists were also analysed against an ID v6.7) . For eacExpression levels of each gene located within candidate regions for a mastitis trait were obtained from a transcriptomic atlas of 406 RNA-seq libraries, representing all major organ systems; the atlas was based on a BFxT sheep cross  and alsoSalmonella minnesota Re 595 ) at a concentration of 100 ng/ml. Salmonella is a TLR4 agonist; Staphylococcus aureus, which is the main cause of mastitis in dairy sheep, is known to activate both TLR2 and TLR4 [e) greater than 9. All RNA-Seq libraries from the sheep transcriptional atlas included in this study were TruSeq stranded mRNA libraries (125 bp paired-end). Libraries were prepared by Edinburgh Genomics (https://genomics.ed.ac.uk/) using the TruSeq mRNA Library Preparation kit (Illumina) and sequenced on the Illumina HiSeq v2500 at a depth of >25 million reads per sample.To generate RNA for library preparation, BMDMs were differentiated for 7 days in recombinant human colony stimulating factor and treated with LPS . Expression level is then estimated directly  by quantifying exact matches between reads and k-mers. Expression is reported per transcript as the number of transcripts per million, and is summarised to the gene-level as described previously [Expression levels for all samples from both the transcriptomic atlas and the milk somatic cell transcriptome were estimated using Kallisto v0.42.4 . Rather embl v84 ; n = 23,eviously .Differential expression analysis was run on the LPS time series data using the Kallisto output with the R/Bioconductor package ‘Sleuth’. Heatmaps were drawn using the heatmap.2 function from the R package gplots v3.0.1. Additional differential expression analyses were conducted between sheep with high, moderate and low levels of SCC, after adjusting for breed (Churra and Assaf). Relevant data for this study are described by Suarez-Vega et al. . Least sμ = overall mean, B = fixed effect of breed i, g = fixed effect of the mean expression of gene j, e = random residual effect.Furthermore, the effect of the expression level of each gene (located within a candidate region for mastitis resistance) on SCC and milk yield was assessed using the following linear model:P ≤ 0.0125. The analyses were conducted with ASReml v4.0 [The significance threshold in this analysis was set at 0.05. Since genes were located within 5 QTL regions, an FDR adjustment for multiple testing was applied and the significance threshold was subsequently re-set to eml v4.0 .In order to identify whether differences in TFBS are associated with differences in expression level of the genes found in the candidate regions for mastitis resistance, we extracted and compared the corresponding 1 kb upstream sequence in all genes from both meat and dairy sheep. Specifically, we extracted the sequences from the six sheep used to create the BFxT transcriptomic atlas and six sheep sequenced as part of the Sheep HapMap Project  (two Chuhttps://genomics.ed.ac.uk). The six libraries were run on one lane of the Illumina HiSeq2500 System to generate in total approximately 220 M + 220 M reads resulting in a 10-fold coverage per sample. Details of the re-sequencing methodology within the HapMap sheep project can be found in Kijas et al. [® 7.0 Suite, with default parameters, which predicts TFBS in DNA sequences using a library of positional weight matrices [The six atlas sheep (BFxT) had been fully re-sequenced. Illumina Tru-Seq Nano 350 gel-free libraries (125 bp paired end) were prepared, for each sample, by Edinburgh Genomics  as well as Lacaune, Churra and Sarda sheep from previous studies. (XLSX 13 kb)Additional file 2: Figure S1.Principal component analysis results for Chios, Lacaune, Churra and Sarda sheep. Red dots represent Chios (present study), green dots Churra, blue dots Lacaune and orange dots Sarda sheep (previous studies). (DOCX 20 kb)Additional file 3: Table S2.Descriptive statistics of milk somatic cell count (SCC), California mastitis test (CMT) and total viable bacterial count in milk (TVC) in Chios sheep. (DOCX 12 kb)Additional file 4: Table S3.Genetic correlations among mastitis measures in different lactation stages in Chios sheep. (DOCX 12 kb)Additional file 5: Figure S2.P-values are plotted against the expected P-values for each trait. Q-Q plots are not presented for TVC and SCC in mid lactation since no significant results for these traits were identified. (DOCX 248 kb)Q-Q plots displaying the genomic association results for the mastitis traits studied in Chios sheep. Q-Q plots for milk somatic cell count (SCC) in early (A), late (B) and overall (C) lactation; for California mastitis test (CMT) in early (D), mid (E), late (F) and overall (G) lactation; for total viable bacterial count in milk (TVC) in early (H), late (I) and overall (J) lactation; for clinical mastitis occurrence (CM) in early (K), mid (L), late (M) and overall (N) lactation; Observed Additional file 6: Table S4.r2) for the significant SNP markers identified in the genomic association analyses of mastitis resistance in Chios sheep. (XLSX 13 kb)Linkage disequilibrium (LD) estimation  for SNP markers associated significantly with mastitis resistance in Chios sheep. LD patterns are shown for chromosomes 2 (A), 3 (B), 5 (C), 16 (D) and 19 (E). LD blocks are marked with triangles. The significant markers are illustrated with a blue arrow. The candidate regions for mastitis resistance identified in previous studies and in Chios sheep (present study) are marked with dashed lines. The strongest LD signals are in red and the weakest in white. (DOCX 1060 kb)Additional file 8: Table S5.Genes located in the candidate genomic regions for mastitis resistance in Chios sheep. (XLSX 15 kb)Additional file 9: Table S6.Annotation of SNPs identified to have a significant association with mastitis resistance traits. (XLSX 16 kb)Additional file 10: Table S7.Functional annotation clustering analysis of the genes located in the candidate regions for mastitis resistance. (XLSX 16 kb)Additional file 11: Figure S4.Expression level of genes, located in the mastitis resistance candidate regions, across both mammary glands and immune cell lines/tissues. Expression level is estimated as the mean TPM (transcripts per million) of all (5) experimental replicates and is represented here as a Z-score per cell line/tissue. Data is obtained from two transcriptomic atlases; one of Scottish Blackface x Texel (BFxT) sheep and one of Texel sheep. (PNG 67 kb)Additional file 12: Figure S5.Expression level of genes, located in the mastitis resistance candidate regions, as extracted from the Churra/Assaf milk somatic cell transcriptome analysis. Expression level is estimated as the mean TPM (transcripts per million) of all (5) experimental replicates and is represented here as a Z-score per individual animal. (PNG 54 kb)Additional file 13: Table S8.Transcription factor binding site analysis using the MATCH programme. (XLSX 41 kb)Additional file 14: Table S9.List of SNP markers comprising the mastitis-specific custom-made array used in the genomic association analysis of Chios sheep. (XLSX 51 kb)Additional file 15: Table S10.Expression levels of genes located in the candidate regions for mastitis resistance identified in Chios sheep. The gene expression levels presented here are across all the body tissues and cell lines, as extracted from the two (BFxT and Texel) sheep transcriptomic atlases. Details on the expression profile of each gene across the transcriptomic atlas are also provided. (XLSX 71 kb)


Recently, there is increasing interest to detect associations between rare variants and complex traits. Rare variant association studies usually need large sample sizes due to the rarity of the variants, and large sample sizes typically require combining information from different geographic locations within and across countries. Although several statistical methods have been developed to control for population stratification in common variant association studies, these methods are not necessarily controlling for population stratification in rare variant association studies. Thus, new statistical methods that can control for population stratification in rare variant association studies are needed. In this article, we propose a principal component based nonparametric regression (PC-nonp) approach to control for population stratification in rare variant association studies. Our simulations show that the proposed PC-nonp can control for population stratification well in all scenarios, while existing methods cannot control for population stratification at least in some scenarios. Simulations also show that PC-nonp’s robustness to population stratification will not reduce power. Furthermore, we illustrate our proposed method by using whole genome sequencing data from genetic analysis workshop 18 (GAW18). Recently, there is increasing interest to detect associations between rare variants and complex traits. The variant by variant methods used to detect associations of common variants may not be optimal for detecting associations of rare variants due to allelic heterogeneity as well as the extreme rarity of individual variantsBurden tests1234781011126X2 to X2/λ and assumes X2/λ follows a chi-square distribution, where the inflation factor λ can be estimated using genotypes at genomic markers. PC-linear approach summarizes the genetic background or ancestry information through the PCs of genotypes at genomic markers. The PCs can be further used to eliminate the effect resulting from population stratification through linear regressions. MLM approach corrects for a wide range of sample structures by explicitly accounting for pairwise relatedness between individuals.All the aforementioned methods are population-based methods for unrelated individuals. It has been long recognized that, for population-based association studies, population stratification can seriously confound association results15171821et al.et al.et al.et al.et al.et al.et al.26Although several methods for controlling for population stratification have been developed for common variants, it remains unclear whether these methods are equally effective for rare variants. Because rare variants have typically arisen recently, they tend to show greater geographic clustering or more latent subpopulations than common variants that are typically older. The more geographic clusters or latent subpopulations, the more difficult it will be to control for population stratification. Mathieson and McVeanIn this article, we propose a PC based nonparametric regression (PC-nonp) approach to control for population stratification in rare variant association studies. PC-nonp adjusts population effects of both trait values and genotypes at candidate loci for PCs of genotypes at genomic markers by applying nonparametric regressions. We use extensive simulation studies to evaluate the performance of the proposed method PC-nonp and compare the performance of PC-nonp with that of GC and PC-linear developed for common variants and recently proposed biased urn permutation test (BiasePerm)n unrelated individuals. Suppose that each individual has been genotyped at a candidate locus (single variant or multiple variants) and at L genomic markers. Let yi, xi, and pi denote the trait value, genotypic score at the candidate locus (weighted sum of genotypic scores if there are multiple variants), and the first k PCs  of genotypes at genomic markers of the ith individual. The PCs of genotypes at genomic markers are good summary measures of ancestry or genetic background. PC-linear is probably the most popular method to control for population stratification. However, this method is based on linear combinations of PCs. Furthermore, recently developed BiasePermConsider a sample of In this article, we propose a PC based nonparametric regression (PC-nonp) approach that adjusts population effects of both trait values and genotypes at candidate loci for PCs of genotypes at genomic markers by applying nonparametric regressions. That is,μ1(·) and μ2(·) are regression functions with unknown forms and will be estimated using smoothing techniques. Let ith individual after adjusting for population effects. We can construct association tests based on the residuals.where K(·) be a kernel function with mode at 0. The kernel estimators of μ1(pi) and μ2(pi) are given byMany methods have been developed to estimate the unknown regression function, including local linear method29303234pj =  is the first k PCs for the jth individual, H =  is the smoothing parameter, and ith individual. Thus, we can consider residuals ith individual after adjusting for population stratification.respectively, where In this study, we use the quartic kernelh1 = ... = hk = h. Then,For computational consideration, we assume that Tscore = U2/V, where Tscore asymptotically follows a chi-square distribution with one degree of freedom (df)xi can be a weighted combination1xi is the genotypic score of a single variant and a regional test in which xi is the weighted combination of genotypes at the variants in a genomic regionTo test association between trait values and genotypes based on h is critical to kernel estimates of regression functions32et al.h. This method is based on the genotypes at a set of genomic markers. Suppose there are L genomic markers. We perform PC-nonp single-variant test for all the L genomic markers and denote P1, …, PL as the associated P-values. If population stratification is well controlled for, P-values P1, …, PL should follow a uniform distribution under the null hypothesis of no association. Let Fn be the empirical distribution function of the P-values P1, …, PL and F be the uniform distribution function. The Kolmogorov test statistic P-values P1, …, PL and the uniform distribution are. We propose to choose h* that minimizes the Kolmogorov test statistic, i.e.,We have so far assumed a given smoothing parameter in the kernel estimates. It is well known that choosing a proper value for smoothing parameter h* can be obtained by a simple grid search across a range of h. We divide the interval [0, ∞)into subintervals 0 ≤ h1 < … < hS−1 < hS < ∞. Then, h* increases linearly with S. However, h* needs to be calculated only once. We can use this h* to calculate the residuals of the nonparametric regression for trait values and genotypes at each variant. Let k denote the number of PCs used. In this study, we use hs = 2s−23)/(5+k)2 GCλ and lth genomic marker. (3) PC-linearWe compare the performance of the proposed test with that of the following four tests. (1) Uncorrected: this test is also based on the score test statistic k0 subpopulations and populations with spatially structured populations. In each set of simulations, we consider both qualitative and quantitative traits. To generate a qualitative disease affection status, we use a liability threshold model based on a continuous phenotype (quantitative trait). An individual is defined to be affected if the individual’s phenotype is at least one standard deviation larger than the phenotypic mean. This yields a prevalence of 16% for the simulated disease in the general population. In the following, we describe how to generate genotypes and how to generate a quantitative trait in the two sets of simulations.We consider two sets of simulations: populations with http://www.1000genomes.org). GAW17 data contain genotypes of 697 unrelated individuals at 24,487 variants. The distributions of MAF at rare variants (MAF < 0.01) and MAF at common variants of 24,487 variants are given in This set of simulations is based on allele frequencies at 24,487 variants calculated from the empirical Mini-Exome genotype data provided by the genetic analysis workshop 17 (GAW17). The genotypes of GAW17 data set are extracted from the sequence alignment files provided by the 1000 Genomes Project for their pilot3 study /Fst and (1 − p)(1 − Fst)/Fst, where Fst is the Wright’s measure of population subdivisionFst = 0.01). For each variant, we accept k0 subpopulations if MAF < 0.01) and at the common variants for k0 = 5 are given in To generate genotypes of individuals in a population with To evaluate type I error, we generate trait values independent of genotypes by using the model:yij denotes the trait value of the jth individual in the ith subpopulation, εij follows a standard normal distribution, and μi is the population mean of the ith subpopulation. In this study, if k0 ≤ 2, we set μ1 = 0 and μ2 = μ; otherwise, we set μ = 5 if k0 = 20; otherwise μ = 2.where nT variants (possibly both rare and common variants) in a genomic region. We randomly choose nc from the nT variants as causal variants . For the jth individual in the ith subpopulation, let xijl denote the genotypic score of the jth individual in the ith subpopulation at the lth causal variant. We assume that all the nc causal variants have the same heritability such that rarer variants have larger effects. Under this assumption, the disease model is given byTo evaluate power, we consider βl are constants and their values depend on the total heritability.where K0 × K0 grid squares. Then, we generate genotypes by starting with a number of individuals and their locations on the grid. We work backward in time to generate random genealogical events. Each event is either a coalescence of two lineages or a migration of a single lineage from one square to another. The relative rates of coalescence and migration depend on the population-scaled migration rate M and the number and distribution of lineages on the grid  + εi, where ϕ(i) =  if the ith individual originates from grid square l, j; Rl,j is the nongenetic risk in grid square l, j; εi is a standard normal random number; and β is a constant. We use the following three models to determine the value of Rl,j. Model 0: no population stratification in which Rl,j = 0 for all l and j. Model 1: a small and sharp spatial distribution in which Rl,j = 1 if l0 ≤ l ≤ l0 + 3 and j0 ≤ j ≤ j0 + 3 for l0 = j0 = 6, or 20 − l0 = j0 = 6, or l0 = j0 = 14; Rl,j = 0 otherwise. Model 2: a wide and smooth spatial distribution in which l0 = j0 = 6. In this study, we use the following parameters: K0 = 20, M = 0.01, and β = 2.To generate quantitative traits under null hypothesis, let nT variants in a genomic region. We randomly choose nc from the nT variants as causal variants. For an individual, let xl denote the genotypic score at the lth causal variant. Under the assumption that all the nc causal variants have the same heritability, the trait value for an individual is generated byUnder alternative hypothesis, we assume that there are y0 is the trait value generated under null hypothesis.where Kol(h). We generate trait values and genotypes at 10,000 variants under simulation set 1 for k0 = 5 and k0 = 10 and under simulation set 2 models 1 and 2. Under each of the four scenarios, we calculate Kol(h) for different values of h. The relationships between Kol(h) and −log(h) under the four scenarios are given in Kol(h) under the four scenarios are all bowl shaped and thus have minimum. The histograms of 10,000 P-values of the proposed test for different values of h are given in h is large, population effects are not adjusted enough and thus the number of small P-values are more than expected; when h is small, population effects are over adjusted and thus the number of large P-values are more than expected; when h minimizes Kol(h), the distribution of P-values is very close to the uniform distribution.We first perform simulation studies to evaluate the existence of the minimum of Kolmogorov test statistic P-values. For all other tests, we use asymptotic distributions to evaluate P-values. For 10,000 replicated samples, the 95% confidence intervals (CIs) for type I error rates of nominal levels 0.01 and 0.001 are  and , respectively.We use 10,000 replicated samples to evaluate type I error rates. For BiasedPerm, we use 5,000 permutations to evaluate k0 = 1 in simulation set 1) and in the case of no population stratification (model 0 in simulation set 2). Type I error rates are given in To evaluate type I error rates, we first want to see the performance of the asymptotic distributions we used. For this purpose, we perform simulations under null hypothesis in a homogenous population  we do not know how many PCs should be used and (2) increasing the number of PCs may decrease the power.Type I error rates under structured populations in simulation set 1 for Type I error rates under spatially structured populations in simulation set 2 for models 1 and 2 are given in k0 = 1 and k0 = 10 in simulation set 1 and under models 0 and 2 in simulation set 2, in which all tests except Uncorrected can control for population stratification well. Power comparisons under k0 = 1 and k0 = 10 in simulation set 1 are given in To evaluate if PC-nonp’s robustness to population stratification will reduce power, we perform simulation studies to compare power using regional tests under Power comparisons under models 0 and 2 in simulation set 2 are given in The data set for GAW18 includes whole genome sequencing (WGS) data of 959 individuals (464 directly sequenced and the rest imputed) from 20 Mexican American pedigrees from San Antonio, Texas. There are 21–76 individuals in each pedigree. Phenotype data include sex, age, year of examination, systolic and diastolic blood pressure (SBP and DBP), use of antihypertensive medications, and tobacco smoking at up to four time points.10 of the four tests and expected log10 under the assumption of uniform distribution of P-values are given in Since Mexican American population is admixture population, association studies based on unrelated individuals from this population may be subjected to bias due to population stratification. For our purpose, we extract 132 genetically unrelated individuals from the 20 pedigrees with phenotypes and WGS data and select SBP as the trait of interest while take sex, age, use of antihypertensive medications, and tobacco smoking as covariates. For WGS data, we only consider one chromosome (chromosome 17). Among the 132 unrelated individuals, there are 404,032 SNPs on chromosome 17. Since the sample size is small, we only consider the 41,754 uncommon SNPs with MAF between 0.02 and 0.05 instead of including rare SNPs. We randomly draw 10,000 SNPs from the 41,754 SNPs without replacement and test association between the phenotype and each of the 10,000 SNPs using each of the four tests: Uncorrected, GC, PC-linear, and PC-nonp. We repeat the drawing procedure 4 times with re-drawing 10,000 SNPs from the 41,754 SNPs. Quantile-quantile plots of the observed −logWith the development of next-generation sequencing technology, there is increasing interest to detect associations between rare variants and complex traits. Many statistical methods have been developed for detecting rare variant associations. However, these methods may be subject to bias due to population stratification and, as pointed out by Mathieson and McVeanyi and xim as the trait value and genotypic score at the mth variant of the ith individual. Let yi = μ(pi) + εi and xim = μm(pi) + εim, where i = 1, …, n and m = 1, …, M. Based on the residuals a1 and a2 evaluated at the sample MAF for the mth variant in the data. To apply our method to CMC, suppose that M variants can be classified as Sr groups of rare variants and Sc individual variant sites. Define indicator variables xis for all individuals and the Sr groups of rare variants, where xis = 1 if minor alleles at any variant in the sth group of the ith individual are present; xis = 0 otherwise. Let S = Sr + Sc and define s = 1, …, Sc) as the genotypic score of the ith individual at the sth individual variant site. Let yi = μ(pi) + εi and xis = μs(pi) + εis, where i = 1, …, n and s = 1, …, S. Based on residuals T2 test because Although we describe our proposed method using a single-variant test and a weighted sum regional test, our method can be applied to most existing rare variant association tests such as CMCet al.Zhang How to cite this article: Sha, Q. et al. A Nonparametric Regression Approach to Control for Population Stratification in Rare Variant Association Studies. Sci. Rep.6, 37444; doi: 10.1038/srep37444 (2016).Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.