Fusarium virguliforme, has spread to northern soybean growing regions in the US causing significant yield losses. The objectives of this study were to identify loci underlying variation in plant responses to SDS through association mapping (AM) and to assess prediction accuracy of genomic selection (GS) in a panel of early maturing soybean germplasm. A set of 282 soybean breeding lines was selected from the University of Minnesota soybean breeding program and then genotyped using a genome-wide panel of 1536 single-nucleotide polymorphism markers. Four resistance traits, root lesion severity (RLS), foliar symptom severity (FSS), root retention (RR), and dry matter reduction (DMR), were evaluated using soil inoculation in the greenhouse. AM identified significant peaks in genomic regions of known SDS resistance quantitative trait loci cqSDS001, cqRfs4, and SDS11-2. Additionally, two novel loci, one on chromosome 3 and another on chromosome 18, were tentatively identified. A ninefold cross-validation scheme was used to assess the prediction accuracy of GS for SDS resistance. The prediction accuracy of single-trait GS (ST-GS) was 0.64 for RLS, but less than 0.30 for RR, DMR, and FSS. Compared to ST-GS, none of multi-trait GS (MT-GS) models significantly improved the prediction accuracy due to weak correlations between the four traits. This study suggests both AM and GS hold promise for implementation in genetic improvement of SDS resistance in existing soybean breeding programs.Sudden death syndrome (SDS), caused by The online version of this article (doi:10.1007/s11032-015-0324-3) contains supplementary material, which is available to authorized users. Glycine max (L.) Merr.], caused by Fusarium virguliforme  of soybean [F. virguliforme in the form of chlamydospore or macroconidia can persist in crop residue and soil for many years , and only a few of them have been consistent across mapping populations from different genetic backgrounds , cqSDS001 on linkage group D2 (chromosome 17), cqRfs1, cqRfs2, cqRfs3 on linkage group G (chromosome 18), and cqRfs6 on linkage group N (chromosome 3) were repeatedly mapped in multiple populations  resistance to SDS is complex. A total of 58 QTL have been reported as providing resistance to SDS in bi-parental mapping populations  Rafalski  providesA new marker-based approach known as genomic selection (GS) could potentially be an alternative strategy to stack numerous genes to achieve durable SDS resistance. Rather than utilizing only molecular markers in tight association with targeted QTLs, GS has been developed with the aim of directly predicting genetic value for quantitative traits by taking advantage of all available genome-wide marker information   followed by autoclaving and cooling. The cooled sorghum was then inoculated with 15 × 5 mm blocks of PDA infested with 2-week-old cultures of the Somerset #1A isolate. Bags were incubated at room temperature with normal fluorescent room lighting for 30 days. The contents of each bag were mixed daily to ensure uniform infestation of the sorghum throughout the bag. At the time of soybean planting, the growth media was inoculated with a 1:20 (volume/volume) ratio of infested sorghum inoculum to media. The uninoculated control treatment contained only growth media. Each entry was planted in a Jumbo Junior  square pot containing 800 ml of soil. After planting, the pots were placed in the greenhouse, watered to field capacity daily, and maintained at 22 °C with 14 h daylight.In spring 2013, a total of 279 soybean lines (seeds of three lines were unavailable) were evaluated for SDS resistance in the greenhouse using the inoculation procedure of Luckew et al. . An isolThe greenhouse experiment was conducted as six separate plantings because of space limitations. The six plantings were conducted consecutively under the same greenhouse conditions. Each planting consisted of 30–50 soybean lines with five inoculated replications plus one uninoculated replication for each line. Two check cultivars: ‘McCall’ (susceptible) and ‘MN0302′ (resistant) were included in each planting. For each planting, each plant was evaluated for four symptoms or responses associated with SDS by the same experienced evaluator 4 weeks after planting. These observations included: root lesion severity (RLS), foliar symptom severity (FSS), root retention (RR), and dry matter reduction (DMR).F. virguliforme infection ranging from 1 (no lesion) to 10 (most severe lesion development): 1 = no lesions visible on taproot, 2 = lesions on 10 % of the taproot, 3 = lesions on 20 % of the taproot, 4 = lesions on 30 % of the taproot, 5 = lesions on 40 % of the taproot, 6 = lesions on 50 % of the taproot, 7 = lesions on 60 % of the taproot, 8 = lesions on 70 % of the taproot, 9 = lesions on 90 to 100 % of the taproot, and 10 = lesions on >90 % of the taproot of the taproot or the taproot is completely missing.RLS is a measure of the severity of root lesion development caused by F. virguliforme /(root dry weight of uninoculated plant) × 100. DMR = 100 − (shoot dry weight of inoculated plant)/(shoot dry weight of uninoculated plant) × 100.Percentage of root or shoot dry weight change caused by y = u + L + ε within each planting and performed analysis of variation (ANOVA) with the PROC ANOVA in Statistical Analysis System (SAS) Version 9.4 , where y was one of the four trait ratings of each plant, u was the intercept, L was the effect of soybean line, and ε was the residual. The effect of line × replication was used as the error term to test significance of the effect of line. The phenotypic value of each soybean line was represented as the mean of trait ratings across five replications for each trait and used the phenotypic values for subsequent AM and GS modeling. Scatter plots were constructed based on the pair-wise correlation between the phenotypic values of each pair of traits.The ratings of each trait were then fitted into a linear regression model: y = Xα + Pβ + Kγ + ε was used, where y is the vector of phenotypic values, X is the vector of SNP marker genotypes, α is the coefficient of marker effect being estimated, P is the matrix of first three principal components from PCA accounting for the population structure plus the covariate vector of experimental plantings, β is the coefficient of principal components and experimental plantings, K is the additive relationship matrix estimated based on SNP genotypes accounting for genetic kinship among the individuals, γ is the vector of random effects corresponding to genetic kinship, and ε is the vector of random effects corresponding to residuals. The variances of γ and ε are Var(γ) = 2KVg and Var(ε) = VR, respectively, where K is the genetic kinship, Vg is the genetic variance, and VR is the residual variance. False discovery rate (FDR) of 0.05 was used to correct for multiple comparisons in AM using package ‘QVALUE’ in R . Manhattan plots were created based on the AM results with SNPEVG  for SDS resistance, the same set of phenotypic and genotypic data was used in a ninefold cross-validation study. Specifically, 279 soybean lines first were randomly divided into nine subsets. In each fold, eight subsets of lines (248 lines) were used as training sets and the remaining subset (31 lines) was a validation set. In the training set, the marker effects were simultaneously estimated by fitting a statistical model to both phenotypic and genotypic data. The marker effects were then used to predict the genetic values of individuals in the validation set. Prediction accuracy was calculated as the correlation between marker-based prediction and phenotypic values. The cross-validation process was repeated nine times (nine folds), with every subset of soybean lines used exactly once as the validation set.y is the vector (n × 1) of phenotypic observations of n individuals, X is the design matrix (n × r) for fixed planting effects, b is the vector (r × 1) of planting effects, Z is the design matrix (n × m) for additive effects of SNP markers, α is the vector (m × 1) of additive effects of SNP markers, and e is the vector (n × 1) of residuals. The variances of α and e are Im is the m × m identity matrix, In is the n × n identity matrix. A computationally efficient method, ridge-regression best linear unbiased prediction (RR-BLUP) was employed to solve the mixed model. Previous GS studies suggested slight differences between different genomic prediction algorithms including G-BLUP , Bayesian approaches, and machine learning algorithms  model and multi-trait genomic selection (MT-GS) were evaluated, and their accuracies for predicting SDS resistance were compared. For ST-GS, a mixed linear model was constructed to estimate marker effects of phenotypic traits: y is the matrix (n × t) of phenotypic observations for t traits of n individuals, It is the identity matrix (t × t), X is the design matrix (n × r) for fixed planting effects for each trait, b is the matrix (r × t) of planting effects for t trait, Z is the design matrix (n × m) for additive effects of SNP markers for each trait, α is the matrix (m × t) of additive effects of SNP markers for t trait, e is the matrix (n × t) of residuals, and α and e are G0 is the covariance matrix (t × t) of additive effects, A is the additive genetic relationship matrix (n × n), R0 is the covariance matrix (t × t) of residuals, and In is the identity matrix (n × n). The marker effects of each trait were simultaneously estimated by solving the mixed model through REML method implemented in R package ‘rrBLUP’ (R Development Core Team i) is the genotype variance for SNPi and α is the vector (m × 1) of additive effects for SNPi. The additive genetic variance and the residual variance were estimated. Ten types of MT-GS models were developed: RLS_FSS model for RLS and FSS; RLS_RR model for RLS and RR; RLS_DMR model for RLS and DMR; FSS_RR model for FSS and RR; FSS_DMR model for FSS and DMR; RR_DMR model for RR and DMR; RLS_FSS_DMR model for RLS, FSS, and DMR; RLS_FSS_RR model for RLS, FSS, and RR; RR_FSS_DMR model for RR, FSS, and DMR; and FT model for all four traits. A notched boxplot was made to compare the prediction performance of MT-GS models to ST-GS models for each trait. The notch marks the 95 % confidence interval for the medians. In the notched boxplot, the medians significantly differ if two boxes’ notches do not overlap.MT-GS models were developed by fitting the phenotypic observations of multiple traits (t) simultaneously in a multivariate mixed linear model: The effect of marker numbers on GS accuracy was also determined through nine-fold cross-validation by including random samples of 96, 192, 384, and 768 SNPs from the full marker set. Within each fold, the analysis was repeated 100 times to avoid sampling bias for markers. All prediction accuracies were estimated with R package ‘rrBLUP’ (R Development Core Team p < 0.05) in all plantings, except for FSS in plantings 4 and 6, and RR and DMR in planting 5  for each of four SDS resistance traits was conducted within each planting. ANOVA showed the effect of soybean genotypes was significant .In general, soybean lines showed a wider range of responses to SDS for both RR and DMR than RLS and FSS scores . The phenotypic data density of RLS was more evenly distributed than that of the other three traits . RLS scores ranged from 2.4 to 10 with a total of 49 lines exhibiting scores less severe than the resistant check ‘MN0302’ . FSS scores ranged from 1 to 8 with a total of 81 lines that did not develop any foliar symptoms plus another 43 less severe than ‘MN0302’ . The range observed in RR was 0–1141 % with a total of 69 lines more resistant than ‘MN0302’ . A total of 29 lines did not show any dry matter reduction plus another 64 lines with DMR less severe than ‘MN0302’ . Based on all four traits associated with SDS resistance, 11 soybean lines consistently exhibited symptoms less severe than that of the resistant check ‘MN0302’ and have potential to be used as breeding parents in the SDS resistance improvement program .r = 0.47    and SDS14-2 (on chromosome 18). For each of the five loci , RR and DMR peaks were coincident with each other .Association mapping (AM) was performed for RLS, FSS, RR, and DMR. The QQ plot indicated the model we implemented for AM was sufficient to control the false positive . We identified two and eight significant (qFDR < 0.05) SNP markers for DMR and RR, respectively, but none for the other two traits  in most plantings indicating an overall reliability of phenotypic data   for RLS and FSS with AM. This might indicate that the genetic variation of RLS and FSS captured in the population is associated with numerous causative genes each with a small effect. In this case, genome-wide selection as implemented through GS is expected to be more effective than MAS because GS would enable breeders to select candidate lines with higher levels of cumulative resistance to SDS conferred by numerous small effect loci. Estimates of prediction accuracy for RLS were as high as 0.64 Fig. , which iSDS resistance breeding is further complicated by the existence of two apparently distinct resistance mechanisms involved in expression of root versus foliar responses to SDS  for FSS, RR, and DMR was comparatively low (<0.3) Fig. . In a siF. virguliforme


First, we identify regions in which cases share more segments identical-by-descent around a putative causal variant than do controls. Second, we use a two-stage mixed-effect model approach to summarize the single-nucleotide polymorphism data within each region and include them as covariates in the model for the phenotype. We assess the impact of linkage disequilibrium in determining identical-by-descent states between individuals by using markers with and without linkage disequilibrium for the first part and the impact of imputation in testing for association by using imputed genome-wide association studies or raw sequence markers for the second part. We apply the method to next-generation sequencing longitudinal family data from Genetic Association Workshop 18 and identify a significant region at chromosome 3: 40249244-41025167 (p-value = 2.3 × 10 In genetic association studies, joint analysis of multiple single-nucleotide polymorphisms (SNPs) can be more powerful than separate SNP analysis because single markers typically either have small effect sizes (common variants) or minor allele frequencies that are too small to reliably fit models (rare variants) . If the In this article, we investigate the contribution that linkage-based methods, such as IBD mapping, can make to association mapping to identify rare variants in next-generation sequencing data. In the first part of our analysis, we use the methods of Browning and Thompson  to identTo assess the impact of linkage disequilibrium (LD) on our analysis, we present results from estimating IBD probabilities using markers with and without LD. We assess the impact of imputation by analyzing both imputed dosage genome-wide association studies (DOS) and whole genome sequence (WGS) data. Table We consider data from 939 individuals from 20 families; 464 are directly sequenced individuals and imputed WGS data, based on existing genome-wide association studies (GWAS) data, are available for their family members. We restrict our work to real genotypic data from chromosome 3. For each individual, we have information on age at examination and current tobacco smoking for up to 4 time points. We use the binary trait hypertension diagnosis at the first time point for selection of regions with excess IBD sharing and the quantitative trait diastolic blood pressure (DBP) for the phenotype model.AllMark data set, and 1 containing only 784 LD-pruned GWAS markers, the NoLD data set. From both data sets we eliminate SNPs with minor allele frequencies (MAFs) <5% because shared alleles that are assumed to be rare represent strong evidence for IBD and can distort results if this assumption is violated [We construct all possible case-case (CaCa) and case-control (CaCo) pairs, such that individuals within pairs are unrelated. This results in 9229 CaCa pairs and 10080 CaCo pairs. We estimate the IBD state using 2 data sets: one containing approximately 50,000 GWAS markers, which we refer to as the To compute the IBD states between pairs of individuals, we use Thompson's method  with ibdβ, the pointwise pairwise probability of IBD, and α, the overall rate of change of IBD state along a chromosome. The choice of these parameters defines the time-depth of the IBD that is sought [In ibd_haplo, one needs to specify a value for parameters of the latent IBD process s sought . For ther, obtained from the first stage, are added as covariates into the model for the phenotype to test for region effects. Below, we describe in brief the phenotypic model used in the second stage.After the regions have been selected, we use the two-stage approach of Tsonaka et al to test for their association with the longitudinal phenotype . In the DBPijt be the diastolic blood pressure for individual j from family i at time point t, where i = 1, . . . , N , j = 1, . . . , ni, t = 1, . . . , 4, and ni is the number of individuals in family i. We use the following linear mixed model for each region r:Let xijt is the vector with covariate values for age and smoking status, ijr ebis the EB estimates of the region r, obtained from the first stage, and ijr sis the number of rare variants within region r; uij is the random family effect and eijt is a normally distributed residual with a 4 × 4 covariance matrix to model the correlation among 6 repeated measurements. We use a multivariate Wald statistic with 2 degrees of freedom to test the null hypothesis of no region effect; that is, H0: β2 = β3 = 0.where Table −3 as the significance level for AllMark and 7 × 10−3 for NoLD.After selecting the regions, we tested their association with the longitudinal phenotype by fitting a linear mixed model to DBP with the EB estimates per region, smoking status, and age as covariates. To further increase our power, we considered a second model, where we adjusted also for the sum of rare variants. We used 2 different genotype data, DOS with imputed genotypes on 939 individuals and WGS with complete genomics on 464 individuals. To account for multiple testing, we used a Bonferroni correction, using a significance level of alpha divided by the maximum number of independent regions tested for each data set; that is, 7 for the NoLD and 8 for the AllMark. We used 6 × 10−3). When WGS was used instead of DOS, the variance of the estimates increased and the signal was no longer significant. When the number of rare variants was removed from the model, the region again reached significance .No significant results were found when the candidate regions were selected using the AllMark data (results not shown). Table We have presented a method that combines linkage and association-based mapping to identify rare variants in next-generation sequencing data. Initially, we identify regions with an excess of IBD between case-case as compared to case-control pairs. Subsequently, we use a two-stage approach to summarize the regions via an EB estimate of the genetic variation and test for region effects. The two-stage approach captures the correlation between SNPs within regions by using random effects. These types of approaches can be more powerful than methods that ignore the dependency structure between the SNPs . The appOne main advantage of this method, as compared to an association-only approach , is thatIn this article, we do not present results for type I error or power. However, Tsonaka et al.  and HouwWe found significant results only when the candidate regions were selected using the NoLD and DOS data. One reason for the better performance of the NoLD data, as compared to the AllMark data, could be the presence of LD in the latter. LD leads to increased rates of false positive IBD results , which c−11). On the other hand, when using WGS, the estimate becomes much smaller, When the NoLD and WGS data were used, the signal of the region found using DOS was no longer significant. This power loss could be a result of the smaller sample size in the WGS data, which leads to increased variances of the parameter estimates (results not shown). The same happens for the estimates of the genetic variance. On one hand, using the DOS data we estimate CADM2, which is 146 kilobase (kb) on the right of the region we identified, is associated, among other phenotypes, with blood pressure and body mass index [−5), rs13074417 , and rs4859048 .Using the NCBI database , we founss index . More spWe identified a significant region when IBD states were determined using LD-pruned markers and association with phenotype was tested using the imputed genotypes of 939 individuals. When the raw sequence data of 464 individuals was used, the signal was significant only after the number of rare variants from the phenotype model were removed.The authors declare that they have no competing interests.BB participated in the design of the study, performed the statistical analysis, interpreted the results, and wrote the manuscript. HWU participated in the design of the study and in the interpretation of the results. RT developed the statistical method. SB participated in the design of the study and helped to draft the manuscript. QH participated in the analysis of the data and the interpretation of the results. JJHD conceived of the study, participated in its design and coordination, and helped draft the manuscript. All authors read and approved the final manuscript.


The inheritance of most human diseases and agriculturally important traits is controlled by many genes with small effects. Identifying these genes, while simultaneously controlling false positives, is challenging. Among available statistical methods, the mixed linear model (MLM) has been the most flexible and powerful for controlling population structure and individual unequal relatedness (kinship), the two common causes of spurious associations. The introduction of the compressed MLM (CMLM) method provided additional opportunities for optimization by adding two new model parameters: grouping algorithms and number of groups.This study introduces another model parameter to develop an enriched CMLM (ECMLM). The parameter involves algorithms to define kinship between groups . The ECMLM calculates kinship using several different algorithms and then chooses the best combination between kinship algorithms and grouping algorithms.Simulations show that the ECMLM increases statistical power. In some cases, the magnitude of power gained by using ECMLM instead of CMLM is larger than the improvement found by using CMLM instead of MLM.The online version of this article (doi:10.1186/s12915-014-0073-5) contains supplementary material, which is available to authorized users. Genome-wide association studies (GWAS) are widely used in human genetics research to identify genes associated with complex diseases and in agricultural research to identify genes associated with quantitative traits such as yield and productivity ,2. The eThese issues are particularly problematic in recent GWAS because the number of polymorphisms can reach millions, but the number of phenotyped and genotyped individuals is rarely more than hundreds of thousands. For candidate gene studies, pre-requisite knowledge is necessary, for example, the location of the candidate genes. In this case, single-locus approaches are a necessary step before further analyses using the multi-locus or candidate gene approaches. Thus, the single-locus approach is still the mainstream method in GWAS.However, advances in genotyping technology have allowed extremely dense genetic marker mapping and the associated computing time has become a major concern for genetic researches. Simultaneously, using large numbers of markers has also increased concerns about false positives -11 and tFalse positives are easy to control, but only at the expense of true positive discovery or statistical power. For example, a stringent association test threshold is an effective way to control the false positive rate, but the numbers of true positives are reduced at the same time. A desirable solution is to reduce false positives without compromising statistical power. This solution is critical because the inheritance of most human diseases and agriculturally important traits is controlled by many genes, which individually have small effects or rare alleles ,13.A number of statistical methods have been developed to eliminate spurious association between phenotypes and testing markers and to increase statistical power in GWAS. One of the causes of spurious association is population structure or stratification. In this case, the population can be partitioned into subpopulations. Then, association tests can be performed within the subpopulations or by using an estimate of population membership as a covariate in a linear or logistic model . A similSpurious association can also be caused by differences in relatedness between pairs of individuals. This effect can be reduced using a general linear model (GLM) to estimate the proportion of genes identical by descent between any pair of individuals and excluding closely related individuals ,19. AlteThis study introduces another parameter for model optimization: defining the relationship among groups in the CMLM. Statistical power is examined through simulations. The results showed that statistical power is further improved through this enriched compressed MLM (ECMLM) method.y is a vector of a phenotype; β represents unknown fixed effects, including population structure and marker effects; u is a vector of size s (the number of groups) for unknown random polygenic effects following a distribution with mean of zero and covariance matrix of K is the group kinship matrix with element kij representing the relationship between group i and j, and X and Z are the incidence matrices for β and u, respectively, and e is a vector of random residual effects that are normally distributed with zero mean and covariance I is the identity matrix and With Henderson’s notation , a CMLM g is a clustering algorithm; s is the number of groups; and ϕ is the operation used to calculate group kinship kij from individual kinship ϕ is the new parameter added in this study. The general formula for the derivation of pair-wise kinship coefficients is as follows:ϕ was defined as the average algorithm in the previous study [The likelihood of equation  can be eus study . Here wementclass2pt{minimArabidopsis) where the UPGMA algorithm is used to cluster individuals into groups. The model fit was reflected by twice negative log likelihood (−2LL). Here, we define the compression level as the average individual number in each group. Different compression levels  were applied. Variations of model fit due to each group kinship algorithm were observed for all species and at some compression levels between 1 and 16  in four species .The joint use of group kinship and individual grouping enlarged the parameter space for the CMLM. We examined three group kinship algorithms  and eight hierarchical clustering algorithms. The eight clustering algorithms were: UPGMA; un-weighted pair-group centroid (UPGMC); complete linkage (COM); Lance-Williams flexible-beta method (FLE); McQuitty’s similarity analysis, which is also called weighted pair-group method using arithmetic averages (WPGMA); weighted pair-group method using centroid (WPGMC); single linkage (SIN), which is also called nearest neighbor; and Ward’s method (WAR). Each combination was examined in the four species  to find the best compression level, kinship, and cluster algorithms. This process took 80 minutes  to perform ECMLM using The statistical power of a method corresponds to model fit. We compared the statistical power of the ECMLM method with three other methods: GLM, MLM, and CMLM. The ECMLM was performed using the best of 24 combinations between the three group kinship algorithms and the eight clustering algorithms that cluster individuals into groups across all compression levels. The CMLM reported previously used the average group kinship and the UPGMA clustering algorithm to cluster individuals into groups across all compression levels. MLM and GLM used the minimum and the maximum compression levels, respectively. Each individual was treated as a single group in the MLM. All individuals were clustered as one group  in the GLM. In these two cases, both the clustering algorithm and the group kinship algorithm would have no effect.P-values on the observed phenotype before the QTN effect was added. The observed phenotypes are body mass index in human, Orthopedic Foundation for Animals (OFA) score in dog, and flowering time in both maize and Arabidopsis. Statistical power improvements with the ECMLM method were observed compared to other methods  to an observed phenotype, then calculating the proportion of detected QTNs. The threshold was determined from the distribution of s Figure . ImproveArabidopsis with SD = 0.75  was also examined for these four methods. The size of the QTN effect is expressed in the unit of phenotypic standard deviation (SD). The observed phenotypes are body mass index in human with SD = 0.08, OFA score in dog with SD = 0.3, and flowering time in both maize with SD = 0.4 and Arabidopsis at 10°C. We found that different PCs have little effect on statistical power in MLM, CMLM, and ECMLM.Comparison of statistical power for the four models  using different numbers of PCs was also investigated  and the third (variances or ratio) parameters relate to the standard MLM. The last three parameters relate to CMLM. Two of them - clustering methods and number of groups - were investigated in a previous study. The current study focused on a sixth parameter: ways of defining group kinship Figure . SimilarThe essential element in cluster analysis is to define the similarity between groups. Many clustering algorithms are named based on the property of similarity. Consequently, group kinship algorithms share the footprints from the individual clustering algorithms. For example, with the maximum algorithm, the kinship between two groups is defined as the maximum kinship between an individual in one group and an individual in another group. Therefore, the maximum algorithm is named the single linkage in cluster analysis, also called nearest neighbor. Single linkage possesses the best theoretical properties .The opposite of single linkage is complete linkage clustering (furthest neighbor method), which sets the similarity between two groups equal to the smallest similarity between an individual of one cluster and an individual of another cluster. This method tends to produce very tight clusters of similar cases and corresponds to using the minimum algorithm. The minimum algorithm gave no advantage over others on the data examined and, therefore, was excluded from this study.In the average method, the kinship between two groups is the average of the all the individual pair-wise kinships between the groups. The average method used in the CMLM corresponds to the average linkage in cluster analysis . AverageThe median method in the CMLM does not correspond to any typical clustering method. Instead of using the median, the centroid is commonly used for cluster analysis. Among the three group kinship algorithms we investigated, the median algorithm never performed the best for any trait from the four species. The other two algorithms (average and maximum) switched back and forth, competing for the best in conjunction with clustering algorithms to group individuals.Future studies that test other clustering algorithms are needed. We only examined eight  of many algorithms that cluster individuals into groups. For example, we did not test any non-hierarchical clustering algorithms ; all clustering algorithms used in this study are hierarchical clustering algorithms.We found a huge variation in model fit among the various combinations of the three group kinship algorithms and the eight clustering algorithms. In general, the optimum combinations were trait- and species-specific Figure . OptimizThe improvement in model fit by introducing a sixth parameter is accompanied by improvement in statistical power for GWAS. Some improvements are large. The improvement in statistical power achieved by using ECMLM on the human dataset instead of CMLM is greater than the power gained by using MLM instead of GLM or by using CMLM instead of MLM . Hip dysplasia was measured as the Norberg angle (a measure of hip congruency) and the OFA hip score. All dogs were genotyped with 23,500 SNPs at genome-wide coverage and 1,000 SNPs were randomly sampled for this study.The dog dataset was sampled from a dataset used for mapping quantitative trait loci underlying canine hip dysplasia ,28 and aThe maize dataset was composed of phenotypes, genotypes (553 SNPs), and a population structure (Q matrix) calculated for 277 inbred lines . The pheArabidopsis dataset included 199 landraces genotyped by 216,130 SNPs [The 130 SNPs . We randP-value at the bottom 5% quantile was used as the empirical threshold [We added a QTN effect to the observed phenotype. We assigned the QTN to each marker, one at a time. The resulting simulated phenotypes retained the original genetic architecture, such as population structure relatedness . The prohreshold .p) of the polymorphism at the QTN: π = 1/(1 + 1/p(1-p)d2) [Arabidopsis (a maximum of d = 1.5) according to sample sizes. To facilitate comparison between datasets, we listed π at the allele frequency of p = 0.3. The genetic effect was assigned to all SNPs, one at a time, to produce replicates across all SNPs.The QTN effect was represented in the unit of phenotypic standard deviation. The percentage of the total variation explained by the QTN (π) is a function of allele substitution difference (d) and sample frequency ((1-p)d2) . DiffereArabidopsis. Population structure was represented by the fractions of subpopulation in maize using Structure software. The population structure of Arabidopsis was represented by the first two PCs derived from the SNPs. Previous study indicates that models incorporating both structure and kinship perform better than when including them separately [Observed and simulated phenotypes were analyzed using Proc Mixed in SAS . Variancparately .et al. [Individuals or their corresponding groups were fit as a random effect. The kinship among individuals was estimated from the genetic markers by the approach of Loiselle et al. . The indet al. . Eight het al.  were exaECMLM has been implemented in GAPIT (R package). Source code and support documents  are available at GAPIT .All the datasets analyzed here were from previously published datasets. This study did not involve taking actual samples from humans or animals.


Genetic epistasis is important in regulating biological function and is considered to explain part of the ‘missing heritability,’ which involves marginal genetic effects that cannot be accounted for in genome-wide association studies. Thus, the study of epistasis is of great interest to geneticists. However, estimating epistatic effects for quantitative traits is challenging due to the large number of interaction effects that must be estimated, thus significantly increasing computing demands. Here, we present a new web server-based tool, the Pipeline for estimating EPIStatic genetic effects (PEPIS), for analyzing polygenic epistatic effects. The PEPIS software package is based on a new linear mixed model that has been used to predict the performance of hybrid rice. The PEPIS includes two main sub-pipelines: the first for kinship matrix calculation, and the second for polygenic component analyses and genome scanning for main and epistatic effects. To accommodate the demand for high-performance computation, the PEPIS utilizes C/C++ for mathematical matrix computing. In addition, the modules for kinship matrix calculations and main and epistatic-effect genome scanning employ parallel computing technology that effectively utilizes multiple computer nodes across our networked cluster, thus significantly improving the computational speed. For example, when analyzing the same immortalized F2 rice population genotypic data examined in a previous study, the PEPIS returned identical results at each analysis step with the original prototype R code, but the computational time was reduced from more than one month to about five minutes. These advances will help overcome the bottleneck frequently encountered in genome wide epistatic genetic effect analysis and enable accommodation of the high computational demand. The PEPIS is publically available at  A new mixed-model was recently developed for mapping DNA regions that are associated with variations in observable traits, known as quantitative trait loci. This model incorporates multiple polygenic covariance genetic structures and has been used to successfully predict yield in hybrid rice. However, quantitatively examining interactions between multiple genetic elements, i.e. epistatic effects, across the entire genome is computationally demanding, which greatly hinders real-world applications of the model, especially in evaluating large populations and multiple genetic markers. In order to overcome these computational challenges, we developed a tool known as the PEPIS—Pipeline for estimating EPIStatic genetic effects in genome wide. We implemented the PEPIS in efficient C/C++ programming language and successfully incorporated several highly efficient open-source mathematical operation and optimization libraries to perform the computational demanding mathematical matrix operations required for these types of analyses. Furthermore, the computational demand is distributed across multiple nodes on our networked Linux computing cluster in order to increase the speed of analyses. Our PEPIS tool will thus help overcome the bottleneck in genetic epistasis analysis. PLOS Computational Biology Software Article.This is a Epistasis, the interaction among multiple genetic loci, contributes significantly to phenotypic variation associated with the expression of polygenic complex traits. Epistatic effects have emerged as a possible explanation for ‘missing heritability,’ that is, marginal genetic effects that cannot be accounted for in a genome-wide association study (GWAS). Examinations of epistatic effects may also enhance understanding of the hierarchical architecture of gene interactions and genetic pathways .Depending on the characteristics of a given study population, variations in the epistatic effects associated with a quantitative trait can be classified based on a number of different variance components, such as additive by additive, additive by dominance, dominance by additive, and dominance by dominance, for example . The relGWAS has played important roles in the identification of possible connections between single-nucleotide polymorphisms (SNPs) and various diseases . HoweverObtaining a thorough understanding of the genetic architecture of a quantitative trait is particularly important in plant and animal breeding in order to develop optimal breeding strategies and obtain maximum genetic gains . To datewww.r-project.org) scripts would take more than one month.A considerable amount of attention has been given to mapping of quantitative trait loci (QTL) to examine epistatic effects –15. Xu eMotivated by the challenge presented by the tremendous computational demand associated with epistatic QTL mapping, we conducted a thorough investigation of the model developed by Xu et al. and re-implemented the model algorithm using C/C++, resulting in the development of a web server-based tool named PEPIS (Pipeline for estimating EPIStatic genetic effects). The PEPIS employs parallelized kinship matrix calculations and main- and epistatic-effect genome scanning. Large computational analyses are divided and allocated to computational nodes on our networked Linux clusters. Furthermore, an open-source C++ linear algebra library, Armadillo , was utij in bin k is numerically coded into two variables, as follows:Zjk and Wjk represent additive and dominance indicators, respectively, and A (the first homozygote), H (heterozygote), and B (the second homozygote) indicate the three genotypes.As the genetic model and statistical method proposed by Xu et al. ,17 servey be an n×1 vector for the quantitative trait values of all n individuals. It can be expressed by the following complete epistatic model for m bins,Let Xβ represents non-genetic effects and ak and dk represent additive and dominance effects, respectively, for bin k. The terms (aa)kk', (ad)kk', (da)kk', and (dd)kk' represent additive by additive, additive by dominance, dominance by additive, and dominance by dominance effects, respectively, for bins k and k'.where In this model, the terms y is E(y) = Xβ, and the variance matrix of y isσ2 represents the residual error variance.When the genetic effects are treated as normally distributed random variables with a mean of zero and a common variance across all markers or marker pairs, we have a mixed-model results. Let K matrix corresponds to a marker-generated kinship matrix, and its value can be calculated utilizing the formulas reported by Xu et al. . Let K = T, then we have Var(y) = V = (Kλ + I)σ2.The variance matrix shown in Eq  can thusGiven the six variance ratios, a complete polygenic structure for a target quantitative trait can be examined in detail. One-dimensional (1D) genome scanning for main effects and two-dimensional (2D) genome scanning for epistatic effects can be employed to estimate individual marker/bin (main) effects and marker/bin pair interaction (epistatic) effects. Here, the individual main and epistatic effects correspond to two and four degrees of freedom LRT test, respectively. In theory, the four degrees of freedom LRT test needs four distinguishable genotypic forms. However, it is rare but possible that only three or less distinguishable genotypic forms occur for a marker/bin pair. This will lead to less than four degrees of freedom for the LRT test for epistatic effects. If we adopt a random model approach, there will be no problem for parameter estimation even if less than four genotypic forms exist. In other words, the random model does not depend on full rank of the design matrix.We developed the PEPIS for rapid epistatic QTL mapping analyses. The PEPIS is composed of two primary sub-pipelines. Sub-pipeline 1 is used for kinship matrix calculations, and sub-pipeline 2 is used for polygenic QTL mapping and integrates three related analysis modules: one for polygenic variance component analysis, another for genome scanning for main effects, and the third module for genome scanning for epistatic effects. The four modules (sub-pipeline 1 and the three modules of sub-pipeline 2) are designated ‘km_cal’, ‘pc_anal’, ‘gs_main’, and ‘gs_epis’, respectively. All the modules were coded in C/C++ and compiled into four separate executable command line programs. Several perl and cshell script files were then developed to function as a wrapper to streamline the complete pipeline.Fig 1 illustrates the overall flow of polygenic QTL mapping analyses in the PEPIS.When coded genotype data are provided, module km_cal calculates and delivers the corresponding kinship matrices. When phenotypic quantitative trait data are provided, module pc_anal estimates and delivers the six polygenic variances utilizing both the quantitative trait data and the available kinship matrices. Following the performance of various information aggregation procedures, including kinship matrix weighing and matrix eigen decomposition, modules gs_main and gs_epis calculate and return 1D LRT values for all markers (bins) and 2D LRT values for all marker (bin) pairs, respectively. To increase the flexibility of analyses, the PEPIS allows users to run only a portion of the pipeline according to the input data and user-configured parameters . Such configuration flexibility allows users to utilize specific PEPIS-generated information with their own statistical genetics models.www.mathworks.com) without considering speed. However, as our goal in developing the PEPIS was to speed up the analysis of genotypic data to identify epistatic effects, we utilized C/C++ and a C++ linear algebra package, thus enabling highly efficient processing.A complete analysis requires processing of a large volume of genotypic matrix data and necessarily involves complex mathematical matrix operations, such as transposition, inversion, determinant calculation, and eigen decomposition, which can be easily prototyped in R or MATLAB (http://www.robertnz.net/nm_intro.htm)  = K. Considering this symmetric feature, the multiplication time for Ka and Kd is O(mn2). The multiplication time for Kaa, Kad, Kda, and Kdd is O(m2n2). These estimations clearly demonstrate the enormity of the multiplication demand associated with kinship matrix calculations, especially when both the individual and marker/bin numbers are large. However, the procedure used to calculate each matrix cell value is the same; thus, all As our goal was to facilitate rapid epistatic QTL mapping, we first needed to analyze the computational complexity of the model and resolve the fundamental problems associated with the computationally demanding nature of these analyses. If we suppose that the number of individuals is represented by matrices , each kim times two degrees of freedom LRT estimation, and the epistatic-effects genome scanning module requires m times two degrees of freedom LRT estimation and The polygenic variance component analysis module needs essentially only one optimization for a seven-parameter log-likelihood estimation. The main-effects genome scanning module requires Table 1. Currently, the PEPIS is configured to efficiently utilize ~500 central processing unit (CPU) nodes in our Linux clusters for parallel computations.As demonstrated above, the computationally intensive modules for kinship matrix calculations and genome scanning for main and epistatic effects can be parallelized to increase the speed and efficiency of the analyses. The strategy utilized in the PEPIS for parallel high-performance distributed computing is summarized in The PEPIS is a web-based program developed in C/C++ to facilitate efficient and rapid epistatic QTL mapping. We verified that the PEPIS could meet our performance expectations by analyzing the same IMF2 rice population genotypic and field phenotypic trait data sets examined by Xu et al. .n×m = 278×1,619 matrix. The Zhenshan 97 genotype was coded as ‘A,’ the Minghui 63 genotype as ‘B,’ and the heterozygote as ‘H.’ We downloaded the genotype and phenotype data from the website specified by Zhou et al. [The IMF2 population described by Hua et al. ,24 consiu et al. .n×m = 278×1,619 matrices and then submitted to the PEPIS. Simultaneously, the YIELD, KGW, GRAIN, and TILLER quantitative phenotypic data were stored as a n×1 = 278×1 column vector for each trait and submitted to the PEPIS. Upon clicking the submit icon, the PEPIS allocates the entire processing job to the available computer nodes distributed across our networked Linux clusters. Once all of the distributed computing jobs are completed, the results are returned as a pop-up page, and users are given the option to download the results from each analysis step. Using the same IMF2 population rice data, the PEPIS returns the same result at each step when compared with the original prototype script developed by Xu et al. [The coded additive and dominance genotypic data were stored as two u et al. , but redFig 2. The results returned include the six ‘gzip’ files corresponding to the six polygenic kinship matrices, as well as three '.txt' files corresponding to the results of polygenic variance component analysis and genome scanning for main and epistatic effects. The analysis results are explained in detail below.The user interface for data submission and return of results is shown in Fig 3. Substantial differences between the polygenic structures were observed for the different traits. For example, additive genetic variance accounted for 73% of trait variance for KGW, whereas epistatic genetic variance was the major contributor to variance for YIELD.Based on the results of polygenic variance component analyses, the polygenic structure of a target quantitative trait can be dissected. Pie charts illustrating the polygenic variance component ratios for the traits YIELD, KGW, GRAIN, and TILLER are shown in Fig 4. The LRT statistic can be used to declare the statistical significance for each marker/bin, herein, if we set a statistic threshold which also called p value, the marker/bin above the threshold suggest an association with the trait, further, the marker/bin under a narrower peak means a higher resolution and indicates a more specific association to the trait.The 1D LRT distributions across all markers/bins were plotted based on the results of genome scanning for main-effect QTL mapping. Plots of the main-effect LRTs for traits YIELD, KGW, GRAIN, and TILLER across the complete rice genome are shown in Fig 5 shows the 2D epistatic-effect LRTs for the traits YIELD, KGW, GRAIN, and TILLER across the complete rice genome. Due to the symmetrical nature of the data, only a lower triangular matrix is shown for each trait instead of the entire square matrix for all of the The 2D LRT distributions across all marker/bin pairs were also calculated based on the results of genome scanning for epistatic-effect QTL mapping. Fig 3, greater detail regarding the genetic composition of these traits can be obtained from the 1D main-effect LRT data illustrated in Fig 4 and the 2D epistatic-effect LRT data illustrated in Fig 5. The high-intensity profile peaks in Fig 4 suggest that genetic loci are associated with the target traits examined in this study, for example, in Fig 4(B), two high-intensity profile peaks located in chromosome 3 and chromosome 5 are two predominant genetic loci for trait KGW. Similarly, the high-intensity pixel points in Fig 5 suggest that a marker/bin pair is associated with each of the target traits, for example, the vertical scale bars in in Fig 5(A) represented by a number of pixel points with comparatively high intensities and vertically scattered between marker/bin 1000-marker/bin 1600, suggest that there are numerous genetic marker interaction pairs. These pairs could be used to construct a more complex marker-marker interaction/regulation network for the trait of YIELD.In summary, although the overall polygenic structure of several hybrid rice target traits can be obtained based on the data illustrated in PEPIS was dedicatedly developed for epistatic genetic estimation. In generally, it has four independent modules including kinship matrix calculation, polygenic component analysis, genome scanning/mapping for main and epistatic effects. Currently, several software tools, such as TASSEL , GCTA 2,28, PLINS1 File, which include 11 sub directories and each contains three '.txt' files, corresponding to the genotypic additive Z Matrix, dominance W Matrix, and vector of phenotypic values. The additive Z Matrix and dominance W Matrix are randomly generated but constrained by formula 1. We submitted each of the 11 data sets to PEPIS and recorded the running time for kinship matrix only and the whole running time for epistatic effect estimation and association mapping. Two scenarios of the simulation experiment were considered, and one was to fix the number of bins but vary the sample size , the other was the opposite. The running times for kinship matrix calculations are shown in S1 Table and the overall running times for epistatic effect estimation and association mapping are shown in S2 Table.Although PEPIS is equipped with a parallel strategy in a distributed Linux computing cluster, it is useful to perform a benchmark evaluation on the computation performance of PEPIS at different data scale. To accomplish this, we specifically generated a series of simulated data sets by varying bin/marker numbers and sample size. These simulated data sets are available in S1 Table, we can see that 1) the running time for calculating the main effect kinship matrix increases moderately with the increase of the number of bins; and 2) the running time for calculating the entire set of polygenic kinship matrices increases intensely with the increase of sample size and number of markers/bins. Such observations are consistent with our previous complexity analysis for kinship matrix calculation. The computational burden for main effect kinship matrix calculation is on the order of O(mn2), while the computational burden for epistatic effect kinship matrix calculation is on the order of O(m2n2).where m and n are the number of markers/bins and the sample size, respectively.From S2 Table, we observe the followings: 1) the running time for epistatic effect estimation and association mapping increases intensely with the increase of the sample size and the number of markers/bins; and 2) the running time increases faster with the increase of sample size compared with the increase of bin number. Our in-depth investigation revealed that the module for polygenic variance component analysis takes significant amount of time when the sample size is very large, because the module is essentially an optimization procedure for a seven-parameter log-likelihood estimation, which takes the six polygenic kinship matrices as a whole input and cannot be parallelized.From It is well known that increasing the marker density and the sample size can further increase the resolution of QTL mapping and reduce the uncertainty of inferred genotypes. However, the high density markers can result in a huge number of marker pairs for epistasis detection. In PEPIS, the module for polygenic variant component analysis is still a bottleneck if the sample size is more than 5,000.In summary, with the efficient algorithm implementation in C/C++ and deployment of parallel strategy, PEPIS has a powerful computational capability and is able to carry out epistatic effect analysis and association mapping when the sample size in the scale of several thousands and the number of markers/bins in the scale of twenty thousand. At these scales, it would require several years to be completed using the original prototype R programs .http://bioinfo.noble.org/PolyGenic_QTL/. We are committed to maintaining and improving the specific function modules per user comments and suggestions.The PEPIS pipeline, the source code and the test data are freely available at Fig 5 are suggestive of the presence of marker interaction pairs that correspond to a biologically meaningful gene regulatory network. Therefore, we are also planning to develop an LRT -based genetic statistical network analysis module that will be incorporated into our publically available high-performance expression-based gene regulatory network analysis web-server (http://plantgrn.noble.org/GPLEXUS/) following validation [The current version of the PEPIS can be configured to perform kinship matrix calculations, polygenic component analyses, the 1D LRT estimations for main-effect QTL mapping, and the 2D LRT estimations for epistatic-effect QTL mapping upon submission of the coded genotypic and phenotypic data. However, a more user-friendly and efficient visualization of the input genotypic data and the analysis results returned at each step would be very useful. We are therefore planning to develop a visualization engine that will allow for more efficient display of the input genotypic data and the polygenic QTL mapping results returned at each analysis step. Furthermore, the high intensities for a number of the pixel points shown in lidation .Difficulties associated with handling high dimensional SNP data and the inability to estimate epistatic effects constitute a significant challenge in GWAS. Reducing the dimensions of SNP data based on biological information is critical and as such should be the first fundamental step in estimating epistatic effects. Xu pioneered a groundbreaking methodology for inferring breakpoints using high density SNP marker data from bi-parental populations and constructed a bin-based genetic marker data . The segBinning markers in bi-parental populations is straightforward . If the S1 FileEleven sub directories are included, and each contains three '.txt' files corresponding to the additive genotypic Z Matrix, the dominance W matrix, and the phenotypic vector.(ZIP)Click here for additional data file.S1 TableTwo scenarios are tested corresponding to A) Fixing sample size at 1000 while varying the number of bins from 1,000 to 40,000; and B) Fixing the number of bins at 1,000 while varying the sample size from 1,000 to 40,000.(PDF)Click here for additional data file.S2 TableTwo scenarios are tested corresponding to A) Fixing the sample size at 1,000 while varying the number of bins from 1000 to 20,000; and B) Fixing the number of bins at 1,000 while varying the sample size from 1,000 to 10,000.(PDF)Click here for additional data file.


Rare variants have been proposed to play a significant role in the onset and development of common diseases. However, traditional analysis methods have difficulties in detecting association signals for rare causal variants because of a lack of statistical power. We propose a two-stage, gene-based method for association mapping of rare variants by applying four different noncollapsing algorithms. Using the Genome Analysis Workshop18 whole genome sequencing data set of simulated blood pressure phenotypes, we studied and contrasted the false-positive rate of each algorithm using receiver operating characteristic curves. The statistical power of these methods was also evaluated and compared through the analysis of 200 simulated replications in a smaller genotype data set. We showed that the Fisher's method was superior to the other 3 noncollapsing methods, but was no better than the standard method implemented with famSKAT. Further investigation is needed to explore the potential statistical properties of these approaches. During the past five years, genome-wide association studies (GWAS) have rapidly become a standard method for discovering susceptible genes for a variety of complex diseases . Up to np values by fitting a mixed effects model for each variant, and then apply 4 noncollapsing algorithms to obtain the gene-wise association p values. Collapsing (or burden) methods combine variant information by assuming consistent direction of effects across variants. None of the methods considered here adopt this assumption, although some  do combine variant information.As part of Genetic Analysis Workshop 18 (GAW18), simulated phenotypic data, based on a real sequencing data set, were provided to the scientific community to evaluate and compare statistical genetic methods for rare variants association mapping. We consider a two-stage, gene-based method to detect association signals for both common and rare variants. We first obtain significance A mixed linear model was fitted for each variant as described in previous literature . The modQ;ν is the fixed effects of Q;Z is the variable that evaluates familial relatedness ; and µ is the random effects coefficient for Z that corrects the polygenic impact.where Y is the quantitative trait of interest (we used first-visit systolic blood pressure [SBP] data were used in this study); × is the genotype; β is the fixed effects of the genotypes; and Q represents the population structure variables. In this study, we chose the first 10 principal components from principal component analysis (PCA) for p values by fitting the mixed linear model, four noncollapsing algorithms were modified and applied to the data set to obtain the gene-wise association p values. The algorithms of the 4 methods are summarized as followed:After obtaining the variant-wise p values within a specific gene were chosen as the gene-wise association p values.1. Naïve method. The most significant variant-wise 2. Fisher's method . The geni pis the p value for variant i, and k is the total number of variants within a specific gene. Because many variants are highly correlated, the basic assumption of independent tests for Fisher's method is violated. Fisher's formula will not have a chi-square distribution, so we assessed the significance via permutation analysis.where p value was summarized by the following equation:3. Simes' method . The geni pis the p value for variant i, and k is the total number of variants within a specific gene.where p values within each gene via a Kolmogorov-Smirnov-like process in which running sums are accumulated. The equation is given as:4. GSEA method ,11. The N is the total number of variants, r(j) is the jth largest statistic values, H Nis the variant number of a given gene, S is any given gene, P is the parameter that gives a higher weight to variants with extreme statistic value, arbitrarily set to 1 in this study, and R Nis given by:where p value was calculated as the fraction of all permutations whose highest statistics  in all genes is higher than a given gene. In addition to the 4 noncollapsing algorithms introduced above, we also included 2 standard rare variants analysis methods: SKAT [Statistical significance and adjustment for multiple hypothesis testing were assessed by a 1000-permutation-based procedure. A family-wise error rate (FWER) procedure was used to adjust for multiple hypothesis testing. The FWER is a highly conservative correction procedure that seeks to ensure that the list of reported results does not include even a single false-positive gene. In this study, the FWER ds: SKAT  and famSds: SKAT  in our aThe chromosome 3 sequencing data were analyzed only for phenotype replicate number 1 because of a huge computational burden. The sequencing data were annotated by ANNOVAR. Intergep values by fitting a univariate linear model with genotype as the only predictor of the residuals. This method may introduce potential bias to the permuted statistics and p values comparing to directly fitting the full model. To quantify this potential bias, we randomly chose 1429 variants and calculated the percentage difference of the −log10 scaled p values obtained from directly fitting a full model and from the 2-step permutation procedure proposed in this paper.To preserve the familial structure, a permutation-of-residuals procedure was applied for the 1000 permutations ,16. Firshttp://cran.r-project.org/web/packages/kinship2/index.html) was used to calculate the kinship coefficient matrix for our data set. The R package coxme (http://cran.r-project.org/web/packages/coxme/index.html) was implemented for fitting the mixed linear model. The R package SKAT (http://cran.r-project.org/web/packages/SKAT/index.html) was implemented for rare variant analysis with SKAT. The R source code for famSKAT was downloaded (http://www.bumc.bu.edu/linga/research/publications/famskat/) and implemented for rare variant analysis. Receiver operating characteristic (ROC) curves were made and compared among the 4 algorithms and two standard methods.Genotypes were coded as dominant, that is, the genotypes with 1 or 2 minor alleles were coded as 1, while genotypes with 2 major alleles were assigned 0. Variants with minor allele frequency >0.3 in genome-wide association data set were selected for PCA. We used Eigenstrat 3.0 for this analysis . The R pp values of these 87,190 variants is shown in Figure The data consisted of 1237 genes with 87,190 variants that passed the annotation criteria were extracted from the sequencing data set of chromosome 3 for 849 subjects. After fitting the mixed linear model, the Q-Q plot and histogram of Data for the 22 true-positive genes with 1098 variants were extracted and used for analysis with 200 simulated phenotype replicates. The statistical power information for the 6 methods was summarized and is presented in Table p values (or statistics) by applying noncollapsing algorithms, compared to ordinary variants collapsing methods, is that it is a method free of the assumption that all the causal variants from a gene have effects in the same direction. This assumption may not be held in many scenarios even though it is the assumed in many existing rare variants association mapping procedures.The noncollapsing methods introduced in this paper have been broadly used in testing the significance of biological pathways in GWAS data sets . When weAnother advantage of this research is the utilization of residuals-of-permutation procedure ,16. CondFrom the ROC curves in Figure MAP4 was successfully identified to be significant for all simulated 200 replicates. All six methods achieved 100% power for this gene. This result is reasonable because, according to the "answer sheet", MAP4 has the most "causal variants" and these variants have a relatively larger effect size comparing to the variants within other genes. However, this result was obtained when we only analyzed 22 genes. For a genome-scale analysis, the significant signals may be missed as a consequence of correction for multiple comparisons. We have analyzed the whole genotypic data set of chromosome 3 with simulated phenotypic replicate number 1 . The result indicated that only naïve method and the 2 standard methods identified gene MAP4 to be significant. For the other 21 genes, the largest power was 0.275, which was achieved by SKAT for LOC152217.From the power analysis results in Table Several previous researchers have already applied the noncollapsing methods proposed in this paper to conduct gene-based analysis ,19. Howep value combination step can be completed much faster (~30 minutes). Because a lot of the non collapsing algorithms require permutation procedures to create null distribution of the statistics, it is somewhat difficult to implement them on a genome-wide-scale data set. In addition, many noncollapsing algorithms cannot be utilized for a gene-based association test directly without proper modifications. The choice of parameters in noncollapsing algorithm for rare variant association detection is more an art than a science. Finally, adjustment for multiple hypothesis testing is another important issue that needs to be addressed.Compared to the permutation process, the Our results indicate that the FWER method is too conservative. For the future work, hierarchical modeling combined with the Markov chain Monte Carlo method may provide better solution to the multiple hypothesis testing problems .Our findings suggest that all the four new methods we proposed along with the standard method implemented with famSKAT were poor in statistical power. In sum, more research is still needed in the statistical method of association mapping for rare variants in the future.The authors declare that they have no competing interests.TXZ and JPR designed the overall study. TXZ and YRX conducted statistical analyses. TXZ and YRX drafted the manuscript. All authors read and approved the final manuscript. TXZ and JPR revised the manuscript critically.


Large genome-wide association studies have reported hundreds of genetic markers associated with lipid levels. However, the discovery and estimated effect of variants at these loci, derived from samples of exclusively European descent, may not generalize to the majority of the world populations. We examined the collective strength of association among these loci in a diverse set of U.S. populations from the Multi-Ethnic Study of Atherosclerosis. We constructed a genetic risk score for each lipid outcome based on previously identified lipid-associated genetic markers, and examined the relationship between the genetic risk scores and corresponding outcomes. We discover this relationship was often moderated by race/ethnicity. Our findings provide insight into the generalizability and predictive utility of large sample size meta-analyses results when leveraging data from a single population. We hope these findings will encourage researchers to investigate genetic susceptibility in more diverse populations and explore the source of such discrepancies. Until then, we caution clinicians, genetic counselors, and genetic testing consumers when interpreting genetic data on complex traits. Lipids, including high-density lipoprotein (HDL) cholesterol, low-density lipoprotein (LDL) cholesterol, total cholesterol (TC), and triglycerides (TG), are frequently measured biomarkers used to evaluate an individual’s risk for cardiovascular disease,2. BloodHere we expand upon a previous investigation of lipid levels to inves-16), LDL (p = 2.82 x 10-3), TC (p = 5.34 x 10-13), and log TG (p<2.2 x 10-16). Genetic risk scores were calculated for all study individuals based on genome-wide significant markers previously identified in the Teslovich et al.[-16). In general, African Americans had a substantially lower mean genetic risk score than other races with respect to HDL, LDL, and TC , TC (p = 3.77 x 10-2), and TG (p = 1.48 x 10-8), though not LDL (p = 0.33).In nearly all instances we observed a significant linear correlation between genetic risk scores and the corresponding observed lipid level p<0.001;  with thens 0–2%; . Indeed,-3). African Americans had a general trend towards association across each outcome. This suggests other lipid-associated markers may be associated with the alternative trait in this population. However, only LDL and TC demonstrated significant associations across multiple racial/ethnic populations: three and two out of the four racial/ethnic populations, respectively. We also observed poor consistency of the individual effects. The HDL genetic risk score was positively correlated with TC in African Americans and Hispanics after controlling for other factors. In all other cases, no significant individual effects were observed across more than one population . In all populations except African Americans, lipid genetic risk scores were associated with the National Cholesterol Education Program (NCEP) 10-year coronary heart disease risk. This appeared to be most strongly influenced by a negative association with the HDL genetic risk score  in many instances. In the future we expect high dimensional models that incorporate wide clinical and genomic data will be more optimal when assessing common disease/complex trait risk/prediction. A critical, but common, assumption we made is that the effects of each marker are independent and each individual contribution can be summed together. This fails to account for non-independence between markers. Though we only modeled one marker per gene, thus eliminating the effects of LD, non-independence can also result from gene-gene interactions or population stratification. Concerning the latter, if common diseases and complex traits are a result of a polygenic architecture of thousands of causal variants, it is hIt is important that we reiterate the purpose of this study. The overarching goal of our work was to examine if the primary results from a large meta-analysis of over 100,000 individuals from a single population can generalize across all populations. Our work was not designed to reproduce the best possible genetic risk scores from results available from complementary sources  and/or For highly polygenic traits, our efforts to control, stratify, or eliminate population differences have, perhaps, limited our genomic understanding to an insufficient number of discoveries in only specific populations. Instead, we should explore the notion that many causal markers may have considerable allele frequency differences or variable effects across populations. If this is true, we will continue to fail to identify such variants by restricting our studies to uniform populations. Instead, genetically diverse admixed populations may reveal a wealth of genomic insight which may complement our understanding of why complex traits demonstrate such profound differences across populations.R2>0.8) even in non-European populations[We acquired genetic information on most of the markers used in this analysis through imputation. While the majority of markers were quite common , and thupulations, there iGenotype and phenotype data was obtained on study participants in the Multi-Ethnic Study of Atherosclerosis (MESA) Classic cohort n = 6,358) through access via the database of Genotypes and Phenotypes . MESA participants were genotyped on the Affymetrix Genome-Wide Human SNP Array 6.0. The MESA cohort and study design has been extensively characterized through ,24. Indip<0.0005), or had exceedingly rare alternative alleles . The remaining genetic data were pre-phased[Prior to imputation, genetic markers were excluded which demonstrated high missingness (>0.05), failed Hardy—Weinberg equilibrium  over a model that only included covariates Click here for additional data file.S2 Fig(DOCX)Click here for additional data file.S1 Tablep-values test equal allele frequencies across populations.(CSV)Click here for additional data file.S2 TableR2 is the change in variation explained over a model only including age, age2, and sex covariates. Individual parameter estimates and corresponding p-values are presented.The overall p-value assesses joint association and delta (CSV)Click here for additional data file.S3 Tablee) triglycerides; HYPTN = hypertension (yes); 10YR CHD = National Cholesterol Education Program (NCEP) 10 year coronary heart disease risk; FRAM RISK = Framingham Risk Score (FRS); META SYN = metabolic syndrome (yes); BMI = body mass index; T2D = type-2 diabetes (yes).HLD = high-density lipoprotein cholesterol; LDL = low-density lipoprotein cholesterol; TC = total cholesterol; log TG = logarithm (base (DOCX)Click here for additional data file.S4 Table(CSV)Click here for additional data file.


Salix) are important as short‐rotation biomass crops for bioenergy, which creates a demand for faster genetic improvement and breeding through deployment of molecular marker‐assisted selection (MAS). To find markers associated with important adaptive traits, such as growth and phenology, for use in MAS, we genetically dissected the trait variation of a Salix viminalis (L.) population of 323 accessions. The accessions were sampled throughout northern Europe and were established at two field sites in Pustnäs, Sweden, and at Woburn, UK, offering the opportunity to assess the impact of genotype‐by‐environment interactions (G × E) on trait–marker associations. Field measurements were recorded for growth and phenology traits. The accessions were genotyped using 1536 SNP markers developed from phenology candidate genes and from genes previously observed to be differentially expressed in contrasting environments. Association mapping between 1233 of these SNPs and the measured traits was performed taking into account population structure and threshold selection bias. At a false discovery rate (FDR) of 0.2, 29 SNPs were associated with bud burst, leaf senescence, number of shoots or shoot diameter. The percentage of accession variation (Radj2) explained by these associations ranged from 0.3% to 4.4%, suggesting that the studied traits are controlled by many loci of limited individual impact. Despite this, a SNP in the EARLY FLOWERING 3 gene was repeatedly associated (FDR < 0.2) with bud burst. The rare homozygous genotype exhibited 0.4–1.0 lower bud burst scores than the other genotype classes on a five‐grade scale. Consequently, this marker could be promising for use in MAS and the gene deserves further study. Otherwise, associations were less consistent across sites, likely due to their small Radj2 estimates and to considerable G × E interactions indicated by multivariate association analyses and modest trait accession correlations across sites (0.32–0.61).Willow species ( Carr      to obtain a limited number of potentially important single nucleotide polymorphism (SNP) markers in target genes. In the nonmodel tree species, the lack of sufficient resources has, hitherto, made large‐scale genomewide association mapping efforts unfeasible.Association mapping analyses of growth and phenology traits have previously been performed with varying success in a number of tree species, for example per se, it is also important to determine the extent to which these associations are dependent on the local environment  is a pioneer outcrossing shrub or small tree that is frequently grown as a short‐rotation coppice (SRC) for bioenergy purposes in cool‐temperate regions  corresponding to 80 kg N ha−1 was applied. The plants were cut back in winter 2009/2010. Further details about the germplasm collection are found in Berlin et al.  were also included to test for possible associations with plant architecture, following previous work indicating that MAX genes may influence growth traits in willows  and primers were designed for amplification of the willow orthologue. Primer3 software v1.1.3  from 24 genetically diverse accessions selected on the basis of previously generated marker data  and combined with regions of good fit to poplar to generate a single reference sequence per gene . Reads were then mapped to the references using Mosaik and SNPs called using samtools v0.1.18  and RRes (Rothamsted Research). At SLU, candidate gene homologues were first identified in the et al. ), were retained as candidates for further genotyping. From SLU, 593 candidate SNPs were selected for genotyping together with 175 additional SNPs used in a previous linkage study of Berlin et al. . GenotypP. trichocarpa genome sequence and the best poplar gene hit then used in a BLASTN search against a preliminary genome assembly of S. viminalis genotype NWC663 (unpublished data). Positive willow contigs were then aligned to their orthologous poplar gene model and used to design primers (using Primer3) for PCR amplification of the candidate gene set in the same set of 24 diverse S. viminalis accessions as used at SLU. Where possible, primers were designed in untranslated regions of the gene to minimize the amplification of paralogous loci. A maximum amplicon size of 5 kb was used, with multiple primer sets designed for longer genes if required. Preformulated PCR Master Mix  was used for all PCRs with primers synthesized and desalted by Sigma. Following amplification, products were run on a 1% agarose gel and bands of the expected size were excised and gel‐purified using QIAquick gel extraction kits in 96‐cell format . Eluted DNA from all successfully amplified genes was then pooled in equimolar amount per individual, barcoded and sequenced on an Illumina GAIIx sequencer at the University of Bristol Genomics Facility. Reads were demultiplexed, mapped back to the target reference sequences using the Bowtie short‐read aligner  was then carried out at the Genome Centre, Barts and The London, Queen Mary's School of Medicine and Dentistry, London. In addition to the SNPs, genotype data from 35 microsatellite (SSR) markers previously developed  for the sample population was estimated using the genotypic data for the 35 SSRs and all SNPs that were polymorphic (1377) jointly . K was estimated in accordance with Loiselle et al.  effects, respectively, p is the vector of random effects of two‐dimensional spatial position, and e is the random residual. The design matrices Xb and Xc link the respective block and accession effect vectors to their corresponding elements in the observation vector. All effects were considered to be independent except for the spatial position effects (p), which were set to follow a two‐dimensional first‐order autoregressive correlation structure  and asreml  to the phenotypic data: c was considered to be fixed rather than random because the accessions originated from genetically and phenotypically divergent populations. Accession distributions may thus deviate considerably from the normal distribution, thereby violating an assumption on which the prediction of random effect relies , accession estimator heritability (Hc2) was also estimated for each trait and for that purpose c was instead considered to be random. Hc2 may serve as an indicator of accession estimator precision and was calculated from the estimated accession variance (σc2) and error variance (σe2) components as: n is the harmonic mean of the number of measured plants per accession.When developing the best linear unbiased estimators (BLUE) for the accessions, μ^) into a new observation variable (yes=1μ^+c^), a second series of LMM analyses was conducted. For this, 316 and 322 successfully genotyped and unique accessions, present at Pustnäs and Woburn, respectively, were included (315 in common). Also the SNPs were further checked with respect to their minor allele frequency (MAF) and to the LD between markers which was estimated using a maximum likelihood method , and ees is the vector of the random residual effects. The design matrix S constitutes the individual genotypes of the analysed SNP as separate and independent factors, implying genetic effects of the form g=[gAAgAagaa]T for the genotypes AA, Aa and aa, respectively. F and Z link the respective individual ancestry proportion and additive chip effect to its corresponding observation. All effects were considered to be statistically independent except for the chip additive genetic effects whose variance was assumed to be structured as Var(u)=2σA2K where σA2 is the chip additive genetic variance and K is the kinship matrix. Due to unsuccessful individual genotyping, 3.8% of the genotype data in S was missing for the average SNP (see Table S1 for details). Such occurrences were dealt with by excluding records with missing genotypes from analysis. Association mapping analysis  was estimated through TASSEL. Eventual inflation of test statistics as a result of unaccounted population structure and/or kinship was assessed by examining the dependency between observed P‐values sorted by magnitude and the corresponding expected P‐values under the null hypothesis of no association .Combining the accession estimator and the estimated trait mean  from the model. Thus, for each trait, the narrow‐sense chip heritability adjusted for population structure (hs2) was calculated from the estimated chip additive genetic and residual variance components  as: For association analysis of SNPs, the full model Eqn  was usedFq or Zu terms were superfluous, these were subjected to significance testing omitting the individual SNP term . For the traits where the tests indicated a reduced model to be preferable (see Table S2), we used that model to redo the association mapping analysis. The results were, however, very similar to those of the full model, and for the sake of consistency, only the full model association results are further treated. Furthermore, to estimate accession correlations between traits (rs) thereby assessing the occurence of genotype‐by‐environment (G × E) interactions, multivariate model analyses  and thus to obtain q‐values. The smoother option of ‘qvalue’ was used, and the lower cut‐off (λ) values were set within the range 0–0.8 with a step size of 0.1 to keep the variance of π0 estimates at reasonably low levels  when reported in isolation . Simulated data were designed to mimic the presence of artificial SNP effects with a prespecified percentage of explained variance but with the objective of re‐estimating this parameter based on the simulated data regardless of the prior knowledge  and corresponding SNP effects (g^adj) were estimated . Using the data from these simulations, the statistical power for detecting SNP–trait associations at FDR = 0.2 given Radj2 values in the range of 0–10% was also estimated for each trait.It is known that markers subjected to intensive selection based on hypothesis test values frequently show severely overestimated  Beavis, . This biSalix purpurea genome using blast , and the best hits were located in SapurV1A.1320s0030 and SapurV1A.0213s0120, respectively. Based on these S. purpurea sequences, new PCR primers were designed , resequencing of fragments covering these loci was performed for ten and six association mapping accessions, respectively, that represented all three possible genotype classes. The DNA sequences containing ELF3b‐5128 and FLD‐1186 were aligned to the recently available All traits for spring and autumn phenology were highly variable at both Pustnäs and Woburn during the different measurement years Table . Less phHc2) and narrow‐sense chip heritabilities (hs2) are presented in Table Hc2 estimates (0.88–0.96), indicating excellent estimator precision. The corresponding Hc2 estimates for bud burst in spring and biomass traits were also generally high (0.55–0.93) except for bud burst at Woburn 2010 (0.24). Several hs2 estimates were also substantial (10 of 17 higher than 0.2), indicating that the marker‐estimated additive kinship term explained appreciable proportions of the population structure‐adjusted accession variation. Structure‐adjusted accession correlations between Pustnäs and Woburn  and were noticeably higher within the subpopulation clusters indicated by the previous study .Broad‐sense accession estimator heritabilities Hc and narr 0.55–0.9 except fp)‐values fell very close to the line of identity . The percentage variance explained (R2) ranged from 1.4% to 7.8% for all associations of at least suggestive nature, but ranged only from 0.3% to 4.4% when adjusted for threshold selection bias (Radj2).In univariate analyses, 11 significant (FDR level at 0.05) and 23 suggestive (FDR level at 0.2) SNP–trait associations were detected encompassing in total 29 different SNPs Table . The leaq < 0.2) in univariate analyses, only ELF3b‐5128 was found to be suggestively associated with the same trait (bud burst) at both Pustnäs and Woburn and at more than one assessment year , indicating a degree of association consistency . Nonetheless, even accounting for the multivariate analysis, the overall association pattern was not very consistent, especially as nine of the 19 associated SNPs exhibited associations interacting with years and/or sites (P < 0.001), thus supporting the existence of G×E interactions at the level of individual SNPs.Among the 29 SNPs suggestively associated with traits  showed that a hypothetical true SNP–trait association exhibiting an Radj2 of 2% would unlikely be detected in any of the traits studied (power<0.22) except only for LS10 in Woburn (power ≈ 0.72). The fact that 17 associations of such, or even lower, Radj2 magnitudes nonetheless were detected for traits other than LS10 suggests that there may be several more small magnitude trait associations among our candidate SNPs that were not detected in this analysis. To some extent, the limited statistical power at the Radj2 level of 2% is also consistent with the poor association consistency observed across trials in univariate analyses.Assessments of the statistical power for association detection at EARLY FLOWERING 3) was suggestively or significantly associated with bud burst for two measurements at Pustnäs and one at Woburn  scores (by 0.4–1.0) compared to the overall mean in both trials  with bud burst in the multivariate analysis because the direction of the SNP effect was consistent across years and sites, while the effect magnitudes appeared to vary. The same rare ELF3b‐5128 genotype was also associated with higher LS10 scores (by 0.2–0.8) compared to the overall mean  was significantly associated with mean and maximum shoot diameter at Pustnäs and with leaf senescence at Woburn  should primarily be compared with the BB10 Hc2 rather than hs2, as dissection of the ELF3b‐5128 genotypic effects showed that ELF3b‐5128 makes a contribution to the additive genetic variance of less than 0.2%. Consequently, the Radj2 estimates of rare allele recessive effect SNPs such as ELF3b‐5128 and FLD‐1186 are predominantly nonadditive also in terms of their genetic variance contribution.For the significant ELF3b‐5128, FLD‐1186 and other influential associations , genotypic effects appeared to be recessive in nature with respect to the rare allele and the genotype effects of the heterozygote and common allele homozygote appeared not to differ  with the observed suggestive or significant associations, it was possible to evaluate whether all our associations could be considered as independent or whether any of the associations could be  explained by alternative SNPs. For eight associations within the genes DT827847, PU08629, DT1122704, APR1, NCD3, UNF2 and ZIP, eleven alternative SNPs were found to exhibit P‐values below 0.05 for the same type of trait, thus indicating association tendencies. All these alternative SNPs were in strong LD (r2 in the range 0.53–0.97) with the original suggestive/significant associations and were also located within the same genes. In particular, the two UNF2 SNPs both suggestively associated with LS10 in Woburn (Table r2 = 0.90), thus likely reflecting one single association rather than two. LD estimates were consistently low (r2 ≤ 0.16) between all other SNPs suggestively/significantly associated with traits, suggesting that they are much more likely to be independent than the UNF2 pair.By studying the effects of SNPs in strong LD , suggesting that the association might only be relevant in a rather limited genetic background. Moreover, this association may have been produced by one or several alternative loci situated at a greater distance than the population LD estimates for FLD‐1186 with other SNPs would indicate (r2 < 0.13). On the other hand, the two accessions of the rare ELF3b‐5128 AA genotype belonged to different subpopulation clusters (eastern Europe and western Russia) and showed a very low kinship (θ = 0.028). The extent of LD for this particular association should not be greater than that suggested by the regular LD estimates (r2 < 0.08) between ELF3b‐5128 and any other candidate SNP.Nonetheless, when evaluating the potential association mapping resolution of the significant rare allele recessive effects exhibited by ELF3b‐5128 and FLD‐1186, estimates of whole population LD may not constitute an adequate measure as the extent of LD may be considerably greater for more closely related sets of individuals. For instance, the two accessions homozygous for the rare Salix on adaptive growth and phenology traits have made substantial contributions to our knowledge about their genetic architecture. However, the usefulness of any markers detected by these studies for mas , but the chip heritabilities for most traits were substantial even when this population structure was accounted for. Consequently, we took the approach of including both population structure parameters and fine‐scale individual kinship in our association mapping model to minimize the probability of detecting false‐positive associations.Previous QTL studies in n et al.  indicateet al., To facilitate the study of G × E effects and conditional neutrality of SNPs, two quite contrasting sites were used in this study. The Swedish trial site (Pustnäs) is situated in an area in which many commercial willow bioenergy plantations are grown, and the growth data approximately reflect the production conditions in central Sweden. Although in an area of the south‐east UK which is predominantly arable for agriculture, the trial at Woburn presents an example of more challenging conditions due to the limited water‐ and nutrient‐holding capacity of the sandy, well‐drained soils at that site. However, the environmental conditions at the two trials were also different in photoperiod, temperature and precipitation, as well as soil texture  and even less when selection threshold bias was taken into account (<5%). Similar observations have been made in several other association mapping studies in tree species (Eckert FDR‐q < 0.2) resulted in rather strict P‐value thresholds  was observed to be suggestively/significantly associated across both trials, years and analysis methods. There are several reasonable explanations for the limited association consistency. First, it should be acknowledged that the criteria for multiple testing correction . ELF3 is involved in the circadian clock that is crucial for proper plant responses to changes in the photoperiod  and also deserves further study. However, because the effect of this association stemmed from two accessions that were indicated with close kinship, causal factors situated far from the position of the FLD gene may have produced this association, thereby limiting the use of FLD‐1186 as a mas marker.Among the associations observed in this study, the SNPs ELF3b‐5128 and FLD‐1186 are notable as they showed suggestive/significant associations to several sites/traits and were further supported by exhibiting common‐type significant trait associations at least once in multivariate analyses. The existence of both SNPs and the veracity of their corresponding rare allele homozygous genotypes were verified by resequencing the corresponding fragments of these genes using the genome sequence of , et al.  observedn et al.  also obsA‐allele for ELF3‐5128 and FLD‐1186, respectively) severely limit the extent to which even large genotype effects may explain the total accession variance , which has been implicated in plant light perception , known to be important for winter dormancy and flowering . This is particularly true for SNPs located in genes that have some relevant functional information. For example, PtPHYB2‐3897 was located in the orthologous coding sequence of the poplar n Smith,  and showet al., et al., S. viminalis for bioenergy and biofuel purposes.In this study, some traits such as bud burst and leaf senescence were measured repeatedly. If such measurements were taken with greater regularity within each season, it would be possible to detect SNP associations with the temporal developments of each trait, thus taking into account their dynamic behaviour , SBP1‐3964 (b) and PtPHYB2‐3897 (c) on leaf senescence at both trials.Click here for additional data file.Supplementary Material and Methods. (A) Salix reference sequence assembly, (B) Significance testing of structure model terms, (C) Multivariate analyses, (D) Adjusting for threshold selection bias by simulation and (E) Derivation of Rps2.Click here for additional data file.Table S1. Characteristics for all SNPs included in the association mapping analysis.Click here for additional data file.Table S2. Significance tests of genetic structure model terms and suggested optimised model.Click here for additional data file.