Big datasets, accumulated from biomedical and agronomic studies, provide the potential to identify genes that control complex human diseases and agriculturally important traits through genome-wide association studies (GWAS). However, big datasets also lead to extreme computational challenges, especially when sophisticated statistical models are employed to simultaneously reduce false positives and false negatives. The newly developed fixed and random model circulating probability unification (FarmCPU) method uses a bin method under the assumption that quantitative trait nucleotides (QTNs) are evenly distributed throughout the genome. The estimated QTNs are used to separate a mixed linear model into a computationally efficient fixed effect model (FEM) and a computationally expensive random effect model (REM), which are then used iteratively. To completely eliminate the computationally expensive REM, we replaced REM with FEM by using Bayesian information criteria. To eliminate the requirement that QTNs be evenly distributed throughout the genome, we replaced the bin method with linkage disequilibrium information. The new method is called Bayesian-information and Linkage-disequilibrium Iteratively Nested Keyway (BLINK). Both real and simulated data analyses demonstrated that BLINK improves statistical power compared to FarmCPU, in addition to remarkably reducing computing time. Now, a dataset with one million individuals and one-half million markers can be analyzed within three hours, instead of one week using FarmCPU. Biomedical innovations have outpaced computing innovations since the completion of the human genome project , 2. GenoPopulation structure is typically incorporated as a fixed effect in the general linear model (GLM), which is computationally efficient. Initially, population structure was derived as the proportions of individuals belonging to sub-populations , 13. SevThe first milestone that reduced this computational burden was the development of the efficient mixed model association (EMMA) . Prior tThe second milestone was the use of empirical Bayesian estimation of population parameters such as genetic and residual variances or their ratio. This method is based on the assumption that each testing marker contributes only a small proportion of total genetic variance. Thus, population parameters for the testing markers can be approximated by the estimates from a reduced model without fitting each marker , 21. DevThe third milestone was the compressed MLM (CMLM) that clusters individuals into groups based on kinship . The comThe fourth milestone was a method called factored spectrally transformed linear mixed model (FaST-LMM) , which uThe fifth milestone was GRAMMAR (Genome-wide association using linear and logistic mixed models and regression)-Gamma , a methoWith the exception of CMLM, the primary aim of the above milestones was to improve computing speed. The statistical power of each of these milestones remains similar to the conventional MLM  because The second adjustment to improve statistical power employs kinship that is not only specific for traits but also specific for testing markers , 27. KinThe third adjustment, known as the multi-locus mixed-model (MLMM) approach, applies elimination of the kinship . In addiP value from each bin is selected as the candidate pseudo QTN. These candidate pseudo QTNs are determined by a random effect model (REM). The candidate pseudo QTNs are first ranked by P value. Then, the best combinations between the different bins and the number of candidate pseudo QTNs are determined by REM. Finally, the two types of models (FEM and REM) are performed iteratively until no change occurs in the selection of pseudo QTNs.Recently, a fourth adjustment was developed, called the fixed and random model circulating probability unification (FarmCPU) . FarmCPUDespite these valuable advancements, more innovative computing tools and analysis methods are needed. For example, although FarmCPU boosts statistical power in GWAS, its REM process remains computationally demanding. Additionally, the bin approach from SUPER requires that all QTNs be evenly distributed throughout the genome, which is rarely true. Furthermore, only one QTN can be selected as a covariate even if multiple QTNs are located in the same bin, which limits statistical power. Thus, a critical need still exists for a method that can increase both computing efficiency and statistical power.We developed a new statistical method that was inspired by this critical need and builds upon our previous method, FarmCPU. In the new method, we use Bayesian information criteria (BIC) in a FEM to replace REML in the REM and we use linkage disequilibrium information to replace the bin method. As a result, we have completely eliminated the computationally expensive REM and the requirement that QTNs be evenly distributed throughout the genome Fig. . We nameWe implemented the BLINK algorithm in two statistical software packages; one was written in R and the other in C. The R package was designed for the popularity of R users. The C package was designed for computational efficiency. We named the two packages BLINK-R and BLINK-C, respectively. The results from the two packages are identical . The difArabidopsis and mouse). Additionally, we conducted association studies on real phenotypes to assess the flowering time trait in maize . Finally, real genotype and phenotype data were duplicated to synthetically create a big dataset to compare observed computing times of BLINK-C, BLINK-R, PLINK, and FarmCPU  and type I error Fig. . To retaize Fig. . Enrichmize Fig. . Real phize Fig. , includiCPU Fig. .Pvalue distribution of all non-QTN-bins. The relationship between statistical power and FDR or type I error is described by the receiver operating characteristic (ROC) curves  did not control inflation as well as the Q + K (kinship) model [We conducted GWAS on flowering time in maize using the four methods . PLINK exhibited strongly inflated ues Fig. . For exap) model . Includip) model .P values. One way to solve the problem is to remove the related individuals. With a kinship cutoff of 0.5, the number of individuals was reduced from the original 2,279 individuals to 1,218 individuals. This pruning strategy not only reduced the sample size and statistical power consequently but also retained substantial inflation of P values. There were still 211 SNPs that passed the 1% threshold after Bonferroni multiple test correction. BOLT-LMM, FarmCPU, and BLINK controlled inflation well. Results from BLINK and FarmCPU indicated that more than 99.9% of SNPs were not associated with flowering time after the adjustments by the associated SNPs. BLINK, BOLT-LMM, and FarmCPU had much better control on inflation of P values across the genome than GLM implemented in PLINK.In addition to population structure, the cryptic relationships among individuals also contributed to the inflation of P values, BLINK identified more associated SNPs than FarmCPU. FarmCPU identified 14 SNPs that passed the Bonferroni multiple test threshold (α = 0.01). In contrast, BLINK not only revealed 9 of these 14 SNPs but also identified 40 additional loci that passed the Bonferroni threshold. The significant SNPs identified by BLINK included the SNPs that are 2 Kb from ZmCCT, 441 kb from ZCN8, and 568 kb from Vgt1—the three genes that have been previously cloned  . FOAM saThe nine genetic loci identified by both FarmCPU and BLINK and the 40 genetic loci uniquely identified by BLINK were significantly enriched on the 1,003 flowering time genes identified by FOAM. The flowering time gene regions were defined as 50 Kb upstream and downstream of the 1,003 genes. These regions occupied about 3% of the maize genome. The strength of the enrichment was indicated by the difference between the observed number of genetic loci hitting the FOAM flowering time gene regions and the expected number under the null hypothesis that genetic loci were selected randomly. The detailed derivations of the expected null distributions are illustrated in the Methods section.Among the nine associated loci identified by BLINK, four were located in the flowering time gene regions. The chance to have four or more overlaps was less than 1% if the nine loci were randomly selected. The five genetic loci unique to FarmCPU were not enriched, but the 40 genetic loci unique to BLINK were significantly enriched. Among these 40 associated loci,8 were located on the flowering time gene regions. The chance to have 8 or more overlaps was below 5% if these 40 loci were selected randomly  and the number of pseudo QTNs and to optimize the genetic-to-residual variance ratio with p iterations. FarmCPU's computing time is tbp(c+t)2N for model selection and (c+t)2MN for association tests; total computing time is (M+tbp)(c+t)2N.In PLINK, association analysis of t pseudo QTNs. Consequently, the iterations are eliminated for optimizing the genetic-to-residual variance ratio. BLINK has a computing time of (c+t)2N for selecting pseudo QTNs. The total computing time for BLINK is (M+t)(c+t)2)N+(c+t)2N. The number of common covariates (c), pseudo QTNs (t), bin sizes (b), and iterations (p) are much smaller than both M and N. These scalars remain constant regardless of M and N sizes. Therefore, the computing time complexity is MN with respect to big O for all three methods  (Table BLINK replaced REM with FEM for the model selection of K) Table .We compared the two BLINK packages' (C and R) computing times for analyzing big datasets with PLINK  and FarmAmong the four packages compared above, BLINK-C can fully use modern computer architecture with multiple central processing unit cores for parallelization. We further examined the efficiency of BLINK-C on multiple-core computer systems. We tested BLINK-C on computers with core numbers ranging from 2 to 12 under Linux and Mac . ResultsInspired by the critical need for computational efficiency and statistical power in big dataset analysis and by the recently developed GWAS method, FarmCPU, we developed a faster and more powerful method. By substituting REML in FarmCPU's REM with BIC in a FEM and by replacing the bin approach with LD, we achieved optimization in one dimension (number of pseudo QTNs) instead of two dimensions (number of pseudo QTNs and bin size). The optimization of the genetic-to-residual variance ratio was also eliminated by substituting REML with BIC, which directly solves residual variance without iterations. These improvements not only reduced computing time but also simultaneously reduced false positives and false negatives.In both FarmCPU and BLINK models, markers are tested one at a time, with pseudo QTNs added as covariates to control false positives and reduce false negatives. FarmCPU selects these pseudo QTNs by using REM. Pseudo QTNs are used to derive kinship among individuals. The model chooses a set of pseudo QTNs to derive a kinship that provides the maximum likelihood . BecauseThe FarmCPU method uses bins as pseudo QTNs, according to the SUPER GWAS method , 29. BotThe selection of pseudo QTNs is influenced by the threshold that determines whether a pair of SNPs is highly correlated. The current default setting used in BLINK-C and BLINK-R is 70% (Pearson correlation coefficient). We evaluated all of BLINK's default settings in all five populations . AlthougNevertheless, BLINK produced fewer false positives and identified more true positives than the most recently developed GWAS method, FarmCPU. BLINK outperformed FarmCPU  and PLINM genetic markers, one at a time. Pseudo QTNs are included as covariates to simultaneously control false positives and reduce false negatives. Specifically, the first FEM can be written as follows: i is the observation on the ith individual; Si1, Si2, …, Sik are the genotypes of k pseudo QTNs, initiated as an empty set; b1, b2, …, bk are the corresponding effects of the pseudo QTNs; Sij is the genotype of the ith individual and jth genetic marker; dj is the corresponding effect of the jth genetic marker; and ei is the residual having a distribution with a mean of zero and a variance of P values for all M testing markers.The BLINK method conducts two FEMs and one filtering process, which selects a set of pseudo QTNs that are not in LD with each other as covariates. The entire sequence runs repeatedly until all genetic markers are tested and the selection of pseudo QTNs is optimized. The first FEM tests The second FEM is employed to optimize the selection of pseudo QTNs. Specifically, the second FEM can be written as follows: P values are output in equation (Equations  and 2)  2) diffeP value and correlation. All markers are sorted first and then filtered out if their P values are larger than a threshold . Of the m SNPs remaining, if their correlation, r (Pearson correlation), with the first SNP (S1*) is larger than a threshold (0.7), they are also removed. This process is repeated to select S2*, S3*, …, until the last SNP, St*, is selected  SNPs and 8,807 individuals. Individuals were randomly selected to amplify sample size to 10,000, 20,000, 50,000, 100,000, 200,000, 500,000, and 1,000,000. The number of SNPs remained the same, at approximately one-half million. The function of creating synthetic datasets has been added into BLINK to allow a user to generate the synthetic dataset. The R demo code is illustrated in GitHub  to explaComputing speed comparisons between BLINK, FarmCPU, and PLINK 1.9 were conducted on the same computer. Parallel computing performance was tested on computers with different operating systems and machine configurations .P value of a bin was represented by the most significant SNP in the bin. Statistical power was defined as the proportion of QTNs detected for each different level of FDR. Type I error was derived from the empirical null distribution of non-QTN bins.The real genotypes of the five species were used to simulate phenotypes to examine statistical power under different levels of type I error and FDR. The simulated phenotypes had a heritability of 75%, controlled by a variable number of QTNs that were sampled from all real SNPs. Two scenarios, with and without restriction, were applied to the sampling of SNPs. The restriction was that a QTN must be within a 300 Kb distance of another QTN. The QTNs had effects that followed a normal distribution. These QTNs were summed together as the total additive genetic effect for each individual, according to its real genotype. The variance of additive genetic effect was calculated across all individuals. A normally distributed residual effect was assigned to each individual. The variance of the residual effect was assigned accordingly, so that the proportion of additive genetic variance equaled heritability. Genomes were divided into different bin sizes . Bins were classified as QTN bins if they contained at least one QTN, otherwise, as non-QTN bins. The P value of a bin was represented by its most significant SNP. A bin was considered a QTN bin if it contained at least one QTN, otherwise, a non-QTN-bin. A non-QTN-bin with a P value that passed a threshold was counted as a false-positive bin. A QTN-bin with a P value that passed the same threshold was counted as a true-positive bin. The proportion of QTNs identified under different thresholds was calculated as statistical power. For all levels of statistical power, the proportion of non-QTN-bins was calculated as FDR. Type I error was derived from the empirical null distribution of all non-QTN bins. Furthermore, ROC curves were used to compare statistical power under different levels of FDR and type I error. The AUC was calculated with a starting point of zero and an ending point of one for FDR or type I error.The numbers of false and true positives were counted based on bins, as described in our previous study . Bin sizProject name: BLINKhttp://zzlab.net/blinkProject home page: https://github.com/Menggg/BLINKGitHub repository: Operating systems: Mac OS and LinuxPrograming Language: C, R and OpenCLLicense: GNU General Public License version 3.0.RRID: SCR_016288GigaScience GigaDB repository [The download URLs of public datasets used in this study are available in the Materials and Methods section. Genotype data and snapshots of the code are also available in the pository .https://github.com/Menggg/BLINKThe R code scripts used to generate testing data during this study are available in GitHub, Table S1. Properties of real genotypes and parameters of phenotype simulation.Table S2. The information of operating system and machine configuration of computers for computing speed evaluation.Table S3. The comparison of command lines between BLINK and PLINK.Figure S1. Population structure revealed by the first three principal components. The principal components (PC) were derived from all the available markers in each of the five species. Pair-wise relationship is displayed by the left column (PC1 vs. PC2), middle column (PC1 vs. PC3) and the right column (PC2 vs. PC3).Figure S2. The distribution of real phenotypes data in maize, Arabidopsis thaliana, mouse and pig.Figure S3. Proportion of case and control for lung cancer. The dataset contained a total of 8807 samples, including 4962 lung cancer cases and 3845 controls.Figure S4. Association studies of flowering time in Arabidopsis thaliana. Four GWAS methods were used, GLM (performed by PLINK), BOLT-LMM, FarmCPU, and BLINK. Flowering time at 16°C was measured on 193 Arabidopsis thaliana individuals, genotyped with 216,131 SNPs. GLM included the first three PCs as covariates to control population structure. The names of flowering time candidate genes with significant SNPs nearby were labeled on the BLINK plot. The distances between significant SNPs and candidate genes were also labeled. All candidate genes’ information came from The Arabidopsis Information Resource (http://www.arabidopsis.org/index.jsp).Figure S5. Association studies of lung cancer in human. Four GWAS methods were used, Logistic Regression (performed by PLINK), FarmCPU, BOLT-LMM, and BLINK. The East Asian lung cancer population included 8807 samples; each sample was genotyped with 629,968 SNPs . The names of lung cancer candidate genes  with significant SNPs nearby were labeled on the BLINK plot. The distances between significant SNPs and candidate genes were also labeled.Figure S6. Association studies of weight growth intercept in mouse. Four GWAS methods were used, GLM (performed by PLINK), FarmCPU, BOLT-LMM, and BLINK. The population included 1940 samples; each sample was genotyped with 12,226 SNPs . GLM included the first three PCs as covariates to control population structure. The names of weight growth intercept candidate genes and QTL with significant SNPs nearby were labeled on the BLINK plot. The distances between significant SNPs and candidate genes/QTL were also labeled. All QTLs’ information came from Mouse Genome Informatics (URL: http://www.informatics.jax.org/).Figure S7. Association studies of last rib backfat thickness in pig. Four GWAS methods were used, GLM (performed by PLINK), FarmCPU, BOLT-LMM, and BLINK. The population included 820 samples; each sample was genotyped with 64,212 SNPs . GLM included the first three PCs as covariates to control population structure. The names of backfat thickness candidate genes and QTL with significant SNPs nearby were labeled in the BLINK plot. The distances between significant SNPs and candidate genes/QTL were also labeled. All QTLs’ information came from the Pig Quantitative Trait Locus Database .Figure S8. ROC plot of Fig. Figure S9. ROC plot of Fig. Figure S10. Statistical power and area under curve to detect un-clustered causal genes. Statistical power was defined as the proportion of simulated QTNs detected at cost defined by either False Positive Rate (FDR) or Type I error. The two types of ROC curves are displayed separately for FDR (a) and Type I error (b). The AUC is also displayed separately for FDR (c) and versus Type I error (d). Four GWAS methods  were compared with phenotypes simulated from real genotypes in five species . The simulated phenotypes had a heritability of 75%, controlled by 500 QTNs for human, 100 QTNs for maize and mouse, and 50 QTNs for Arabidopsis thaliana and pig. These QTNs were randomly sampled from the available SNPs without restriction. The number of false and true positives were counted based on 10KB-sized bins.Figure S11. ROC plot of Fig. S10 with 1 KB window size to count false and true positives. Number of false and true positives were counted based on 1 KB-sized bins.Figure S12. ROC plot of Fig. S10 with 1 bp window size to count false and true positives. Number of false and true positives were counted based on 1 bp-sized bins.Figure S13. The performance of three model selection criteria. The three model selection criteria are Bayesian Information Criterion (BIC), Akaike Information Criterion (AIC), and Extended Bayesian Information Criterion (EBIC). The performance was evaluated as statistical power vs. False Discovery Rate (FDR). Statistical power was defined as the proportion of simulated Quantitative Trait Nucleotides (QTNs) detected at different levels of FDR. The simulated QTNs were sampled from the real genotypes in five species . The simulated phenotypes had a heritability of 75%, controlled by 500 QTNs for human, 100 QTNs for maize and mouse, and 50 QTNs for Arabidopsis thaliana and pig. These QTNs were randomly sampled from the available Single Nucleotide Polymorphism (SNPs) with the restriction that every two QTNs were clustered within 300 Kb distance. BIC overperformed other two model selection criteria.Figure S14. Impact of cutoff to exclude correlated markers on statistical power. The impact was evaluated as statistical power at different levels of False Discovery Rate (FDR). Statistical power was defined as the proportion of simulated Quantitative Trait Nucleotides (QTNs) detected at different levels of FDR. The simulated QTNs were sampled from the real genotypes in five species . The simulated phenotypes had a heritability of 75%, controlled by 500 QTNs for human, 100 QTNs for maize and mouse, and 50 QTNs for Arabidopsis thaliana and pig. These QTNs were randomly sampled from the available Single Nucleotide Polymorphism (SNPs) with the restriction that every two QTNs were clustered within 300 Kb distance. The cutoff was varied from 0.1 to 0.9 for excluding genetic markers sorted on the strength of association with phenotypes. Higher cutoff leads to more markers as covariates in the model for next iteration of association tests.Figure S15. Identical P values by using BLINK C version and R version. The P values were the association tests on real phenotypes in four species. The phenotypes are (a) last rib backfat thickness (pig), (b) lung cancer (human), (c) weight growth intercept (mouse), and (d) flowering time (Arabidopsis). The P values are displayed as –log10.Figure S16. Snapshot of random selected Manhattan plots out of 100 replicates. The Manhattan plots were based on the P values by using BLINK on phenotypes simulated from real genotypes in five species . The simulated phenotypes had a heritability of 75%, controlled by 500 QTNs for human, 100 QTNs for maize and mouse, and 50 QTNs for Arabidopsis thaliana and pig. These QTNs with gray dots and circles were randomly sampled from the available Single Nucleotide Polymorphism (SNPs) with the restriction that every two QTNs were clustered within 100 Kb distance. The green lines indicated the Bonferroni multiple test threshold.Figure S17. Effects of number of principal components (PCs) and kinship pruning. Fitting two PCs and their products had much worse control of P value inflation due to population stratification compared with fitting ten PCs for association study on maize flowering time. The inflation was further improved by kinship pruning in PLINK at cutoff of 0.5, which reduced number of samples from 2279 to 1218. The number of significant SNPs (Bonferroni cutoff of α = 0.01) were reduced from 48,194 SNPs with two PCs and their product, to 2671 SNPs with ten PCs, and to 211 SNPs with ten PCs plus kinship pruning.AUC: area under curve; BIC: Bayesian information criteria; BLINK: Bayesian-information and linkage-disequilibrium iteratively nested keyway; CMLM: compressed MLM; EM: expectation and maximization; EMMA: efficient mixed model association; EMMAX: EMMA eXpedited; FarmCPU: fixed and random model circulating probability unification; FaST-LMM: factored spectrally transformed linear mixed models; FDR: false discovery rate; FEM: fixed effect model; FOAM: F-one association mapping; GEMMA: genome-wide efficient mixed-model association; GLM: general linear model; GWAS: genome-wide association studies; LD: linkage disequilibrium; ML: maximum likelihood; MLM: mixed minear model; MLMM: multi-locus mixed-model; P3D: population parameters previously determined; PC: Principal Component; QTL: Quantitative Trait NucleotidePCA: principal component analysis; QTL: QTN: quantitative trait nucleotide; REM: random effect model; REML: restricted maximum likelihood; ROC: receiver operating characteristic; SNP: single-nucleotide polymorphism; SUPER: settlement of MLM under progressively exclusive relationship; USDA: US Department of Agriculture.Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the funding agencies. All datasets analyzed herein were published previously. This study did not involve samples from humans or animals.The authors declare that they do not have competing financial interests.This material is based upon work that is supported by an Emerging Research Issues Internal Competitive Grant from the Agricultural Research Center in the College of Agricultural, Human, and Natural Resource Sciences at Washington State University; the Washington Grain Commission (endowment and award 126593); the National Science Foundation (award DBI 1661348); the National Institute of Food and Agriculture; and the USDA (awards 2018–70005-28792 and 2016–68004-24770).Z.Z. conceptualized the study and wrote the manuscript. The concepts were implemented by M.H. in C language (BLINK-C) and by Y.Z. in R language (BLINK-R). M.H., X.L., Y.Z., and R.M.S. performed the data analyses.GIGA-D-18-00064_Original_Submission.pdfClick here for additional data file.GIGA-D-18-00064_Revision_1.pdfClick here for additional data file.GIGA-D-18-00064_Revision_2.pdfClick here for additional data file.GIGA-D-18-00064_Revision_3.pdfClick here for additional data file.Response_to_Reviewer_Comments_Original_Submission.pdfClick here for additional data file.Response_to_Reviewer_Comments_Revision_1.pdfClick here for additional data file.Response_to_Reviewer_Comments_Revision_2.pdfClick here for additional data file.Reviewer_1_Report_ -- Stefania Angelucci4/8/2018 ReviewedClick here for additional data file.Reviewer_1_Report_Revision_1 -- Stefania Angelucci9/3/2018 ReviewedClick here for additional data file.Reviewer_2_Report_ -- Rene Zahedi4/27/2018 ReviewedClick here for additional data file.Supplemental FilesClick here for additional data file.


Drosophila melanogaster for nine traits in two sexes. We incorporated the transcriptome data into genomic prediction via two methods: GTBLUP and GRBLUP, both combining single nucleotide polymorphisms (SNPs) and transcriptome data. The genotypic data was used to construct the common additive genomic relationship, which was used in genomic best linear unbiased prediction (GBLUP) or jointly in a linear mixed model with a transcriptome-based linear kernel (GTBLUP), or with a transcriptome-based Gaussian kernel (GRBLUP). We studied the predictive ability of the models and discuss a concept of “omics-augmented broad sense heritability” for the multi-omics era. For most traits, GRBLUP and GBLUP provided similar predictive abilities, but GRBLUP explained more of the phenotypic variance. There was only one trait  in which the predictive ability of GRBLUP (0.23) was significantly higher than the predictive ability of GBLUP (0.21). Our results suggest that accounting for transcriptome data has the potential to improve genomic predictions if transcriptome data can be included on a larger scale.Gene expression profiles potentially hold valuable information for the prediction of breeding values and phenotypes. In this study, the utility of transcriptome data for phenotype prediction was tested with 185 inbred lines of  Prediction of genetic values has been a key problem in quantitative genetics. Since Meuwissen et al.  publishePhytophthora sojae, Loh et al. , a semi-parametric prediction method, was introduced by Gianola et al.  to the fDrosophila melanogaster is a widely used model organism for biological research in genetics, physiology, microbial pathogenesis, and life history evolution, and it has been demonstrated that the architecture of Drosophila quantitative traits is dominated by extensive epistasis  is a community resource for analysis of population genomics and quantitative traits. It consists of more than 200 fully sequenced inbred lines derived from the Raleigh population, USA  is a community resource for analysis of population genomics and quantitative traits. It consists of 205 fully sequenced inbred lines derived from 20 generations of full sibling inbreeding of a single outbred population in Raleigh, North Carolina, USA  was used for the imputation of missing SNP genotypes  and two 0.25-in ceramic beads . For details on fly husbandry, RNA extraction, RNA sequence annotation and quality control see  in 185 DGRP lines was quantified using Affymetrix In total, nine traits, which were measured on females and males separately were used: startle response (STR), starvation resistance (STV), alcohol sensitivity and tolerance (AST), food intake (FI), and olfactory perceptions to five chemical odorants: olfactory perceptions to 2-Heptanone (OP2H), Methyl Salicylate (OPMS), l-Carvone (OPIC), 1-Hexanol (OP1H), Ethyl Butyrate (OPEB). These phenotypes are line means or medians of repeated measurements in different ways, and are treated as response variables in our statistical model. For startle response (starvation resistance), there were on average 40 ± 4 (52 ± 11) measurements for females, and 40 ± 4 (52 ± 11) measurements for males, the line medians were taken in several replicates for each trait .The whole genome sequence data, gene expression data of 185 DGRP lines, and phenotype data of nine traits are available on To remove the gender effect in prediction, we performed the subsequent analyses with female and male data separately. Predictions of phenotypes were done with three basic approaches and two combined methods. The basic approaches were genomic BLUP (GBLUP) to predict phenotypes using genotype data, transcriptomic BLUP (TBLUP) predicting phenotypes using transcriptome data with a linear kernel, and RKHS predicting phenotypes using transcriptome data with a Gaussian kernel  of marker i. Moreover, Z denotes the MAF adjusted marker matrix with entries (0−2pi) and (2−2pi) for genotypes AA and aa, respectively.where Drosophila lines were used as predictor variables. The statistic model is:In this approach, transcriptome data of the 185 E = RR′ which is a linear kernel calculated from an n × m matrix R of standardized gene expression levels from n lines and m genes. The standardization of gene expression levels was conducted by calculating xij is the expression level of gene j in line i, j across all lines, and sj is the standard deviation of gene expression level of gene j.where Analogously, to the previously described approaches, the statistical model was:K being the genetic covariance matrix  and the optimum bandwidth parameter h is found by grid-search and cross-validation.The only difference between GTBLUP and GRBLUP is that in GRBLUP we replace The omics-augmented broad sense heritability was defined as the proportion of phenotypic variance explained by whole genome SNP markers and other omics data,where (1) SNP-based genomic narrow sense heritability for GBLUP  SNP and gene expression data-augmented broad sense heritability for GTBLUP  ranged from 0.25 to 0.99 and are generally high. On average across all traits, they are slightly higher for females (081) see . It shouIn GTBLUP and GRBLUP, we integrated transcriptome data into genomic prediction. The only difference between these two methods is that two different kernels were used to construct the relationship matrix based on transcriptome data. For the SNP and gene expression data-augmented heritability, th sexes . Only thThe predictive abilities of the nine traits obtained with the 5 statistical models for females and males are shown in Except for one trait (OPEB) in females, there was no significant difference of predictive abilities between GRBLUP and GBLUP. For the trait OPEB in female, GRBLUP (0.23 ± 0.012) gave a higher predictive ability than GBLUP (0.208 ± 0.012). Both GRBLUP  and GBLUP  provided better predictive abilities on average in all traits than GTBLUP  for female and male. It is worth noting that predictive abilities between males and females for all models were found to be remarkably different for six out of nine traits . In females, the predictive abilities of three models  varied slightly among all nine traits with a range between 0.139 ± 0.012 (OP1H in GTBLUP) and 0.24 ± 0.013 (STV in GRBLUP), while in males the predictive abilities of these three models have a more significant variation ranging from 0.045 ± 0.014 (OPMS in GTBLUP) to 0.326 ± 0.014 (FI in GRBLUP). The correlation coefficient between predictive abilities in females and males across all traits and models is 0.623 . The corectively . The banDrosophila genomic prediction studies have shown that there is a high degree of genotype by sex interaction in some traits. Ober et al.  is “closer” to phenotypes than genomic information, and thus should help providing better predictions of phenotypes than genomic information. However, unlike the DNA sequence, the transcriptome information is not stably inherited and measurements of transcriptome abundance are affected by choice of tissue, time of sampling and experimental conditions. In this study, predictive abilities of RKHS obtained on 9 traits were relatively low , and were much lower than predictive abilities obtained with GBLUP using SNP data. A similar result was also shown in maize, where predictive abilities of transcriptomic prediction were always lower than the genomic prediction when comparing both using eight statistical models  for some traits, such as STV, AST, OP2H in females and STV, AST, FI, OPIC in males, while the predictive abilities for these traits were relatively low. The reason could be the small sample size of lines and this result was consistent with the previous study for starvation resistance and startle response which the predictive abilities were 0.239 ± 0.012 and 0.23 ± 0.012, respectively. Ober et al.  showed tE = RRT for modeling transcriptome data in genomic prediction.We incorporated transcriptome data with genomic prediction using GRBLUP which combine the standard GBLUP and the RKHS method. From an RKHS point of view, the genomic relationship matrix G in GBLUP can be viewed as a parametric kernel that only captures genetic values based on an additive genetic relationship among individuals. The Gaussian kernel is a non-parametric kernel which may pick up genetic signals regardless of the underlying genetic architecture. Choosing the most suitable bandwidth parameter h can provide an optimal u et al.  observedIn our result, there was only one trait  for which the predictive ability of GRBLUP (0.23) was higher than the predictive ability of GBLUP (0.21). This indicated that predictive ability can be improved when combining transcripts with SNPs using GRBLUP, but it depends on the traits. For the rest of the traits for both males and females, the SNP and gene expression data-based heritability We constructed a semiparametric prediction model (GRBLUP) with two kernels combining SNP and transcriptome data. The parametric G kernel was used to capture the additive genetic part, and the Gaussian kernel is a non-parametric kernel which was used to pick up non-additive genetic effects and biological regulation effects regardless of the underlying genetic architecture. In our study, GRBLUP and GBLUP provided similar predictive ability, but GRBLUP could capture more phenotypic variance components explained by transcriptome data. The better goodness of fit of GRBLUP in general did not translate into a better predictive ability. It should be noted, though, that sample size was small and gene expression was not measured at one time point and in one specific tissue functionally linked to the trait of interest. However, including transcriptomic data can increase predictive ability, as was shown for the trait OLED in females. We conclude that adding more specifically collected transcriptome data has the potential to improve genomic predictions in larger scale applications.All authors were involved in the design of the study. ZL performed the model validations and wrote the manuscript, with contributions of HS. NG and JM participated in discussing the statistical models. All authors commented on the manuscript and read and approved the final version.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Crossbreeding is widely used in pig production because of the benefits of heterosis effects and breed complementarity. Commonly, sire lines are bred for traits such as feed efficiency, growth and meat content, whereas maternal lines are also bred for reproduction and longevity traits, and the resulting three-way crossbred pigs are used for production of meat. The most important genetic basis for heterosis is dominance effects, e.g. removal of inbreeding depression. The aims of this study were to (1) present a modification of a previously developed model with additive, dominance and inbreeding depression genetic effects for analysis of data from a purebred sire line and three-way crossbred pigs; (2) based on this model, present equations for additive genetic variances, additive genetic covariance, and estimated breeding values (EBV) with associated accuracies for purebred and crossbred performances; (3) use the model to analyse four production traits, i.e. ultra-sound recorded backfat thickness (BF), conformation score (CONF), average daily gain (ADG), and feed conversion ratio (FCR), recorded on Danbred Duroc and Danbred Duroc-Landrace–Yorkshire crossbred pigs reared in the same environment; and (4) obtain estimates of genetic parameters, additive genetic correlations between purebred and crossbred performances, and EBV with associated accuracies for purebred and crossbred performances for this data set.Additive genetic correlations (with associated standard errors) between purebred and crossbred performances were equal to 0.96 (0.07), 0.83 (0.16), 0.75 (0.17), and 0.87 (0.18) for BF, CONF, ADG, and FCR, respectively. For BF, ADG, and FCR, the additive genetic variance was smaller for purebred performance than for crossbred performance, but for CONF the reverse was observed. EBV on Duroc boars were more accurate for purebred performance than for crossbred performance for BF, CONF and FCR, but not for ADG.Methodological developments led to equations for genetic (co)variances and EBV with associated accuracies for purebred and crossbred performances in a three-way crossbreeding system. As illustrated by the data analysis, these equations may be useful for implementation of genomic selection in this system.The online version of this article (10.1186/s12711-019-0486-2) contains supplementary material, which is available to authorized users. Crossbreeding is widely used in pig production because of the benefits of heterosis effects and breed complementarity. Commonly, sire lines, for example Duroc or Pietrain lines, are bred for production traits such as feed efficiency, growth and meat content, whereas maternal lines, for example Landrace and Yorkshire lines, are also bred for reproduction and longevity traits. The production traits give value to farmers in the finishing units, whereas the reproduction and longevity traits give value to farmers in the sow units. To benefit from heterosis effects in all production units, a crossbreeding system that is often used is a terminal three line system where two maternal lines are mated to produce crossbred sows, which then are mated to a sire line to produce three-way crossbred pigs for the production of meat. For example, in the DanBred production system, the maternal lines Danbred Landrace (L) and Danbred Yorkshire (Y) are mated to produce two-way crossbred Landrace–Yorkshire (LY) sows, which are then mated to sire line Danbred Duroc (D) boars, resulting in three-way crossbred Duroc-Landrace–Yorkshire (DLY) pigs.Heterosis is caused by non-additive gene actions  and is oNon-additive genetic effects are usually not incorporated into models for genetic evaluation for at least two reasons. First, to accurately estimate dominance genetic variance and dominance genetic effects based on pedigree-based relationships, very large data sets with a high proportion of full-sibs are required . Second,For genomic evaluation in a crossbreeding system with two purebred lines and a crossbred F1 population, Vitezica et al.  introducThe aims of this study were to (1) present a modification of the model of Vitezica et al.  and XianIn this section we present successively (1) the model for genomic evaluation in the terminal crossbreeding system with purebred sire line and three-way crossbred animals; (2) an equivalent single nucleotide polymorphism (SNP) effects model and equations for genetic variances and covariance; (3) equations for EBV with associated accuracies; (4) the data used in this study; and (5) details of the data analysis.p for purebred, is the sire line part of the model, and the second equation, with subscripts c for crossbred, is the three-way crossbred part of the model. For the sire line part of the model, vector The bivariate model with phenotypes on the sire line and three-way crossbred pigs considered as two correlated traits is:n is the total number of animals, k is the number of SNPs, and superscript T refers to the transpose of a matrix. The dominance genomic relationship matrix is:The model is a modification of the model in Vitezica et al.  and Xianentclass1pt{minimaThe variance–covariance matrices of the genetic effects defined above are not immediately suitable for fitting the model using standard genetic evaluation software. However, the variance–covariance structure of the genetic effects in the model can equivalently be described as:mentclasspt{minimaa et al. , except mentclasspt{minimaInbreeding depression is incorporated in the model as a fixed effect, as in Xiang et al. , where tNote that for the three-way crossbred part of the model, additive and dominance effects for the three-way crossbred performance are assumed independent of the origin of the allele (sire line or crossbred sow population). An alternative would be a breed of origin effects model , 14, butFor the derivation of equations for genetic variances and covariances, and for EBV with associated accuracies that are presented in the following subsection, it is more suitable to express the genotypic effects mentclass2pt{minimThe inbreeding depression effects, mentclass2pt{minimj denote the jth element of vectors, the resulting expression for the additive genetic variance is:The classical additive genetic variance is not the parameter mentclasspt{minimamentclasspt{minimag et al. , here weF measures departure from Hardy Weinberg equilibrium, and equals h being the frequency of the heterozygous genotypes (with the definition that The additive genetic variance for crossbred performance (mating sire line with sows from another population) is according to Xiang et al.  obtained in Eqs. , 9), \documentThe additive genetic covariance between purebred and crossbred performances is according to Xiang et al.  obtained in Eqs. , 8) and and9\docBreeding values (BV) of sire line boars for purebred performance are obtained from the allele substitution effects shown in Eq.  as \docuBreeding values of sire line boars for three-way crossbred performance are obtained from the allele substitution effects of sire line boars for crossbred performance as shown in Eq. , and canfined in . Thus, fThe data set consisted of Danbred Duroc and Danbred DLY pigs reared in the same time period at the test station Bøgildgaard in Denmark, and measured for phenotypes on production traits. All pigs had known pedigree, and sires of both the Duroc and the DLY populations were active breeding boars in the DanBred Duroc population , which creates a genetic relationship between the Duroc and the DLY pigs used in this study. The Duroc pigs were part of the routine breeding program in which owners of nucleus herds select up to three male pigs from litters with a high index (average of indices for parents) for entering the test station. The DLY pigs were produced at a commercial sow herd, for which 698 litters of DLY pigs were produced with the intention of having four pigs  from each litter in the test. The Duroc pigs came from 13 nucleus herds, and the DLY pigs came from one sow herd, resulting in a low biosecurity level at the test station due to the mixing of pigs from many herds with different disease status. At the test station, feed was given ad libitum in feeders with individual recordings on feed intake and each pen was planned to contain 14 pigs. During the study, changes of feed composition at the test station were made twice, but in all cases the feed was optimised according to Danish nutrient recommendations . The expThe pigs were genotyped with either the 8.5K GGP-Porcine LD Illumina Bead SNP array or the GeneSeek Genomic Profiler (GGP) Porcine HD 70K SNP array . For simplicity, and to avoid issues with effects of imputation errors, only SNPs that overlapped between the two SNP arrays were used here. Animals with parentage error due to the Duroc sire  were removed, and otherwise genotypes with such mismatches were declared missing. The SNPs were quality controlled using the following criteria: SNPs with a call-rate lower than 90% or with a minor allele frequency (MAF) lower than 0.01 were excluded. After quality control, 6316 SNPs were retained. The genotype of an animal was only retained if the call rate was higher than 90%. Following this quality control, 1.2% of genotypes were missing, and these were imputed using Fimpute .The following four traits were analysed: ultra-sound recorded backfat thickness (BF), overall conformation score (CONF), average daily gain (ADG), and feed conversion ratio (FCR). BF was measured as the average of ultrasound measurements in mm at four positions of the back, recorded at the end of the test period, CONF was measured as a score ranging from 2 to 4 at the end of test period, ADG was total weight gain in kg during the test period divided by number of days of the test period, and FCR was total feed intake in kg divided by total weight gain in kg during the test period. One animal with an unusually low end weight of 43 kg was removed from the study, and for one animal that had an unusually large feed intake given the weight gain, this feed intake was changed to missing. A summary of the data is in Table For the analysis of traits BF, CONF, ADG or FCR observed on the Duroc sire line and DLY crossbreds, as presented in the subsection on data, we used the bivariate model , where vThe allele frequencies of sire line and crossbred sows and heterozygote genotype frequencies in crossbred sows all enter into Eqs. , 8) and and8) anmentclasspt{minimaon 2.1-5 . AccuracTable The results in Table Based on estimates of model parameters and allele frequencies, genetic parameters were computed using Eqs. , 8) and and8) anEBV of Duroc boars for purebred and crossbred performances are in Fig. mentclass2pt{minimmentclass2pt{minimAccuracies of EBV for purebred and crossbred performances of Duroc boars are in Fig. We present a model for genomic evaluation for a terminal three-way crossbreeding system when information is available on the purebred sire line and three-way crossbred pigs. This model is a modification of the models studied in Vitezica et al.  and XianThe model in this paper includes both additive and dominance genotypic effects, which are allowed to differ between purebred and crossbred animals, but correlated. The estimates of correlation between additive genotypic effects, For BF, FCR and ADG, the additive genetic variance was smaller for purebred performance than for crossbred performance. This may be related to a difference in how purebred and crossbred pigs were selected for entering the test station: crossbred pigs were selected randomly from litters (about 4 pigs from each litter) from sires with a low, medium or high index, whereas purebred pigs were selected among litter mates (1–2 pigs from a litter) from litters with a high index by the owners of the nucleus herds. The owners of the nucleus herds had clear economic interests in having the best performing boars at the test station. Therefore, the selection of purebred pigs entering the test station was not random, and very likely this selection was not independent of the phenotypic records observed in the study, which likely resulted in a reduced genetic variation, in particular for ADG.Guo et al.  analysedo et al.  was abouo et al.  was subso et al.  was stanThe estimated additive genetic correlations between purebred and crossbred performances, nd Calus ). In ourADG was the only trait for which there was a statistically significant inbreeding depression effect, and only in the Duroc population. This is not surprising since inbreeding depression is particularly important for traits that have been under strong selection .For the traits BF, CONF and FCR, accuracies of EBV for Duroc boars for purebred performance were higher than those for crossbred performance, while they were simlar for ADG. Several factors influence accuracy, such as information on relatives and heritability. As mentioned in the Results section, seven of these Duroc boars had Duroc offspring, which increases the accuracies of purebred performance for these boars. More generally, the Duroc boars all had own records on purebred performance, and not on crossbred performance, which increases the accuracies of EBV for purebred performances for all traits. In terms of heritability  and consider these as additional correlated traits, i.e. instead of a bivariate model as used here, a four-variate model would be used. Such a model would be an extension of the model in Christensen et al.  and SeviIn this paper, we present a model for genomic evaluation in a terminal three-way crossbreeding system when information on the purebred sire line and three-way crossbred pigs is available. This model is a modification of previously reported models, accounts for both additive and dominance genotypic effects, and provides flexibility through genetic correlations for these effects between purebreds and crossbreds. Equations were derived for additive genetic parameters as functions of parameters in the model, allele frequencies, and genotype frequencies, for EBV for purebred and crossbred performances as functions of estimated effects in the model, allele frequencies and genotype frequencies, and for accuracies of EBV as functions of allele frequencies, and genotype frequencies and submatrices of the inverse coefficient matrix in the mixed model equations. The methods presented in this paper will be useful for practical implementation of genomic selection for both purebred and crossbred performances in a three-way crossbreeding system.Additional file 1. Investigation of predictive performance.


To discover the genome-wide eQTL mediation effects combing genomic and transcriptomic profiles, it is necessary to develop novel computational methods to rapidly scan large number of candidate associations while controlling for multiple testing appropriately. Here, we present eQTLMAPT, an R package aiming to perform eQTL mediation analysis with implementation of efficient permutation procedures in multiple testing correction. eQTLMAPT is advantageous in threefold. First, it accelerates mediation analysis by effectively pruning the permutation process through adaptive permutation scheme. Second, it can efficiently and accurately estimate the significance level of mediation effects by modeling the null distribution with generalized Pareto distribution (GPD) trained from a few permutation statistics. Third, eQTLMAPT provides flexible interfaces for users to combine various permutation schemes with different confounding adjustment methods. Experiments on real eQTL dataset demonstrate that eQTLMAPT provides higher resolution of estimated significance of mediation effects and is an order of magnitude faster than compared methods with similar accuracy.Expression quantitative trait locus (eQTL) analyses are critical in understanding the complex functional regulatory natures of genetic variation and have been widely used in the interpretation of disease-associated variants identified by genome-wide association studies (GWAS). Emerging evidence has shown that  Understanding the complex functional natures of genome variants has been the focus of many studies in recent years, which provides us with advanced insights into phenotype variability and disease susceptibility . Vast gecis-acting) and distant gene transcriptions (trans-acting), defined by the physical distance between an eQTL and a gene, usually using 1 Mb as a threshold or on different chromosomes for trans-acting associations  and might affect the gene expression directly through affecting transcription factor (TF) binding process , cis-eGene (C) and trans-eGene (T), represented by a trio (L→C→T), a recently proposed work which aims to test the significance of the effect of cis-eGene on trans-eGene controlled by the genotype of L and confounders , the objective is to test the significance of β1. In practice, this requires performing a large amount of association tests in order to scan all possible candidate trios due to related variants in linkage disequilibrium (LD). Thus, it will result in a large number of nominal statistics, i.e., P values, and multiple testing has to be considered in order to control the false discovery rate. A traditional solution is to use Bonferroni correction method, which multiplies the nominal P value with the total number of tests to get an adjusted P value. However, the Bonferroni method has been proven overly stringent in genomic area due to the fact that a large number of tests are not independent because of variants in LD, and this method will result in a lot of false negatives   . Second,, or BB) , allelic, or BB) , and bio, or BB) . In the  testing . HoweverP values at any significance level in a short running time; and 3) flexible choices of different confounding factors adjustment methods. In addition, eQTLMAPT provides flexible interfaces for users to combine different features and perform the proper permutation scheme based on their practical needs. Experiments on a real eQTL dataset demonstrate that eQTLMAPT is an order of magnitude faster than GMAC, and its estimated significance has a much higher resolution than the compared method.In this work, we present eQTLMAPT, an R package which improves upon GMAC  by implecis-eGene mediators of trans-eQTLs in whole genome, we developed eQTLMAPT, an R package to perform mediation analysis with multiple permutation schemes and flexible covariate adjustment strategies. The core regression models we used in mediation analysis is similar to the model used in the recently proposed method, GMAC (G represents the genotype of single nucleotide polymorphism (SNP)L; C, and T represent gene expression levels of cis-eGene and trans-eGene, respectively; Cov represents covariates; and ϵ represents the error term following normal distribution. For the trio , we assume L is significantly associated with C and T by testing β1 ≠ 0 and β2 ≠ 0 in the linear models, with β estimated by least-squares fitting. The statistic of mediation analysis here is to test the mediation effect of cis-eGene C on trans-eGene T while controlling for the effects of eQTL L, covariants Cov. The null hypothesis is H0:β3 = 0.To efficiently identify od, GMAC . The modz = β3/se, in Equation 3, where se represents the standard error of β3. Second, to account for multiple testing in assessing the significance of the mediation effect, we perform within-genotype group permutations of cis-eGene transcripts C to empirically characterize the null distribution of mediation effects . The purpose of within-genotype group permutation is to break the potential mediation effects from C to T within each genotype group  while keeping the cis-eQTL and trans-eQTL associations. The adjusted empirical P value of mediation test would finally be calculated by comparing the observed mediation statistic z with the permutation statistics Z0 under the null.Our method can be separated into two main steps: first, we calculate the nominal association statistic, Z0, and provide users with flexible choices, we implemented three permutation schemes in our package: 1) fixed permutation scheme, which generates N permutation datasets ; 2) adaptive permutation scheme, which prunes the permutation process when there are too many null statistics better than the observed z statistic ; and 3) GPD approximation, which models the tail of the null distribution via a drastically reduced number of null statistics and estimates P value with higher resolution (Model the Tail of the Null Distribution Using GPD). To deal with complex hidden confounding effects, we also adopt an adaptive confounder adjustment method (Confounding Factors Adjustment).To obtain the null distribution of mediation effects, i.e., t method  and a fiL,C,T) we aim to test are not independent due to the fact that multiple SNPs are correlated because of LD. Traditional multiple testing correction methods like Bonferroni and Benjamini–Hochberg correction, which give a global significance threshold based on all nominal P values, prove to be overly stringent and may result in false negatives in such correlated genomic analyses. Thus, we adapt permutation-based testing approaches to assess the significance in association test for each trio  (Equation 3). Permutation test is a widely used non-parametric method in many bioinformatics applications. It generates a null statistic distribution by random permutations and then assesses how likely the observed statistic obtained in the nominal association originates from the null distribution.The associations of trios  by Equation 3, where se is the standard error of β3. Given a fixed number of N, we perform N times permutations within-genotype groups for cis-eGene C by randomly permuting sample labels in each genotype group, i.e., AA, AB, and BB. It will generate N null mediation statistics, denoted by i∈. If M null statistics in Z0 are stronger than the observed statistic |z|, the empirical P value is assessed by Equation 4, where pseudo-count 1 is added to avoid meaningless denominator.Assume the nominal mediation statistic P value has lower bound limitation that N to get precise P value estimates for strong mediation effects with smaller P values, which will tremendously increase the computational costs. For example, if the true P value is 10−6 for a trio, at least 1 million permutations should be performed to achieve the precise P value. But for most trios, with true P values larger than 10−3, 1 million permutations would be a waste of resources because thousands of permutations could lead to precise P values. To solve this problem, we implemented an adaptive permutation strategy in eQTLMAPT to prune permutations once we observe too many null statistics stronger than the nominal statistic z of mediation analysis.The strategy of fixed permutation scheme is direct, easy to implement, and adopted by most permutation testing approaches. However, the adjusted α = 0.05 for example, and a maximum permutation times N, in case of indefinitely running the process, we define the pruning threshold K = α*N, and usually K << N. For each trio , if we observe more than or equal to K null statistics that N, the permutations process will be stopped. Suppose Γ times of permutations are executed in total and M null statistics are found to be stronger than the observed statistic |z|, the adjusted P value is given by Equation 5.The basic idea of adaptive permutation strategy is to perform more permutations for significant trios while decreasing the number of permutations for insignificant trios. This is because insignificant trios could be assessed with fewer permutations than significant ones. By setting a significance level, N = 10,000 and α = 0.05, then K = 500, and assume we have performed 800 times of permutation for a trio and find K null statistics stronger than nominal statistic z. Then, we stop performing further permutations and the final adjusted P value = 501/801. In this case, only 800 times permutations are needed instead of 10,000 times in the fixed permutation scheme. This strategy tremendously reduces the number of permutations required for insignificant trios; however, the lower bound of adjusted P value still exists, which is 1/(N + 1). To solve the lower bound problem, we approximate the tail of null statistics distribution by generalized Pareto distribution and estimate the small P values at any significance level without the limitation of lower bound.For example, given P values especially in large-scale genomic analyses, where huge numbers of associations are simultaneously tested. To determine precise small P values at any significance level without performing all possible permutations, we implemented a P value approximation method based on GPD, which has been widely used in modeling extreme values , we suppose both z and Z0 are sorted in decreasing order, i.e., i<j. Define Nexc as the number of exceedances , and Y0⊂Z0, and exceedance threshold z0 > t, if z0∈Y0. Then, we calculate z0−t for each element z0∈Y0 to get a vector of exceedances X0 are used to fit the tail of the null distribution modeling by GPD. The GPD has cumulative distribution function (CDF) shown in Equation 6.In our case, given permutation statistics set a and k are scale parameter and shape parameter, respectively, and the range of x requires k > 0, and x ≥ 0 for k ≤ 0. If x falls out of these ranges, the GPD estimated P values will be zeros, i.e., a and k in F(x) given X0. The goodness-of-fit test of the Anderson–Darling statistic is used to evaluate whether the exceedances follow the GPD , then iteratively reduce Nexc by 10 until a good fit is achieved. Besides, the GPD approximation can only be used when the nominal mediation statistic z is in the range of extreme permutation null statistics (tail of null distribution). For example, if z is in the middle of the null distribution, this method cannot be applied. To specify, let M be the number of permutation values that exceed the test statistic z, if M < N*α, α = 0.01 in default, GPD approximation will be performed; otherwise, fixed permutation scheme will be performed. The detailed methods have been described in The presence of heterogeneous known or latent unmeasured covariates that affect genotype and phenotype (gene expression in our context) is a major source of bias in the mediation analysis, which needs to be adjusted. The common sources of covariates, such as batch effects, age, sex, postmortem interval (PMI), RNA integrity number (RIN), and population stratification, are associated with either samples or individuals. The latent unwanted covariates can be identified by methods like principal component analysis (PCA) , surrogaCov variable in Equation 3 when performing mediation analysis. The second way is proposed in GMAC  in default]. Then, for each trio , only a small number of PCs will be selected from H for adjustment based on the correlations between PCs and C,T. And experiments demonstrated that this adaptive covariates selection method improved power and precision in mediation analysis   and MemoP < 10−6, informative missingness test P < 10−9, and with minor allele frequency (MAF) < 0.05 seperately. To QC in subject level, we removed subjects with call rate <95%, with outlying heterozygosity rate based on heterozygosity F score (beyond 4*sd from the mean F score), and with gender mismatch. We also performed IBS/IBD filtering: pairwise identity-by-state probabilities were computed for removing both individuals in each pair with IBD > 0.98 and one subject of each pair with IBD > 0.1875. To test for population substructure, we performed PCA using smartPCA in ENGINSOFT (We applied PLINK2 (v1.9beta)  and in-hNGINSOFT .4Y1. Sample outliers with problematic gene expression profiles were detected and removed based on hierarchical clustering  by scaling gene length (union of exon length) and sequencing depth. We removed samples with gender mismatch by checking gender-specific expression genes XIST and RPSustering . Genes wcis/trans-eQTL mapping using additive linear model. In cis-eQTL analysis, variants (SNPs and indels) within 1 M upstream and downstream from the TSS were tested for association with gene expression traits. And variants beyond the ±1M window were associated with the gene expression traits in trans-acting manner. For cis-eQTL results, a significance level of false discovery rate (FDR) ≤0.05 was used. And for trans-eQTL results, we adopt a global significance level P < 1 × 10−8 because of the tremendous amount of trans-associations and weak trans-eQTL effects.MatrixEQTL  was usedN = 10,000, α = 0.05) were applied for all candidate trios , where eQTL L was significantly associated with cis-eGene C  and trans-eGene T . For performance comparison, mediation analyses were performed in multiple scenarios described in the Results section.For biological discovery, mediation analyses with adaptive permutation scheme and GPD approximation (ROSMAP Dataset and Preprocessing), 26,662 gene transcripts and 6,736,714 variants (including SNPs and indels) of 334 subjects were left for eQTL analysis. We detected 3,195,073 significant cis-eQTL associations, representing 5,711 unique cis-eGenes and 60,758 unique cis-eQTLs, and 145,153 trans-eQTL associations, representing 1,382 trans-eGenes and 66,847 unique trans-eQTLs, under significance thresholds of FDR ≤ 0.05 (corresponding P < 1 × 10−3) and P < 1 × 10−8 for cis- and trans-eQTL associations, respectively. Seventy-five percent of trans-eQTLs were also identified as cis-eQTLs, which is similar to previous findings  representing 6,217 unique gene pairs  were derived from significant cis- and trans-eQTL associations. For multiple correlated variants linked to each gene pair, we used permutation schemes introduced in Material and Methods to control for multiple testing, and for genome-wide unique gene pairs, we used a FDR procedure to control for multiple testing.After stringent quality controls for both RNA-seq and genotyping data  from candidate trios, we selected the most significant cis-eQTL for cis-eGene C, resulting in 6,217 trios. Mediation analyses with fixed permutation scheme  and adaptive permutation scheme  were both performed on those 6,217 trios. Empirical P values Pfixed and Padaptive were shown in r = 0.999, indicating the two schemes have similar precision. While fixed scheme always executed 10,000 times of permutations for each tested trio, adaptive scheme significantly reduced the permutation times, as shown in the histogram in We first compared adaptive permutation scheme implemented in our package with fixed permutation strategy which was commonly adopted by traditional methods, including GMAC . For eacP values with fewer number of permutations compared with traditional fixed permutation strategy  trios with fixed permutation P values were less than or equal to 0.01 . And then we rerun mediation analyses for those trios with GPD approximation under fixed permutation schemes with N = 1,000, 5,000, and 10,000. The reason that we only select trios with P ≤ 0.01 is because only permutation P values at the tail of null distribution can be estimated by the GPD approximation method (see Model the Tail of the Null Distribution Using GPD). P values versus P values derived from the fixed permutation scheme , and we can see that GPD-estimated P values have higher resolution than fixed permutation scheme. For instance, GPD-estimated P values range from 10−2 to 10−8, while fixed permutation-derived mediation P values range from 10−2 to 10−3, when N is set to 1,000. And GPD-estimated P values are much smaller than fixed permutation-derived P values, which demonstrates that the GPD approximation method has the ability to detect mediation effect more accurately with higher significance resolution.Using generalized Pareto distribution to model the tail of null distribution of permutation statistics could derive more precise empirical strategy . To testP value equal to 0.01 under the fixed permutation scheme with N = 100. It is reasonable to suppose that the significance is likely to be underestimated because of the small N (Pfixed ≤ 0.01). Then we rerun the mediation analyses for those 1,000 trios with N set to 10,000, where Pfixed ≤ 10−4. The density plot of P values of those 1,000 trios derived under the fixed permutation scheme  was shown in −2 and 10−3 were shown. The peak around 10−2 indicates some trios have true significance level around 10−2. However, the larger peak centers around 10−3 indicate that the significance of a large number of tests is underestimated when N = 100. Then we asked whether using GPD approximation strategy can derive P values proxy for true P values even when N was still set to 100. We extracted trios with significance levels between  interval    to contrP ≤ 0.05) and similar number of significant trios with FDR ≤ 0.25 (Storey and Tibshirani multiple-test controlling method). The Venn diagram in P ≤ 0.05 or FDR ≤ 0.25) detected by GMAC can be discovered by eQTLMAPT with N = 10,000, 5,000, 1,000, and even 500. For example, among the 113 significant trios with FDR ≤ 0.25 detected by GMAC, 110 (97%) can be discovered by eQTLMAPT with N = 10,000, and 104 (92%) can be discovered by eQTLMAPT with N = 500. With the similar ability in discovering significant trios, eQTLMAPT is about 90, 40, 8, and 4 times faster than GMAC when N = 500, 1,000, 5,000, and 10,000, respectively  and trans-eQTLs (P ≤ 1 × 10−8). From the NHGRI GWAS catalog (July 2019), 70,971 unique SNPs, reportedly associated with traits and genotyped in ROSMAP dataset, were downloaded (r2 > 0.3) using PLINK and ROSMAP genotype data, 30,894 independent trait-associated SNPs were left, of which, 16,398 SNPs had GWAS P ≤ 5 × 10−8 and 14,496 SNPs had GWAS P ≤ 5 × 10−8, respectively. Among SNPs with GWAS P ≤ 5 × 10−8, 28% were cis-eQTLs compared with 18% in SNPs with GWAS P ≤ 5 × 10−8 . Using the same GWAS enrichment method, we found GWAS SNPs were more likely to have mediation effects . Among those cis-mediators, we found 14 are TFs, including ZNF488, ZSCAN26, ZNF254, TBX1, FOXS1, ZFP57, ZNF568, ZNF260, ZNF14, GTF2I, ZFX, CSDC2, GTF2IRD2B, and GTF2IRD2. For example, we observed the trio , where TBX1 is the cis-eGene and MSC is the trans-eGene, and MSC has been predicted to be the target of the transcription factor TBX1 in brain tissue and central nervous system  (The 519 trios with suggestive permutation s system . This inmanTFDB) .trans-eQTLs work on distant genes. The eQTL mediation analysis is a promising tool to uncover the mechanisms underlying trans-eQTLs. In order to discover the eQTL mediation effects in whole genome, millions of candidate associations of  trios need to be tested, which requires the computational methods to control for multiple testing appropriately. In practice, there are hundreds of variants on average associated with eGenes in both cis- and trans-manner, which result in huge numbers of candidate trios. For example, in the ROSMAP dataset, nearly 1 million candidate trios need to be tested, which only represent 6,217 unique  pairs. To determine the genome-wide significance of a nominal testing statistics, we need to account for two multiple-testing levels: multiple genetic variants are tested per  pair, and multiple  pairs are tested genome-wide. We used permutation test to correct for the former and FDR estimation to control for the latter.There has been intense efforts to identify causal genes and other biomarkers such as RNA, protein, and microbiota underlying complex diseases . One of P value resolution, which is limited by a lower bound. And there is no efficient built-in permutation scheme in current tools aiming at analyzing eQTL mediation effect. To fill this gap, we present eQTLMAPT, which implements a fast and accurate eQTL analysis method with efficient permutation procedures to control for multiple testing. eQTLMAPT can correct for the multiple correlated variants tested via three different permutation schemes: the fixed permutation scheme, the adaptive permutation scheme, and the generalized Pareto distribution (GPD) approximation, which models the null distribution of no mediation effects using GPD trained from a few permutation statistics and could accurately estimate the adjusted P values without the limitation of lower bound. These strategies implemented in eQTLMAPT greatly accelerated the efficiency of multiple test controling in mediation analyses and provided users higher resolution of estimated significance which would help them distinguish the best signals.The traditional permutation scheme, which runs a fixed number of permutations, has to balance the time cost and the P ≤ 0.05), representing 351 unique cis-eGenes. Among those cis-mediators, we found 14 are TFs, including ZNF488, ZSCAN26, ZNF254, TBX1, FOXS1, ZFP57, ZNF568, ZNF260, ZNF14, GTF2I, ZFX, CSDC2, GTF2IRD2B, and GTF2IRD2. This proves that TFs might play a role in the mediation effects. We also tried to replicate these significant trios with mediation effects in the GTEx dataset analyzed by P ≤ 0.05 in multiple tissues. For example, the gene pair  was observed with mediation effects in multiple tissues including brain putamen, fibroblast, colon, esophagus, lung, muscle, pancreas, pituitary, skin, thyroid, and vagina. And the significance of the mediation effect can reach 2 × 10−7 in GTEx muscle tissue. This might suggest a common trans-eQTL regulatory mechanism across tissues.In the analyses of the ROSMAP dataset, we detected 519 trios with suggestive mediation effects , microRNA, snRNA, antisense RNA, and pseudogene, were also detected. The top 3 gene classes are protein coding, pseudogene, and lncRNA genes. Although many studies have shown that non-coding RNAs play key roles in the complex regulatory networks in cell system, most of their functions are still missing . Source code and comprehensive documentation of eQTLMAPT are freely available to download at https://github.com/QidiPeng/eQTLMAPT.Genotype and RNA-seq data of ROSMAP study (in control use): Synapse platform .The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Pyrus spp.) hybrid populations are often used to develop novel cultivars. Pear cultivar breeding is a lengthy process because of long juvenility and the subsequent time required for reliable fruit phenotyping. Molecular techniques such as genome-wide association (GWA) and genomic selection (GS) provide an opportunity to fast-forward the development of high-value cultivars. We evaluated the genetic architecture of 10 pear fruit phenotypes (including sensory traits) and the potential of GS using genotyping-by-sequencing of 550 hybrid seedlings from nine interrelated full-sib families. Results from GWA suggested a complex polygenic nature of all 10 traits as the maximum variance explained by each marker was less than 4% of the phenotypic variance. The effect-size of SNPs for each trait suggested many genes of small effect and few of moderate effect. Some genomic regions associated with pear sensory traits were similar to those reported for apple – possibly a result of high synteny between the apple and pear genomes. The average (across nine families) GS accuracy varied from 0.32 (for crispness) to 0.62 (for sweetness), with an across-trait average of 0.42. Further efforts are needed to develop larger genotype-phenotype datasets in order to predict fruit phenotypes of untested seedlings with sufficient efficiency.Interspecific pear ( Pyrus varies according to different studies, but commercial breeding has mainly focussed on three species: P. communis (European pear), and two Asian pears namely P. pyrifolia and P. x bretschneideri1. High flavour, buttery and juicy texture are among the key characteristics of European pears, while Asian pears generally have crisp texture and subtle flavour. Breeding programmes in New Zealand and elsewhere have combined Asian and European pears to develop crisp, juicy and highly flavoured hybrid cultivars.Pear is currently grown commercially in almost every continent of the world. The number of catalogued species in the genus 2. One way of reducing the juvenility period is by growing seedlings in the glasshouse to accelerate the growth rate before planting them into the orchard. New genomic technologies also offer the possibility of accelerating and increasing efficiencies and effectiveness of breeding programmes for new pear cultivars. Most pear fruit traits, such as texture, size, storage ability, resistance to scuffing, flavour and aroma are reported to be affected by numerous loci with small-to-moderate effects6.Pear breeding programmes generally involve mating of selected parents to create hybrid seedling populations, selection amongst seedling populations, and testing of best performing seedlings. Traditional pear breeding is an expensive and lengthy process primarily because seedlings grown on their own roots typically have a long juvenile period. Reduction of generation time is a focus of many breeding programmes as this has the largest influence on the time taken for new products to reach market7. The traditional MAS is best suited for monogenic or oligogenic traits whereas GS is ideal for traits controlled by many loci with small-to-moderate effects. Therefore, a two-stage selection strategy combining MAS and GS has the potential to accelerate breeding cycles and improve the efficiency of fruit breeding programmes8. There are reports of GWAS in germplasm populations of Japanese pear P. pyrifolia9 and Pyrus spp.5, but the evaluation of GS has only been attempted in Japanese pear6.A traditional MAS scheme ignores the contribution of all other genes, and thus could result in a lower response to selection, especially for polygenic traits. High-speed and reduced cost of genotyping technologies have facilitated the availability of large number of single nucleotide polymorphic (SNP) markers enabling researchers to study marker-trait associations across the whole genome. High-density genotyping platforms have also facilitated the implementation of genomic selection (GS). GS involves simultaneous estimation of genomewide SNPs effects to predict genomic breeding valuesDespite sensory traits being primary selection criteria for developing new cultivars, there appears to be no report of GWA and GS for pear sensory traits – something that hinders the acceleration of breeding cycles. The main objectives of this study were to conduct GWA to find the candidate genomic regions for pear fruit traits including sensory eating quality traits, and evaluate the potential of GS using a hybrid population derived from crosses between Asian and European pears.The distribution of adjusted fruit phenotypes is shown in Supplementary Fig. G) grouped seedlings largely according to their familial relationships  decay in the genetically related population of 550 individuals showed a high degree of LD even at longer distances between markers. For example, the average r2 for SNPs separated by 0.5 cM, 1.0 cM, and 5.0 cM was 0.29, 0.26, and 0.19, respectively  to 0.72 (p449), and the average relationships among seedlings from different families ranged from 0.30 and 0.52 Fig. . Family p < 0.001) SNP-trait association signals for FIRM and CRIS were identified on LG3 and LG10; for SCUF on LG2, LG4 and LG10; for SHAP on LG11 and LG15; for SOUR on LG1, LG6 and LG13; for SWET on LG4, LG5 and LG13; and for FINT on LG1 and LG8  different from normal and exponential density functions, but fitted best to a gamma distribution . The estimated shape and rate parameters of gamma distribution were about 1.25 and 15.0, respectively, for all traits.Using single-locus GWA, significant  SNPs identified using multi-locus methods MLMM and MRMLM were 79 and 77, respectively, compared to 67 identified from single-locus GWA , resulting in a nine-fold cross validation. The results are displayed using a boxplot graph Fig. . The aveh2) of a selection trait is critical in designing molecular breeding strategies such as MAS and GS8. Using genomic relationship matrix (GRM) in mixed-model equations would provide improved estimation of h2 compared to the pedigree-based relationships10. For a quantitative trait controlled by many genes, the proportion of h2 explained by a SNP would indicate its worthiness for use in traditional MAS. Most traits in this study were under moderate-to-high genetic control, with GRM-based estimates of h2 being comparable with earlier published pedigree-based estimates. For example, h2 of fruit weight (0.68) and acid taste (0.62) were almost identical to those reported by Minamikawa et al.6. The, h2 for fruit firmness in interspecific populations has been reported to be around 0.60–0.7012, slightly higher than that found in this study (0.47). Heritability of FINT in interspecific populations was reported to be 0.5413, which is similar to that observed in this study (0.46). However, h2 of fruit sweetness as reported here (0.16) was half of that observed by Abe et al.14 in hybrid seedlings. In our study, the average (over traits) GRM-based h2 was 0.50, which was slightly lower than the pedigree-based estimate (0.60). In general, GRM-based h2 were shown to be in the similar range to pedigree-based estimates in various species17, which is consistent with results from this study.Understanding of the heritability  in our study is almost identical to that reported (0.18) by Minamikawa et al.6 in full-sib progenies of Japanese pear. The average r2 between the adjacent SNPs was 0.47, which is higher than that reported for pear (0.33)6 and apple (0.32)17.LD is a measure of associations between SNP alleles and the alleles at QTL18. High LD between the adjacent SNPs also plays a key role in improved accuracy of GS19.The higher magnitude of short-range and long-range LD in our study could be a result of the genetic structure of the seedling population. Clustering patterns of families Fig.  reflecte20. Similar to our study, there are reports of QTLs for pear fruit shape index on LG2 and LG1121. Cao et al.22 reported a large effect QTL for apple fruit shape index on LG11, suggesting an orthologous region between apple and pear genomes. A QTL for AVFW was mapped on LG721, which agrees with our results, but some studies4 mapped QTLs for AVFW on different linkage groups – suggesting this trait has a complex polygenic nature. As might have been expected given the high genetic correlation between FIRM and CRIS, these traits shared the largest-effect SNPs located on LG10 and LG3. Genomic locations on LG10 and LG3 have previously been shown to be associated with fruit firmness in apple23. QTLs influencing apple and pear fruit traits  have been mapped on the same LGs of apple and pear genomes24 – further evidence for high synteny between pear and apple genomes26.Peak association signals for various traits were located close to genomic regions that have been previously identified. For example, putative QTLs identified for SCUF on LG2, 4 and 10 are in agreement with earlier reports27. In agreement with results of this study, QTLs for SSC have been mapped on LG44 and LG521. A high correlation was reported between malic acid content and SOUR taste in apple fruit27, hence common QTLs for these fruit traits would be expected. Minamikawa et al.6 mapped a large-effect QTL for acid content in pear fruit on LG6, which supports our GWA results. Unlike our study, there are no QTLs reported for SOUR taste on LG1 and LG13 – suggesting a possibility of population-specific QTL28. Similar to apple FINT QTL29, significant marker-trait associations were observed for pear FINT on LG1 and LG8. SNPs within the apple genes MdCXE4 (LG1) and MdMYB44 (LG8) were shown to be associated with apple FINT and/or fruit acid content30. Further work is needed to identify and test the functionality of putative candidate genes underpinning marker-trait associations reported in this study, and evaluate their synteny with apple genes.The same SNP marker was found to be associated with SWET and JUIC on LG5. A significant genetic correlation 0.41) between these traits  of the significant SNPs, especially the large-effect SNPs, were common between the two approaches in our study (Supplementary Table p < 0.001) used to identify significant SNPs in our study is low and the false-discovery rate adjusted p-values were >0.05 for most traits. These results emphasise that an independent validation of the significant SNPs would be required irrespective of the GWA method used.Multi-locus GWA could be more powerful than single-locus GWA because they account for LD between SNPs, and a small-effect locus may be more apparent when other large-effect loci are already fitted in the model7. Consequently, the distribution of estimated SNP effects should resemble the distribution of the underlying QTL effects. In this study, the maximum variance explained by a SNP for any trait was very small (<4%) and there were very few SNPs with ASE higher than 0.50 PSD, so gamma distributions best fitted to the observed SNP effects for all fruit traits. These results indicated that many QTLs of small effect and only a few moderate effect QTLs control the fruit traits investigated in this study. Hayes et al.33 also reported a gamma distribution of QTL effect sizes in dairy and pig breeding programmes; this is supported by similar observations in other species35. Our results suggest that the response from MAS in most pear fruit phenotypes would be small – hence GS could be a better selection tool. MAS would still be useful for traits controlled by major genes, such as pear red skin colour36.Understanding the genetic architecture of pear fruit phenotypes is facilitated by having information regarding the distribution of QTL effects. The variance explained by a SNP is mainly a function of the size of the QTL associated with the significant marker and LD between the marker and QTL7. In fruit breeding programmes, the traditional MAS for major gene traits  is followed by orchard testing for polygenic traits – hence the time required for developing new cultivars is not shortened to any great extent. A two-step approach, i.e. MAS for monogenic traits followed by GS for polygenic traits, obviates the need for Stage-1 seedling testing, hence fast-forwarding the development of new cultivars8. This strategy has been evaluated in the New Zealand-based PFR apple breeding programme, and it could be adapted to pear breeding programmes as well.The fundamental difference between MAS and GS is that the former only utilizes the SNPs that are significant in a GWAS, whereas the latter uses high-density genome-wide SNPs so that all QTLs are expected to be in LD with one or more SNPs. In the case of GS, potentially all the genetic variance for a trait can be tracked because the marker effect does not need to exceed a pre-determined significance threshold to be used to predict breeding value37, suggesting that genomic prediction accuracies of sensory and instrumental measures could be similar. Minamikawa et al.6 used instrumental measures of some of the sensory traits used in our study. The average prediction accuracy of traits , that were common with Minamikawa et al.6, was almost identical (cf. 0.46). For the across-family validation scheme implemented in this study, the genetic relationships between the training and validation families and population-level LD are among the key drivers of genomic prediction accuracy. Genomic predictions were reported to be most accurate when models were trained with some individuals from the validation families because of the close relation between training and validation sets38. Prediction accuracies of pear fruit phenotypes could be improved further by increasing the training population size, and also by combining parental and breeding populations6.Various studies on apple have shown that the correlation between sensory and instrumental measures of firmness, acidity, and sweetness were about 0.75, 0.80 and 0.50, respectively39. Training data would need to be obtained from different sites and years in order to develop robust genomic prediction models for pear fruit traits. A common approach for the evaluation of GS in fruit breeding programmes is to cross-validate using validation samples from the same generation40. The accuracy of predicting phenotypes of successive generations would be lower than within-generation accuracy due to marker-QTL LD decay19. Meuwissen et al.41 suggested that Bayesian GS models could outperform GBLUP because they capture marker-trait LD that persists in the successive breeding cycles. However, various studies43 based on empirical data suggested very little or no advantage of Bayesian GS models for most traits, so GBLUP method is widely adopted for GS in commercial breeding programmes45.When the training and validation samples are observed independently over different sites/environments, prediction accuracies can be lower depending on the magnitude of genotype-by-site interaction47, fruit breeding programmes need to develop unique multi-generation genotype-phenotype datasets to evaluate persistence of accuracy of genomic predictions over several generations under different environmental conditions. Based on the accuracy of GS in our study, we conclude that it shows strong potential to accelerate the pear breeding cycle by making selections prior to extensive fruit-quality phenotyping. Thus, a GWAS-GS combination could be an effective tool for increasing the efficiency of pear breeding programmes.Similar to animal and plant species36. A subset of the third-generation families created during 2007–08 using best seedlings from the second-generation were used for this study. A total of 12 second-generation selections were used as parents to create nine families and details of mating design and relatedness among the 12 parents were reported earlier36.An interspecific pear breeding programme at Plant & Food Research Limited (PFR) New Zealand was initiated in 1986 using commercial cultivars of European, Chinese and Japanese pear as parents. Second-generation populations were created in 1996 using the best selections produced from the first-generation hybrid families as parentset al.36, a random subset of seedlings that reached a minimum height after growing in a field nursery for 1 year were propagated and planted in PFR’s orchard in Motueka during 2011. Fruit were harvested during the fruiting season in 2015 and 2016 and a random sample of six fruit from each seedling was stored for 28 days at 3 °C, then a further 1 day at 20 °C before evaluation36. Sensory traits including firmness (FIRM), crispness (CRISP), juiciness (JUIC), sweetness (SWET), sourness (SOUR) and flavour intensity (FINT) were evaluated on a scale from 0 (=lowest) to 9 (=highest) by two trained assessors, and one overall score for each trait was given to each seedling. Russet (RUSS) was scored visually on a scale from 0 (=lowest) to 9 (=highest). Scuffing (SCUF) was rated on a 0–9 scale  following the method described by Brewer et al.48. Fruit shape index (SHAP) was measured using a two dimensional shape chart5 and fruit weight (AVFW) was measured as the average weight of the six fruits.As described earlier by Kumar 36. Resulting SNPs were used to construct a consensus linkage map as reported earlier by Kumar et al.36. Briefly, a total of 16 paternal and maternal maps were constructed using Joinmap v4.1 software, and then common SNPs shared by at least three maps were selected as a bridge to merge the maps using R package MergeMap. Finally, 7,509 high quality SNPs were mapped and used for GWA and GS in this study.Protocols for DNA extraction, genotyping-by-sequencing (GBS) library preparation, and variant calling were reported earlier as the SNP data used in this study are those used for fine-mapping of pear red skin colour gene in the same nine families17:Phenotypes adjusted for fixed effects (e.g. year and assessor effect) were used for estimation of variance components and best linear unbiased prediction (BLUP) of additive effects using the following modely is a vector of adjusted phenotypes; μ is an intercept, 1n is a vector of 1 s; Z is the known design matrix relating to a, the unknown vector of random additive genetic effects with a ~ N49 was obtained using all available SNPs, and Eq. (50. Estimates of variance components derived from Eq. (h2) as the ratio of additive  and the remning eight families were used as a training population (TP). GBLUP of seedlings in a VP were predicted by fitting Eq.  where thr2), derived from allele frequencies at a pair of loci, were calculated between SNPs located on the same linkage group36. The r2 estimates were corrected for population structure and cryptic relatedness, and then the average across LGs were plotted against pairwise genetic distance using R package LdcorSV51. The LD decay curve was fitted using a standard logarithmic function. Genetic architecture, in terms of allele substitution effects (ASE) of genome-wide SNPs, of each trait was investigated using the single-locus unified mixed linear model (MLM) approach that accounts for family structure and cryptic relationships as implemented in R package GAPIT52. The following MLM was implemented for single-locus GWA52:y is a vector of adjusted phenotypes; β is an unknown vector containing estimates of fixed effects ; X and Z are the known design matrices relating to β and a 49, and ε is a vector of independent random deviates with variance 53. The proportion of phenotypic variance explained by each SNP was also calculated using R package GAPIT.Estimates of LD (31) which accounts for population structure and cryptic relatedness and uses a stepwise regression with forward inclusion and backward elimination of SNPs as fixed cofactors; second, a multi-locus random-SNP-effect mixed linear model (MRMLM32) which fits all SNPs simultaneously as random effects and also accounts for population structure and cryptic relatedness (https://cran.r-project.org/web/packages/mrMLM.GUI/index.html).Multi-locus GWA methods could potentially be more powerful than single-locus GWA especially for complex traits. Two multi-locus methods were evaluated in this study; first, a multi-locus mixed model (MLMMSupplementary Figures and Tables