Gossypium hirsutum L. represents the largest source of textile fibre, and China is one of the largest cotton‐producing and cotton‐consuming countries in the world. To investigate the genetic architecture of the agronomic traits of upland cotton in China, a diverse and nationwide population containing 503 G. hirsutum accessions was collected for a genome‐wide association study (GWAS) on 16 agronomic traits. The accessions were planted in four places from 2012 to 2013 for phenotyping. The CottonSNP63K array and a published high‐density map based on this array were used for genotyping. The 503 G. hirsutum accessions were divided into three subpopulations based on 11 975 quantified polymorphic single‐nucleotide polymorphisms (SNPs). By comparing the genetic structure and phenotypic variation among three genetic subpopulations, seven geographic distributions and four breeding periods, we found that geographic distribution and breeding period were not the determinants of genetic structure. In addition, no obvious phenotypic differentiations were found among the three subpopulations, even though they had different genetic backgrounds. A total of 324 SNPs and 160 candidate quantitative trait loci (QTL) regions were identified as significantly associated with the 16 agronomic traits. A network was established for multieffects in QTLs and interassociations among traits. Thirty‐eight associated regions had pleiotropic effects controlling more than one trait. One candidate gene, Gh_D08G2376, was speculated to control the lint percentage (LP). This GWAS is the first report using high‐resolution SNPs in upland cotton in China to comprehensively investigate agronomic traits, and it provides a fundamental resource for cotton genetic research and breeding. Gossypium spp.) is an economically important crop around the world, providing the most natural fibre for the manufacture of textiles. Gossypium contains more than 50 recognized species belonging to eight genome groups , G. arboreum (A2), G. hirsutum (AD1) and G. barbadense (AD2), have been domesticated and cultivated. G. hirsutum L. . In addition, high‐end textile products using natural fibre have become increasingly popular. Therefore, it is important to improve the yield and fibre quality concurrently. In recent years, extensive studies have focused on forward and reverse genetics molecular biology methods to reveal genes for traits in upland cotton.Cotton  mapping is an effective tool that has been widely used to dissect the genetic architecture of complex quantitative traits. It can also be effectively used in marker‐assisted selection (MAS) in cotton breeding. Traditional molecular markers, such as restriction fragment length polymorphism (RFLP), amplified fragment length polymorphism (AFLP) and simple sequence repeat (SSR), have played important roles in previous QTL mapping in cotton. Based on biparental recombinant populations, large numbers of intraspecific and interspecific genetic linkage maps have been developed and used in QTL mapping by linkage analysis in cotton  is an alternative tool for detecting QTLs and a promising genetic method for the dissection of complex traits in plants  clarify the genetic structure and linkage disequilibrium (LD) level of upland cotton in China, (ii) explore the genetic architecture of 16 agronomic traits and (iii) provide information about candidate QTL regions and genes that control the corresponding traits.The first 63K cotton array (CottonSNP63K Array) and a cotton HDGM based on this array were developed and published in 2015 (Hulse‐Kemp The 503 inbred upland cotton accessions were genotyped using the CottonSNP63K chip with a 63 058 SNP array. There were 61 928 (98.21%) SNPs reserved when the call rate was set at 90%. In addition, 31 069 (49.27%) polymorphic SNPs were screened out by the method of deleting the SNPs with minor allele frequencies (MAFs) <5%. After further exclusion, 11 975 SNPs showed high polymorphism and genotyping quality by match‐loading with the HDGM that contained 19 191 SNP markers. These SNP markers were used to assess population structure, relative kinship and GWAS analysis and covered 3854.3 cM of the entire cotton genome with an average density of one SNP per 0.32 cM, with the average distance ranging from 0.20 cM (Chr24) to 0.59 cM (Chr21) among different chromosomes. The average polymorphism information content (PIC) of all SNP markers was 0.332, varying from 0.279 (Chr08) to 0.369 (Chr01). In addition, the gene diversity varied from 0.32 (Chr08) to 0.44 (Chr01), with an average value of 0.39 for the entire genome (Table r2) were used to investigate the extent of LD calculated within a 0–10 cM window. The LD decay distance for the 503 accessions between all SNP markers was ~6.1 cM when the value of the cut‐off for r2 was set at 0.1 and was ~2.2 cM when the value of the cut‐off for r2 was set at 0.2  to more than 10 cM  when r2 = 0.1. Four pairs of homologous chromosomes, Chr01‐Chr15 (5.0–5.5 cM), Chr03‐Chr17 (3.5‐4.0 cM), Chr05‐Chr19 (>10 cM) and Chr08‐Chr24 (5.0–5.5 cM) (r2 = 0.1), showed similar LD decays. However, obvious LD differentiation, defined as a difference in the two LD decay distances >3 cM, was found among five pairs of homologous chromosomes: Chr02‐Chr14, Chr06‐Chr025, Chr09‐Chr23, Chr10‐Chr20 and Chr12‐Chr26.In this study, squared correlations of allele frequencies  increased continuously from K = 1 to K = 10 without an obvious inflexion point  of three subpopulations was greater than that of the clusters divided by breeding stage and geographic origin. The FST values ranged from 0.19 to 0.41 (three subpopulations), −0.09 to 0.02 (four breeding stages) and −0.04 to 0.09  matrix is another important factor for GWAS. The average pairwise relative kinship coefficient was 0.084. Pairwise relative kinship values of 0 accounted for 56.13% of all kinship coefficients. In addition, kinship values from 0 to 0.2 accounted for more than 80% of all pairwise kinship coefficients  to 91.8% , with an average h2 of 78.9%. The coefficient of variation (CV) ranged from 0.79%  to 12.33%  (Table P < 0.001) were observed in G and E. This result suggested that both the genotype and environment had important effects on these traits. In addition, the main effects were decided by the genotype. Furthermore, significant variations were observed in six traits in G × E by the P value <0.001  or P value <0.05 (FP: flowering period and EBN: effective boll number)  Table . An analr) Table . The sixP < 0.01 or 0.05). The 16 traits, except FSBN, SW and FE (fibre elongation), showed significant discrepancies between each pair of the four breeding stages , PC1 and PC2, accounted for 74.9% of the phenotypic variation. The P‐PCA scatterplot illustrated that the spots for Group 1, Group 2 and Group 3 overlapped with each other. The four breeding stages and seven geographic origins had recognizable differential distributions among different groups Figure a–c. In aes Table c. Moreoves Table b. This rConsidering the false positive in trait–marker associations, six common models for association analysis (the naive model (GLM), the Q model (GLM (Q)), the PCA model (GLM(PCA)), the K model (MLM(K)), the PCA + K model (MLM(PCA + K)) and the MLM model (Q + K)) were compared using a quantile–quantile (Q‐Q) plot Figures , and thep) > 4.078 (Bonferroni correction P < 1/11 975). A total of 324 significant SNPs were detected for all traits using the Q + K model , LW, LP, FS and SF, were identified at more than 10 loci. According to the correspondence between the genetic and genome maps, 158 of 160 QTLs were assigned on physical regions with an average length of 6.36 Mb using BLAST, and a total of 29 390 genes were located in the riveted regions  had positive effects on FP (2.1%) and WGP (3.2%), but a negative effect on LP (−0.9%). The QTL effect for population phenotype was defined as the average of the cumulative allele effect. Interestingly, the QTL effects were discrepant among Group 1, Group 2 and Group 3. In contrast, there were minor relative differences in QTL effects among S1, S2, S3 and S4 and YtRR, YRR and NIR Table .qGhPH‐c10 and qGhLW‐c10‐1 were located at the close loci of ~76.16 and 76.55–76.6 cM, respectively. The 39 regions covered 84 QTLs and constructed association networks for different traits. In most cases, the regions linked to multiphenotypic traits were found among the same types of traits, that is growth period‐related , yield  or fibre quality traits . For example, there were nine QTL regions related to two growth period traits, FP and WGP. Twenty‐three regions were associated with the same type traits, and 12, 9 and 2 for growth period‐related, fibre quality and yield traits, respectively. However, several QTL regions were detected that contained significant associations for different types of traits. Four regions were identified with associations for multiple traits between fibre quality and yield traits  using BLAST, were 450 kb and 150 kb when r2 = 0.1 and 0.2, respectively. The qGhLP‐c24 accounted for 3.99% of the variation in LP, and the minimum P‐value was 0.000076. Combining the LD heatmap, the candidate region was finally focused between two SNPs, i20550Gh (63 485 399) and i04707Gh (63 648 326) ] and seed (10 DPA and 20 DPA). These genes were supposed to affect related traits; for example, the genes preferentially expressed in fibre may be involved in fibre yield and quality. After screening, 336 genes were obtained and were hypothesized to be the most likely candidate genes for 86 QTLs Tables . There wGASL3 and GhGASL5, were riveted on D04 loci associated with FSBN and FP. Several genes were on more than one site and located in the QTL that controlled different traits. For example, GhACT1 has three loci that control SF or LW. The functions of 19 genes could explain the corresponding traits directly or indirectly. For example, four GhGASLs located in five QTL regions that controlled FSBN, FP, SF and LW were reported as targets of GA regulation that could control seed germination and storage mobilization, stem elongation, flower initiation, pollen and fruit growth and root development.The genes with known functions could also afford evidence for candidate gene identification. Nineteen known genes that had been reported to function in cotton fibre were located in 25 identified QTL regions using BLAST Table . In detaet al., et al., et al., et al., G. hirsutum cultivar collection from China has high levels of genotypic and phenotypic diversity. The 503 accessions cover the five cotton‐growing regions in China. A few accessions were introduced from the Soviet Union (SU) and the United States (USA), such as Deltapine, Coker and Stoneville, which are the founder accessions for cotton breeding in China and have made significant contributions to Chinese cotton production. The relatively larger sample size ensured sufficient genetic variation, and the sample size was similar to the GWAS population sizes used for Arabidopsis thaliana ; Yuanyang, Henan Province ; Shihezi, Xinjiang ; and Korla, Xinjiang —are located in the three main cotton belts in China. In addition, the four sites belong to four climate classifications, Cfa (warm temperate and fully humid), Cwa (warm temperate and desert), BSk (cold and arid steppe) and BWk (cold and arid desert), respectively , the Southern China Region (SCR), USA and SU) and four breeding stages  but were not completely separated according to geographic origin and breeding stage. Three major cotton belts, YtRR, YRR and NIR, composed 88.47% of the whole panel. Three similar genetic backgrounds were present among the groups of YtRR, YRR and NIR, along with S1, S2, S3 and S4   mutant plants have lower steryl glycoside and acyl steryl glycoside levels and reduced seed size in Arabidopsis , 324 significant SNPs and 160 QTLs were identified and associated with 16 agronomic traits. Pleiotropy and close connections were identified and established by a traits network. A number of genes are prospective candidate genes, and further work will be conducted to investigate them. Our study can be applied in genetic research and breeding in cotton.G. hirsutum inbred accessions was assembled for genotyping and phenotyping. The collections were derived from five main China cotton belts: the YtRR (141), the YRR (225), the NIR (79), the NSEMR (28) and the SCR (4), and historically introduced varieties and germplasm resources lines from abroad: the SU (6) and the USA (20) .A multiple‐environment experiment was designed for phenotyping to eliminate the environmental effect Table . A totalG. hirsutum and 17 954 putative interspecific SNP markers from other cotton species  >5%.The 503 accessions were grown in glasshouses for DNA sampling. Genomic DNA was extracted from the young leaf tissue. DNA was standardized at 50 ng/mL for each accession and was processed according to Illumina protocols. The DNA was then hybridized to the CottonSNP63K array, which contains 45 104 putative intraspecific SNP markers from the cultivated cotton species , et al.  developeet al., K value) was set from 1 to 10. The natural logarithms of probability data (LnP(K)) and the ad hoc statistic ΔK were calculated using STRUCTURE HARVESTER: a website and program for visualizing STRUCTURE output and implementing the Evanno method . The ΔK was set as the determinant factor for evaluating the optimal value of K  model with STRUCTURE 2.3.4 software  when the number of SNP markers was n.Phenotype–genotype association analysis and the allele effect calculations were performed using TASSEL version 5.0. Six models, a naive general linear model (GLM), the general linear model adjusted using the Q‐matrix (GLM (Q)) and PCA‐matrix ), the mixed linear model adjusted using the K‐matrix (MLM (K)), the mixed linear model correcting for both Q‐matrix and K‐matrix (MLM (Q + K)) and the mixed linear model correcting for both PCA‐matrix and K‐matrix (MLM (PCA + K)), were employed to reduce errors from population structure and relative kinship. Bonferroni‐corrected thresholds at α  =  1 were set as the significance of associations between SNP markers and the traits  was performed. Haploview 4.2 was used to generate an LD map based on physical location . More, the G. hirsutum gene expression database  decay for each of the 26 chromosomes. (a) LD decay for 13 At‐genome chromosomes: Chr01‐Chr13. (b) LD decay for 13 Dt‐genome chromosomes: Chr14‐Chr26.Figure S2 Distribution of the three subpopulations in 7 region‐clusters and 4 breeding period‐clusters. (a) Percentages stacked column chart for 7 regions: YRR, YtRR, NIR, NSEMR, SCR, USA and SU. (b) Percentages stacked column chart for four breeding periods: S1, S2, S3 and S4.Figure S3 Round tree for the 503 accessions. The outer ring represents three subpopulations; the inner ring represents 7 regions.Figure S4 Principal component analysis of 503 accessions based on phenotype. (a) Scatter plot for three subpopulations: Group 1, Group 2 and Group 3. (b) Scatter plot for 7 regions: YRR, YtRR, NIR, NSEMR, SCR, USA and SU. (c) Scatter plot for four breeding periods: S1, S2, S3 and S4.Figure S5 Summary of GWAS results for flowering period (FP). (a) Phenotype histogram for FP. (b) Q‐Q plots for FP using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for FP GWAS results. The threshold value was set at p < 10−4.078.Figure S6 Summary of GWAS results for the entire growth period (WGP). (a) Phenotype histogram for WGP. (b) Q‐Q plots for WGP using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for WGP GWAS results. The threshold value was set at p < 10−4.078.Figure S7 Summary of GWAS results for first fruit spur height (FFSH). (a) Phenotype histogram for FFSH. (b) Q‐Q plots for FFSH using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for FFSH GWAS results. The threshold value was set at p < 10−4.078.Figure S8 Summary of GWAS results for fruit spur branch number (FSBN). (a) Phenotype histogram for FFBN. (b) Q‐Q plots for FSBN using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for FSBN GWAS results. The threshold value was set at p < 10−4.078.Figure S9 Summary of GWAS results for first fruit spur branch number (FFSBN). (a) Phenotype histogram for FFSBN. (b) Q‐Q plots for FFSBN using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for FFSBN GWAS results. The threshold value was set at p < 10−4.078.Figure S10 Summary of GWAS results for plant height (PH). (a) Phenotype histogram for PH. (b) Q‐Q plots for PH using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for PH GWAS results. The threshold value was set at p < 10−4.078.Figure S11 Summary of GWAS results for effective boll number (EBN). (a) Phenotype histogram for EBN. (b) Q‐Q plots for EBN using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for EBN GWAS results. The threshold value was set at p < 10−4.078.Figure S12 Summary of GWAS results for seed weight (SW). (a) Phenotype histogram for SW. (b) Q‐Q plots for SW using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for SW GWAS results. The threshold value was set at p < 10−4.078.Figure S13 Summary of GWAS results for lint weight (LW). (a) Phenotype histogram for LW. (b) Q‐Q plots for LW using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for LW GWAS results. The threshold value was set at p < 10−4.078.Figure S14 Summary of GWAS results for lint percentage (LP). (a) Phenotype histogram for LP. (b) Q‐Q plots for LP using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for LP GWAS results. The threshold value was set at p < 10−4.078.Figure S15 Summary of GWAS results for fibre upper half mean length (FUHML). (a) Phenotype histogram for FUHML. (b) Q‐Q plots for FUHML using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for FUHML GWAS results. The threshold value was set at p < 10−4.078.Figure S16 Summary of GWAS results for fibre strength (FS). (a) Phenotype histogram for FS. (b) Q‐Q plots for FS using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for FS GWAS results. The threshold value was set at p < 10−4.078.Figure S17 Summary of GWAS results for micronaire value (MV). (a) Phenotype histogram for MV. (b) Q‐Q plots for MV using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for MV GWAS results. The threshold value was set at p < 10−4.078.Figure S18 Summary of GWAS results for fibre uniformity (FU). (a) Phenotype histogram for FU. (b) Q‐Q plots for FU using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for FU GWAS results. The threshold value was set at p < 10−4.078.Figure S19 Summary of GWAS results for short fibre percentage (SF). (a) Phenotype histogram for SF. (b) Q‐Q plots for SF using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for SF GWAS results. The threshold value was set at p < 10−4.078.Figure S20 Summary of GWAS results for fibre elongation (FE). (a) Phenotype histogram for FE. (b) Q‐Q plots for FE using GLM, GLM (Q), GLM (PCA), MLM (K), MLM (PCA+K), and MLM (Q+K). (c) Manhattan plot for FE GWAS results. The threshold value was set at p < 10 −4.078.Figure S21 Association loci effects in different clusters.Figure S22 Pleiotropic effects of GWAS QTLs. Orange represents overlapped QTLs for different traits, and blue represents the closely linked QTLs or different traits.Figure S23 Network‐dissected QTL regions containing associations for agronomic traits. (a) Growth period traits, plant height (PH) and yield traits. (b) Growth period traits, PH and fibre quality traits. FP: flowering period; WGP: whole growth period; FFSH: first fruit spur height; FFSBN: first fruit spur branch number; PH: plant height; LW: lint weight; LP: lint percentage; FUHML: fibre upper half mean length; FS: fibre strength; FU: fibre uniformity; SF: short fibre, FE: fibre elongation.Click here for additional data file.Table S1 Information for 503 upland cotton accessions.Table S2 Number of accessions and gene diversity levels in different clusters.Table S3 Population genetic differentiation statistics (FST) between pairwise clusters.Table S4 Data acquisition for sixteen traits from four places in 2012 and 2013.Table S5 Descriptive statistics for phenotypic variations and broad‐sense heritability for 16 traits.Table S6 ANOVA for 16 traits from multi‐environments in 2012 and 2013.Table S7a Phenotypic comparisons among Group 1, Group 2 and Group 3 by ANOVA and LSD.Table S7b Phenotypic comparisons among S1, S2, S3 and S4 by ANOVA and LSD.Table S7c Phenotypic comparisons among YtRR, YRR and NIR by ANOVA and LSD.Table S8 Shared numbers of significant SNPs associated with 16 traits in six models.Table S9 The 324 significant SNPs associated with 16 traits.Table S10 Information for 160 QTLs and the number of genes in the QTL regions.Table S11 Number of significant SNPs and QTLs for traits and chromosomes.Table S12 The SSR markers identified in Nie's report located in our QTL regions.Table S13 Association loci effects in different clusters.Table S14 The 17 genes located in the candidate region of qGhLP‐c24 and its annotation in Arabidopsis thaliana.Table S15 Expression profiles of genes located in the QTL regions in different tissues.Table S16 Preferential expression of genes in different tissues.Table S17 Information regarding the reported genes located in the QTL regions.Click here for additional data file.


The association studies on human complex traits are admittedly propitious to identify deleterious genetic markers. Compared to single-trait analyses, multiple-trait analyses can arguably make better use of the information on both traits and markers, and thus improve statistical power of association tests prominently. Principal component analysis (PCA) is a well-known useful tool in multivariate analysis and can be applied to this task. Generally, PCA is first performed on all traits and then a certain number of top principal components (PCs) that explain most of the trait variations are selected to construct the test statistics. However, under some situations, only utilizing these top PCs would lead to a loss of important evidences from discarded PCs and thus makes the capability compromised.To overcome this drawback while keeping the advantages of using the top PCs, we propose a group accumulated test evidence (GATE) procedure. By dividing the PCs which is sorted in the descending order according to the corresponding eigenvalues into a few groups, GATE integrates the information of traits at the group level.Simulation studies demonstrate the superiority of the proposed approach over several existing methods in terms of statistical power. Sometimes, the increase of power can reach 25%. These methods are further illustrated using the Heterogeneous Stock Mice data which is collected from a quantitative genome-wide association study.Overall, GATE provides a powerful test for pleiotropic genetic associations.The online version of this article (doi:10.1186/s12864-017-3928-7) contains supplementary material, which is available to authorized users. A lot of human complex traits are highly correlated due to genetics, environmental influences and interactions among them, such as, low density lipoprotein and triglycerides, serum calcium and phosphorus, serum prostate specific antigen and prostate cancer –3. Identp-values with some p-value combination procedure to construct an omnibus test. Fisher-combined p-values  and −2 lns follow the Chi-squared distribution with 2 degrees of freedom. Simulation studies show that GATE outperforms TATES, MultiPhen and mCPC under most scenarios in terms of power. Sometime more than 25% power increase can be achieved  first perform the PCA on all traits and calculate the n unrelated individuals enrolled from a source population in a genetic study. For the ith individual, let yij be the observation values of the jth trait and denote its genotype at a SNP locus by gi, i=1,2,⋯,n, j=1,2,⋯,m, where m is the number of traits of interest. Denote Y=(yij)n×m and G=τ. Let j,j1,j2=1,2,⋯,m. By the singular value decomposition, Δ can be written as Δ=QΛQτ, where Λ is a diagonal matrix with diagonal elements being λ1,λ2,⋯,λm (λ1≥λ2≥⋯≥λm≥0) and Q is an orthogonal matrix with columns being the eigenvectors. Denote Z=YQ, which is called the principal component matrix whose columns correspond to all principal components. Let zj be the jth column vector of Z. So the relationship between the traits and the genotype can be transformed into εj is the residual error term independently following from a normal distribution with mean of zero and unknown variance of σ2. The null hypothesis that there is no association between the genetic variant and phenotypes becomes H0:β1=β2=⋯=βm=0. Denote the Wald test statistic for βj=0 by Tj, j=1,2,⋯,m. Then T1,T2,⋯,Tm are independently and identically distributed and follow from the standard normal distribution asymptotically under the null hypothesis.Suppose that there are H0, a natural choice is the Fisher’s combined test denoted as m degrees of freedom (DF) asymptotically. Notice that T1,T2,⋯,Tm are sorted by the descending order of the eigenvalues λ1≥λ2≥⋯≥λm. Aschard et al. (2014) proposed to use s is the smallest integer satisfying j=1,2,⋯,m, and Fd(·) is the cumulative distribution function of the centralized chi-squared distribution with d DFs. As pointed out in the later simulations, using 0.8 to determine s is not robust and mCPC could loss power substantially. Sometimes such power loss can be more than 25% , we can first construct a combined statistic as Fd(·) is the cumulative distribution function of the centralized Chi-squared distribution with d DFs and K DFs. It should be noted that when K=1, although the DF of the distribution of HK(·) is the cumulative distribution function of mt≤m,t=1,2,⋯,K. Note that when K=1, GATE is reduced to FCT and becomes mCPC when K=2,m1=s. Hence GATE is expected to have more broader application than FCT and mCPC.To test see Fig.  below. Ip-value of GATE, we propose to adopt the following resampling procedure. Since the distribution function of the statistic η(0). Set a large number B, for example B=10000;Calculate GATE based on the observations, denote it by b from 1 to B, generate m random variables which are i.i.d. from the standard normal distribution and denoted as For H with the Estimate the distribution function b from 1 to B, using ηb) and assume that Hardy-Weinberg equilibrium holds in the general population on the SNP locus. Thus the corresponding genotype frequencies are Pr(G=0)=(1−p)2, Pr(G=1)=2p(1−p) and Pr(G=2)=p2. It should be noted that in reality, the latent variables are unobservable. The correlations among phenotypes rely on the coefficients β=τ and γ=τ, which measures the strength of the association between the genetic marker and the latent variables and the association between the latent variables and the traits, respectively. The proportion of the variance of the ith phenotype explained by the genetic variant is i=1,2⋯,m.Since the effect of a causal genetic variant on the phenotypes can be indirect and direct , here weU1,U2,⋯,UL are L latent variables that are independently normally distributed with mean 0 and variance 1, G and ε1,ε2,⋯,εm are defined as above, γik and βi are coefficients, i=1,2,⋯,m, k=1,2,⋯,L. The proportion of the variance of the ith phenotype explained by G can be calculated by i=1,2⋯,m. These two simulated schemes are illustrated in Fig. For Model 2, the genetic markers are directly associated with the phenotypes and their genetic effects are independent of the latent factors. The relationships are mentclass2pt{minimTo compare our proposed method with the existing methods, we generate datasets from the indirect and direct association model, respectively. The detailed setups are as follows.L=m/4. Besides, we assume that every four phenotypes subject to one common latent factor and the respective effects are the same. Then Model 1 becomes Yi=Ui/4⌉⌈γi+εi, Ui/4⌉⌈=Gβi/4⌉⌈+ei/4⌉⌈, i=1,2,⋯,m, where ⌈i/4⌉ denote the smallest integer that is greater than i/4, and i1/4⌉=⌈i2/4⌉, i1,i2∈{1,2,⋯,m}. Actually, there are L different values for γ and denote them by k=1,2⋯,L. For a meaningful comparison, we simulate m=20,100 correlated traits under four patterns of correlation structures: (1) uniform low correlation; (2) uniform strong correlation; (3) a gradient of moderate to low correlations; (4) a gradient of strong to moderate correlations. Thus the derived correlation matrix under the indirect trait model is L-block diagonal. Denote the correlation matrix among m phenotypes by i=1,2,⋯,L are m/L×m/L positive definite matrices.We set the number of latent factors to be smaller than that of phenotypes: γi, i=1,2,⋯,m to ensure the non-zero elements of the correlation matrix Δ match the above four structures under the null hypothesis (i.e. β1=β2=⋯=βL=0). For uniform low and high correlation structure, we let all γi,i=1,2⋯,m be equal to 0.5 and 2 which results in a uniform correlation matrix with equal correlation coefficient of 0.2 and 0.8, respectively. On the other hand, we consider a list of monotone decreasing values for γ to construct a gradient correlation matrix. When m=20, we have L=5 and set Δ belongs to the third pattern of correlation matrix and have the biggest non-zero correlation coefficient of 0.500 (moderate) and the smallest value of 0.038 (low). We set m=20. We denote the obtained four correlation structures for the indirect association model by S1, S2, S3, and S4, respectively. The detailed settings of γi, i=1,2⋯,m corresponding to the above four correlation structures for m=20 are presented as follows: We specify different values for the latent variable coefficients Δ is calculated under the null hypothesis (β1=β2=⋯=β5=0). Similarly, we simulated 100 correlated phenotypes with the above third and forth correlation structures through letting i=1,2⋯,25, respectively. In addition, we provide the detailed settings of correlation structures when 100 correlated phenotypes are considered in Additional file The correlation matrix Yi=Uγi+Gβi+εi, i=1,2,⋯,m, where G is the genotype vector. Similarly, we simulate m=20,100 correlated traits under four above patterns of correlation structures. All γi, i=1,2⋯,m are specified as 0.5 and 2, respectively, for the uniform correlation matrix  with equal correlation coefficient of 0.2 and 0.8. In order to construct a gradient correlation matrix, we consider a list of monotone decreasing values for γ. When γ=τ, γi=1.00−0.05(i−1), i=1,2⋯,m−1, the resulting correlation matrix Σ belongs to the third pattern (denote by S11) with the values decreasing from left to right and have the biggest correlation coefficient of 0.48 (moderate) and the smallest value of 0.005 (low). We set γ=τ, γi=1.50−0.05(i−1), i=1,2⋯,m−1, to get the fourth pattern of correlation structure (denote by S12) with the biggest value of 0.68 and the smallest value of 0.25. Four patterns of correlation structures used for m=20 when the phenotypes are sampled from Model 2 are γi=0.50, i=1,2,⋯,20; Δ=(δst)20×20,δss=1, δst=0.2 when s≠t;γi=2.00, i=1,2,⋯,20; Δ=(δst)20×20,δss=1, δst=0.8 when s≠t;γi=1.00−0.05(i−1),i=1,2,⋯,20; γi=1.50−0.05(i−1),i=1,2,⋯,20; For the direct association model, the effect of latent variables is independent with that of genetic variants. Without loss of generality, we consider the structure that all the phenotypes are related to one common latent variable. Then Model 2 becomes β1=β2=⋯=β20=0). And we also summarize the settings of four corresponding correlation structures  for m=100 which is provided detailedly in the Additional file Likewise, these correlation matrices are calculated under the null hypothesis , dividing all single Wald test statistics T1,T2,⋯,Tm into 2 groups will lead to the smallest DF. Hence, we deduce that the GATE with K=2 will have better power performance than the other selections of K. Furthermore, we conduct some simulation studies to explore the performances of GATE under different selections of K. The simulations results are summarized in the Additional file K∈{1,2} to other existing methods.The selection of In order to test the performance of the proposed GATE aprroach, four existing methods including TATES , MultiPhFirstly, we assume the correlated phenotypes are sampled from the indirect association model and explore the performances of the above five tests.m=20, all of the five tests can control the type I error correctly with their empirical values being close to the nominal significance level. For example, when MAF=0.15,m=20, the empirical type I error rates of TATES, MANOVA, MultiPhen, mCPC, and GATE for the phenotypes with the correlation matrix of S3 are 0.051, 0.051, 0.049, 0.051, and 0.052, respectively. However, when the number of simulated phenotypes is large (m=100), MultiPhen always has inflated type I error rates. For instance, when the phenotypes are generated from the indirect association model with the correlation matrix of S5, the empirical type I errors of MultiPhen for MAF=0.05,0.15,0.30,0.50 are 0.111, 0.087, 0.096, and 0.114, respectively. So we exclude it in the following comparisons of power for 100 phenotypes.Table λ=20%,40%,60%,80%,100% of the phenotypes that are associated with the genotype are considered. Denote the number of the associated traits by k (=λm). Without loss of generality, we assume that the first k phenotypes are associated with G. Besides this, we consider the scenarios that the phenotypes are randomly selected to be associated with the genotype and the corresponding results are presented in Additional file Next, we compare the powers of the TATES, MANOVA, MultiPhen, mCPC, and GATE under the nominal significance level of 0.05. Under each scheme of the correlation structures, 5 levels of association including n=1,500,λ=60% and Σ belongs to S3, the empirical powers of TATES, MANOVA, MultiPhen, mCPC, and GATE are 0.324, 0.309, 0.312, 0.286, and 0.453, respectively. GATE is sightly less powerful than mCPC and TATES when the non-zero correlation coefficients are uniformly equal to 0.8 (S2) and the gap between them narrows as the proportion of associated phenotypes increases. For example, under the correlation matrix of S2 and MAF=0.15,n=1,500, the empirical powers of TATES for λ = 20%, 40%, 60%, 80%, 100% are 0.166, 0.313, 0.396, 0.495, and 0.580, respectively, and those of GATE are 0.118, 0.238, 0.346, 0.486, and 0.605. We also find that MANOVA and MultiPhen usually have similar performance when the number of phenotypes are not too large.Figure m=100 are similar to those for m=20. In most scenarios, GATE performs better than the other three methods. From Fig. n=1,500,λ=60% under the correlation matrix of S7, the powers of TATES, MANOVA, mCPC, and GATE are 0.251, 0.428, 0.409, and 0.683, respectively. When the non-zero correlation coefficients are uniformly large (S6), the mCPC performs slightly better than the GATE. This occurs because when the correlations among different phenotypes are strong and a relatively large number of analysed traits are analyzed, the test n=1,500,λ=60% under the correlation matrix of S7, the powers of mCPC and GATE are 0.408 and 0.693, respectively.The empirical powers of four test (excluding MultiPhen) for 100 phenotypes which are simulated from the indirect trait model with the correlation schemes of S5, S6, S7, and S8 are summarized in Fig. Next, we assess the performance of the proposed test compared with those of the other four tests when the correlated phenotypes are sampled from the direct association model.m=20, all five tests can control the type I error rates correctly because their empirical type I error rates are close to the nominal level. For example, when MAF=0.30, the type I error rates of TATES, MANOVA, MultiPhen, mCPC, and GATE under the correlation structure of S11 are 0.049, 0.054, 0.055, 0.052, and 0.054, respectively. Similarly, when the number of simulated phenotypes is large (m=100), MultiPhen has inflated type I error rates, while the other four tests maintain correct type I error rates. For instance, when m=100, the type I error rates of MultiPhen under the correlation structure of S15 for MAF=0.05,0.15,0.30, and 0.50 are 0.106, 0.102, 0.118, and 0.103, respectively.In this section, we compare the type I error rates of TATES, MANOVA, MultiPhen, mCPC, and GATE when multiple phenotypes are generated from direct association model. Table λ=20%,40%,60%,80%,100% of the phenotypes that are associated with the genetic variant are considered and the number of the associated traits is k. Here we report the results under the scenario where the first k phenotypes are associated with the genetic variant. Additional empirical power results for the cases that the associated phenotypes are randomly selected with equal probability are available in Additional file Next, we compare the power of the five tests when multiple phenotypes are simulated from Model 2. Under each scheme of the correlation structures, 5 levels of association: m=20, we find that MANOVA and MultiPhen usually have similar performances. For example, when MAF=0.30 and the correlation structure is S9, the powers of MANOVA and MultiPhen for λ=20%,40%,60%,80%, and 100% are , , , , and , respectively. When the correlations among different phenotypes are equal (S9 and S10) and the number of the associated phenotypes are relatively small (λ<60%), MANOVA and MultiPhen are two most powerful tests and they have similar power performances. In other situations, GATE performs better than MANOVA and MultiPhen. For example, under the configuration of S9 and MAF=0.15, the powers of the MANOVA, MultiPhen and GATE tests for λ=40% are 0.742,0.726 and 0.710, and their corresponding powers for λ=100% are 0.501, 0.475 and 0.798, respectively. When the correlations among different phenotypes are nonuniform (S11 and S12), the GATE performs better than the other four methods in most cases. When λ is relatively small, mCPC outperform slightly than GATE. However, when λ becomes large, the powers of the GATE exceed those of mCPC significantly. And in some cases, the power increase can reach 25%. For example, when the correlation structure is S11 and MAF=0.15, the powers of mCPC and GATE for λ=80% are 0.372 and 0.621, respectively. Moreover, when there exist strong correlations among phenotypes (S10), TATES suffers significant loss of power. For example, when MAF=0.05,n=1,500, and the rate of associated phenotypes with S10 is 60%, the powers of the TATES, MANOVA, MultiPhen, mCPC, and GATE are 0.192, 0.980, 0.981, 0.969, and 0.975, respectively. Hence, the proposed GATE is the most robust test against different levels of pleiotropy and strengthes of correlation. Figure m=20.Figure The mouse is an important model organism which can provide information on gene functions in mammals. Its use has been proved to be a powerful approach to understanding the genetic architecture of human disease and fundamental mammalian biology . To furtp-value of GATE. Under the nominal significance level of 0.05, the adjusted significance level for a single test is 0.05/250= 2×10−4 from the Bonferroni correction for multiplicity. On a whole, the number of identified SNPs that are associated with all the 52 phenotypes by GATE is more than those by the other four methods. Among 250 SNPs, there are 125 SNPs are detected by GATE, while 80, 114, 118, and 116 SNPs are detected by TATES, MANOVA, MultiPhen, and mCPC, respectively. Among the 125 SNPs, there are 7 SNPs that are only detected to be significantly associated with all the analyzed phenotypes by the proposed methods. The p-values of these SNPs for five methods are presented in Table p-value of GATE is always the smallest and smaller than the adjusted significance level 2×10−4. For example, for rs13483499, the p-value of GATE is 6.30×10−5 which is smaller than those of TATES (3.27×10−3), MANOVA (1.79×10−4), MultiPhen (3.34×10−4) and mCPC (1.63×10−4). Besides, some of the identified SNPs have been found to have implications on the analyzed phenotypes in the literature. For example, Valdar et al. 17], onlyp-value as the test statistic. Basically, TATES can be viewed as a function of p-values. The function is the combination of linear operator and minimum operator. For GATE, it utilizes the cumulative distribution function, log, quadratic-form and summation functions. These four functions are commonly used in constructing the test statistics in hypothesis testing, which is expected to integrate the information over a wide range of scenarios than other functions. The simulations show that GATE is more robust than TATES under most of the considered scenarios.GATE is also an extension of TATES who uses the minimum of weighted Covariates or confounding factors including the gender, age, environment factors can be of great importance in assessing the associations between the genetic variants and complex traits. Adjusting for covariates in genetic association studies have two motivations: one is correcting for the bias of the genetic effect estimates, and another is improving statistical power. For example, the hidden population structure can not be ignored in population genetic association studies and a failure to consider the population stratification might lead to many false positive findings. So it is a routine for researchers to correct for the population stratification in the genome wide association studies. Fortunately, the proposed GATE procedure can be directly applied to multiple traits association studies with covariates by adding the covariates in the association studies of the single PC and genetic variant.GATE is an efficient and robust procedure for association studies between multiple traits and a single SNP, which holds the key to understanding the genetic architecture of complex diseases. GATE is implemented based on the principal components (PCs) of multiple correlated traits. In contrast with the traditional PCA, GATE utilizes all obtained PCs instead of some selected PCs. This is because that low variances may possess important evidences of association and combines them in a group manner. Extensive numerical analyses show the superiority of the proposed approach over several existing methods in terms of statistical power.


Xiphophorus birchmanni, to investigate effects on behavior, size, growth, and survival. We then applied quantitative genetic analyses to determine (i) whether competition leads to phenotypic and/or genetic integration of behavior with life history and (ii) the potential for IGE to constrain life history evolution. Size, growth, and survival were reduced at high competition. Male dominance was repeatable and dominant individuals show higher growth and survival. Additive genetic contributions to phenotypic covariance were significant, with the G matrix largely recapitulating phenotypic relationships. Social dominance has a low but significant heritability and is strongly genetically correlated with size and growth. Assuming causal dependence of growth on dominance, hidden IGE will therefore reduce evolutionary potential.Competition for resources including food, physical space, and potential mates is a fundamental ecological process shaping variation in individual phenotype and fitness. The evolution of competitive ability, in particular social dominance, depends on genetic (co)variation among traits causal  or consequent  to competitive outcomes. If dominance is heritable, it will generate both direct and indirect genetic effects (IGE) on resource‐dependent traits. The latter are expected to impose evolutionary constraint because winners necessarily gain resources at the expense of losers. We varied competition in a population of sheepshead swordtails,  Xiphophorus birchmanni and explore the extent that genetic and environmental effects, notably including the degree of competition itself, shape the multivariate phenotype. Our goals are to assess the extent to which competition leads to phenotypic and/or genetic integration of behavioral and life‐history traits, and to evaluate the potential for constraining IGE on the latter.An individual's phenotype is determined by its genotype and the environment it experiences throughout life. Competition with conspecifics for resources  is one important environmental factor known to have large effects on phenotypic traits including growth  between traits subject to resource‐dependent trade‐offs Wilson . PositivHowever, genetic variance in dominance, or competitive ability, is also expected to generate indirect genetic effects (IGE). IGE occur when the phenotype of one individual is causally dependent on the genotype of another, and are inevitable in the case that dominance is genetically variable. IGEs can have important implications for predicting evolutionary responses, and in particular are predicted to dampen the response of resource‐dependent traits when they arise from competitive interactions. This is because selection on these trait(s) is expected to result in the correlated evolution of a more competitive social environment that offsets the expected phenotypic change Hadfield . ConsequXiphophorus birchmanni. Swordtails have been widely used in studies of social dominance  will be insufficient to predict selection responses. More specifically, G will give an upwardly biased expectation of the potential for adaptive evolution because it does not account for IGE that will necessarily arise if dominance is heritable.In our experiment we use density manipulations to increase the expected intensity of competition in the sheepshead swordtail. After testing for competition effects on mean growth, personality, and survival, we estimate the among‐individual and genetic covariance structures between traits related to social dominance. We manipulate competition by subjecting a captive bred generation of fish to contrasting low (L) and high (H) competition treatments in both early and later life. We hypothesise that high competition , particularly if experienced in early life, will reduce growth rates and negatively impact fitness components . Having shown direct effects of competition on phenotypic expression, we use a multivariate modeling approach to estimate the relationships among traits at the individual and additive genetic levels. We predict that personality differences will predict social dominance with bolder individuals tending to be dominant. We also expect that social dominance will positively predict growth and survival. If heritable variation for dominance is present, then similar correlation structure is expected at the (direct additive) genetic level. Importantly, if this is this case the (direct) genetic (co)variance structure among resource‐dependent traits  were caught from the Río Coacuilco, in the municipality of San Felipe Orizatlán, Hidalgo, Mexico, and imported to the United Kingdom. Between August 2010 and May 2011, these fish were crossed to produce 384 offspring comprising 61 families nested within a half‐sib structure . Families of ≤8 individuals were raised in one half of a brood tank . Full‐sib families of >8 individuals were divided equally across two partitions (in different tanks). Groups of six brood tanks , henceforth referred to as stacks shared a single recirculating water supply. Offspring were fed twice daily . At an average age of 16 weeks (range 12–27) and length 27 mm (20–35), fish were tagged below the dorsal fin with coloured elastomer and assigned to mixed family groups (n = 8). Each group was subject to one of two initial density treatments; low (L) density groups were housed in a full tank (37 × 37 × 22 cm), while high (H) density groups in a partitioned tank . Six stacks were set up sequentially (each comprising four L and four H groups) as sufficient fish (64 per stack) reached sufficient size to enable individual identification by use of elastomer tags. Sex ratio was not controlled as external sexing of juveniles is not possible. All groups received the same food ration  twice daily. After 28 weeks (subsequently referred to as part 1), density treatments were reversed for four randomly chosen groups within each stack. All groups were maintained for a further 22 weeks (part 2 of the study). Thus within each stack, four density regimes were experienced , with two groups per regime. Natural mortality over the course of the experiment resulted in some variation in group size  through time, although survival was high (368 of 384) over the first density treatment period .Briefly, one hundred adult fish  for use as a proxy for boldness.Behavioral data were collected on boldness and dominance. Boldness was determined using open field trials (OFT) described fully in Boulton et al. . IndividSocial dominance was assayed for males only using in‐tank observations (ITO). Behavior of each male in each group was recorded for five minutes, at up to five occasions during the experiment: two at the initial density treatment during part 1 of the study (weeks 18 and 25), and three at the final density treatments during part 2 of the study . Remaining males at the end of part 2 were pooled with others from their stack in a large tank (45 × 120 × 30 cm) containing previously unencountered stock females and observed on a minimum of five further occasions (consecutive days where possible). Within groups, focal males  were watched sequentially in a haphazard order by a recorder seated in front of the tanks in full view of the fish. Fish were accustomed to researcher presence and our judgement was that this did not impact behavior.X. birchmanni  and dominance are not equivalent, the former is often used to assert the latter  and live mass (WT) were measured at tagging (measure 1) and subsequent four‐weekly intervals. Up to 13 measures were made on each fish . We also recorded longevity as the age at death in days , and right‐censored to age at the end of part 2 of the experiment .F‐tests were used for inference on fixed effects in the univariate models, with sequential dropping of nonsignificant terms (but retaining main effects in the presence of significant interactions). Starting fixed effects were included to control for potentially confounding effects, and to test for density treatment effects. Inference on random effects was by likelihood ratio test (LRT). We follow Self and Liang  for tests of a single variance component. For comparing models differing in more parameters  we adopted the more conservative strategy of setting DF equal to the number of additional covariance components in the more complex model.Following graphical exploration of the raw data, density treatment, and genetic effects on behavior, size, and growth were tested using (univariate) linear‐mixed effect models, including pedigree‐based animal models  and late life density . Thus ELD:LLD defines a factor specifying the full regime . Since LLD treatment cannot influence phenotypic observations made during part 1 of the experiment, effects were fitted only to part‐specific data (denoted by subscript) where appropriate. part1ELD, part2LLD, and part2ELD:LLD were therefore fitted for activity, dominance score, SL, and WT.To test density treatment effects two‐level factors were defined corresponding to early life density (stack (a six‐level factor), sex , sex ratio (SR), and group size (GS). Group size (GS) and sex ratio (SR) experienced were defined for each individual (i) rather than each group. GS was defined as the geometric mean number of fish in i’s group, averaged across months up to and including the observation, and included to control for effects of mortality (reducing group size from the starting n = 8). SR was similarly defined as the geometric mean (across previous months) of the proportion of i's tank mates that are mature males ; order ; and observation type (within‐group during main experiment versus in larger tank after) for dominance score. For size traits (SL and WT) we included Measure (a factor with 13 levels) and Measure:Sex, allowing sex‐specific average growth patterns over the 13 months. A linear effect of absolute Age (zero‐centred) was also included to account for variation in age among fish entering the experiment.All starting models included fixed effects of SL and WT this model was extended to include 1st order (linear) random regressions on age (zero centred on the mean age of 294 days) for both additive and permanent environment effects  to phenotypic variance (VP) conditional on fixed effects using a simple repeat measures mixed model containing identity and (for SL and WT only), identity x age as random effects. Narrow sense heritabilities, h2 were estimated from the animal models as the ratio of VA to VP. We did not generate R or h2 estimates for Growth (as inferred from either SL or WT) because among‐individual and additive genetic variances in reaction norm slopes are estimated from the random regression models but residual variances are not.Random additive genetic and permanent environment effects were fitted using a standard repeat measures animal model  or one was also tested for using a univariate animal model and the same fixed predictors with an addition linear effect of age at Measure 1. We assumed a Gaussian error structure to obtain an estimate of heritability on the observed  data scale, but note that statistical inferences from this model parameters should be treated caution as a consequence.Right censored age of death data were analyzed using a proportional hazards regression model implemented by coxph in the R library survival  assuming Gaussian residuals. The 6 × 6 ID matrix was first estimated among the set of observed traits  and the two growth traits (SLGrowth and WTGrowth modeled using random regressions). Observed traits were scaled to standard deviation units to facilitate convergence and fixed effects included on each trait as determined from univariate analyses. A random effect of individual identity was included on all traits. This model was compared to one where ID was constrained to be a diagonal matrix  as an overall test of among‐trait covariance. The original model was then extended to include survival  as an additional response variable. Survival is observed once only, such that VI and VR are not separable and the latter was therefore fixed to zero. This partitions all variance in survival  to the extended ID structure. Observed survival was divided by the mean to convert to relative fitness. The covariance estimates between each trait and relative fitness in ID can then be interpreted as the ordinary selection differentials of quantitative genetic theory  so power was maximised by estimating G from a pooled‐sex multivariate animal model. We note that if G x Sex interactions occur, the resultant estimate of G can be viewed as an average of sex‐specific matrices. Trait‐specific fixed effects were included as before, with random additive genetic and permanent environment effects on all traits. For SL and WT first‐order random regressions of age were used. The full model fit was compared to one with a diagonal G matrix assumed as a global test for genetic covariance among traits and individual COVA estimates were scaled to genetic correlations (rG) to facilitate interpretation. Given a lack of VA in the univariate analysis, we did not expand this analysis to include (relative) survival.The above steps were repeated using male and female data separately to qualitatively check whether pooling sexes for multivariate genetic analysis was sensible, and determined whether selection was similar across sexes. SL and WT); 1235 observations of activity in OFT; and 1385 observations of male dominance score. Visual comparison suggests mean growth trajectories are similar across sexes ; 4175 age‐specific measures of size  = 0.635 (0.113), F1,2583.6 = 31.5, P < 0.001) and WT (part1ELD L coefficient (SE) = 0.050 (0.020), F1,2583.6 = 31.5, P < 0.001). Significant effects of part2LLD and (Part2LLD:ELD) on WT were also detected (Table part2WT yields effect size estimates (relative to expected mean for HH fish) of LL –0.038 (0.039) , LH –0.165 (0.039), HL 0.024 (0.033). Thus, the significant effect of LLD on WTpart 2 is driven largely by a negative impact of switching from L to H at the end of part 1.Effects of density treatment were not obvious from visual inspection of behavioral data (not shown), while univariate models confirmed no significant density treatment effects on sex effects are present on mean size (SL and WT) and average growth trajectory (manifest as Measure:Sex effects) and GS effects positively influenced SL. SR was not a significant predictor of any trait. Among‐stack differences were present for all traits except dominance score. For activity, significant trial effects were driven by greater mean activity at trials 2–4 relative to trial 1, while a negative effect of order was also present. These were already known from prior analysis of OFT data ) and male dominance score (R = 0.242 (0.032); Table 2activity = 0.093 (0.056), h2Dominance = 0.066 (0.052)) but VA estimates significant at P < 0.05 (Table SL and WT) and first‐order random regression models yielded significantly better fits than simple repeated measures models. Thus there is among‐individual variance in both size and growth  and RWT = 0.839 (0.011) at 294 days (the mean observed age in the dataset). At this age h2SL = 0.247 (0.106) while h2WT is lower (though not significantly so) at 0.144 (0.076).Univariate models provided statistical support for among‐individual variance , underpinned by genetic effects for all traits . Testing random effects in univariate models confirmed significant repeatability of Stack, SR, and GS .Two hundred fish 52.8%) remained alive at the end of part 2 measure 13). Observed survival to measure 13 was higher in females  with treatment‐specific rates (sexes combined) of LL = 59.1%, LH = 46.9%, HL = 55.9%, and HH = 49.5%. Testing of the treatment effects in the proportional hazards regression predicts that, conditional on other model effects, survival is lower in LH and HH and higher in HL relative to LL , and sex‐specific analyses . Length and weight are strongly positively correlated with each other in the pooled sex estimate of ID (Table growth traits. In other words random intercepts (size at mean age) and slopes (growth) were positively correlated in the random regression models. For males, dominance score is positively correlated with all measures of size and growth but, counter to predictions, is negatively associated with activity. The correlation between activity and dominance score is nominally significant based on Z score ≥ 1.96 (rI = –0.410 (0.104)).Comparison of unstructured and diagonal models revealed significant among‐trait covariance in dominance score is a male limited trait, and excluding relationships with survival (see below) the correlation structure in ID is qualitatively similar in the two sexes (Table S2) and G was thus estimated from a pooled analysis. A diagonal genetic matrix was supported over a null model , and the fully unstructured matrix was significantly better again . The first comparison corroborates the presence of genetic variance, the second provides evidence of among‐trait genetic covariance structure . Thus, for example, we find a strong negative genetic correlation between activity and dominance score (rG = –0.845 (0.361)). The former is also negatively genetically correlated with size and growth, for the latter the correlation structure is positive  phenotype‐fitness covariance . Selection differentials, S ). Thus large, fast growing, dominant  individuals showed higher survival, but activity does not predict fitness. In males, the phenotype‐fitness associations were similar (Table S2), and significant . However, single sex analysis provided no statistical support for significant selection on female traits .Finally, expanding the multivariate model used to estimate The aims of this study were to ascertain the effects of experimentally manipulated competition on growth, personality, and survival in the sheepshead swordtail, and to investigate among‐individual and genetic covariance structures between traits related to social dominance. We found evidence of reductions in size and growth at high competition as predicted, but no evidence of density effects on personality. Between traits there was significant among‐individual covariance structure, with personality (boldness) predicting social dominance, size, growth, and survival, though not all relationships matched our a priori predictions. Genetic covariance was also found between traits and we detected low, but statistically significant genetic variance in male dominance that has important consequences for the evolution of traits dependent on competitive outcomes. In what follows we first discuss the density effects on phenotype and patterns of phenotypic covariance before addressing these evolutionary implications in more detail.As predicted, we found evidence that density  influenced phenotypes and fitness. For example, size and growth rates were lower in early life at high density, consistent with the widespread reporting of density‐dependent growth rates in fishes  fitness are defining features of competition found ubiquitously across taxa. Thus the reduced growth and survival at higher density are consistent with our density treatment having manipulated the level of competition as intended. We previously reported a weak trend toward later and smaller maturation at high early life density in these fish, highlighting the fact that other aspects of life history are also impacted . Repeatability of activity, used here as a proxy for boldness, was known from prior analysis of this data . Although dominance score is observed for males only, this finding agrees with previous work on the same population where resource acquisition during dyadic interactions was used to assay dominance in both sexes , with bolder fish being more susceptible to avian predation  and male dominance score, as well as in size and growth (as measured by standard length and weight). The presence of genetic variance means that there is scope for adaptive evolution variance. Therefore, there is evidence for genetic variance in boldness , genetic correlations with other traits examined suggest that this could have important evolutionary consequences. This is because if the resources won by a focal individual in competition depend on its own genotype, it follows that they will also be influenced by the genotypes of competitors, giving rise to indirect genetic effects  ‐ COVA·VA(x)−1 ·COVA. This yields, for example, values of VA|Dom/VA of 38% and 42% respectively for size and growth as measured by live weight.A second important result, and one more germane to our study rationale, is that we found evidence for heritable variation in male vice versa. We have no way to assess causality from our data although there are a wealth of contest studies in Xiphophorus showing that body size is a key predictor of contest outcome . In fact, a second generation of breeding was initially planned to allow further investigation of IGEs, as well as testing for G x E across the density treatments. However, this was precluded by poor reproductive success of surviving fish both during and following the current experiment. We therefore note that more complex quantitative genetic models are thus unlikely to provide further insight.A second caveat to note is that we did not explicitly model or estimate IGEs on traits presumed consequent to competitive outcomes . Ideally this would have allowed us to verify the expected consequences of heritable dominance for downstream traits (as discussed above). Although in principle it is relatively straightforward to estimate IGEs within a variance partitioning animal model framework  and dominance, the sign of this relationship was negative counter to our predictions. Thus, fish that were bolder actually tended to be less dominant . This is compatible with the premise that personality is an important determinant of social dominance, but the direction of the relationship is something of an anomaly when set against the wider context of empirical studies of boldness. We also found evidence of genetic (co)variance underpinning observed phenotypic variation. Thus there is genetic integration between boldness, dominance, size, and growth, and these aspects of phenotype will not evolve independently under selection. Of particular significance is the conclusion that dominance is both heritable and genetically correlated with size and growth. Provided growth depends on the outcome of competition, heritable dominance leads to an expectation of indirect genetic effects that will act as constraints on selection responses.In summary, this study sought to investigate the direct effects of social competition on phenotype and fitness, test for among‐individual variation in competitive ability  and investigate the multivariate genetic architecture linking traits putatively causal and consequent to dominance. We found that higher levels of competition caused reductions in growth and survival but had no effect on average behavior. Associate Editor: T. FlattHandling Editor: M. NoorTable S1. Estimated coefficients from univariate animal models of each trait.Table S2. Estimates among‐individual (ID) matrices for males (MID) and females (FID) separately.Figure S1. Observed size  and relative growth  by age for female (red), male (green) and fish of unknown sex (blue).Click here for additional data file.


Brassica napus L.) is one of the most important oil crops in the world. However, the yield and quality of rapeseed were largely decreased by clubroot (Plasmodiophora brassicae Woronin). Therefore, it is of great importance for screening more resistant germplasms or genes and improving the resistance to P. brassicae in rapeseed breeding. In this study, a massive resistant identification for a natural global population was conducted in two environments with race/pathotype 4 of P. brassicae which was the most predominant in China, and a wide range of phenotypic variation was found in the population. In addition, a genome-wide association study of 472 accessions for clubroot resistance (CR) was performed with 60K Brassica Infinium SNP arrays for the first time. In total, nine QTLs were detected, seven of which were novel through integrative analysis. Furthermore, additive effects in genetic control of CR in rapeseed among the above loci were found. By bioinformatic analyses, the candidate genes of these loci were predicted, which indicated that TIR-NBS gene family might play an important role in CR. It is believable that the results presented in our study could provide valuable information for understanding the genetic mechanism and molecular regulation of CR.Rapeseed ( Brassica napus L., AACC, 2n = 38) is one of the most important and widely cultivated oil crops, which derived from the hybridization of two basic diploid species in U-triangle, Brassica rapa  and Brassica oleracea , several clubroot-resistant cultivars including Gelria R, Siloga, Debra, and Milan White were found , CRQTL-YC, CRQTL-GN_1, and CRQTL-GN_2 were identified in B. oleracea  dry soil. The soil humidity which reached to the extent of kneading with clumps and touching with scatter was advisable, and then was sealed to ferment for 2 days in dark. The hole tray was filled with nutritional soil and then irrigated enough water. About 3 g fermentative soil per hole was placed, and sowed two seeds in the fermentative soil. The artificial inoculation followed a randomized complete block design with three replications, and inoculated 24 seeds for each replication. The plants grew under a 16 h L/8 h D photoperiod at an average temperature of 20–25°C.All of the 472 accessions were planted in field infected seriously by n1+2n2+3n3)/(NT × 3)] × 100, where n1 to n3 were the number of plants with different disease severity of 1–3 scale and NT represented the total number of identified plants, respectively. An incidence rate (IR) was calculated with a formula IR (%) = [(n1+n2+n3)/NT] × 100. The broad-sense heritability was calculated as 2h= σ2g/(σ2g+σ2ge/n+σ2e/nr). Where σ2g is the genetic variance, σ2ge is the interaction variance of the genotype with environment, σ2e is the error variance, n is the number of environments and r is the number of replications. The estimates of σ2g, σ2ge, and σ2e were obtained from the analysis of variance (ANOVA) procedure in SPASS 20.0 and the frequency distribution of phenotype was carried out using Excel 2010. The multiple comparisons of phenotypic average value from different origins were tested with package of SAS 9.2 at a significance level of P < 0.05. Also, the difference analysis of genotypic effects with containing different No. of favorable SNPs was performed online1.At flowering time of IF trials and 50 days after inoculation in greenhouse, plants were all taken out with roots and the disease severity was assessed using a 0–3 scale as follows, 0 = no clubs, 1 = a few small clubs, 2 = moderate clubbing, and 3 = severe clubbing . A diseaBrassica 60K Illumina Infinium® HD Assay SNP arrays by Emei Tongde, Co. (Beijing) according to the manufacturer’s protocol2. The quality control process of SNP array genotyping data was the same to our previous study (B. napus (Darmor-bzh) reference genome by E-value threshold of 1E–15, and blast matches to multiple loci with the same top E-value were not considered to be mapped successfully. Totally, 19,945 SNPs were assigned to B. napus chromosomes.The young leaves from the representative plants of all accessions were collected, and extracted their genomic DNAs using a modified cetyltrimethyl ammonium bromide CTAB,  method. us study . The LD us study . The prius study  were use10(p) for each SNP from marker-trait associations was compared with the expected distribution in a quantile-quantile plot (Q-Q plot). The significance of associations between SNPs and traits was based on a threshold P < 5.01 × 10-5 . Meanwhile, to avoid ignoring the effects of minor loci, the threshold P was lowered to 2/N = 1.00 × 10-4. Q-Q plots and Manhattan plots were drawn using the R package qqman3.To evaluate the effects of population structure , the trait-SNP association analysis was performed using TASSEL 4.0 with six models: (i) GLM, controlling for nothing, (ii) Q model, controlling for Q, (iii) PCA model, controlling for PC, (iv) K model, controlling for K, (v) Q+K model, controlling for both Q and K, (vi) P+K model, controlling for both PC and K. The GLM, Q, K, and PCA models were performed using a general linear model with the following equation: y = Xα+ e. The Q+K and P+K models were performed using a mixed linear model with the following equation: y = Xα+Kμ+e. In above equations, y represented phenotype, X represented genotype, α was a vector containing fixed effects, K was the relative kinship matrix, μ was a vector of random additive genetic effects, and e was the unobserved vector of random residual. The distribution of observed -logB. napus reference genome (Darmor-bzh) with a threshold of 1E–10. By that, we could ensure the homoeologous fragments’ physical positions of QTLs/genes reported in previous studies on B. napus genome.To compare the QTLs detected in this study with previous studies on CR, the sequences of QTLs/genes which have been reported in previous studies were collected and performed BlastN or e-PCR on the r2 > 0.4 with the most significant SNP. To ascertain regions that potential candidate genes of each QTL identified in this study maybe located, we considered the flanking markers that outside and adjacent to the LD block were considered as the candidate region’s boundary. To further identify the candidate genes and the participated pathways of the identified loci, the gene function annotation and enrichment analysis were carried out. All B. napus genes which located in the candidate regions were searched against the NCBI non-redundant (Nr) protein database using BlastP with an E-value ≤ 1E–05. The gene ontology (GO) terms of the B. napus genes were annotated by merging the Blast2GO and InterPro annotation results . In IF environment, DI ranged from 9.26 to 75.00, with an average value of 31.39 ± 5.92 (± SE); IR ranged from 18.18 to 100% with an average value of 48.11 ± 8.10% (± SE). The broad-sense heritability (2h) of DI and IR in IF were 64.8 and 55.0%, respectively (Table 1). In GH environment, DI was from 36.11 to 100.00, with an average value of 84.77 ± 9.67 (±SE); IR was from 58.26 to 100%, with an average value of 96.49 ± 7.64% (± SE). Moreover, the 2h of DI and IR in GH were 78.2 and 69.4%, respectively (Table 1). The results thus indicated that the stability of CR is high, especially in a control environment. The correlation analysis of these two indicators between the two environments showed that the correlation reached to highly significant level, but the value of which (0.24 and 0.19) is too low , different country (China and abroad), ecotype , and breeding eras in China  on CR in the population, the accessions were classified by the above categories, respectively  from P+K model was the closest to the expected distribution for all four traits, which led to a low level of false-positive signals  and two mixed liner models  for IF-DI (DI in IF), IF-IR (IR in IF), GH-DI (DI in GH), and GH-IR (IR in GH), respectively. As the Q-Q plot showed, the distribution of observed -logP < 5.01 × 10-5) and potential association SNP  were introduced in the study. If a confidence interval of a locus included one or more sSNPs, we called it significant association locus. Similarly, if a confidence interval of a locus did not have sSNP, and only included two or more pSNPs, we called it potential association locus. In total, seven significant association loci and two potential association loci were identified with these two indicators .To avoid ignoring the micro effective locus, significant associated SNP  was identified, with a peak SNP (highest significant) Bn-scaff_17521_1-p419499 which contributed to 4.81% of phenotypic variance. One potential association locus on BnC09 (named MCR-C9) was detected, with a peak SNP Bn-scaff_15576_1-p660538 which explained 4.21% of phenotypic variance (Table 2). For IF-IR, one significant association locus on BnA04 (named MCR-A04) was mapped, with a peak SNP Bn-A04-p16156157 which contributed 4.72% to phenotypic variance. It was noteworthy that MCR-C3 in IF-DI was also detected in IF-IR. For GH-DI, one significant association locus SCR-C6 was detected on BnC06 with a peak SNP Bn-scaff_16064_1-p26703 which explained 5.26% of phenotypic variance. Another significant association locus SCR-A10a was identified on BnA10 with a peak SNP Bn-A10-p16087066 which accounted for 4.71% of phenotypic variance. For GH-IR, three significant association loci were identified in total. Among that two loci were both on BnC04 (named SCR-C4a and SCR-C4b), which explained 5.72 and 4.54% of phenotypic variance, respectively. The rest one named SCR-A10b was on BnA10, with a peak SNP Bn-A10-p3966740 which explained 4.87% of phenotypic variance. In addition, one potential association locus SCR-C3 was mapped on BnC03 with a peak SNP Bn-scaff_18559_1-p166394 which accounted for 4.12% of phenotypic variance (Table 2).For IF-DI, one significant association locus on BnC03 . The genotype and phenotypic variance (2R) of each QTL was substituted by that of each peak marker of the corresponding QTL. On this basis, 472 accessions were grouped into three classes which contained 1, 2, and 3 favorable alleles in IF, respectively . Because of too less accession contained in one group if all accessions were grouped by containing 1–6 favorable alleles in GH, all these six loci were merged and divided into three groups which contained 1–2, 3–4, and 5–6 favorable alleles to statistical analyze the DI in GH . The results indicated that the more favorable alleles  had, the lower DI (more resistance to P. brassicae) was demonstrated . In IF, the DI of the accessions held three resistant alleles can be reduced by 20.62% compared with that of holding one resistant allele. Similarly, the DI of the accessions held 5–6 resistant alleles can be reduced by 10.56% compared with that of only holding 1–2 resistant alleles in GH. The above results revealed that the CR was mainly controlled by additive effect. Therefore, the CR for B. napus can be improved by polymerization of all identified resistant alleles.To understand the effects of allelic variations on CR in rapeseed, the combined effects of DI of different QTLs were analyzed in IF and GH, respectively  to search their homoeologous regions in B. napus . In B. rapa, two homoeologous regions of CRa were identified on BnA03  and BnC07 , respectively . Three homoeologous regions of Crr1 were found on BnA08 , BnC03 , and BnC07 , respectively . One homoeologous region of CRb was identified on BnA03 . Two homoeologous regions of Rcr1 were found on BnA03  and BnC07  in B. napus. Homoeologous regions of other QTLs in B. rapa were not identified, because of their limited marker information or the physical positions of forward and reverse markers linked with the QTLs were not on the same chromosome in B. napus genome. Because of the limited reports on CR studies in B. oleracea, only three QTLs CRQTL-YC, CRQTL-GN_1, and CRQTL-GN_2 were searched. Two homoeologous regions of CRQTL-YC were identified on BnA03 and BnC03 respectively. Also, two homoeologous regions of CRQTL-GN_1 were detected on BnA02  and BnC02 . And one homoeologous region of CRQTL-GN_2 was mapped on BnC03 .To compare the GWAS results in this study with the previous reported CR QTLs/genes, the sequences of CR genes and markers of QTLs related to CR were collected in B. napus according to the providing markers information in previous reports. Markers sequences of five QTLs reported on previous study in B. napus were searched and the physical position of PbBn-01:07-1 and PbBn-korp-4 were confirmed successfully. The physical positions of these two QTLs, BnA03  and BnC06 , were proved to be accordance with the previous reports. However, the physical positions of other three QTLs were not accordance with the previous studies. For example, the QTL PbBn-a-1 located on BnA08 was mapped on BnC03, and the PbBn-01:07-3 located on BnC03 was mapped on BnA03 in this study .Moreover, it was attempted to ensure the QTL physical position in Figure 3). The QTL SCR-C3 identified in GH was not located on the confidence intervals of one homoeolog of Crr1, but was close to it. We considered it as an old CR locus. The same situation was on SCR-C6, which closed to the physical position of PbBn-korp-4. The other four QTLs identified in this study were novel. So the GWAS had stronger ability to detect new loci on a given trait.It was found that the three QTLs identified in IF in our study were all novel, in which regions there were no homoeologous regions of CR QTLs reported in previous studies (SCR-C6 (10 TIR-NBS genes) and MCR-C9  were participated in the function of ADP binding. The results indicated that the TIR-NBS gene family may associated with CR.To predict the candidate genes of CR QTLs identified in this study, the candidate region  for each QTL was confirmed. There were 542 predicted genes in all nine candidate regions, 471 genes of which had functional annotations in total. To gain insights into the functionality of above genes, we performed GO enrichment analysis using Blast2GO . The resP. brassicae in B. rapa were identified in B. napus genome . For example, there were five SNPs involved in the MCR-C3 LD block, the candidate gene region was 21.72–21.92 Mb (194.6 Kb) in BnC03, where 44 genes were included, and four of which were the homoeologous genes of four DEGs identified in B. rapa . Also, seven SNPs were involved in the LD block for SCR-C3, which the candidate region was 57.86–58.10 Mb (339.3 Kb) in BnC03, and 16 genes were predicted in this region. Only one homoeologous gene BnaC03g68270D of DEGs corresponding in B. rapa was searched . Similarly, 9, 1, 2, 1, 6, 2, and 4 candidate genes obtained from the B. rapa homoeologous genes were identified in the candidate regions of MCR-A4, SCR-A10a, SCR-A10b, SCR-C4a, SCR-C4b, SCR-C6, and MCR-C9, respectively . But the same QTL between two environments were not detected, which may because that we performed the evaluation at the seedling stage in GH while at the harvest stage in IF. There were maybe different genetic mechanisms between vegetable stage and adult stage in rapeseed. It also gave us an ideal that it can improve the rapeseed CR both in vegetable stage and adult stage by polymerization of all identified CR loci/genes by MAS. The identified CR QTLs were showed additive effect , which provided theoretical basis for putting forward the resistant breeding strategy of polymerization of multiple loci, similarly. The SNPs detected in this study could be transformed to PCR markers  can be used for CR improvement through distant hybridization between these species . Therefore, we will have two approaches to improve the rapeseed CR. Firstly, all QTLs/genes which come from B. napus and closely relate species can be polymerized by MAS or interspecific hybridization  had more potential on CR improvement . Researchers also found that there were more resistant loci in C-genome than A-genome of B. napus in the study of QTL mapping of resistance to Sclerotinia sclerotiorum  related to CR were collected in these three species . Most of the reported CR QTLs/genes were mapped on B. napus genome through bioinformatic analyses. The integration results illustrated the CR hotspots also existed in B. napus, which were the regions of BnA03 (21–23 Mb), BnC03 (1–2 Mb), and BnC07 . It was interesting that the top of BnA03 was homoeologous with the top of BnC03, and the bottom of BnA03 was homoeologous with the bottom of BnC07 . NBS-LRR proteins could recognize the specialized pathogen effectors of ETI . Therefore, TIR-NBS-LRR gene family is most likely to play an important role in the process of the clubroot disease, which needs to deeply study in Brassica. Besides that, plant hormones, mainly salicylic acid (SA), jasmonic acid (JA), auxins and cytokinins, also played a role in compatible interactions between Arabidopsis and P. brassicae (Brassica (CRb were used for RNA-Seq in B. rapa, not two accessions or varieties. Therefore, these 30 candidate genes identified through the data from RNA-Seq in B. rapa just only provided clues for candidate genes prediction and understanding the molecular regulation of CR in B. napus. However, it needs to carry out more experiments to obtained more evidence to identify or evidence these candidate genes. For example, the RNA-Seq analysis can be carried out by using of some CR and susceptible lines from the natural population in this study.So far, there were more than 70 m plants . Most ofR family . It was rassicae . It alsoBrassica . It was XW conceived the study. LL and XW designed the experiments. FL performed the genotyping of the association panel. LL, BC, and KX organized the implementation of field trials. LL, YL, FZ, HL, QH, XX, and TZ participated in the phenotyping of CR. LL and FL analyzed the data. LL and XW wrote the paper. YL, FL, JH, and XW participated in the modification of the manuscript. All the authors have read and approved the publication of the manuscript.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.


We also examine the crucial issue of the impact of linkage disequilibrium (LD) on effect sizes and parameter estimates, both analytically and in simulations. We apply this approach to meta-analysis test statistics from two large GWAS, one for Crohn’s disease (CD) and the other for schizophrenia (SZ). A scale mixture of two normals distribution provides an excellent fit to the SZ nonparametric replication effect size estimates. While capturing the general behavior of the data, this mixture model underestimates the tails of the CD effect size distribution. We discuss the implications of pervasive small but replicating effects in CD and SZ on genomic control and power. Finally, we conclude that, despite having very similar estimates of variance explained by genotyped SNPs, CD and SZ have a broadly dissimilar genetic architecture, due to differing mean effect size and proportion of non-null loci.Characterizing the distribution of effects from genome-wide genotyping data is crucial for understanding important aspects of the genetic architecture of complex traits, such as number or proportion of non-null loci, average proportion of phenotypic variance explained per non-null effect, power for discovery, and polygenic risk prediction. To this end, previous work has used effect-size models based on various distributions, including the normal and normal mixture distributions, among others. In this paper we propose a scale mixture of two normals model for effect size distributions of genome-wide association study (GWAS) test statistics. Test statistics corresponding to null associations are modeled as random draws from a normal distribution with zero mean; test statistics corresponding to non-null associations are also modeled as normal with zero mean, but with larger variance. The model is fit via minimizing discrepancies between the parametric mixture model and resampling-based nonparametric estimates of replication effect sizes and variances. We describe in detail the implications of this model for estimation of the non-null proportion, the probability of replication in  de novo samples, the local false discovery rate, power for detecting non-null loci, and proportion of variance explained from additive effects. Here, we fit this model by minimizing discrepancies with nonparametric estimates from a resampling-based algorithm. We examine the effects of linkage disequilibrium (LD) on effect sizes and parameter estimates, both analytically and in simulations. We validate this approach using meta-analysis test statistics (“z-scores”) from two large GWAS, one for Crohn’s disease and the other for schizophrenia. We demonstrate that for these studies a scale mixture of two normal distributions generally fits empirical replication effect sizes well, providing an excellent fit for the schizophrenia effect sizes but underestimating the tails of the distribution for Crohn’s disease.We describe in detail the implications of a particular mixture model  for effect size distributions from genome-wide genotyping data. Parameters from this model can be used for estimation of the non-null proportion, the probability of replication in  A SNP is often declared significant if the p-value of its test statistic surpasses a Bonferroni-inspired threshold of 5 × 10−8. Note, within this typical GWAS hypothesis testing framework, the effect size for a given SNP computed from massively univariate test statistics is a weighted combination of effects from all SNPs that it is in linkage disequilibrium (LD) with , and the power to detect susceptibility loci for a given study sample size. These parameters are also closely related to the proportion of the phenotypic variance explained by the additive effects of common variants and upper limits on the accuracy of polygenic risk scores  and replication z-scores Zr,i[k], for i = 1, …, N. The Zi[k] are then binned into intervals h on the interval [−c, c) for fixed c > 0, where m = 1, …, M. Let k, to obtain smoothed estimatesn and Z are the effective sample size and the z-score of the discovery sample, and nr and Zr are the effective sample size and the z-score of the replication sample. E{δ∣Z = z} and E{δ2∣Z = z} and hence δ given training the z-scores.More specifically, to obtain nonparametric estimates of θ by utilizing Eqs (θ) into quadratic estimating equations that solve for parameter estimates minimizing the differences between the empirical and model-based replication expectations and variances. For the scale mixture of normals model, Eqs  over the effective sample size of the discovery sample (nρ). The advantage of varying ρ is the ability to observe the effects of changing sample size on the effect size distribution and finite-sample replication rates.For a given model zing Eqs  and 17)E(θ) and del, Eqs  and 14)E(θ) and ρ = 0.5 for the Crohn’s disease data, and we vary ρ between 0.3 and 0.5 in the schizophrenia data. Monte Carlo simulations in the ρ = 0.50). For all analyses, the bin width h was chosen such that there were M = 201 bins equally-spaced bins spanning the range of z-scores. The values for c are chosen to span the entire range of observed z-scores in the given analysis. For the Crohn’s disease example c = 17, for schizophrenia example c = 12, and in the simulations c = 10. Note, the mean allele frequency pi could be incorporated by binning with respect to effect allele frequency in addition to binning by discovery z-score; however, in practice this appears to have little effect on estimates.In the real applications below, we keep θ using a simplex algorithm to produce estimated values z-scores, and the local false discovery rate.z-scores, at https://sites.google.com/site/covmodfdr/.The resampling and fitting algorithm is available in R and Matlab scripts, along with code to generate synthetic sub-study GWAS S1 TextS1 Text Sections 1–2 provide statistical derivations of the effects of linkage disqequilibrium (LD) on massively univariate regression estimates from GWAS. Section 3 outlines how posterior effect size estimates from the mixture model could be used in polygenic risk score estimation. Section 4 provides details and results from Simulation Studies, as described in the main text. Section 5 gives the authorship list for the Psychiatric Genetics Consortium Schizophrenia Working Group.(PDF)Click here for additional data file.S1 Figz-scores for schizophrenia (SCZ) SNPs from nonparametric estimates (blue curves) and from mixture model based fit (green curves) for varying proportions of discovery and replication sample sizes. Top row: Left to right, 10%, 20%, and 30% in training sample. Middle row: Left to right, 40%, 50%, and 60% in training sample. Bottom row: Left to right, 70%, 80%, and 90% in training sample.Mean replication (EPS)Click here for additional data file.S2 FigExpected posterior effect sizes for normal mixture model for different parameter values. Note, when (TIF)Click here for additional data file.S3 Figπ1 = {0.01, 0.02, 0.05}, σ1 = {0.0, 0.001, 0.01} and σ2 = {0.01, 0.05, 0.1}.Simulated empirical(red) and predicted conditional mean were shown in the first column. The second column shows simulated empirical(red) and predicted conditional variance. The third column shows boxplots of parameter estimates across all 50 iterations of the simulation study. The parameter values were set by expanding a grid with (EPS)Click here for additional data file.S4 Figπ1 = {0.01, 0.02, 0.05}, σ1 = {0.0, 0.001, 0.01} and σ2 = {0.01, 0.05, 0.1}.Simulated empirical(red) and predicted conditional mean were shown in the first column. The second column shows simulated empirical(red) and predicted conditional variance. The third column shows boxplots of parameter estimates across all 50 iterations of the simulation study. The parameter values were set by expanding a grid with (EPS)Click here for additional data file.S5 Figπ1 = {0.01, 0.02, 0.05}, σ1 = {0.0, 0.001, 0.01} and σ2 = {0.01, 0.05, 0.1}.Simulated empirical(red) and predicted conditional mean were shown in the first column. The second column shows simulated empirical(red) and predicted conditional variance. The third column shows boxplots of parameter estimates across all 50 iterations of the simulation study. The parameter values were set by expanding a grid with (EPS)Click here for additional data file.S6 Figπ1 = {0.01, 0.02, 0.05}, σ1 = {0.0, 0.001, 0.01} and σ2 = {0.01, 0.05, 0.1}.Simulated empirical(red) and predicted conditional mean were shown in the first column. The second column shows simulated empirical(red) and predicted conditional variance. The third column shows boxplots of parameter estimates across all 50 iterations of the simulation study. The parameter values were set by expanding a grid with (EPS)Click here for additional data file.S7 Figπ1 = {0.01, 0.02, 0.05}, σ1 = {0.0, 0.001, 0.01} and σ2 = {0.01, 0.05, 0.1}.Simulated empirical(red) and predicted conditional mean were shown in the first column. The second column shows simulated empirical(red) and predicted conditional variance. The third column shows boxplots of parameter estimates across all 50 iterations of the simulation study. The parameter values were set by expanding a grid with (EPS)Click here for additional data file.