Eucalyptus grandis individuals for association with 3 terpene traits, 1,8-cineole, γ-terpinene, and p-cymene. A multilocus mixed model analysis identified 21 SNP markers for 1,8-cineole on chromosomes 2, 4, 6, 7, 8, 9, 10, and 11, that individually explained 3.0%–8.4% and jointly 42.7% of the phenotypic variation. Association analysis of γ-terpinene found 32 significant SNP markers on chromosomes 1, 2, 4, 5, 6, 9, and 11, explaining 3.4–15.5% and jointly 54.5% of phenotypic variation. For p-cymene, 28 significant SNP markers were identified on chromosomes 1, 2, 3, 5, 6, 7, 10, and 11, explaining 3.4–16.1% of the phenotypic variation and jointly 46.9%. Our results show that variation underlying the 3 terpene traits is influenced by a few minor loci in combination with a few major effect loci, suggesting an oligogenic nature of the traits.Terpenes are an important group of plant specialized metabolites influencing, amongst other functions, defence mechanisms against pests. We used a genome-wide association study to identify single nucleotide polymorphism (SNP) markers and putative candidate genes for terpene traits. We tested 15,387 informative SNP markers derived from genotyping 416  Eucalyptus trees is rich in a group of plant specialized metabolites known as terpenes (5), mono- (C10), sesqui- (C15), di- (C20), tri- (C30), and tetra-terpenes (C40) (5 precursors isopentenyl diphosphate (IPP) and its isomer dimethylallyl diphosphate (DMAPP). In plants, the mevalonate (MVA) pathway, in the cytosol, produces IPP from acetyl-CoA, some of which is converted to DMAPP by the isopentenyl diphosphate isomerase (IPPI) enzyme (10H16), FPP, formed by 1 DMPP and 2 IPP’s, is the precursor of sesquiterpenes (C15H24) and finally, GGPP, formed by 1 DMPP and 3 IPP’s, is the precursor of diterpenes (C20H32). The 3 molecules GPP, FPP, and GGPP are the substrates used by 1 type of enzymes known as terpene synthases, which catalyze the formation of all terpenes known  compared to any other plant sequenced to date. The TPS family has been split into 7 groups/subfamilies according to phylogenetic analysis namely TPS-a, -b, -c, -d, - e/f, -g, and -h  has been causing severe losses on susceptible Eucalyptus species or populations within species, resulting in complete failure of some commercially grown clones  and susceptible E. camaldulensis x E. grandis (GC) clones , salicylic acid (SA), abscisic acid (ABA), and jasmonic acid (JA) regulate terpene biosynthesis pathways  markers and genomic regions on chromosomes 3, 7, and 8 that contained putative candidate genes for resistance against L. invasa  study is to identify and understand the genetic architecture underlying the phenotypic variation  from individuals of Melaleuca alternifolia. Results showed strong positive correlations between some traits, for example, γ-terpinene and p-cymene had a correlation of 0.71. p-cymene followed by hydroxylation of p-cymene. p-cymene. A hierarchical clustering dendrogram of terpene traits in E. grandis showed that γ-terpinene and p-cymene clustered close together , Mtunzini, and Nyalazi in KwaZulu Natal, South Africa as shown in E. grandis trials were established in August 2012. In October 2013, when the trees were 14-months old, trees were inspected for L. invasa infestation. Symptoms were scored visually using the following scale: 0—not infested, 1—infested showing evidence of oviposition but no gall development, 2—infested with galls on leaves, mid-ribs, or petioles, and 3—stunting and lethal gall formation. Each tree was thus categorized as either 0, 1, 2, or 3. Environmental and trial design information of the study population, the number of families, phenotyped and genotyped individuals are shown in −1). The leaf discs were shipped for terpene identification and profiling to the Australian National University (ANU).The genetic material used in this study consisted of 416 p-cymene used in this study were obtained from E. grandis trees using gas chromatography on an Agilent 6890 GC and detected with an Agilent 5973 Mass Spectrometer at ANU as described by −1, followed by a ramp to 250 °C at 5 °C·min−1, where it remained for 4 min. The peaks were identified by comparing mass spectra to reference spectra in the National Institute of Standards and Technology library  and verified major peaks with authentic standards. One sample from each individual was reanalyzed on the final day of chromatography to confirm that retention times remained stable. The area under each peak was measured manually with the help of MSD Chemstation Data Analysis  and converted to a relative concentration by comparison with the internal standard, tetradecane. The Pearson’s correlation coefficients (P ≤ 0.05) was calculated in RStudio v4.0.3 to assess the correlation between each pair of traits based on their concentrations (mg g−1 DW) using a method similar to Three terpene traits 1,8-cineole, γ-terpinene, and E. grandis leaf samples using the Machery-Nagel NucleoSpin 96 Plant II kit . The integrity of all DNA preparations was validated by 1% agarose TBE gel electrophoresis and spectrophotometry using a NanoDrop Spectrophotometer . All samples were genotyped using the EUChip60K SNP chip by GeneSeek . Intensity data were first assessed using GenomeStudio V2011.1  to recluster genotypic classes as described by R) value of the heterozygote cluster > 0.2, and mean normalized theta of the heterozygote cluster between 0.2 and 0.8. SNPs that did not pass these cutoff criteria were removed from further analyses. SNP genotypes were assessed to retain markers ascertained in at least 80% of samples with minor allele frequency (MAF) more than 0.01. Two alternative SNP datasets were also generated by retaining only SNPs with MAF ≥ 0.01 and MAF ≥ 0.05, respectively.DNA was isolated from 416 r2) using Expectation-Maximization algorithm in SNP & Variation Suite (SVS) v8.4.1 . Average LD was computed for the 11 chromosomes and determined as the x-axis intercept of the fitted nonlinear regression of pairwise LD at r2 = 0.2 against physical distance (in kb) in SVS v8.4.1. LD pruning using a threshold of r2 = 0.5 was applied on the significant SNP markers by computing the correlations of allele frequencies of significant SNP markers in SVS v.8.4.1. Phenotypic variance was estimated as Linkage disequilibrium (LD) was evaluated by computing the squared allele frequency correlation coefficient , β is the vector of the fixed effect coefficients (intercept site), Z = Z is an incidence matrix for the random effects of individual trees, µ is the vector of random effect coefficients , ε is the vector of residual effect coefficients. The variance of the random effects was assumed to be P-values obtained using the MLMM approach. A genome-wide adjusted false discovery rate (FDR) set at P ≤ 0.05 to correct for multiple testing using the Benjamin–Hochberg procedure and a more stringent genome-wide level of Bonferroni procedure was implemented to control for type I error at P ≤ 0.05.GWAS was performed using a stepwise Efficient Mixed-Model Association eXpedited (EMMAX) regression  and treatment (infested and uninfested) as independent variables to determine the effect of gene expression in response to L. invasa oviposition. Data (available at plantgenie.org) was obtained from a previous transcriptomics study comparing host responses in a resistant and susceptible background  for RNA-Seq analysis. The reads were mapped to the E. grandis genome, assembled into transcripts, and the fragments per kilobase of transcripts per million mapped reads (FPKM) were calculated. Significant differential expression between uninfested and infested samples of each genotype as well as between the uninfested samples and infested samples (i.e. Egr-uninfested vs. GC-uninfested and Egr-infested vs. GC-infested) was calculated  (−1 DW (average 2.3 mg g−1 DW) (p-cymene ranged from 0 to 16.8 mg g−1 DW (average 1.0 mg g−1 DW)  for the terpenes at family level across the locations is shown in Supplementary Table 1. There was a moderate positive correlation  between γ-terpinene and p-cymene. Weak positive correlations were observed between p-cymene and 1,8-cineole  and 1,8-cineole and γ-terpinene  (Supplementary Table 2).Within the 416  g−1 DW) , γ-terpi g−1 DW) , and p-c g−1 DW) . The conE. grandis trees with 64,639 SNP markers produced a total of 15,387 and 13,770 informative markers in at least 80.0% of the population for MAF ≥ 0.01 and MAF ≥ 0.05, respectively. The resulting SNP density was 1 SNP marker per 39 kb. With LD decaying to r2 = 0.2 within 3.35 kb on average (E. grandis genome (Supplementary Table 3).Two alternative SNP datasets with different MAF thresholds were used to investigate whether removing lower frequency SNPs had an impact on GWAS results. The genotyping of 416  average , the 15,E. grandis half-sib population. The first 2 principal components explained 5.88% and 5.39% of the genetic variance, respectively  were used to assess the number and magnitude of observed associations between genotyped SNPs and trait under study, compared to the association statistics expected under the null hypothesis of no association. Observed association statistics, −log10P adjusted values calculated from them, are ranked in order from smallest to largest on the y-axis and plotted against the distribution that would be expected under the null hypothesis of no association on the x-axis. Since the underlying assumption in GWAS is that the vast majority of assayed SNPs are not associated with the trait, strong deviations from the null suggest a very highly associated locus.The Quantile-Quantile plots for the 3 terpene traits 1,8-cineole , γ-terpinene , and 10  = 3.4 threshold (black dashed line) after multiple testing, the 2 alternative marker datasets MAF ≥ 0.01 and MAF ≥ 0.05 for the 3 traits 1,8-cineole, γ-terpinene, and p-cymene did not show any difference in results of the significant SNP marker associated with the trait, therefore we report on the data set MAF ≥ 0.01. The association analysis of 1,8-cineole found 21 SNP markers on chromosomes 2, 4, 6, 7, 8, 9, 10, and 11, and explained between 3.0% and 8.4% of the phenotypic variation (P ≤ 0.05 (red dashed line) only 1 SNP marker was associated with the trait. Association analysis of γ-terpinene found 32 significant SNP markers .With a genome-wide adjusted FDR correction –logariation . Jointly markers . These S markers . The SNPE. grandis reference genome, putative candidate genes of the significant SNP markers were identified within the 100 kb (50 kb upstream and 50 kb downstream) of the significant SNP associations with the 3 terpene traits 1,8-cineole, γ-terpinene, and p-cymene. The number of genes identified were 166, 270, and 216 (Supplementary Tables 4–6), respectively. The study reported on genes we suggest could play a role in terpene biosynthesis. Eucgr.F00752, a candidate gene for p-cymene encodes geranylgeranyl diphosphate synthase (ggpps) tagged by the SNP marker EuBR06s10733103 and explains 3.66% of the phenotypic variation. Eucgr.H01393, a candidate gene for 1,8-cineole encodes isopentenyl pyrophosphate isomerase 2 (ippi) tagged by the SNP marker EuBR08s17426789 explains 8.48% of the phenotypic variation. A cluster of UDP-glycosyltransferase genes  were identified for γ-terpinene and 1,8-cineole traits, the SNP markers EuBR02s63883704 and EuBR08s15122213 tagging these gene were found on chromosome 2 and 8 explaining 3.76% and 3.97% of the phenotypic variation, respectively, were identified. Our study identified ubiquitin protein ligase genes Eucgr.A01473, Eucgr.A01934, Eucgr.A01937, Eucgr.B03986, Eucgr.G03167, leucine-rich repeat protein kinase Eucgr.B02179, Eucgr.G02950, Eucgr.J02704, Eucgr.J02705 as well as an expansin gene Eucgr.A01806. The SNP makers tagging these genes explained between 3.47% and 4.88% of the phenotypic variation. Genes related to phytohormone were identified, for example, Eucgr.G01437 and Eucgr.K02605 encoding an abscisic acid-responsive  family protein. The SNP markers EuBR07s24890821 and EuBR11s33763867 tagging these 2 genes explained 4.33% and 4.88% of the phenotypic variation. Eucgr.G01954 a candidate gene for p-cymene encodes jasmonate-ZIM domain protein (JAZ). The SNP marker EuBR07s35438867 tagging this gene was found on chromosome 7 explaining 4.06% of the phenotypic variation. The genes Eucgr.G01952 for p-cymene and 2 genes Eucgr.K00658 and Eucgr.K00660 for γ-terpinene each encode auxin family protein. The SNP markers tagging these genes explained 4.06% and 3.91% of the phenotypic variation, respectively.Based on the detailed annotation for the E. grandis (Egr) and susceptible E. camaldulensis × E. grandis (GC) from an independent experiment  (E. grandis (Supplementary Tables 4–6). SNP markers explaining more than 5% of the phenotypic variation were observed for the 3 traits 1,8-cineole, γ-terpinene, and p-cymene with the strongest association of the SNP marker EuBR01s17197369, explaining 15.46% and 16.08% of the phenotypic variation for γ-terpinene and p-cymene, respectively (Supplementary Tables 5 and 6). The results suggest that terpene traits are influenced by a few large-effect loci in combination with minor effect loci .The SNP markers explained variation that was generally within the range of SNP markers for other quantitative traits in p-cymene on position 33,057 Mb is approximately 14 kb away from the SNP marker in chromosome 1 on position 33,044 Mb tagging a terpene synthase gene EgranTPS063. Our study did not find any terpene synthase gene within 100 kb of the SNP marker EuBR01s28423985. Although, the results suggest that these SNP markers could be associated with the same genetic factor, this is impossible to test since different species, that is, E. grandis versus E. polybractea, different genotype platforms, whole-genome sequencing versus EUChip60K SNP chip were used in the 2 studies, and the window region for candidate gene search was different between the 2 studies.A previous GWAS by E. grandis genome. The use of previously generated gene expression data in our study (The identification of the candidate genes in a window region (100 kb) around significant SNP markers was based on the observed LD decay in this study and the known gene density in the geranylgeranyl diphosphate synthase (ggpps), Eucgr.H01235 located near SNP EuBR06s10733103. The enzyme GGPPS uses IPP and DMAPP as substrate and produces GGPP, the substrate for di-terpene synthases and for tetraterpene biosynthesis. One study found a significant association between GGPPS and beta-carotene (a tetraterpene) in Sorghum bicolor , Eucgr.H01393, was located near the significant marker EuBR08s17426789. The enzyme IPPI isomerizes IPP to its structural isomer DMAPP, thereby changing the ratio of IPP: DMAPP, which determines which prenyldiphosphate synthase  is most active since they all use different ratios of IPP: DMAPP , a member of the CYP707A gene family encoding a protein with ABA 8′-hydroxylase activity and Eucgr.K02605  encoding abscisic acid  family protein are involved in ABA synthesis. ABA was suggested to initiate defence in host plants  gene was upregulated in the JA pathway upon wounding . JA is considered the major regulator of plant resistance to insect oviposition . Oates eA. thaliana ortholog AT1G12820 (AFB3), Eucgr.K00658 with putative A. thaliana ortholog AT2G04850, and Eucgr.K00660 with putative A. thaliana ortholog AT5G47530 encode for auxin-responsive family proteins. L. invasa may cause the observed changes in gene expression, there is no concrete evidence to support this (Piptadenia gonoacantha (Fabaceae) played a role in influencing the size, shape, different direction of cell expansion and these changes may influence gene expression.Auxins regulate several plant processes including growth and development as well as defence against pest and pathogen attack . In otheort this . HoweverE. grandis. We showed that variation in the 3 terpene traits 1,8-cineole, γ-terpinene, and p-cymene may be influenced by at least a few loci of minor and a few of major effect, suggesting an at least oligogenic nature of the traits . The results from the study provide a list of SNP markers and putative candidate genes, for example, IPPI and GGPPS that were previously reported to be involved downstream of the MVA and MEP pathways in terpene production , the associations that we detected adds to our understanding of the genetic architecture that underlies terpene traits in https://figshare.com/s/92751d332fd45a1b58f1. Data of the results obtained from the GWAS analysis was used to generate the figures Manhattan plots, LD decay, and PCA plot. Gene expression data were obtained from a previous study by https://www.ncbi.nlm.nih.gov/bioproject/PRJNA305347. Resistant infested , resistant uninfested , susceptible infested , and susceptible uninfested . The FPKM is available at https://figshare.com/s/92751d332fd45a1b58f1.All datasets used in GWAS analysis are available at G3 online.Supplemental material is available at


Through simulations, we show that our method improves upon state-of-the-art association mapping and enrichment approaches across a wide range of genetic architectures. We then further illustrate the benefits of BANNs by analyzing real GWA data assayed in approximately 2,000 heterogenous stock of mice from the Wellcome Trust Centre for Human Genetics and approximately 7,000 individuals from the Framingham Heart Study. Lastly, using a random subset of individuals of European ancestry from the UK Biobank, we show that BANNs is able to replicate known associations in high and low-density lipoprotein cholesterol content.In this article, we present Biologically Annotated Neural Networks (BANNs), a nonlinear probabilistic framework for association mapping in genome-wide association (GWA) studies. BANNs are feedforward models with partially connected architectures that are based on biological annotations. This setup yields a fully interpretable neural network where the input layer encodes SNP-level effects, and the hidden layer models the aggregated effects among SNP-sets. We treat the weights and connections of the network as random variables with prior distributions that reflect how genetic effects manifest at different genomic scales. The BANNs software uses variational inference to provide posterior summaries which allow researchers to simultaneously perform ( i) mapping with SNPs and (ii) enrichment analyses with SNP-sets . We illustrate the benefits of our method over state-of-the-art approaches using extensive simulations. We also demonstrate the ability of BANNs to recover novel and previously discovered genomic associations using quantitative traits from the Wellcome Trust Centre for Human Genetics, the Framingham Heart Study, and the UK Biobank.A common goal in genome-wide association (GWA) studies is to characterize the relationship between genotypic and phenotypic variation. Linear models are widely used tools in GWA analyses, in part, because they provide significance measures which detail how individual single nucleotide polymorphisms (SNPs) are statistically associated with a trait or disease of interest. However, traditional linear regression largely ignores non-additive genetic variation, and the univariate SNP-level mapping approach has been shown to be underpowered and challenging to interpret for certain trait architectures. While nonlinear methods such as neural networks are well known to account for complex data structures, these same algorithms have also been criticized as “black box” since they do not naturally carry out statistical hypothesis testing like classic linear models. This limitation has prevented nonlinear regression approaches from being used for association mapping tasks in GWA applications. Here, we present Biologically Annotated Neural Networks (BANNs): a flexible class of feedforward models with partially connected architectures that are based on biological annotations. The BANN framework uses approximate Bayesian inference to provide interpretable probabilistic summaries which can be used for simultaneous ( P-values or posterior inclusion probabilities (PIPs) which lend statistical evidence on how important each variant is for explaining the overall genetic architecture of a trait. However, this univariate SNP-level mapping approach can be underpowered for “polygenic” traits which are generated by many mutations of small effect  and PIP(g) ≡ Pr. We use this information to prioritize statistically associated SNPs and SNP-sets that significantly contribute to the broad-sense heritability of the trait of interest. With biologically annotated units and the ability to perform statistical inference on explicitly defined parameters, our model presents a fully interpretable extension of neural networks to GWA applications. Additional details and derivations of the BANNs framework can be found in By using these point mass mixture distributions in Eqs  and 3),,3), we agression , 56 to tG = 2,816 SNP-sets to be included in the simulation study.In order to assess the performance of models under the BANNs framework, we simulated complex traits under multiple genetic architectures using real genotype data on chromosome 1 from ten thousand randomly sampled individuals of European ancestry in the UK Biobank  see Mat. Next, wH2 = 0.2 and 0.6); the proportion of broad-sense heritability that is being contributed by additive effects versus pairwise cis-interaction effects (ρ = 1 and 0.5); and the percentage of enriched SNP-sets that influence the trait . We use the parameter ρ to assess the neural network’s robustness in the presence of non-additive genetic effects between causal SNPs. To this end, ρ = 1 represents the limiting case where the variation of a trait is driven by solely additive effects. For ρ = 0.5, the additive and pairwise interaction effects are assumed to equally contribute to the phenotypic variance.After the annotation step, we assume a linear model to generate quantitative traits while varying the following parameters: broad-sense heritability . For the latter, GWA summary statistics are computed by fitting a single-SNP univariate linear model (via ordinary least squares) after quality control to obtain: effect size estimates, standard errors, and bination .P = 0.05/36518 = 1.37 × 10−6 at the SNP-level and P = 0.05/2816 = 1.78 × 10−5 at the SNP-set level, respectively); while, Bayesian methods are evaluated according to the median probability model (PIPs and posterior enrichment probability ≥ 0.5) , where rj =  is a vector of correlation coefficients between the j-th SNP and all other SNPs in the study; and a list of G-predefined SNP-sets (B) A partially connected Bayesian neural network is constructed based on the annotated SNP groups. In the first hidden layer, only SNPs within the boundary of a gene are connected to the same node. Similarly, SNPs within the same intergenic region between genes are connected to the same node. Completing this specification for all SNPs gives the hidden layer the natural interpretation of being the “SNP-set” layer. (C) The hierarchical nature of the network is represented as nonlinear regression model. The corresponding weights in both the SNP (θ) and SNP-set (w) layers are treated as random variables with biologically motivated sparse prior distributions. Posterior inclusion probabilities PIP(j) ≡ Pr and PIP(g) ≡ Pr summarize associations at the SNP and SNP-set level, respectively. The BANN-SS framework uses the same variational inference procedure that is used when we have access to individual-level data. Browser . (B) A p(PDF)Click here for additional data file.S2 FigH2 = 0.2 with only contributions from additive effects . We show power versus false positive rate for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the selected enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S3 FigH2 = 0.2 with only contributions from additive effects . In these simulations, traits were generated while using the top ten principal components (PCs) of the genotype matrix as covariates. We show power versus false positive rate for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the selected enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S4 FigH2 = 0.6 with only contributions from additive effects . In these simulations, traits were generated while using the top ten principal components (PCs) of the genotype matrix as covariates. We show power versus false positive rate for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the selected enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S5 FigH2 = 0.2 with equal contributions from additive effects and epistatic interactions . We show power versus false positive rate for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S6 FigH2 = 0.6 with equal contributions from additive effects and epistatic interactions . We show power versus false positive rate for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S7 FigH2 = 0.2 with equal contributions from additive effects and epistatic interactions . In these simulations, traits were generated while using the top ten principal components (PCs) of the genotype matrix as covariates. We show power versus false positive rate for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S8 FigH2 = 0.6 with equal contributions from additive effects and epistatic interactions . In these simulations, traits were generated while using the top ten principal components (PCs) of the genotype matrix as covariates. We show power versus false positive rate for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S9 FigH2 = 0.2 with only contributions from additive effects . We show precision versus recall for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the selected enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S10 FigH2 = 0.6 with only contributions from additive effects . We show precision versus recall for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the selected enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S11 FigH2 = 0.2 with only contributions from additive effects . In these simulations, traits were generated while using the top ten principal components (PCs) of the genotype matrix as covariates. We show precision versus recall for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the selected enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S12 FigH2 = 0.6 with only contributions from additive effects . In these simulations, traits were generated while using the top ten principal components (PCs) of the genotype matrix as covariates. We show precision versus recall for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the selected enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S13 FigH2 = 0.2 with equal contributions from additive effects and epistatic interactions . We show precision versus recall for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the selected enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S14 FigH2 = 0.6 with equal contributions from additive effects and epistatic interactions . We show precision versus recall for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the selected enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S15 FigH2 = 0.2 with equal contributions from additive effects and epistatic interactions . In these simulations, traits were generated while using the top ten principal components (PCs) of the genotype matrix as covariates. We show precision versus recall for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the selected enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S16 FigH2 = 0.2 with equal contributions from additive effects and epistatic interactions . In these simulations, traits were generated while using the top ten principal components (PCs) of the genotype matrix as covariates. We show precision versus recall for two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We then set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the selected enriched SNP-sets, respectively. To derive results, the full genotype matrix and phenotypic vector are given to the BANNs model and all competing methods that require individual-level data. For the BANN-SS model and other competing methods that take GWA summary statistics, we compute standard GWA SNP-level effect sizes and P-values (estimated using ordinary least squares).  Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10).  Competing SNP-set mapping approaches include: RSS [Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Note th(PDF)Click here for additional data file.S17 FigH2 = 0.2 with only contributions from additive effects set . We consider two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be ∼ 1% and ∼ 10% of all SNPs located within the enriched SNP-sets, respectively. Results are shown comparing the posterior inclusion probabilities (PIPs) derived by the BANNs model fit with individual-level data on the x-axis and  SuSiE [ RSS [ℓ = 3000). The blue horizontal and vertical dashed lines are marked at the “median probability criterion”  [Quantitative traits are simulated to have broad-sense heritability of C) SuSiE  and  RSS  on the yhan 0.5) . True po(PDF)Click here for additional data file.S18 FigH2 = 0.2 with only contributions from additive effects set . We consider two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. In these simulations, traits were generated while also using the top ten principal components (PCs) of the genotype matrix as covariates. Results are shown comparing the posterior inclusion probabilities (PIPs) derived by the BANNs model fit with individual-level data on the x-axis and  SuSiE [ RSS [ℓ = 3000). The blue horizontal and vertical dashed lines are marked at the “median probability criterion”  [Quantitative traits are simulated to have broad-sense heritability of C) SuSiE  and  RSS  on the yhan 0.5) . True po(PDF)Click here for additional data file.S19 FigH2 = 0.6 with only contributions from additive effects set . We consider two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. In these simulations, traits were generated while also using the top ten principal components (PCs) of the genotype matrix as covariates. Results are shown comparing the posterior inclusion probabilities (PIPs) derived by the BANNs model fit with individual-level data on the x-axis and  SuSiE [ RSS [ℓ = 3000). The blue horizontal and vertical dashed lines are marked at the “median probability criterion”  [Quantitative traits are simulated to have broad-sense heritability of C) SuSiE  and  RSS  on the yhan 0.5) . True po(PDF)Click here for additional data file.S20 FigH2 = 0.2 with equal contributions from additive effects and epistatic interactions . We consider two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. Results are shown comparing the posterior inclusion probabilities (PIPs) derived by the BANNs model fit with individual-level data on the x-axis and  SuSiE [ RSS [ℓ = 3000). The blue horizontal and vertical dashed lines are marked at the “median probability criterion”  [Quantitative traits are simulated to have broad-sense heritability of C) SuSiE  and  RSS  on the yhan 0.5) . True po(PDF)Click here for additional data file.S21 FigH2 = 0.6 with equal contributions from additive effects and epistatic interactions . We consider two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. Results are shown comparing the posterior inclusion probabilities (PIPs) derived by the BANNs model fit with individual-level data on the x-axis and  SuSiE [ RSS [ℓ = 3000). The blue horizontal and vertical dashed lines are marked at the “median probability criterion”  [Quantitative traits are simulated to have broad-sense heritability of C) SuSiE  and  RSS  on the yhan 0.5) . True po(PDF)Click here for additional data file.S22 FigH2 = 0.2 with equal contributions from additive effects and epistatic interactions . We consider two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. In these simulations, traits were generated while also using the top ten principal components (PCs) of the genotype matrix as covariates. Results are shown comparing the posterior inclusion probabilities (PIPs) derived by the BANNs model fit with individual-level data on the x-axis and  SuSiE [ RSS [ℓ = 3000). The blue horizontal and vertical dashed lines are marked at the “median probability criterion”  [Quantitative traits are simulated to have broad-sense heritability of C) SuSiE  and  RSS  on the yhan 0.5) . True po(PDF)Click here for additional data file.S23 FigH2 = 0.6 with equal contributions from additive effects and epistatic interactions . We consider two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. In these simulations, traits were generated while also using the top ten principal components (PCs) of the genotype matrix as covariates. Results are shown comparing the posterior inclusion probabilities (PIPs) derived by the BANNs model fit with individual-level data on the x-axis and  SuSiE [ RSS [ℓ = 3000). The blue horizontal and vertical dashed lines are marked at the “median probability criterion”  [Quantitative traits are simulated to have broad-sense heritability of C) SuSiE  and  RSS  on the yhan 0.5) . True po(PDF)Click here for additional data file.S24 Fig shows the comparison of BANNs SNP layer with SuSiE [ shows the comparison of BANNs SNP-set layer with RSS [This experiment follows largely from previous work , 65. Herth SuSiE , and Click here for additional data file.S25 Figπθ and πw as additional parameters in the variational EM algorithm. In the normal BANNs setup, we initialize L different models with varying priors for inclusion probabilities specified over a grid πθ = 1/J and πw = 1/G as an analogy to the “single causal variant” assumption frequently used in fine mapping [πθ/1 − πθ = ∑j ∑kαjk/∑j ∑k(1 − αjk), and πw/1 − πw = ∑gαg/∑g(1 − αg). Results here are shown using simulations with the self-identified “white British” ancestry cohort from the UK Biobank on synthetic traits that have broad-sense heritability H2 = 0.6 with sparse genetic architecture. Each plot combines results from 100 simulated replicates Click here for additional data file.S26 FigH2 = 0.2 with different levels of contributions from additive effects and epistatic interactions. We consider two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. Panels  show heritability estimates on simulations with genetic data from individuals who self-identify as being of “white British” ancestry in the UK Biobank; while, panels  show heritability estimates on simulations with genetic data from individuals who more broadly identify as being of European ancestry. True heritability values are shown as the dashed grey horizontal lines. The root mean square error (RMSE) between the BANNs model estimates of the PVE and the true values are also provided.In this work, we define PVE as the total proportion of phenotypic variance that is explained by sparse genetic effects (both additive and non-additive) . Here, q(PDF)Click here for additional data file.S27 FigH2 = 0.6 with different levels of contributions from additive effects and epistatic interactions. We consider two different trait architectures:  sparse where only 1% of SNP-sets are enriched for the trait; and  polygenic where 10% of SNP-sets are enriched. Panels  show heritability estimates on simulations with genetic data from individuals who self-identify as being of “white British” ancestry in the UK Biobank; while, panels  show heritability estimates on simulations with genetic data from individuals who more broadly identify as being of European ancestry. True heritability values are shown as the dashed grey horizontal lines. The root mean square error (RMSE) between the BANNs model estimates of the PVE and the true values are also provided.In this work, we define PVE as the total proportion of phenotypic variance that is explained by sparse genetic effects (both additive and non-additive) . Here, q(PDF)Click here for additional data file.S28 Fig(A) body mass index (BMI) and body weight, (B) percentage of CD8+ cells and mean corpuscular hemoglobin (MCH), and (C) high-density and low-density lipoprotein  cholesterol. Posterior inclusion probabilities (PIP) for the input layer weights are derived from the BANNs model fit on individual-level data and are plotted for each SNP against their genomic positions. Chromosomes are shown in alternating colors for clarity. The black dashed line is marked at 0.5 and represents the “median probability model (MPM)” threshold [i) 0.09 and 0.08 for BMI, (ii) 0.39 and 0.40 for body weight, (iii) 0.51 and 0.48 for percentage of CD8+ cells, (iv) 0.34 and 0.32 for MCH, (v) 0.34 and 0.28 for HDL, and (vi) 0.15 and 0.15 for LDL.Traits are grouped based on their category and include: hreshold . Here, w(PDF)Click here for additional data file.S29 Figg) ≥ 0.5  the database of Genotypes and Phenotypes (dbGaP) and  the GWAS Catalog (2019). Nearly all enriched categories are related with  HDL and  LDL, respectively. Note that in LDL, the BANNs framework identified the gene APOB as having a high PIP = 0.976. There have been hypotheses connecting LDL to cognitive traits [APOB has been shown to be related to cerebrospinal fluid and memory [(D) are also relevant.Here, SNP-set annotations are based on gene boundaries defined by the NCBI’s RefSeq database in the UCSC Genome Browser . Unannothreshold ). We takhreshold , 91 to ie traits , 141, and memory –144. The(PDF)Click here for additional data file.S30 Figg) ≥ 0.5  the database of Genotypes and Phenotypes (dbGaP) and  the GWAS Catalog (2019). Nearly all enriched categories are related with  HDL and  LDL, respectively.Here, SNP-set annotations are based on gene boundaries defined by the NCBI’s RefSeq database in the UCSC Genome Browser . Unannothreshold ). We takhreshold , 91 to i(PDF)Click here for additional data file.S31 FigGene1-Gene2 in the legend. Gene set enrichment analyses for these SNP-sets can be found in S31 Fig. Stars (★) denote SNPs and SNP-sets that replicate findings from our analyses of HDL and LDL in the Framingham Heart Study (see Posterior inclusion probabilities (PIP) for the neural network weights are derived from the BANNs model fit on individual-level data and are plotted for each SNP against their genomic positions. Chromosomes are shown in alternating colors for clarity. The black dashed line is marked at 0.5 and represents the “median probability model” threshold . SNPs wi(PDF)Click here for additional data file.S32 Figg) ≥ 0.5  the database of Genotypes and Phenotypes (dbGaP) and  the GWAS Catalog (2019). Nearly all enriched categories are related with  HDL and  LDL, respectively. Note that in LDL, the BANNs framework again identifies the gene APOB as having a high PIP (replicating the finding in the Framingham Heart Study). There have been hypotheses connecting LDL to cognitive traits [APOB has been shown to be related to cerebrospinal fluid and memory [(B) are also relevant . We takhreshold , 91 to ie traits , 141, and memory –144. The(PDF)Click here for additional data file.S33 Figg) ≥ 0.5  the database of Genotypes and Phenotypes (dbGaP) and  the GWAS Catalog (2019). Note that for panel (A), BANNs did not find many enriched SNP-sets with PIPs meeting the “median probability model” threshold and so we used a lower SNP-set threshold (PIP ≥ 0.1) to enable Enrichr to find associated dbGaP categories.Here, SNP-set annotations are based on gene boundaries defined by the NCBI’s RefSeq database in the UCSC Genome Browser . Unannothreshold ). We takhreshold , 91 to i(PDF)Click here for additional data file.S1 TableH2 = 0.2 with only contributions from additive effects set . We consider two different trait architectures: sparse where only 1% of SNP-sets are enriched for the trait; and polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. (Top) Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10). (Bottom) Competing SNP-set mapping approaches include: RSS [P = 0.05/36518 = 1.37 × 10−6 at the SNP-level and P = 0.05/2816 = 1.78 × 10−5 at the SNP-set level, respectively). All results are based on 100 replicates and standard deviations of the estimates across runs are given in the parentheses. Approaches with the greatest power are bolded in purple, while methods with the lowest FDR is bolded in blue.Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Resultsude: RSS . Results(PDF)Click here for additional data file.S2 TableH2 = 0.6 with only contributions from additive effects set . We consider two different trait architectures: sparse where only 1% of SNP-sets are enriched for the trait; and polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. (Top) Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10). (Bottom) Competing SNP-set mapping approaches include: RSS [P = 0.05/36518 = 1.37 × 10−6 at the SNP-level and P = 0.05/2816 = 1.78 × 10−5 at the SNP-set level, respectively). All results are based on 100 replicates and standard deviations of the estimates across runs are given in the parentheses. Approaches with the greatest power are bolded in purple, while methods with the lowest FDR is bolded in blue.Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Resultsude: RSS . Results(PDF)Click here for additional data file.S3 TableH2 = 0.2 with only contributions from additive effects set . We consider two different trait architectures: sparse where only 1% of SNP-sets are enriched for the trait; and polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. (Top) Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10). (Bottom) Competing SNP-set mapping approaches include: RSS [P = 0.05/36518 = 1.37 × 10−6 at the SNP-level and P = 0.05/2816 = 1.78 × 10−5 at the SNP-set level, respectively). All results are based on 100 replicates and standard deviations of the estimates across runs are given in the parentheses. Approaches with the greatest power are bolded in purple, while methods with the lowest FDR is bolded in blue.Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Resultsude: RSS . Results(PDF)Click here for additional data file.S4 TableH2 = 0.6 with only contributions from additive effects set . We consider two different trait architectures: sparse where only 1% of SNP-sets are enriched for the trait; and polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. (Top) Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10). (Bottom) Competing SNP-set mapping approaches include: RSS [P = 0.05/36518 = 1.37 × 10−6 at the SNP-level and P = 0.05/2816 = 1.78 × 10−5 at the SNP-set level, respectively). All results are based on 100 replicates and standard deviations of the estimates across runs are given in the parentheses. Approaches with the greatest power are bolded in purple, while methods with the lowest FDR is bolded in blue.Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Resultsude: RSS . Results(PDF)Click here for additional data file.S5 TableH2 = 0.2 with contributions from both additive and epistatic effects set . We consider two different trait architectures: sparse where only 1% of SNP-sets are enriched for the trait; and polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. (Top) Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10). (Bottom) Competing SNP-set mapping approaches include: RSS [P = 0.05/36518 = 1.37 × 10−6 at the SNP-level and P = 0.05/2816 = 1.78 × 10−5 at the SNP-set level, respectively). All results are based on 100 replicates and standard deviations of the estimates across runs are given in the parentheses. Approaches with the greatest power are bolded in purple, while methods with the lowest FDR is bolded in blue.Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Resultsude: RSS . Results(PDF)Click here for additional data file.S6 TableH2 = 0.6 with contributions from both additive and epistatic effects set . We consider two different trait architectures: sparse where only 1% of SNP-sets are enriched for the trait; and polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. (Top) Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10). (Bottom) Competing SNP-set mapping approaches include: RSS [P = 0.05/36518 = 1.37 × 10−6 at the SNP-level and P = 0.05/2816 = 1.78 × 10−5 at the SNP-set level, respectively). All results are based on 100 replicates and standard deviations of the estimates across runs are given in the parentheses. Approaches with the greatest power are bolded in purple, while methods with the lowest FDR is bolded in blue.Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Resultsude: RSS . Results(PDF)Click here for additional data file.S7 TableH2 = 0.2 with contributions from both additive and epistatic effects set . We consider two different trait architectures: sparse where only 1% of SNP-sets are enriched for the trait; and polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. (Top) Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10). (Bottom) Competing SNP-set mapping approaches include: RSS [P = 0.05/36518 = 1.37 × 10−6 at the SNP-level and P = 0.05/2816 = 1.78 × 10−5 at the SNP-set level, respectively). All results are based on 100 replicates and standard deviations of the estimates across runs are given in the parentheses. Approaches with the greatest power are bolded in purple, while methods with the lowest FDR is bolded in blue.Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Resultsude: RSS . Results(PDF)Click here for additional data file.S8 TableH2 = 0.6 with contributions from both additive and epistatic effects set . We consider two different trait architectures: sparse where only 1% of SNP-sets are enriched for the trait; and polygenic where 10% of SNP-sets are enriched. We set the number of causal SNPs with non-zero effects to be 1% and 10% of all SNPs located within the enriched SNP-sets, respectively. (Top) Competing SNP-level mapping approaches include: CAVIAR [ℓ which fixes the maximum number of causal SNPs in the model. We display results when this input number is high (ℓ = 3000) and when this input number is low (ℓ = 10). (Bottom) Competing SNP-set mapping approaches include: RSS [P = 0.05/36518 = 1.37 × 10−6 at the SNP-level and P = 0.05/2816 = 1.78 × 10−5 at the SNP-set level, respectively). All results are based on 100 replicates and standard deviations of the estimates across runs are given in the parentheses. Approaches with the greatest power are bolded in purple, while methods with the lowest FDR is bolded in blue.Here, quantitative traits are simulated to have broad-sense heritability of : CAVIAR , SuSiE [: CAVIAR , and FIN: CAVIAR . The sofude: RSS , PEGASUSude: RSS , GBJ [27ude: RSS , GSEA [4ude: RSS , and MAGude: RSS . Resultsude: RSS . Results(PDF)Click here for additional data file.S9 Tableℓ which fixes the maximum number of causal SNPs in the model. We display results when this input parameter is high (ℓ = 3000) and when this input parameter is low (ℓ = 10). Note that we implemented BANNs using the Python 3 version of the software, and the timing for its variational algorithm includes inference on both SNPs and SNP-sets. CAVIAR and FINEMAP are set up to work with GWA summary statistics, so their inputs (and timing) are the same irrespective of the sample size.Methods compared include: BANNs, CAVIAR , SuSiE , and FIN(PDF)Click here for additional data file.S10 Tablen = 10,000 individuals. Each table entry represents the average computation time (in seconds) it takes each approach to analyze a dataset of the size indicated. Run times were measured on an Intel i5-8259U CPU with base frequency of 2.30GHz, turbo frequency of 3.80GHz, and memory 16GB 2133 MHz LPDDR3. Here, we used 4 cores for parallelization when applicable. Note that PEGASUS, GBJ, SKAT, and MAGMA are score-based methods and, thus, are expected to take the least amount of time to run. Both the BANNs framework and RSS are regression-based methods. The increased computational burden of these approaches results from its need to do (approximate) Bayesian posterior inference; however, the sparse and partially connected architecture of the BANNs model allows it to scale more favorably for larger dimensional datasets. Note that we implemented BANNs using the Python 3 version of the software, and the timing for its variational algorithm includes inference on both SNPs and SNP-sets.Methods compared include: BANNs, RSS , PEGASUS(PDF)Click here for additional data file.S11 TableJ ≈ 10,000 SNPs and G = 1, 925 SNP-sets from N = 1,814 mice—with specific numbers varying slightly depending on the quality control procedure for each phenotype Click here for additional data file.S12 TableJ ≈ 10,000 SNPs and G = 1, 925 SNP-sets from N = 1,814 mice—with specific numbers varying slightly depending on the quality control procedure for each phenotype Click here for additional data file.S13 TableJ ≈ 10,000 SNPs and G = 1, 925 SNP-sets from N = 1,814 mice—with specific numbers varying slightly depending on the quality control procedure for each phenotype Click here for additional data file.S14 TableJ ≈ 10,000 SNPs and G = 1, 925 SNP-sets from N = 1,814 mice—with specific numbers varying slightly depending on the quality control procedure for each phenotype Click here for additional data file.S15 TableJ ≈ 10,000 SNPs and G = 1, 925 SNP-sets from N = 1,814 mice—with specific numbers varying slightly depending on the quality control procedure for each phenotype Click here for additional data file.S16 TableJ ≈ 10,000 SNPs and G = 1, 925 SNP-sets from N = 1,814 mice—with specific numbers varying slightly depending on the quality control procedure for each phenotype Click here for additional data file.S17 TableGene1-Gene2 in the table. Posterior inclusion probabilities (PIP) for the input and hidden layer weights are derived by fitting the BANNs model on individual-level data. A SNP-set is considered enriched if it has a PIP(g) ≥ 0.5 . We also report the corresponding SNP and SNP-set level results after running SuSiE [Here, SNP-set annotations are based on gene boundaries defined by the NCBI’s RefSeq database in the UCSC Genome Browser . Unannothreshold ). We repng SuSiE  and RSS ng SuSiE  on theseng SuSiE .(PDF)Click here for additional data file.S18 TableJ = 394,174 SNPs and G = 18, 364 SNP-sets from N = 6,950 people. Here, SNP-set annotations are based on gene boundaries defined by the NCBI’s RefSeq database in the UCSC Genome Browser [We analyze  Browser   the(ZIP)Click here for additional data file.S19 TableJ = 394,174 SNPs and G = 18, 364 SNP-sets from N = 6,950 people. Here, SNP-set annotations are based on gene boundaries defined by the NCBI’s RefSeq database in the UCSC Genome Browser [We analyze  Browser   the(ZIP)Click here for additional data file.S20 Tablea) we use the UCSC gene boundary definitions directly, and (b) we augment the gene boundaries by adding SNPs within a ±500 kilobase (kb) buffer to account for possible regulatory elements.The first page compares the overlap of significant SNPs and SNP-sets found by each method according to the “median probability model” threshold  Click here for additional data file.S21 TableJ = 394,174 SNPs and G = 18, 364 SNP-sets used in the Framingham Heart Study analyses. Here, SNP-set annotations are based on gene boundaries defined by the NCBI’s RefSeq database in the UCSC Genome Browser [We analyze the same  Browser . Unannot Browser   the(ZIP)Click here for additional data file.S22 TableJ = 394,174 SNPs and G = 18, 364 SNP-sets used in the Framingham Heart Study analyses. Here, SNP-set annotations are based on gene boundaries defined by the NCBI’s RefSeq database in the UCSC Genome Browser [We analyze the same  Browser . Unannot Browser   the(ZIP)Click here for additional data file.S1 TextSpecifically, we give description of the variational inference algorithm for the BANN framework, the data quality control procedures, simulation setup and scenarios, and additional results for the traits analyzed from the Framingham Heart Study and the UK Biobank.(PDF)Click here for additional data file.


Characterization of genetic variations that are associated with gene expression levels is essential to understand cellular mechanisms that underline human complex traits. Expression quantitative trait loci (eQTL) mapping attempts to identify genetic variants, such as single nucleotide polymorphisms (SNPs), that affect the expression of one or more genes. With the availability of a large volume of gene expression data, it is necessary and important to develop fast and efficient statistical and computational methods to perform eQTL mapping for such large scale data. In this paper, we proposed a new method, the low rank penalized regression method (LORSEN), for eQTL mapping. We evaluated and compared the performance of LORSEN with two existing methods for eQTL mapping using extensive simulations as well as real data from the HapMap3 project. Simulation studies showed that our method outperformed two commonly used methods for eQTL mapping, LORS and FastLORS, in many scenarios in terms of area under the curve (AUC). We illustrated the usefulness of our method by applying it to SNP variants data and gene expression levels on four chromosomes from the HapMap3 Project. This allows us to investigate the associations between SNP genotypes and gene expression levels. Expression quantitative trait loci (eQTLs) are those genetic variants that can explain variation in gene expression levels and can help to elucidate the underlying genetic mechanisms of human complex traits . eQTL mat eQTLs) . cis-eQTn humans .In fact, gene expression levels observed are not only regulated by genetic variants but also influenced by non-genetic factors which are known or hidden, for example, batch effects. Therefore, in eQTL mapping, how to account for confounding factors is an important issue and can influence the detection power of eQTL mapping. Up to now, a number of methods have been proposed to account for confounding factors in eQTL mapping, for example, PANAMA , PEER S, LORS Y, HEFT G, LMM-EH-L1 penalty on the regression coefficient matrix. In addition, LORS includes one unknown matrix with the nuclear norm penalty to account for variations caused by non-genetic factors. LORS, a method based on the low rank sparse regression, was proposed for eQTL mapping in . LORS isTo improve the detection power of eQTL mapping, a number of methods have been developed to incorporate the structure information from SNP variants data and gene expression levels, for example, clustering based on gene expression levels  and geneHowever, there are some drawbacks for most of the aforementioned methods. First, if two SNPs are highly correlated with each other, and one SNP is associated with some genes, but the other SNP is not associated with them, we should not expect that these two SNPs have similar coefficients for those genes. Similarly, if some SNPs are classified into one group, we should not expect that the SNPs within the same group have similar coefficients for common genes. Second, the group structures of SNP data and gene expression data are usually identified by performing clustering on the data, however, clustering is an unsupervised leaning approach, the number of clusters is usually artificially determined. When we use the resulting clusters of SNPs and gene expressions to design the penalty term, it may lead to loss of detection power and even spurious associations. Third, complicated design of penalty term in penalized regression modeling can result in untractable computational bottleneck, especially when dealing with a large volume of data.L1 penalty in LORSEN. In addition, we used the low rank approximation to account for non-genetic factors in LORSEN , the methods based on the L1 penalty can only select one of them. In theory, the Elastic Net penalty can overcome these two drawbacks. For the estimation of the model parameters in LORSEN, we developed an efficient optimization algorithm based on the proximal gradient method : the L1 norm of v is defined as L2 norm  of v is defined as m × n matrix M with the elements Mij, the Frobenius norm of M is defined as σ1, …, σr are the singular values of M and r is the rank of M; and the L1 norm of M is defined as For the convenience of description, we first introduce the following notations used in this paper. For a B is sparse. B, L, μ by solving the optimization problemρ and λ are regularization (tuning) parameters that control the rank of L and the sparsity of B, respectively. When L and μ are fixed, the optimization problem becomes a least absolute shrinkage and selection operator (Lasso) , the Lasso selects at most n SNPs. In the context of eQTL mapping, there are usually a small number of samples available. Even though the proportion of SNPs that are associated with the gene expression levels is small, it is highly likely that the number of SNPs associated with the gene expressions can still be larger than the number of samples. In this case, the L1 penalty on B will fail to identify some SNPs that are associated with the gene expressions. Second, the Lasso tends to select only one variable among a group of highly correlated variables. This can be problematic in eQTL mapping. For example, if two SNPs are in high linkage disequilibrium and both of them are associated with gene expressions, only one SNP will be selected by the Lasso. Furthermore, if two SNPs are in high linkage disequilibrium and only one of them is associated with gene expressions, the selected SNP by the Lasso may not even be associated with gene expressions.In this paper, we follow the same sparsity assumptions used in . First,  (Lasso)  problem d out in , the LasL1 penalty on B can overcome the limitations of the Lasso. Therefore, we propose the following optimization problem to estimate B, L, μ:ρ, λ1 and λ2 are non-negative tuning parameters. For real data sets, it is quite possible that some entries in Y are unobserved (missing). In such scenarios, the missing data will not be used in  is a convex and differentiable function, h(x) is a closed proper convex which means h(x) is a convex function, the epigraph of h(x) is closed and h(x) < +∞ for at least one x and h(x) > −∞ for every x. Furthermore, we assume that ∇g(x), the gradient of g(x), is Lipschitz continuous with constant ℓ, which implies that ∇2g(x)⪯ℓI. Two symmetric matrices of the same dimensions A and B have the relationship A⪯B, if B − A is positive semidefinite. Then we havex0 is an arbitrary point in the domain of f(x) and ⟨⋅, ⋅⟩ represents the inner product of two vectors. Instead of using the optimization problem  is called proximal operator. The iterative process is repeated until the stopping criterion is satisfied or the maximum number of iterations is reached.Problem  can be stL, tB, and tμ are like t used in problem  and ρ‖L‖* plays the role of h(x) in  in . By Lemm h(x) in , at the L and μ, then , where k indexes the iteration. In FISTA, the step size can be constant or be determined by backtracking line search. The algorithm to solve LORSEN with FISTA is described in Algorithm 1. For simplicity, here, only the detailed algorithm with the constant step size is described, but the algorithm using the step size determined by backtracking line search is also provided in our R program (https://github.com/gaochengPRC/LORSEN).To accelerate the computational speed, we used the accelerated proximal gradient method. Specifically, we applied the fast iterative shrinkage-thresholding algorithm (FISTA)  which ke1 and Ω2 such that they have the same dimensions as Ω, Ω1 contains all training entries and Ω2 contains all testing entries. Furthermore, we have Ω = Ω1 + Ω2 and Ω1 ⊚Ω2 = 0. For the consistency, we re-parameterized λ1 and λ2 as λ ⋅ α and λ ⋅ (1 − α), respectively. So the optimization problem , we solve the following optimization problemGiven the values of parameters , L and μ, then we evaluate the parameters by calculating the prediction errorThe solutions are ρ, ρ, ρ, α from a sequence sequentially, thereafter, we performed one-dimensional grid search for λ for each α. Specifically, we generated a sequence of λ values with length nλ decreasing from λ such that Xi is the i-th column of X, and Yj the j-th column of Y. In our R program, we set ϵ = 0.02, nλ = 50 and Sα≔. The optimal parameters were B, L, and μ were then obtained based on the set of optimal tuning parameters.The grid search over three parameters may be too computationally intensive. Therefore, we first found an optimal value for shown in  by meansshown in . Please shown in  to find lgorithm , where Xβi’s by solvingXi is the i-th column of X, βi is a q-dimensional vector for the coefficient of the i-th SNP on q genes, i = 1, 2, …, p. For each gene, we selected the top n SNPs in terms of the absolute values of association coefficients, then we obtained the union of selected SNPs for each gene as the final set of SNPs to be involved in the joint modeling. For HC-Screening, we first obtained association coefficients as above, then calculated the standardized estimates of coefficients. For each SNP, the Higher Criticism (HC) statistic . CEU samples refer to Utah residents with Northern and Western European ancestry from the CEPH collection. After the quality-control , the genotype data of 13,815 SNPs of Chromosome 1 and 2,607 SNPs of Chromosome 21 for n = 165 samples were retained in analysis. To simulate gene expression levels for q = 200 genes over n = 165 samples, we first simulated non-genetic effects of k = 15 hidden factors. We randomly generated nk random numbers from N to form a n × k matrix H, then let Σ = HHT. Uj’s were simulated from N, j = 1, 2, …, q and stacked by column to form a n × q matrix U. ej’s were simulated from N as random noise for j-th gene expression and combined by column to form a n × q random noise matrix e. Then the expression data of q genes over n samples were simulated by Y = XB + U + e, where X is the n × p genotype data matrix. We set the total number of SNPs p = 2000, the number of causal SNPs as 60, 200, or 400. Each causal SNP randomly influences m = 10 (or 50) genes. We simulated nonzero genetic effects from a uniform distribution. For the “weak-dense” scenario, each causal SNP affects m = 50 randomly selected genes and the corresponding values in B were simulated from a uniform distribution between 0.25 and 0.75. For the “strong-sparse” scenario, each causal SNP affects m = 10 randomly selected genes and the corresponding values in B were simulated from a uniform distribution between 1.5 and 2. The different simulation scenarios are summarized in Our simulation is similar to that described in . We firsThe number of selected SNPs and the number of selected causal SNPs from two screening methods under different simulation scenarios are summarized in cis-eQTLs and trans-eQTLs.The area under the curve (AUC) was used to compare the performance between LORSEN and two existing methods, LORS  and FastThe results with the SNP screening before eQTL mapping under different simulation scenarios are presented in Our simulation results showed that LORSEN is more powerful to identify weak signals, while it does not have obvious advantage in identifying strong signals compared to LORS. Therefore, we performed additional simulation studies in which the causal variants have mixed weak and strong effects. Specifically, the half of the causal variants had the weak effects and their effects were generated from a uniform distribution between 0.25 and 0.75, while the other half of the causal variants had the strong effects and their effects were generated from a uniform distribution between 1.5 and 2. The number of causal SNPs was set as 60, 200, or 400. The number of genes affected by one causal SNP was set as 50. The AUCs and corresponding 95% confidence intervals are presented in –12, 10–6, 10–4. From the In addition to AUC, a commonly used measure to assess the performance of methods for eQTL mapping, we also reported the false positive rates (FPRs) based on four thresholds for the regression coefficients: 0, 10A number of conclusions emerge from the results based on our extensive simulation studies. First, the HC-Screening method retains much smaller number of SNPs than the LORS-Screening method. Second, when all the SNPs are not filtered with the SNP screening method and are used in the analysis, LORSEN is the most powerful method to identify weak signals, while it does not have obvious advantage in identifying strong signals compared to LORS and FastLORS. LORSEN still performs the best with the mixture of the strong and weak effects when the number of causal variants is large. Third, when the SNPs are first filtered with the HC-Screening method, FastLORS performs the best in all simulation scenarios. With the LORS-Screening method, LORSEN has the highest detection power in most of simulation scenarios. Fourth, LORSEN outperforms FastLORS and LORS when a large portion of the causal SNPs are rare and when the causal variants have a mixture of positive and negative effects.ftp://ftp.ncbi.nlm.nih.gov/hapmap/genotypes/hapmap3_r3/plink_format/and http://www.ebi.ac.uk/arrayexpress/experiments/E-MTAB-264/, respectively. Because the set of samples with the SNP genotype data and the set of samples with the gene expression data are slightly different, we only kept the samples that have both the SNP genotype data and the gene expression data in the analysis. We removed SNPs with missing values, and performed the LD pruning using PLINK with its default parameters . After the data pre-processing, a total of 160 samples  were included in analysis. The number of SNPs and the number of genes with the expression used in the analysis on chromosome 3 are 4,086 and 1,075, on chromosome 15 are 2,235 and 612, on chromosome 17 are 2,226 and 1,098, on chromosome 20 are 1,863 and 606, respectively. Since the significance tests generally cannot be performed for the penalization based regression models, we focused on the top 100 SNP-probe pairs with the largest absolute regression coefficients. From the Venn diagrams , LORS-LORS (LORS with the LORS-Screening) and HC-FastLORS (FastLORS with the HC-Screening) to the HapMap3 data. Here, we focused on Asian samples (CHB and JPT) in the HapMap3 data and selected four chromosomes for the analysis. SNP genotype data and gene expression data are publicly available, and can be downloaded from diagrams –4, we nodiagrams , LORS-LOdiagrams  which aldiagrams , we clasdiagrams . For eacdiagrams . From Tahttps://seeqtl.org/) records the eQTLs identified from a meta-analysis (consensus eQTLs) from the HapMap human lymphoblastoid cell lines. A total of fourteen SNP-probe pairs were found in seeQTL and were listed in http://www.gtexportal.org/home/) built by the Genotype-Tissue Expression Project (GTEx) (https://www.genome.gov/Funded-Programs-Projects/Genotype-Tissue-Expression-Project) to see if those SNP-probe (gene) pairs are listed as the eQTLs and/or sQTLs (splicing quantitative trait locus). A total of seven SNP-probe pairs were also found in GTEx and were presented in To further validate our findings, we searched an existing database called seeQTL . seeQTL A number of conclusions emerge from the results based on HapMap3 data. First, there is a large overlap between the SNP-probe pairs identified by LORS-LORS and LORS-LORSEN but there is a small overlap between the SNP-probe pairs identified by HC-FastLORS and LORS-LORS (or LORS-LORSEN). Second, LORS-LORS and LORS-LORSEN perform similarly and have higher detection power than HC-FastLORS since LORS-LORS and LORS-LORSEN have identified more SNP-probe pairs that are also found in seeQTL and GTEx. Third, five out of seven SNP-probe pairs identified by both LORS-LORS and LORS-LORSEN and found in seeQTL are also found in GTEx, thus it may be beneficial to combine the results from multiple methods to generate a list of SNP-probe pairs for further investigation.As more human gene expression data become available, fast and efficient statistical and computational methods are needed to fully take advantage of such data to investigate the relationship between genetic variants and gene expression levels to further reveal the genetic mechanisms that underlie human complex diseases. However, most existing methods are built on small-scale samples and not applicable to human-size datasets. In this paper, we proposed a new low rank penalized regression method (LORSEN) to detect eQTLs. We developed a fast and efficient algorithm to solve optimization problems arising from our methods based on proximal gradient methods. Comprehensive simulation studies showed that LORSEN outperformed two commonly used methods, LORS and FastLORS, in many simulation scenarios. From our simulation results, we can briefly conclude that, first, LORSEN is more powerful in detecting eQTLs which are rare and/or have weak effects. This is especially an appealing advantage since it is expected that a portion of causal variants are rare and/or have the weak effects in the real world. Second, LORSEN is more powerful when some causal variants have the positive effects and the other causal variants have the negative effects.Since there are a large number of SNPs and genes to be included in the eQTL mapping and it is expected that only a small portion of SNPs will affect the gene expression levels, a number of pre-screening methods have been developed. In this paper, we used the LORS-Screening  and the There are several limitations for LORSEN. First, as a method based on the penalized regression model, we can rank the SNP-gene pairs in terms of the regression coefficients obtained from LORSEN, but cannot perform the significance test. Second, the computational time of LORSEN depends on many factors such as the number of candidate values of hyperparameters, the initial values of hyperparameters, and the number of samples. The computation was performed parallelly using software R (verson 4.1.1) and 16 cores on a server with 64 Intel(R) Xeon(R) Gold 6130 CPUs @ 2.10 GHz. From It has shown that the incorporation of the SNP correlation and the gene interaction network can potentially increase the power of detecting eQTLs . We expe