The genomics era has brought useful tools to dissect the genetic architecture of complex traits. Here we propose a multivariate reaction norm model (MRNM) to tackle genotype–covariate (G–C) correlation and interaction problems. We apply MRNM to the UK Biobank data in analysis of body mass index using smoking quantity as a covariate, finding a highly significant G–C correlation, but only weak evidence for G–C interaction. In contrast, G–C interaction estimates are inflated in existing methods. It is also notable that there is significant heterogeneity in the estimated residual variances  across different covariate levels, i.e., residual–covariate (R–C) interaction. We also show that the residual variances estimated by standard additive models can be inflated in the presence of G–C and/or R–C interactions. We conclude that it is essential to correctly account for both interaction and correlation in complex trait analyses. Complex traits are often influenced by genetic and non-genetic factors , which are themselves interconnected. Here, the authors develop a method for disentangling genotype–covariate correlation and interaction, and investigate their effects on estimating statistical genetic parameters. However, many so-called environmental risk factors  are themselves complex traits whose variation also reflects both genetic and non-genetic factors. For example, the relationship between smoking and body mass index (BMI) is complex, i.e., common causal genetic variants have biological effects on both traits (pleiotropy or genetic correlation)4 while BMI is also modulated by smoking status (interaction)7 7 Fig. . The rel10, but it is critical to distinguish between them because their biological mechanisms differ, as do their implications. This association/interaction problem has been well posed in the classical twin study approach11, showing that association and interaction can be disentangled and correctly estimated with an appropriate model and sufficient data. Unfortunately, large well-powered data sets with measures on multiple family members are few. However, genome-wide association studies (GWAS) now provide different types of genetically informative data to investigate GCCI. The genomic era has brought useful tools to dissect the genetic architecture of complex traits, where genetic variance and covariance can be estimated based on genome-wide single nucleotide polymorphisms (SNPs) genotyped in large-scale population samples. The increased availability of sufficiently powered data sets, with information on measured genetic and non-genetic risk factors, motivates the need to develop appropriate statistical tools for GCCI analysis. Locus specific interaction at single locus level has been widely studied12. However, it is desirable to estimate whole-genome level interaction, which is accumulated from every locus across the genome and has a direct implication on actual phenotypic modulations.Both correlation (‘association’) and interaction (‘effect modification’) are fundamental in biology13 and agriculture15. The RNM allows environmental exposures to be modelled such that the genetic effects of a trait can be fitted as a nonlinear function of a continuous environmental gradient. The possible modulators of the phenotypes of the trait are not limited to environmental exposures, but can include any covariates, regulated by environmental and genetic factors, which are shared with the phenotypes. In other words, the genetic effect, and therefore the phenotype, of one trait often depends on the phenotype of another trait. This can be modelled by introducing dependence between the phenotype and the covariate, where the covariate represents the phenotype of the modulating trait, with both phenotypes having shared genetic and environmental components.The reaction norm model (RNM) has been developed and applied to genotype–environment interaction analyses in ecology11. Random regression-genomic restricted maximum likelihood (RR-GREML)17 and G–C interaction (GCI)-GREML18 have been used to detect and estimate G–C interaction at the whole genome level for BMI modulated by smoking quantity16. However, the analytical approach used in this study was based on univariate models which did not account for G–C correlation, thereby assuming that there is no correlation between the covariates and the outcomes. This can inflate signals indicating the presence of G–C interaction and lead to biased estimates by the failure to account for the G–C correlation. A further limitation with the existing methods is that these cannot be applied to continuous covariates without an arbitrary stratification into discrete exposure groups16. Importantly, additive models used for the estimation of SNP heritability  may give biased estimates for genetic and residual (error) variance if the trait of interest is moderated by (unknown) covariates due to failure in adequate capture of the interaction effects. It is currently not possible to use RR-GREML or GCI-GREML to assess such bias especially when using continuous covariates.In the context of whole genome analyses of human complex traits, there is currently no approach that can fit GCCI effects to disentangle interaction from correlation at the genome-wide level. Yet, ignoring either the genotype–covariate (G–C) correlation or the G–C interaction may cause biased estimates of variance components which form the basis of SNP-heritability or interaction estimationIn this study, we develop a whole-genome reaction norm model (RNM) that is computationally flexible and powerful when estimating genome-wide G–C interactions for complex traits. We also extend this approach to a whole-genome multivariate RNM (MRNM) framework to capture fully the GCCI effects, jointly modelling pleiotropy and interactions at the genome-wide level. As the proposed methods will be able to more realistically account for the complexity of GCCI effects, we hypothesise that they will lead to a significant reduction in bias and notably improve the estimation of the genetic architecture of complex traits.y) representing the main trait is modulated by a continuous covariate variable (c) as gi and ei are the genetic and residual effects for the ith individual record, and gi can be further decomposed to the zero and first order random regression coefficients, denoted as αi0 and αi1  and ci is the unique covariate value for the ith individual . This model is the same as RR-GREML17 proposed by Maier et al.22 except the fact that ci in our model is a continuous variable, as opposed to a discrete variable. The RNM can be generalised to account for residual heterogeneity or residual–covariate (R–C) interaction, by decomposing ei as follows:αi0, αi1, and ci are defined as above, and τi0 and τi1 are the zero and first order of random regression coefficients for the residual variance (see Methods).We propose an extension of the whole-genome RNM that can estimate G–C interactions, where covariates can be continuous phenotypes of traits correlated with the response. In a simplified form of this model, the response variable  is affected by genetic effects and residual error , and there can be non-negligible correlations between α0 and β, α1 and β, τ0 and ε, and τ1 and ε . We used MRNM to take into account the G–C and R–C correlation.The RNM described above is used to model G–C and R–C interaction effects without accounting for G–C and R–C correlation. As briefly explained in Introduction section, the same genetic factors can affect both the covariate trait and the main trait (response variable), and at the same time, the covariate trait phenotypes can directly modify the main trait. For example, both BMI and smoking have non-zero SNP-based heritability17 and GCI-GREML18), RNM, and MRNM using simulated and real data from the UK Biobank24 , neuroticism score (NEU) and the first principal component of genotypes (PC1) as the covariates. We used SMK and NEU because of their well-known association with BMI26 although the variance and covariance components of the interaction effects were not clearly known. We would expect to see little or no evidence for interaction due to PC1 because BMI was reported to have relatively small interaction across different populations27 and the data used in this study were stringently restricted according to their ancestry (see Methods). In addition, we applied the standard GREML18 and LDSC21 methods to estimate SNP-based heritability for the main response variable (y), where y is modulated by one or more unknown covariates. With this analysis, we are able to explore the potential bias in results obtained by these methods in the presence of non-negligible interactions.We compared the performance of previously published methods (RR-GREMLWe used simulation (see Methods) to quantify type I error rate and power of detecting G–C interaction for the proposed RNM, RR-GREML and GCI-GREML, without considering G–C correlation. As shown in Fig. We also tested if the methods can give unbiased estimates for variance components of random regression coefficients underlying the mechanism of G–C interaction. When G–C interactions were present, RNM gave unbiased estimates, whereas estimates from RR-GREML and GCI-GREML differed from true values , was partly captured and estimated as interaction variance, var(α1 · c), from the model, y = α0 + α1 · c + e. However, MRNM performed notably better in these analyses, being able to control for type I error rate (0.046) in detecting G–C interaction . Under the null (no G–C interaction), in the presence of non-negligible genetic correlations between the main response and covariate variables, we observed spurious signals for G–C interaction in the univariate analysis using the RNM Fig. . This wa rate 0.06 in deteion Fig.  and the ion Fig.  when appα1) for RNM in Supplementary Table In the presence of G–C correlations and G–C interactions, both RNM and MRNM performed similarly in detecting G–C interactions  from the regression of the main response on the covariate  are correlated with the covariate (ci) and that there is interaction (RCCI)  the information of specific covariates. So, it is important to assess the biasness of estimates if interaction effects are not properly modelled either because of the model limitation  or lack of covariate information. With simulated data based on the G–C or R–C interaction model, we showed that both the GREML and LDSC overestimated the residual variance for the main response variable hence underestimating the SNP-based heritability  to compare various models that test interaction using RR-GREML M1) and GCI-GREML (M2), and the proposed approaches RNM (M3-M7) and MRNM (M8-M12)  or GCI-GREML , confirming published results16. However, published methods did not account for G–C correlation or RCCI  = 0.47 with SE = 0.12) compared to those obtained using RR-GREML (M1) and GCI-GREML (M2). It is noted that residual heterogeneity (reflected by R–C interaction) was partly controlled in M1 and M2 as these models adjusted for group differences with the covariate stratified into four discrete groups, which however generated biased estimates as shown in Supplementary Data p-value = 3.26E-02). This was probably due to the fact that the non-negligible G–C correlation  but not GCI-GREML or vice versa (7.73E-06 for M6 and 2.36E-05 for M11). This shows that G–C and R–C interactions are both important and contribute to the shared aetiology between BMI and NEU. As shown in Fig. We also analysed BMI using NEULastly, we used PC1 as the covariate in the same analyses Table , probablThe phenotypes of BMI showed some deviation from a normal distribution (kurtosis = 5.65 and skewness = 1.08). We explicitly tested the normality assumption using simulations, and found that rank-based inverse normal transformed (RINT) phenotypes could remedy type I error inflation due to non-normality    and cov that were not explicitly parameterised in GREML. We did not fit multiple covariates jointly in MRNM because of our focus on SNP-based heritability comparisons based on univariate models  and due to the need to control computational demands.We observed in simulation data that residual variances for a trait estimated from LDSC or GREML were inflated when there were G–C or R–C interactions Fig. , and thit) Table  did not ons Fig. . When coons Fig.  although31 of GCCI and RCCI results across different data subsets is useful and reliable  RNM to estimate genotype–covariate correlation and interactions and residual–covariate correlation and interaction effects for complex traits, which is powerful and reliable. The key findings are summarised as follows:For continuous covariates, the proposed MRNM is a more appropriate model, compared to RR-GREML and GCI-GREML.Covariates can be regulated by genetic and environmental factors that are possibly shared with the main response (GCCI and RCCI effects), which is the most plausible mechanism for many complex traits. It is desirable to model GCCI and RCCI effects appropriately (using multivariate RNM).SNP-based heritability estimates from standard additive models should be carefully interpreted or revisited if covariate information is available.The proposed models can be applied to large-scale biobank data by meta-analysis of results from sub-samples, for which the analyses are computationally feasible.Complex traits are determined by both genetic and environmental effects. Some environmental covariates of complex traits may themselves be determined by genetic and non-genetic factors. Genotype–covariate correlation and interactions (GCCI) and residual–covariate correlation and interaction effects (RCCI) may be important underlying factors shaping complex trait phenotypesThe existing methods for estimation of G–C interactions, i.e., RR-GREML and GCI-GREML, require that the outcome of interest  should be stratified into multiple discrete groups. The discrete grouping ignores the difference of covariate values for the individuals within each group, and results in some loss of information. In contrast, RNM and MRNM use a flexible model to fit a continuous covariate. Based on the analysis method that we believe to be the most appropriate for the data (MRNM) the G–C interaction estimate was much reduced and only borderline significant while R–C interaction was much more significant.In the presence of G–C or R–C interactions, estimated SNP-based heritability of the main response variable by GREML or LDSC could be biased. The biased estimates reflect that the interaction effects are absorbed by residual variance and the overall estimated residual variance was inflated. The residual variance estimated from GREML was significantly higher than that from RNM for the BMI-SMK, BMI-NEU or BMI-SMK/NEU analysis using the real data. Currently reported SNP-based heritabilites estimated based on meta analysis of GWAS summary statistics from diverse study-cohorts tend to be lower when the number of study-cohorts is larger (as a proxy of heterogeneity) Fig. , which c33. While in this example the advice may not be practical for other health reasons, the principle can be used to other traits and diseases with modifiable covariates. The information from the G–C correlation and interaction can be useful for an early intervention  although the magnitude of the effects is relatively small, compared to R–C components. We also investigated NEU and found strong G–C and R–C interactions, indicating the personality trait NEU is a major covariate influencing the environmental factor for BMI as well as revealing a novel genetic architecture of BMI to interact with different levels of NEU  can be possibly controlled by changing the covariate (SMK), provided that the covariate is modifiable. In this example, the implication is that the intervention of increasing smoking could be used to control BMINEU Fig. . Both geNEU Fig. . In otheOur analytical framework could be extended to consider genotype-by-genotype interactions, i.e., epistasis, such that an interaction between the genetic component of a trait and a covariate might be partitioned as interaction variances in the proposed model. However, for the sorts of phenotypes considered here there is no power to disentangle epistasis from other interactions with the data available to us. However, the proposed approach may have important utility in the context of gene-expression, transcriptome and methylation data as novel covariates, for which the specific genetic architecture may be powered to detect epistatic interactions and correlations, and may merit future study. It is possible to use models that fit multiple covariates simultaneously as we did for fitting both SMK and NEU jointly Table , which i11 applied to twin- or pedigree-based data. However, availability of such data is limited, restricting our ability to study GCCI effects for a wide range of complex traits and covariates. For example, phenotypes moderated by ancestry components  cannot be studied by an approach that is based on twins or relatives. It is also difficult to disentangle the genetic and shared environmental effects when using a pedigree-based approach. Standard REML packages  can be used to test the GCCI effects although it is questionable that the classical REML algorithm, which has been optimised for pedigree-based studies, can be computationally tractable when fitting genetic covariance structures based on genomic information17. Therefore, it may be infeasible investigate the GCCI effects using the classical REML packages although they have been applied widely in livestock36 and ecological genetics38 to explore the phenotype-genotype relationship across environmental gradients. When extending analyses to cover large-scale data such as the UK Biobank24, it is essential to develop computationally efficient methods that also correctly capture the GCCI effects, based on genomic information.An alternative approach to disentangle interaction from association is through the classical structural equation modelsk = 1 for both G–C and R–C interactions. Further study is required to validate performance with higher order interactions to generalise the proposed approach. Second, our approaches are flexible, but computationally demanding. For a large data set or when conducting randomisation tests, e.g., permutation or bootstrapping, it may not be computationally feasible to conduct an analysis within a reasonable time. Subsampling and meta-analysis approaches can reduce analysis time, however, the power is notably reduced, compared to when using the whole data , National Information Governance Board for Health & Social Care (NIGB), and Community Health Index Advisory Group (CHIAG). The protocol of Atherosclerosis Risk in Communities Study (ARIC) study has been reviewed and approved by the Institutional Review Boards (IRB) of each participating institution, including the IRBs of the University of Minnesota, Johns Hopkins University, University of North Carolina, University of Mississippi Medical Center, and Wake Forest University. Research Ethics approval was obtained from University of South Australia Human Research Ethics Committee (HREC).k = 1 in the following equations. More general equations with a higher order can be found in Supplementary Note We only consider interaction of order 36, the dependent variable for individual i can be modelled asyi is the phenotypic observation, bi represents fixed effects, gi is the random genetic effect, αi0 and αi1 are the zero and first order of random regression coefficients, ci is the covariate value, and ei is the residual effect for the ith individual. Assuming that each individual has unique covariate value, the variance–covariance matrix of observed phenotypes (yi) isA is the N × N genomic relationship matrix based on genome-wide SNP information, Zi is an incidence matrix for gi, and I is an N × N identity matrix. The terms i. The terms idual i and j , respectively17. The random genetic and residual effect are assumed following a normal distribution with mean as zero and variance as gi, can be regressed on the covariate gradient (reaction norm), which can be efficiently modelled with random regression coefficients. The variance–covariance matrix of random regression coefficients (K) isα0 and α1 are the zero and first order random regression coefficients. The genetic (co)variance matrix of genetic effects between N individuals or N covariate values  is a function of random regression coefficients and polynomials, which can be expressed asΦ is the N × 2 matrix of the zero and first order polynomials of N covariate values, that is To account for phenotypic plasticity and norms of reaction in response to different covariate or environmental conditions among samplesyi and ci, it naively assumes that yi and ci are uncorrelated. For this reason, this model is also referred to as a genotype–covariate interaction (G–C interaction) model.Given that this model does not explicitly parameterise the correlation between yi and ci are uncorrelated is often violated. In a more proper model, the covariate value for individual i is decomposed as ci = μi + βi + εi, where μi is fixed effects, βi is the random genetic effect, and εi is the residual effect. When considering the main response (y) and covariate (c) jointly in a multivariate model, the variance–covariance matrix of observed phenotypes yi and ci isZc is an incidence matrix for the vector of the random genetic and residual effects, β and ε, underlying c. The genetic and residual variances of covariate c are denoted as ith covariate levels , respectively. The random genetic and residual effects of y are the same as defined above. The random genetic and residual effect of c are assumed following a normal distribution with mean as zero and variance as Ky is the same as K defined above, and Ky,c consists of the covariance between β and the random regression coefficients, that isThe naïve assumption of the univariate RNM (or G–C interaction model) that e is the vector of residual effects for the main phenotypes, assuming that var(e) is homogenous across different levels of covariate values, i.e., ε is the vector of residual effects for the covariate, defined as above, and var(ε) is the residual variance of the covariate.The multivariate residual covariance structure isKy,c), therefore, is referred to as a genotype–covariate correlation and interaction (GCCI) model. Importantly, values for cov or cov are often non-negligible. Neglecting these terms can cause confounding between G–C correlation and interaction, thereby generating spurious signals and biased estimates for the interaction. Yet many studies do not account for G–C correlations when estimating and testing G–C interaction16.This model explicitly parameterises covariance between the random regression coefficients for the main phenotypes and the genetic effects underlying the covariate , is homogeneous across different values of the covariate. However, it is often possible that residual–covariate (R–C) correlation and interaction exist, resulting in heterogeneous residual variances across different covariate values. To account for this possibility, MRNM can be further generalised asei, can be also regressed on the covariate gradient, modelled with the zero and first order of random regression coefficients (τi0 and τi1) and polynomial of the covariate.The models we described so far assume that the residual variance for the main phenotypes, var, αij0 and αij1 are the zero and first order random regression coefficient for the jth random effect and cij is the covariate value for the jth random effect. Therefore, this model is a multiple random effects model fitting multiple components22, but it allows the inclusion of interaction effects for each random effect. As in the original multiple random effects model, it is assumed that there is no correlation between the random effects42.RNM can be further extended to include multiple covariates. A model fitting with multiple covariates can be expressed asyi) for this multiple random effects model isAj is the genomic relationship matrix for the jth random effect, ith covariate level for the jth random effect, gij in each random effect (j = 1 ~ x) can be regressed on the covariate gradient in the same manner. The variance–covariance matrix of random regression coefficients for each random effect (Kj) can be written asThe variance–covariance matrix of observed phenotypes variance matrix of individual genetic effects between 17.All models described above can be fitted using MTG2p-value > 0.001, keeping qualified genotyped SNPs. After the standard QC, 583,058 SNPs and 8,291 individuals remained. In addition, we estimated pair-wise relatedness from the remaining SNPs and randomly excluded one individual from each pair with an estimated relatedness greater than 0.05. This relatedness cut-off QC reduced the sample to 7263 individuals.Phenotypic simulation was based on individual genotypes from the GWAS data of the Atherosclerosis Risk in Communities Study (ARIC) cohort. We used autosomes only and applied the standard quality control (QC) to genotypes, which included MAF > 0.01, SNP call rate > 0.95, sample call rate > 0.95 and Hardy–Weinberg Equilibrium Simulation under GCCI model (MRNM G–C model): We simulated phenotypes for the main response (y) and covariate (c) under the GCCI model with the first order interaction effect, i.e., k = 1. In the simulation, we used the following covariance structure for the Ky matrix in Eq. (x in Eq.  as\documα1) set at 0, 0.25, 0.5, 0.75 or 1. For the covariate, ci = μi + βi + εi, var(β) and var(ε) were set at 1.We used a wide range of the G–C interaction with var: In this simulation, we used the following covariance structure for the Ky matrix in Eq.  and var(ε) were set at 1.For the covariate, Ky,c matrix : Similar to the GCCI simulations above, we used values for the Ky matrix in Eq. (x in Eq.  as\documα1) set at 0, 0.25 or 1, and for the covariate, both var(β) and var(ε) were set at 1.We performed simulations with var does not have white British ancestry, (2) has a genotype missing rate > 0.05, (3) whose reported gender does not match with the gender inferred using genotype data, and (4) has a putative sex chromosome aneuploidy. At the SNP level, we excluded SNPs with an INFO score < 0.6, with a MAF < 0.01, with a Hardy–Weinberg equilibrium p-value < 1E-4, or with a call rate < 0.95. We excluded ambiguous or duplicated SNPs. We only used the HapMap 3 SNPs in the main analyses because they are reliable and robust to bias in estimating SNP-based heritability and genetic correlation46. In addition, we excluded individual population outliers with the first or second PC outside six standard deviations of the population mean. For individuals who were in the first and second releases of UK biobank genotype data, we calculated the discordance rate between imputed genotype of the two versions for each individual and for each SNP, and excluded individuals and SNPs with a discordance rate lager than 0.05. We also excluded one individual randomly from any pair with a genomic relationship larger than 0.05. After the QC above, 288,866 individuals and 1,130,918 SNPs remained. Of these remaining individuals 91,472 were from the first release of UK Biobank (denoted as UKBB1) and 66,281 individuals with complete records of covariates were used in the main analyses. The rest of 197,394 individuals were from the second release of UK Biobank (denoted as UKBB2) and 115,053 with complete records of covariates were used in the validation and meta-analyses.Main response variable and covariates: We applied the novel (M)RNM model using BMI as the main response variable to estimate the GCCI/RCCI components with each of several covariates, including pack years of smoking (SMK), neuroticism score (NEU) or the first principal component (PC1) provided by the UK Biobank. We also fitted the model that includes multiple covariates  jointly, i.e., RNM with multiple covariates. For all analyses, covariates were standardised as mean zero and variance 1. Prior to model fitting, we adjusted the main response variable (BMI) for confounders including genotype batch, assessment centre at which participant consented, year of birth, sex, age, diet variation, diet change, the first 15 PCs, SMK, weekly alcohol consumption (ALC) and Townsend deprivation index at recruitment (TDI). In the analyses using NEU and PC1 as the covariates, we further adjusted BMI for NEU to correct the mean difference. The distribution of each covariate is in Supplementary Fig. When including the covariate  as the second trait in a MRNM, it was also pre-adjusted for the confounders in a similar way as for the main trait . For instance, as the second trait in a MRNM, SMK was pre-adjusted for BMI, genotype batch, assessment centre at which participant consented, year of birth, sex, age, diet variation, diet change, the first 15 PCs, ALC, and TDI. NEU was pre-adjusted for BMI, genotype batch, assessment centre at which participant consented, year of birth, sex, age, diet variation, diet change, the first 15 PCs, ALC, TDI and SMK. PC1 was pre-adjusted for BMI, genotype batch, assessment centre at which participant consented, year of birth, sex, age, diet variation, diet change, the first 15 PCs except the first one (PC1), ALC, TDI, SMK and NEU.Detailed information regarding covariates used in the interaction models is described below and that for other confounders used to adjust the main phenotypes is in Supplementary Note Pack years of smoking (SMK): We combined pack years adult smoking as proportion of life span exposed to smoking (UK Biobank data field: 20162) and ever smoked (UK Biobank data field 20160) as SMK. The distribution of SMK is in Supplementary Fig. 16, we stratified SMK into four levels: 8,773 individuals with SMK > 0.8, 9,192 individuals with 0.5 ≤ SMK ≤ 0.8, 11,741 individuals with 0 < SMK < 0.5, and 36,575 individuals with SMK = 0 .Neuroticism score (NEU): The neuroticism score (data field 20127) of a given individual was indexed by the number of ‘yes’s to 12 touchscreen questions that evaluate neurotic behaviours. The distribution of NEU is in Supplementary Fig. The fist principal components of genotype (PC1): PCs were pre-calculated by the UK Biobank. Detailed information regarding the calculation is described elsewhere47. Briefly, PCs-loadings were estimated using fastPCA48 based on 407,219 unrelated individuals and 147,604 markers that were pruned to minimise linkage disequilibrium, onto where all samples were projected, to generate a set of PC scores. For RR-GREML and GCI-GREML, we stratified the sample into four groups based on quartiles of PC1.Meta analyses of real data: The proposed MRNM requires individual-level genotype data, which makes it computationally demanding. As sample size increases , the computing time increases substantially. To complete the analyses within a reasonable timeframe, we used a meta-analysis approach. We performed two sets of meta-analyses, one across two groups within UKBB1 to assess the performance of the meta-analysis, compared to that of the whole UKBB1 data analysis, and the other across UKBB1 and UKBB2.P-values from each group were meta-analysed using the Fisher’s method31. We then compared these p-values with those based on the whole UKBB1 data set.Meta-analyses within UKBB1. We randomly divided the UKBB1 into two groups of equal size (denoted as g1 and g2), and fitted all models mentioned above for each group. 31. For UKBB2, the same pre-adjustment as for UKBB1 was applied to the main response and covariates as the second trait in MRNM.Meta-analyses across UKBB1 and UKBB2. In UKBB2, 197,394 individuals with genotype data passed the QC, of which 94 K have no missing covariates and main response. Similar to meta-analyses within UKBB1, we randomly divided the UKBB2 into two groups of equal size (denoted as G1 and G2), and fitted all models mentioned above for each group. We then meta-analysed the results from G1, G2, and UKBB1 (denoted as G0) using the Fisher’s methodhttp://www.ukbiobank.ac.uk/UK Biobank, https://github.com/bulik/ldscLDSC, http://cnsgenomics.com/software/gcta/GCTA, https://www.cog-genomics.org/plink2Plink1.9, https://sites.google.com/site/honglee0707/mtg2MTG2, Further information on research design is available in the Supplementary InformationReporting SummaryDescription of Additional Supplementary FilesSupplementary Data 1Supplementary Data 2Supplementary Data 3Supplementary Data 4Supplementary Data 5Supplementary Data 6Supplementary Data 7Supplementary Data 8Supplementary Data 9Source Data


The objective of this study was to identify environmental factors strongly associated with and to estimate genetic parameters of reproductive traits in Japanese Black heifers.Data included reproduction records of Japanese Black heifers born between 2004 and 2014. First service non-return rate (NRR) to 56 days from first to successful insemination (FS), number of services per conception (IN), age at first calving (AFC) and gestation length were analyzed with the use of the general linear model. Genetic parameters were estimated with the use of the univariate animal model of the residual maximum likelihood.Averages of reproductive traits over eleven years were assessed, and the effects of farm, year, month, artificial insemination technician and interaction of farm×year on the traits were determined. Estimated heritability of FS was very low and that of AFC was higher than that of the other traits. A close genetic relation was observed among NRR, IN, and FS; however, their heritabilities were very low. AFC shows favorable genetic correlation with IN and FS.Low heritabilities of most reproductive traits in Japanese Black heifers are strongly influenced by farm management practices, and that large residual variances make genetic evaluation difficult. Among the reproductive traits, AFC is potentially more useful for genetic improvement of heifer reproductive traits because it has high heritability and favorable genetic correlations with IN and FS. Variation in the reproductive performance of cattle is dependent on genetic and environmental factors. The observed performance of a trait in cattle is the result of heredity and the effects of the environment where the animal is raised. Genetic improvement of female reproductive traits is currently attracting great interest in genetic evaluation of beef cattle because of the influence of such traits on the productivity of calves. Difficulties in measuring these traits and the low genetic variability result in slow response to selection ,2. ReproMost reproductive traits in heifers are of low heritabilities in dairy cattle –5 and inReproductive traits of heifers and cows need to be evaluated separately, although reproductive traits in heifers genetically correlate strongly with reproductive traits in subsequent parities . TherefoReproduction records of Japanese Black heifers were obtained from the Artificial Insemination Center of Northern Okinawa: data comprised artificial insemination and calving events, five reproductive traits from artificial inseminations and calving from a registry book. The traits studied were first service non-return rate (NRR) to 56 days coded one (1), when no subsequent insemination occurred until 56 days after the first insemination, and zero (0) otherwise; days from first to successful insemination (FS) being the number of days between the first insemination and the successful insemination that resulted in conception; number of services per conception (IN); age at first calving (AFC) defined as the difference between the heifer’s date of birth and the date of birth of her first calf, and gestation length (GL) being the interval in days from the last insemination to the subsequent calving.A data set included reproduction records of heifers born between 2004 and 2014, first insemination between 2005 and 2015, less than 10 services and farms with a minimum of five records. Heifers with incomplete records, embryo transfer donor or recipient, and bearing twin calves were excluded from the data set. Edited records of 2,215 heifers under NRR and 2,047 heifers under FS, IN, AFC, and GL were analyzed. To evaluate the effects of age at first insemination, heifers were classified into three age groups: <16, 16 to 19, and >19 month . Eight aThe general linear model (GLM) procedure of Statistical Analysis System 9.3 (SAS) software  was usedyijklmop is the observation of NRR, FS, IN, and GL, Fi the ith fixed effect of farm, Yj the jth fixed effect of year of insemination, Mk the kth fixed effect of month of insemination, Tl the lth fixed effect of AI technicians, Am the mth fixed effect of age at first insemination, so the oth random effect of sire, and eijklmop the random residual of yijklmop.where The following model was used for AFC:yijklop is the observation of AFC. Fi, Tl, and so are as described in the previous model where two effects were defined differently, Yj the jth fixed effect of year of birth, Mk the kth fixed effect of month of birth and eijklop the random residual of yijklop. The GLM procedure of SAS 9.3 software [where software  was usedsoftware  was usedy = a vector of reproductive traits, b = a vector of fixed effect, X = an incidence matrix for the fixed effects, u = a vector of random genetic additive effect, Z = an incidence matrix for the random effect and e = a vector of random residuals.where, y, u, and e areThe expectations for The variance-covariance structure of random effects wasA is the numerator relation matrix, I the identity matrix, where The averages of the reproductive traits over eleven years were 67%, 1.76, 36.33 days, 25.8 months and 288.22 days for NRR, IN, FS, AFC, and GL, respectively . Farm efThe highest LSM for NRR was observed in 2010, and for IN and FS in 2005. The longest GL was observed in 2010. Heifers born in 2010 showed the highest LSM for AFC . EstimatThe highest and the lowest genetic correlations were between FS and NRR and between FS and GL, respectively, in absolute values. The AFC showed moderate genetic correlation with NRR and GL, but high with IN and FS. The phenotypic correlation between NRR and GL was the lowest, whereas that between FS and AFC was the highest .In the present study, NRR after the first service was slightly lower (67%) than in published results: 74% in Canadian Holstein heifers , and 73.In the present study, analysis of variance showed that farm had a significant effect on all reproductive traits of Japanese Black heifers in Okinawa, indicating that different management practices on farms influenced heifer reproductive traits significantly. Animal management standards on farms has a significant effect on the number of services to Holstein heifers . Better A previous study on Holstein heifers has shown that genetic variance in binary and count traits are lower than those in interval traits ,24. The In this study, heritability of NRR (0.027) was slightly higher than the estimate (0.012) for dairy heifers  with theIn the present study, genetic correlations among pairs of the traits varied substantially: NRR showed negative and moderate genetic correlations with IN (−0.636) and GL (−0.569). Genetic correlation between NRR and IN was slightly lower to those estimated in Chinese Holstein heifers (−0.87) . NRR shoIN and FS showed high genetic correlations with AFC, whereas genetic correlations between AFC and NRR, and AFC and GL were moderate in absolute values. High genetic correlation seems to be due to the pleiotropic effect of genes, where a group of genes may simultaneously have a common genetic effect on these three traits . Heifers that calve early tend to have fewer IN and shorter FS. Therefore, AFC is considered a key trait in the reproductive cycle of and as an indicator of sexual precocity in female cattle. Thus, AFC is potentially useful in genetic evaluation of heifer reproductive traits. Selection of AFC anticipates the achievement of higher genetic development of reproductive traits because of high heritability and favorable genetic correlations with IN and FS. The AFC is genetically correlated with calving interval in Japanese Black cows . HeifersThe present study suggests that low heritabilities of the most reproductive traits in Japanese Black heifers is strongly influenced by farm management practices, and that large residual variances make genetic evaluation difficult. The NRR, IN, and FS demonstrated close genetic correlation with one another, although their heritabilities were very low. Among the reproductive traits, AFC is potentially more useful for genetic improvement of heifer reproductive traits because it has high heritability and favorable genetic correlations with IN and FS.


MPPED2I, GUCY2E, NAV2, and ZBTB16. Genes from the TRIM family were also identified; these genes are potentially related to the regulation of triglyceride levels. Our results suggest that the W-test could be an efficient and effective method to detect gene–methylation interactions. Furthermore, the identified genes suggest an interesting relationship between lipid levels and the etiology of neurological disorders.An increasing number of studies are focused on the epigenetic regulation of DNA to affect gene expression without modifications to the DNA sequence. Methylation plays an important role in shaping disease traits; however, previous studies were mainly experiment, based, resulting in few reports that measured gene–methylation interaction effects via statistical means. In this study, we applied the data set adaptive W-test to measure gene–methylation interactions. Performance was evaluated by the ability to detect a given set of causal markers in the data set obtained from the GAW20. Results from simulation data analyses showed that the W-test was able to detect most markers. The method was also applied to chromosome 11 of the experimental data set and identified clusters of genes with neuronal and retinal functions, including  Genetic variants, such as single-nucleotide polymorphisms (SNPs), have been found to influence risk for human diseases. Recent studies show that epigenetics affect SNPs in genes and subsequently influence the gene function and disease trait . EpigeneGAW20 provided the study data. The study participants were patients with diabetes who had undergone lipid-control treatments with the drug fenofibrate and were recruited from the Genetics of Lipid Lowering Drugs and Diet Network (GOLDN) clinical trial project. The analyzed data sets consisted of a simulated and experimental data sets. The triglyceride (TG) levels were collected at 4 clinical visits, with 2 measurements before treatment and 2 measurements after treatment. Age, sex, smoking status, and location were recorded. Genome-wide association data were sequenced with the Affymetrix Genome-wide Human SNP array 6.0, and DNA methylation profiling was performed with the Illumina Infinium Human Methylation 450 K Bead Chip Array, using the buffy coat harvest from blood samples collected at the second and fourth visits. In the simulated data, the phenotype of the simulated data set was generated using experimental genetic data under a hypothetical model . The TG The TG levels can be used as a measure of drug response. Because common clinical standards regard a 30% decrease in TG levels as an effective control of lipids , we adopThe W-test measures the probability distributional differences for a set of biomarkers between the 2 groups of participants such as the 2 drug-response groups . Under aith category out of total cases or controls, respectively. SEi is the standard error of the log of odds ratios. The test statistics follows a chi-squared distribution with f degrees of freedom. Two parameters, h and f, are estimated using large-sample approximation by drawing smaller bootstrap samples under a null hypothesis. Consequently, the testing distribution is robust for complicated genetic architectures, as it adaptively adjusts to the data structure of the working data [cis-regulation patterns in the gene-methylation data, the SNPs and CpG sites located within a 10-kb genome distance on chromosome 11 were evaluated exhaustively [where mentclasspt{minimamentclass2pt{minimY = SNP + CpG + SNP × CpG, where CpG is a binary variable;LR-m1: : Y = SNP + CpG + SNP × CpG, where CpG is a continuous variable.LR-m2Two types of logistic-regression models were applied as accompanying benchmarks to the W-test. The first logistic-regression model, LR-m1, considered the CpG site as a binary variable like the W-test, and the second logistic-regression model, LR-m2, included the CpG sites as a continuous variable using the original methylation values. Both logistic-regression models incorporated the main and interaction effects of SNP and CpG sites. In short, we denote:The type I error rate is an average false-positive proportion using a permuted phenotype on a pair of gene–methylation markers in 2000 replicates. A total of 140,501 epistatic pairs were tested, and a Bonferroni correction resulted in a significance level of 3.56E-7 at a family-wise error rate of 5%.p values obtained from alternative methods. Generally, the W-test gave smaller p values than LR-m1 in most answer pairs, and also had comparable p values to LR-m2. The top 3 answer pairs were all identified to be significant by the 3 methods. The W-test also found the fourth answer pair  with a p value = 0.0475, which was slightly smaller than the p values from the LR-m1  and LR-m2 . The results suggested that the W-test could be sensitive to small signals with lower heritability. In terms of the performance for the noise pairs, all methods yielded noise p values greater than 0.05. The Type I error rate of the W-test was 2.95%, less than the family-wise error rate of 5%. Meanwhile, the Type I error rates of LR-m1 was 5.40% and of LR-m2 was 5.43%. The results showed that the W-test was able to distinguish between signal and noise in the simulated data set.In the simulated data set, the W-test, LR-m1, and LR-m2 were applied to the given causal and noise pairs. Table Computing time was calculated on a laptop computer with a 1.6 GHz chipset and 4 GB of random access memory using 2000 replicates on 1 pair of markers. The W-test was 4 times faster than logistic regression on a general laptop .MPPED2 , which encoded the protein metallophosphoesterase and was reported to be related to neuronal function [MPPED2 was associated with chronic kidney disease, and knockdown of this gene in zebrafish embryos suggested a role for it in renal function [GUCY2E was ranked fourth and has been reported to function in the central nervous system and retinal [NAV2  is a neuron navigator that induces neurite outgrowth for all-trans retinoic acid, and plays an essential role in the development of the cranial nerve and the regulation of blood pressure in humans [ZBTB16 at 11q23.2  also has been reported as an inhibitor of neurite outgrowth in the adult central nervous system [TRIM5, TRIM6-TRIM34, and TRIM3 , which were highly correlated with TG levels in mice [The W-test was applied to test the gene–methylation interactions for GAW20 experimental data on chromosome 11. No significant interaction pair passed the Bonferroni correction significance level of 3.56E-07 (Table function . Previoufunction . GUCY2E  retinal , 18. NAVn humans . ZBTB16  in mice . The quacis-regulatory region. Existing methods solve the challenge by using stage-wise and integrated analyses, in which the SNPs are separately selected and then the epistatic interactions with CpG sites are jointly evaluated in regression-based approaches [There has been increasing evidence for the contribution of epigenetics in regulating gene expressions implicated in diseases. Previous studies were mainly focused on experimentally studying gene–methylation interactions. In this study, we demonstrated that the W-test can be used as an effective method to identify the epistatic interactions between SNPs and CpG sites in the GAW20 simulated and experimental data sets. One common obstacle in the analysis of epistasis in the genome and epigenome is the large number of pairwise tests, the volume of which is determined by the size of the proaches , 11. Theproaches . HoweverMPPED2, GUCY2E, NAV2, and ZBTB16 as associated with hyperlipidemia. Among these 4 genes, MPPED2 was the most significant; it plays a role in neural development, and genetic variations in this gene are reported to be related to migraines, a common disease of the neural system disease [CUCY2E are reported to be related to retinal disorders [ZBTB16 encodes a protein that is highly expressed in the brain, and polymorphisms in this gene are used as a marker for attention deficit hyperactivity disorder, a neuropsychiatric condition [Application of this method on the experimental data from patients who had undergone treatment for managing TG levels via fenofibrate identified genes that played roles in renal function, the central nervous system, and retinal functions. The enriched signals found in neuronal-related genes suggest that the blood lipid levels could be related to the neurological dysfunction in the brain, which is the most cholesterol-rich organ in the body. By performing an epistatic evaluation between SNPs and CpG sites, we identified  disease . Furtherisorders , 25. ZBTondition . It is i


Saccharomyces cerevisiae, is the primary factor determining industrial bioethanol productivity. Until now, genomic elements affecting ethanol tolerance have been mapped only at low resolution, hindering their identification. Here, we explore the genetic architecture of ethanol tolerance, in the F6 generation of an Advanced Intercrossed Line (AIL) mapping population between two phylogenetically distinct, but phenotypically similar, S. cerevisiae strains (a common laboratory strain and a wild strain isolated from nature). Under ethanol stress, 51 quantitative trait loci (QTLs) affecting growth and 96 QTLs affecting survival, most of them novel, were identified, with high resolution, in some cases to single genes, using a High-Resolution Mapping Package of methodologies that provided high power and high resolution. We confirmed our results experimentally by showing the effects of the novel mapped genes: MOG1, MGS1, and YJR154W. The mapped QTLs explained 34% of phenotypic variation for growth and 72% for survival. High statistical power provided by our analysis allowed detection of many loci with small, but mappable effects, uncovering a novel “quasi-infinitesimal” genetic architecture. These results are striking demonstration of tremendous amounts of hidden genetic variation exposed in crosses between phylogenetically separated strains with similar phenotypes; as opposed to the more common design where strains with distinct phenotypes are crossed. Our findings suggest that ethanol tolerance is under natural evolutionary fitness-selection for an optimum phenotype that would tend to eliminate alleles of large effect. The study provides a platform for development of superior ethanol-tolerant strains using genome editing or selection.Ethanol tolerance, a polygenic trait of the yeast  Saccharomyces cerevisiae has long been studied as a model organism for eukaryotic molecular and cellular biology. S. cerevisiae’s genome was the first eukaryote genome completely sequenced, with haploid genome of 12,068 kb and 5,885 potential protein encoding genes in 16 chromosomes  design is based on crossing two or more genetically distinct pure lines, followed by several generations of intercrossing . RecombiQTL mapping for ethanol tolerance has generally involved industrial strains selected to differ widely in this trait . However5 during the intercross phase. During the interval between sexual generations, there would have been opportunity for one to three generations of vegetative reproduction. Consequently, some differential clonal reproduction may have occurred in the intervals between sexual generations.An AIL population was produced by crossing a haploid laboratory strain, S288c, of known genome sequence, with a heterothallic haploid strain, YE-531, isolated from nature . We choode novo sequencing. Then we used the full genomes of the two founders and 13 publicly available S. cerevisiae genomes of different lineages   . The selFollowing frequency smoothing for location by locally weighted scatterplot smoothing (LOESS) , mappingThe 90 individuals selected in each growth or survival pool were randomly assigned to three independent replicate subpools of 30 diploid individuals (60 chromosomes) each. Averaged across all markers, the variance estimates after LOESS were 0.0010, 0.0034, and 0.0044 among replicates of the control, growth, and survival pools, respectively. These observed values are plausible  for growth and 12.8 kb (0.001–54.8 kb) for survival  and MMP1  were mapped to level of a single gene  of genetic variation for growth and essentially all genetic variation for survival.Allele effects averaged 0.018 (0.012–0.044) optical density (OD) units for growth and 0.357 (0.207–0.794) survival score units for survival  that were also located in genomic sites enabling simple gene manipulation. Based on mapping, we hypothesized that a candidate mutation generating a phenotypic difference between a tail and F6 control  should also cause a difference in the same direction, between the two alleles in an F1 background on a reciprocal hemizygosity analysis (RHA) test .NTH2 × survival) gave exactly equal results, while five differed in the expected direction, but did not reach statistical significance. This may simply be a matter of Type II error.Four of the 10 gene × trait combinations differed significantly in the RHA ). At the end (after 17 h of incubation), for all samples, glucose was not completely utilized so that glucose levels were not a limiting factor.The effect ofaldehyde , the ideThe key to unlocking the full potential of the yeast model for genetic analysis of quantitative traits was the development in our laboratory  and elseGenotyping by deep sequencing ensured that causative mutations are included among the tested markers (except for sites with low sequencing reliability such as repetitive elements), minimizing occasional loss of power of closely linked markers due to hotspots of recombination or sampling variation in proportion of recombination between marker and causative mutation . SequencLocal smoothing by LOESS  eliminatFinally, the use of FDR criterion to set critical thresholds for significance greatly reduced Type II error while controlling Type I error, in this way enabling many more true effects to be uncovered compared to use of a Bonferroni criterion .Drosophila , so that they can be compared across traits . Expressed in this way, QTL effects for the two traits were virtually identical: average effect = 0.121 s.d.u. for growth (range 0.081 to 0.295), and 0.129 s.d.u. for survival (range 0.075–0.288). These values are small, but not negligible. To put them in perspective, average QTL effect was 2.6% of the mean for growth) (0.698 OD units), and 4.5% of the mean for survival . Thus, the cumulative effect of 50 QTLs for growth or 25 QTLs for survival would double the mean growth and survival scores, respectively. On average, the identified QTLs individually explain about 0.7% each, of phenotypic variation. Taken together, they account for a substantial fraction of the phenotypic variation, 34.7% and 72.0% for growth and survival, respectively. The proportion of genetic variation explained would be even greater, depending on the heritability of the trait. Taking heritability = 0.50, for example, the mapped QTL explains almost 70% of genetic variation in growth and essentially all of the genetic variation in survival.A number of very large-scale QTL mapping experiments explored the architecture of complex quantitative traits, using yeast as a model system. Each study involved numerous traits, mainly ability of yeast to grow under various adverse conditions  . GeneralGenerally, the model studies uncovered a much higher proportion of loci of moderate to large effect compared to the present study. This may represent a real difference in the genetic architecture of ethanol tolerance in the wild, which is under strong natural fitness-selection for an optimum that would tend to eliminate alleles of large effect, and the genetic architecture of the model traits studied by MSK1 and APJ1 were found in growth QTLs; PIC2, GIP2, and SSA4 were found in survival QTLs; PFK26 was found in QTLs of both traits  mapped to a level of a single gene   , a non-sprotein) , the prepathways . Here, tADH2, MOG1, MGS1, and YJR154W, were validated by RHA. ADH2 is an alcohol dehydrogenase whose primary function is to oxidize ethanol to acetaldehyde . MAS/GAS is particularly important when phenotyping costs are high or heritability of the trait is low, so that simple phenotypic selection is inefficient.Genetic improvement can be attained by introgressing the positive alleles of the mapped QTLs from one parent to the other. In the past, this could only be done by laborious and time consuming multiple marker crossing and selection introgression designs that are limited to a small number of genes at any one time . In buddFinally, identifying causative QTG and Quantitative Trait Nucleotides (QTN) underlying genetic variation at a QTL, may also identify the causative pathway in which the QTG is operative. This may open further opportunities for improvement by reverse engineering of other genes in that pathway.5 each generation. This amount greatly exceeds the 100 individuals per AIL generation, recommended to minimize sampling effects on recombination proportions across the genome  served as the control group. We studied survival under high ethanol stress and growth under moderate ethanol stress as two distinct traits. To establish the tolerant tail pools for the two traits, a two-stage selection was employed in the F6 generation  ethanol for 22 h was tested by final OD600 nm measurement. Average and standard deviation of final OD600 nm values of the four replicates were calculated for each tested segregant , before and after overnight growth for 22 h. Ten segregants suspected of having ethanol degrading activity were excluded. Of the remaining 290 segregants, the top 90 showing the highest OD values were chosen to form the upper tail.For the creation of the survival upper tail group, the ability to survive in 19% (V/V) ethanol solution for 5 h was tested by a spot assay. Namely, for each replicate, 1 ml final volume of 1 OD cell culture suspended in YPD liquid media mixed with ethanol, was transferred into one well of a 24 multiwell plate. A spot assay at 1- to 100,000-fold dilution was used  to evaluate cell survival ability. The survival-ability test was designed to test the ability to survive for a short time (5 h) under high ethanol stress. Survival score was assigned depending on dilutions showing survival in the spot assay, and selection was based on total score across all four repeats   most tolerant segregants for each trait were chosen for the upper tail. For the statistical analysis (below), for each of the two traits, the 90 selected segregants were divided at random into three independent groups of 30 segregants each.Importantly, although only 90 segregants were eventually taken for genotyping and mapping; selection proportions in the tails were 35% and 30% in the first and the second stage; together 10.5%. However, as the design used here was a 1-tail design , according to the manufacturer’s instructions , was around 1,000 for all pools , and to Sequence alignment was done with BWA-mem algorithm, version 0.7.10 . Only unTo confirm the allele frequencies obtained by deep sequencing, we individually genotyped two of the SNPs in individuals from the control and the tolerant groups, using High Resolution Melting (HRM) analysis  were used for mapping QTLs.Allele frequency estimates were smoothed along each chromosome by location, using LOESS local regression with span (i.e. window) = 80 SNPs and degree = 2. We used the LOESS implementation in R that is based on cloess . The smoBasically, we used a simple single-marker test for marker-trait association. That is, each marker, denoted Mi, was tested individually for association with growth, and separately for association with survival. Simply put, if frequency in the growth or survival pools of alternative alleles at a given marker differ more widely from their frequency in the control pools than expected by chance, we attribute this to association of the alleles at the marker with the tested trait. Thus, taking growth for example as the target trait, the following steps were taken for testing Mi for association with growth.Estimate frequency of the alternative alleles of the target marker Mi in the growth pools  and in the control pools (piC). .Calculate the observed difference, denoted Di = piG − piC, in marker allele frequency between the pools of the target trait (in our case the growth pools) and the control pools.0: marker is not associated with trait value (see later for how SE(Di) was obtained).Obtain the expected difference (termed the “standard error” of Di), denoted SE(Di) under the null hypothesis H1: there is an association between marker and trait, is accepted; with possibility of error  = 0.05. This is termed a CWER, as it is error rate for a single comparison. If one tested many markers all located on a single chromosome, then we could have “chromosome-wise error rate”, which would be the error rate of having even one of the markers on the chromosome falsely associated with trait value. Similarly, there could be an “experiment-wise error rate”.Calculate Zi = Di/SE(Di). If Di is much greater than SE(Di), such that Zi >1.645, we conclude that Di was not generated under the null hypothesis. i.e., the null hypothesis is falsified, and the alternative HThe critical value of Zi = 1.645 for Zi to represent a CWER of P < 0.05 is obtained from tables of the cumulative standard normal curve.(v) If Zi <1.645, we conclude that null hypothesis has not been falsified, and hence we are not justified in claiming that there is an association between marker and trait.In applying these principles to our case, we need to take into account that the estimates of allele frequency in the selected pools and in the control are based on three subpools for each pool or control. The estimates of selected frequency in the selected pools and in the controls are thus based on average of the three subpools. Also, in our case, we do not use a standard error based on theoretical derivation, but an empirical standard error based on the variance among the frequency in the three subpools of the selected pools, and the three subpools of the control pool.However, detailed examination of the any GWAS based on such single marker tests, showed that even significant regions are always a mixture of significant markers and non-significant markers, making it difficult to set boundaries for the QTL region (QTLR). To circumvent this, following The following is a more explicit presentation of the application of the above principles to our specific case.th marker in the jth pool of the kth category,Let pMijk be the LOESS smoothed frequency estimate (eFrequency) of the YE-531 allele of the iwhere,i = 1 to M markers used in the study,j = 1 to 3, is the serial number of the random subpool within its category,k = 1 to 3 is the category, where 1 = Growth selected pools; 2 = Survival selected pools; 3 = Control (F6) pools.Thus, pM111 would be the eFrequency of Marker 1, in subpool 1, of the growth selected pools.Marker-trait association was determined by a single-marker test, separately for growth and survival, where CWER P-value of the ith marker was set equal towhere,Di = Avg(pMi.1) − Avg(pMi.3), for the growth trait pools, andDi.2 = Avg(pMi.2) − Avg(pMi.3), for the survival trait pools.Avg(pMi.1) is the average of pMi taken across the three growth subpools. The dot in the expression Avg(pMi.1) tells us that the average is taken over the three growth subpools.Avg(pMi.2) is the average of pMi taken across the three survival subpools,Avg(pMi.3) is the average of pMi taken across the three AIL-F6 control subpools.SE(D.k) is the standard error taken over all pools and markers in a given category.By definition, the SE of a “treatment” effect is the expected standard deviation under the null hypothesis that the treatment (the above Di.k), has no effect. An empirical estimate of the SE(D.k) was obtained based on the assumption that the variance of marker allele frequency among pools of the same category , is affected by sampling variation and technical variation, but does not represent a real difference in marker allele frequency due to a QTL effect. Thus, it is a measure of sampling variation among pools within a given category under the null hypothesis . SE(D.k)To implement this, for each marker we calculated the variance of the marker allele frequency across the three F6 control subpools. The locally smoothed variance was calculated to obtain VarPijk for each marker, an empirical estimate of the sampling variation among control subpools under the null hypothesis. Similarly, for each marker × trait combination, the variance of marker allele frequency across all three tail subpools was calculated. Then the locally smoothed variance across span of 10% of the markers for growth and survival pools, respectively (VarSijk.1 and VarSijk.2), was calculated.Since Avg(pMi) for the various categories is an average of three subpools, SE2(Dijk) = VarSijk/3 + VarPijk/3, where Dijk = D for the ith marker, in the jth pool, of kth category.To account for the multiple tests involved in the present study , we used the False Discovery Rate (FDR) criterion , with al10P value in a region, and moving out on both sides until a drop of 1.0 of the log10P value is obtained . Note that a single QTL may be considered as two QTLs when two adjacent peaks are observed. We considered QTLs of the two traits as overlap when their 1.0 log-drop boundaries partly overlapped.QTLs boundaries were defined with a 1.0 log-drop support interval procedure as in We compared our strain YE-531, isolated from nature, and the popular laboratory strain S288c with a selected set of 13 commercial and wild strains available in the SGD database . Some ofAllele substitution effect (δ) was calculated for each marker located in a QTL following 2  who contributed so much to the development of the selective DNA pooling and AIL mapping procedures on which this study is based.The whole genome sequencing data have been deposited to SRA at NCBI PRJNA428852.Author ContributionsRH and YK conceived and designed the study. RH, IK, and KB-R performed experimental lab work. GH processed the sequencing raw data. RH, GH, EL, MS, and YK performed statistical analysis and interpretation of results. MP predicted the effect of candidate mutations on the 3D protein structure. RH, GH, EL, MS and YK wrote the manuscript. All authors read and approved the final 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 propose a gene-level association test that accounts for individual relatedness and population structures in pedigree data in the framework of linear mixed models (LMMs). Our method data-adaptively combines the results across a class of score-based tests, only requiring fitting a single null model (under the null hypothesis) for the whole genome, thereby being computationally efficient.APOA5 and ZNF259) near rs964184, whereas neither the other gene-level tests nor the standard test on each individual single-nucleotide polymorphism (SNP) detected any significant gene in a genome-wide scan.We applied our approach to test for association with the high-density lipoprotein (HDL) ratio of post- and pretreatments in GAW20 data. Using the LMM similar to that used by Aslibekyan et al. , our method identified 2 nearly significant genes (Gene-level association testing can be a complementary approach to the SNP-level association testing and our method is adaptive and efficient compared to several other existing gene-level association tests. Genome-wide association studies (GWASs) are considered to be the standard approach to use to detect common genetic variants associated with complex traits. It has become popular to extend the most popular single-nucleotide polymorphism (SNP)-level analysis to gene-level analysis by aggregating multiple SNPs in a gene or other functional unit. As a complement to the standard single SNP-based approach, the gene-level approach can achieve higher reproducibility and power. An additional benefit of the gene-level approach is that a decreased number of hypotheses need to be tested, thereby reducing the burden of multiple testing.The goal of this work is to perform a gene-level association test to detect genes significantly associated with a single trait using the GAW20 data while effectively controlling for the false-positive rate. Note that the candidate gene approach conducted by Aslibekyan et al. was based on the 95 loci drawn from previous studies based on SNP-level association testing , and fouThe Genetics of Lipid Lowering Drugs and Diet Network (GOLDN) study collected pedigree data, motivating the use of LMMs to account for population structures and relatedness as adopted by Aslibekyan et al. . In our yi denotes a quantitative trait for individual i = 1, ⋯, n, Xi = ′ is a vector of q covariates, and Gi = ′ is a vector of p SNPs in a gene for individual i. A LMM is constructed asα and β are the unknown regression coefficient vectors for the corresponding covariates and SNPs, bi and εi are a random intercept and an error term that are independent with each other. We further assume that the error terms εis are independently distributed, but bis are not. Specifically,Suppose that Ψ is a known n × n genetic relationship matrix, which reflects the genetic relatedness among the subjects in the data. The null hypothesis to be tested for association between the group of the SNPs and the trait is H0 : β = 0.where U = ′, to be used to construct various gene-level score-based tests, can be expressed asFitting  LMMs can be computationally demanding. However, using penalized quasi-likelihood (PQL) to fit the model enables us to extract the test statistic for score-based tests including the aSPU test . It is kU and its covariance matrix V under the null hypothesis, which can also be written in a closed form. Because the score vector follows asymptotic normal distribution with mean zero under the null hypothesis, one can use the Monte Carlo method to compute p-values. Note that both U and V depend only on the null model (3), which provides computational efficiency when the number of tests is large as in a genome-wide scan. We can use an R package GMMAT to derive U and V [The aSPU test statistic can be obtained using the score vector  U and V .U and V, and each nonadaptive test has its own advantages and disadvantages. For example, consider these 2 cases: (a) every SNP encoded in a gene is associated with an equal effect size and direction, and (b) only one or a small proportion of the SNPs are associated. The burden test, which takes U1|, ⋯, |Up|} as a test statistic when the variances of the score elements are the same, is advantageous in the second case but not in the first case. Thus, applying a single and nonadaptive score-based test might not be powerful in gene-level analysis. The aSPU test offers a way to combine various score-based tests; it is based on a class of the sum of powered score (SPU) tests indexed by a positive integer γ. Specifically, the SPU(γ) test statistic is.We briefly introduce the idea of the aSPU test here. All score-based association tests require γ = ∞. One can treat γ as a factor that decides the weight on each score element. The aSPU test uses the minimum p value of the SPU tests as the test statistic, which provides a general data-adaptive method to test for associations. The set of γ ∈ {1, 2, ⋯, 8, ∞} was proposed by Pan et al. based on experiences [It is easy to see that the burden test and the sum of squared score (SSU) test are equivalent to the SPU(1) and SPU2) tests respectively. It was also shown that SPU(2) is equivalent to sequence kernel association test (SKAT) with the linear kernel and to Multivariate Distance Matrix Regression (MDMR) with the Euclidean distance (under the framework of LMM) . Further tests reΨ of the random effects reflected the genetic relatedness, where each Ψij was the Pearson correlation coefficient between 2 subjects i and j of 20,000 randomly selected SNPs. Our analysis was based on 821 subjects who did not have missing values in either the trait or the covariates. We only included common variants with minor allele frequencies (MAFs) greater than 0.05. Among those, we randomly imputed missing variants using MAF if the proportions of missing values were less than 1%. It resulted in a total of 595,304 SNPs included in our analysis. For the gene-level analysis, we used hg18 as a reference genome and each gene included the SNPs that were within 10,000 regions upstream or downstream of the gene’s coding region. In total, we included 22,434 genes in our analysis.The LMM we used for the GAW20 data was similar to that used by Aslibekyan et al.; we used the ratio of post- and pretreatment HDL as the trait, and we used age, gender, and study center as covariates. The only difference was the covariance matrix of the random effects. Our covariance matrix γ) and aSPU tests under the LMM. In addition to the SPU(1), SPU(2), and SPU(∞) tests where their theoretical equivalences with other existing gene-level tests are shown in the Methods section, we also performed the gene-level score test and the famSKAT (family-based sequence kernel association test) [APOA4 and ZNF259 on chromosome 11) were close to being significant. However, these 2 genes were detected by the SPU(1) test, suggesting that their association effects were not dominated by a small number of variants. We emphasize the adaptiveness property of the aSPU test by noting gene BUD13 on chromosome 11 and GUCD1 and SNRPD3 on chromosome 22, whose −log10 were not less than 3 by SPU(1), but much larger by the SPU(∞) test (as well as by a few other SPU tests and the aSPU test). We also note that APOA5 and ZNF259 were located nearby as shown in Fig. DDX42 on chromosome 17) almost significant at the genome-wide significance level, but the score test did not detect any loci near rs964184. Similarly, the famSKAT did not detect any significant gene.We conducted the SPU(on test)  using tho the SPU, SPU2),,γ) and a<insert Figure(s) 1 and 2 here>.− 8 as the genome-wide significance level for the SNP-level analysis. Even though rs964184 turned out to be the one most significantly associated with the trait among all the SNPs, its p value was far away from the genome-wide significance level, as shown in Fig. We also compared the gene-based tests to the score test for single variants. We used the usual 5 × 10In GWAS, individuals in pedigree data are not independent, thus motivating the use of  LMMs. We considered a general LMM with a random intercept that reflects the genetic relatedness among the subjects. We then conducted the aSPU test on the genes across the whole genome based on fitting a single null model, and identified 2 genes near SNP rs964184 to be nearly significant. In contrast, none of the SNPs, including SNP rs964184, were nearly significant in a standard single SNP-based analysis.We have demonstrated the applicability and usefulness of our proposed aSPU test in LMMs for association analysis of large pedigree data. Furthermore, our study has confirmed possible advantages and complementary roles of gene-level analyses with the adaptive aSPU test when compared to standard single SNP-based analyses.