Genome-wide association studies (GWAS) have proven successful in predicting genetic risk of disease using single-locus models; however, identifying single nucleotide polymorphism (SNP) interactions at the genome-wide scale is limited due to computational and statistical challenges. We addressed the computational burden encountered when detecting SNP interactions for survival analysis, such as age of disease-onset. To confront this problem, we developed a novel algorithm, called the Efficient Survival Multifactor Dimensionality Reduction (ES-MDR) method, which used Martingale Residuals as the outcome parameter to estimate survival outcomes, and implemented the Quantitative Multifactor Dimensionality Reduction method to identify significant interactions associated with age of disease-onset.To demonstrate efficacy, we evaluated this method on two simulation data sets to estimate the type I error rate and power. Simulations showed that ES-MDR identified interactions using less computational workload and allowed for adjustment of covariates. We applied ES-MDR on the OncoArray-TRICL Consortium data with 14,935 cases and 12,787 controls for lung cancer  to search over all two-way interactions to identify genetic interactions associated with lung cancer age-of-onset. We tested the best model in an independent data set from the OncoArray-TRICL data.BRCA1 , as the top marker to predict age of lung cancer onset.Our experiment on the OncoArray-TRICL data identified many one-way and two-way models with a single-base deletion in the noncoding region of From the results of our extensive simulations and analysis of a large GWAS study, we demonstrated that our method is an efficient algorithm that identified genetic interactions to include in our models to predict survival outcomes. A fundamental aim of studying human genetics is to predict disease risk from genomic data. Genome-wide association studies (GWAS) that used single-locus models by testing each single nucleotide polymorphism (SNP) for association with a phenotype, proved to be instrumental in identifying thousands of genetic variants associated with human traits and disorders –4. HowevWith the arrival of multi-dimensional and complicated genetic data sets, researchers have adapted to this growth by integrating machine learning methods to analyze complex genetic architectures. In genetic epidemiology, a popular series of methods were centered around a machine learning approach adapted to detect gene–gene interactions called the Multifactor Dimensionality Reduction (MDR) method. First introduced by Ritchie et al. (2001), MDR aimed at reducing high-dimensional genetic interacting loci to a one-dimensional binary variable that could be easily classified into high and low risk groups . While MCurrently, there are limited methods capable of identifying genome-wide genetic interactions efficiently with adjustment for covariates when studying age of disease-onset, such as a patient’s age at first diagnosis or recurrence of disease, for large-scale studies due to computational demands. It is important to have reliable estimates on the age of first diagnosis to understand the etiology of the disease and to tailor clinical practices, especially when determining the appropriate starting age for diagnostic screening, such as lung cancer screening . In thisIn this section, we discuss how we improved the computational efficiency without reducing accuracy to develop the ES-MDR method when analyzing SNP interactions  in association with age of disease-onset.ES-MDR improved the efficiency of Surv-MDR and applied the QMDR algorithm to analyze age of disease-onset in association with genetic interactions. Our novel ES-MDR approach used a combination of survival analysis and QMDR for continuous outcome analysis in two steps. In the first step, we started replacing event time and status with Martingale Residuals with covariate adjustment as a new continuous score. In the second step, we applied QMDR to efficiently categorize the genotype combinations into high-risk and low-risk groups. The best model was determined in the same way as QMDR, by using the cross-validated t-test statistic computed from a continuous variable attribute  to determine the best interaction and overall model.Selected K SNPs from all the SNPs in the data set and created a contingency table among every genotype combination of K SNPs.For each multi-locus genotype combination cell, summed the Martingale Residuals between samples with and without each genotype combination.Labeled cells “high-risk” if the sum of the Martingale Residuals was positive; otherwise negative Martingale Residuals were labeled “low-risk”.Pooled all the high-risk labeled cells into one group and all the low-risk labeled cells into another group to create a new one-dimensional variable.The novel algorithm for ES-MDR was performed as follows:Using Martingale Residuals to determine high or low risk group for survival data analysis was comparable to using the log-rank test statistic in Surv-MDR, however, more efficiently when classifying genotype combinations. It can be shown that the sum of the Martingale Residuals is a good surrogate variable of the log-rank test statistics for the purpose of determining high/low risk groups for each genotype combination. Next, we compared the similarities of the equations for Martingale Residuals and the log-rank test statistic. The sign and magnitude of the Martingale Residuals were dependent on the association of SNPs and the hazard function in the following equation:i(t) denotes the number of observed events that occur at each survival time t. The number of expected events was calculated using the cox-proportional hazards model with x as the genetic factor and y as the adjusted covariate. The log-rank test statistic was defined as the following:In this equation, δHere, we show that Martingale Residuals is equivalent to the numerator of the log-rank test statistic. Therefore, the sum of the Martingale Residuals is equal to the log-rank test statistic when the variance is set to 1. This infers that using Martingale Residuals as a substitute for the log-rank test statistic in evaluating genomic combinations associated with survival outcomes could provide the same data reduction and categorization process as Surv-MDR.Our purpose of running a simulation study was to evaluate how well ES-MDR performed and how well it performed compared with Surv-MDR. To demonstrate the strength of the proposed method, two simulations were designed to evaluate the testing score’s null distribution to evaluate the type I error rate and to analyze power.U . Then we simulated n continuous outcomes from a standard normal distribution. The SNP and continuous outcome data were created independently to ensure that there were no associations between SNPs and the outcome. These steps were repeated to create 1000 null data sets for 24 different groups varied by the number of SNPs, sample size, and MAF. As a result, a total of 24,000 data sets were generated. Simulations were conducted in R 3.0.0 . To determine whether the type I error rate was close to 5%, we analyzed the percentage of times that ES-MDR randomly identified two interacting SNPs from a null data set.The first simulation study was created to estimate the 5% type I error threshold by evaluating an empirical null distribution with independent non-interacting SNPs and quantitative outcome values. Here, we created sets of SNPs  with additive coding and sample sizes  in the simulation data. For every combination of m and n, we simulated m SNPs with minor allele frequencies (MAF) drawn from the uniform distribution over the interval The second simulation study was created to evaluate the power of ES-MDR with a data set that included quantitative outcome variables and a pair of functional interacting SNPs and 18 non-interacting SNPs. Surv-MDR was performed to evaluate whether ES-MDR was as effective as Surv-MDR in identifying functional SNPs.The simulation data sets included different penetrance functions that described the probabilistic relationship between the quantitative outcome variable and functional SNPs generated with additive coding. We considered two different MAFs (0.2 and 0.4) and seven different broad-sense heritability statistics  to create a total of 14 unique model combinations, where the two functional SNPs associated with the outcome were evenly distributed across the seven heritability statistics. To create a purely epistatic model, each of the 14 unique models had one or the other functional SNP (MAF 0.2 or 0.4) with no main effects. The 14 allele-heritability frequency combinations were replicated five times to generate 70 models with varying sample sizes that included size (n) = {400, 800, 1600}.fij be an element from the ith row and jth column of a penetrance function. We generated the binary variable from a Bernoulli distribution with the following:Assuming SNP1 and SNP2 were the two functional SNPs. Let We randomly selected 200 high-risk subjects and 200 low-risk subjects from each of the 70 probabilistic models to create one simulated data set. We repeated this simulation 100 times to obtain at total of 7,000 data sets.To generate the survival time, we used the Cox-proportional hazards (Cox ph) model:0(t) is the baseline hazard function with a Weibull distribution using the shape parameter of 5 and the scale parameter of 2. The x is the genetic factor fixed at value 1 for high risk patients and 0 for low risk patients. ß represents the effect size or the log hazard ratio for a one-unit increase in x . The censoring fractions were sampled from the uniform distribution over the interval U  from the Bernoulli distribution, resulting in 40% censoring. Finally, we merged survival time and censoring status with the SNP data.In this equation, ht) denotes the number of expected events that occurred at each survival time t. Assuming a null model with no target effects (ß = 0), this residual is the difference between the observed events and expected number of events. The sign and magnitude of the Martingale Residuals are dependent on the association of SNPs and the hazard rate function. Each individual genotype with a positive Martingale Residual  was classified as high-risk. Otherwise, a negative Martingale Residual was classified as low-risk. For every multi-locus genotype combination of SNPs, we computed the sum of the Martingale Residuals to obtain a new variable that could be used to classify into the high-risk or low-risk group.We used Martingale Residuals in our novel ES-MDR method to classify each multi-locus genotype combination into high-risk and low-risk groups. The Martingale Residual is the stochastic component and in residual form gives the following:To estimate the power of the proposed method, we ran ES-MDR on each of the 7000 data sets and searched for the best model over all possible one- , two- , and three-way  interaction models, using the T-statistic testing score. We also used the 95th percentile of the testing score from the null models as a threshold to guard against any non-significant findings. The power was estimated as the percentage of time ES-MDR correctly included the two functional interacting SNPs in the best model out of each set of 7000 data sets. This significant threshold for the results was at the 0.05 level. For comparison, we ran Surv-MDR on the simulated data to define its power. Training and testing scores for ES-MDR were analyzed using two-fold cross-validation. The rational for using two-fold cross-validation  was thatA total of 533,631 SNPs from 57,775 individuals in the OncoArray-TRICL Consortium population-based study, selected from 29 studies across North America and Europe, as well as Asia, were genotyped using the Illumina OncoArray-500K BeadChip Platform, which included the genome-wide backbone and select loci known to be associated with cancer phenotypes. To facilitate efficient genotyping and minimize variability that might arise from genotyping at multiple sites, genotyping was conducted at the following five institutions: the Center for Inherited Disease Research, the Beijing Genome Institute, the Helmholtz Zentrum München, Copenhagen University Hospital, and the University of Cambridge. Quality control steps described previously were followed for this OncoArray-TRICL data set . The folWe applied ES-MDR to the OncoArray-TRICL Consortium population-based study to identify genetic interactions in association with lung cancer age-of-onset. The OncoArray-TRICL Consortium is a collaboration among world leaders to investigate common causes of cancer susceptibility and progression . Lung caTo build a predictive model that combined the strength of both one-way and two-way models, we took all the SNPs involved in the top 1000 one-way models and all the SNPs from the top 1000 two-way interactions models and applied a penalized Cox regression method to filter and select the best predictive models to evaluate genetic factors associated with age of lung cancer onset. We ranked the test scores from highest to lowest and picked the top SNPs that best predicted lung cancer onset.To construct predictive models linking SNPs to censored survival data, we used the least absolute shrinkage and selection operator (Lasso) penalized estimation for the Cox regression model to select top SNPs that were relevant to patients’ ages of lung cancer onset to create a prediction model with a parsimonious set of SNPs that could provide good prediction accuracy . The LasSurvival plots were generated using the Kaplan–Meier (KM) method to visualize the differences in age of lung cancer onset between high-risk and low-risk groups based on top identified SNPs associated with lung cancer risk. To adjust for additional factors related to patient survival, the Cox ph regression model included adjustment for smoking status as a covariate in the model.f(X), the time-dependent sensitivity and specificity functions were defined as follows:To assess the performance of our model in predicting lung cancer onset at different age intervals, we applied time-dependent receiver operating characteristic (ROC) curve and area under the curve (AUC) to evaluate the predictive performance of the best models, previously introduced by Heagerty et al. . In our t|f(X)) curve for any time t as the plot of sensitivity) versus 1—specificity) with the cut-off point c varying. The AUC is the area under the ROC(t|f(X)) curve, which was denoted as AUC(t|f(X)) [δ(t) is the event indicator at time t. In this study, a larger AUC at time t based on the score function f(X) indicated better predictability of time-to-event at time t as measured by sensitivity and specificity evaluated at time t.We defined the corresponding ROC((t|f(X)) . Here, tIn the first simulation, we determined whether the type I error rate was close to the expected value when there were no SNP interaction effects. Assuming a data set that included 20 non-interacting SNPs and a total sample size of 400, we expected the type I error rate to be 0.05.In Fig. In the second simulation, we estimated the power of ES-MDR with a data set that included quantitative outcome variables and a pair of functional interacting SNPs and 18 non-interacting SNPs. We determined whether the power of ES-MDR was comparable to Surv-MDR in identifying the two functional SNPs. We counted the number of times that the functional SNP pair was correctly identified and divided that number by the total number of data sets (500 for this simulation) to get the estimated success rate.Figure Figure We compared the computing time between ES-MDR and Surv-MDR for 100 simulated data sets, for one-, two-, and three-way interactions, and with ten-fold cross-validation. The computing time for Surv-MDR was 734.5 min versus 2.25 min for ES-MDR, both of which were run on 1 node in the high-performance computing cluster called Discovery with AMD 3.1 Ghz CPU and 64 GB of memory. Discovery uses a Linux RedHat 6.7 operating system and is comprised of 160 computing nodes (3000 + cores), 12.5 TB of memory, and is available to the Dartmouth research community.The main goal was to identify SNPs with main effects and SNP interactions that were associated with lung cancer susceptibility at different ages of disease onset. Using a population-based study, we applied ES-MDR on the OncoArray-TRICL Consortium data with 14,935 cases and 12,787 controls for lung cancer to search over all one-way and two-way interactions to identify genetic interactions in relation to lung cancer age-of-onset. For this study, we included 533,631 genotyped variants and removed SNPs in linkage disequilibrium (LD) > 0.1 .TULP1, FKBP5 (rs6906359), in-between genes GTF2IP1, PMS2P5 (rs149743903), and in the deletion of a noncoding region of BRCA1 (rs749410065) (NC_000017.10:g.41196821delT per Human Genome Variation Society nomenclature), and for a top two-way interacting SNPs in gene regions of BRCA1 (rs749410065) and CBR1, LOC100133286 (rs151043730) displayed a clear separation of curves between the high- and low-risk groups. This demonstrated the efficacy of ES-MDR using Martingale Residuals to differentiate high risk and low risk groups based on genotype variation when evaluating lung cancer age-of-onset. We continued our analysis with a comparison of smoking only and smoking plus SNP models to determine the best performance in predicting lung cancer onset at different ages. We used a common graphical plot called the area under the receiver operating characteristic (ROC) curve, also known as AUC, to measure the performance of our models to discriminate the best parameters at predicting lung cancer onset at different ages based on accuracy. In Fig. Table In this study, we present a novel algorithm to identify genetic interactions associated with the age-of-onset for lung cancer. We demonstrated in two simulation studies that our ES-MDR method was properly controlled for at the 5% type I error rate under the null distribution and improved power to detect causal SNPs. We identified new loci that were biologically plausible for lung cancer onset using the large OncoArray-TRICL data with 27,722 individuals. There are two unique contributions from this study. First, we offer a more computationally efficient algorithm, ES-MDR, a method that analyses survival data by using Martingale residuals in place of survival outcome data. Second, ES-MDR includes the ability to adjust for covariates, such as smoking status, a necessary step to control for confounding factors, whereas existing methods, used for survival analysis such as Surv-MDR, are unable to provide.Using the MDR method to reduce the size of multiple dimensions to a single dimension to identify multi-locus genetic interactions in high-dimensional genomic data sets has been a well-established approach. Richie et al. (2001) first introduced MDR, a non-parametric  and genetic model-free  model, that condensed multiple genetic loci into a single variable in order to categorize genotypes into two groups . The goaLike the original MDR algorithm, Surv-MDR is a non-parametric and genetic model-free method proposed by Gui et al. (2011), and it was developed to allow for the analysis of time-to-event data, such as patient survival time or time to disease relapse . Surv-MDQMDR optimized the MDR algorithm by offering a computationally efficient way to analyze quantitative or continuous trait outcomes. QMDR compared the mean value of each multi-locus genotype to the overall mean and labeled each genotype combination as “high-risk” or “low-risk”. Cross-validation was also implemented in QMDR to identify the optimal set of K SNPs and overall best model. For each K-way interaction, the steps used for a k-fold cross-validation were similar to the Surv-MDR method except for the step to identify the best K-way interaction. In this case, the largest T-test statistic was used instead of the square of the log-rank statistic when identifying the best interaction model. Inspired by the computational capabilities of QMDR to analyze quantitative outcomes associated with genetic variations, we leveraged this method’s straightforward computing efficiency to evaluate survival outcome data for time-to-event analysis.Our approach transformed survival data  into a single variable, Martingale Residuals, to use as a surrogate for time-to-disease and disease status, with application of QMDR for rapid processing of genotype combinations into high and low risk groups. We were able to identify thousands of significant one-way and two-way models using ES-MDR and cross-validation when applied to the lung cancer OncoArray-TRICL data set. We were unable to compare the results of ES-MDR and Surv-MDR, both because Surv-MDR would have taken an extensive amount of time  to conduct a genome-wide genetic interaction analysis using the large OncoArray-TRICL data set, and because the current Surv-MDR algorithm would not allow for adjustment of confounding factors such as smoking status.When searching for SNP interactions using real data, we chose a two-fold cross-validation instead of a ten-fold cross-validation to evaluate the optimal one-way and two-way interaction models as described previously . From thWe explored prediction models that included SNPs that could be used to forecast lung cancer onset. Figure While our novel ES-MDR overcame some of the limitations described in previous methods used to evaluate genetic interactions, it was not without some of its own disadvantages. When analyzing survival data, the method did not directly evaluate survival variables such as time and event status. As a result, when using Martingale Residuals instead of specific survival outcomes data, we might be missing some important information that was needed to identify associations between SNP interactions and survival outcomes. Our QQ plot analysis from real data indicated a strong departure from the null distribution which indicated that there might exist a systematic bias. This result could be due to a combination of the large sample size and continuous outcome. As a result, we used permutation tests to evaluate the results from the OncoArray-TRICL data set. Another limitation came from over parameterizing our models, resulting in many multifactor cells with missing data . This diES-MDR is a powerful alternative to Surv-MDR for identifying interactions, especially at the genome-wide scale. We demonstrated its ability to identify high-order genetic interactions in simulated and real data sets. Although ES-MDR addresses previous limitations of Surv-MDR and other MDR-like methods, there are ways in which this method can be improved. While ES-MDR had greatly improved computing efficiency, genome-wide scans for interactions will still require massive computing resources, especially to analyze higher-order interactions. It will be necessary to optimize the selection of SNPs in predictive models, for example, by selecting genes known to participate in biological and metabolic pathways . This caIn summary, the ES-MDR method provides a way to analyze high-order interactions at the genome-wide scale to advance studies of genetic interactions. We developed a new method that efficiently captures non-linear and high-order interactions for time-to-event analysis.In general, ES-MDR has improved power performance relative to Surv-MDR using simulated data. Based on the noticeable trends, we are confident that with bigger sample sizes, ES-MDR will continue to significantly gain in power to detect functional interacting SNPs without inflating the type I error rate. Providing new and improved methods to analyze epistasis or gene interactions may offer new opportunities to not only explain the missing heritability for complex disease risk, but can also potentially detect new genetic determinants that is important for clinical utility such as disease diagnosis and prognosis.


In genetic association analysis, several relevant phenotypes or multivariate traits with different types of components are usually collected to study complex or multifactorial diseases. Over the past few years, jointly testing for association between multivariate traits and multiple genetic variants has become more popular because it can increase statistical power to identify causal genes in pedigree- or population-based studies. However, most of the existing methods mainly focus on testing genetic variants associated with multiple continuous phenotypes. In this investigation, we develop a framework for identifying the pleiotropic effects of genetic variants on multivariate traits by using collapsing and kernel methods with pedigree- or population-structured data. The proposed framework is applicable to the burden test, the kernel test, and the omnibus test for autosomes and the X chromosome. The proposed multivariate trait association methods can accommodate continuous phenotypes or binary phenotypes and further can adjust for covariates. Simulation studies show that the performance of our methods is satisfactory with respect to the empirical type I error rates and power rates in comparison with the existing methods. Genome-wide association studies (GWAS) intend to find genetic variants such as single nucleotide polymorphisms (SNPs) associated with common traits or with complex diseases , 2. AssoSeveral statistical methods have been developed to identify the association between multivariate traits and a genetic variant , 5. CurrAlthough these new developments keep many benefits, existing methods have some potential limitations . Most cuIn this investigation, we develop the statistical methods for identifying pleiotropic effects of genetic variants on multivariate traits using collapsing and kernel methods with pedigree- or population-structured data. The proposed multivariate trait association method is able to handle binary phenotypes or continuous phenotypes and further can adjust for covariates. Moreover, the proposed multivariate trait association method not only can leverage the dependence on the phenotypes but also can account for the sample relatedness in the pedigree-based or population-based structured data.The rest of the article is organized as follows. In the materials and methods section, we construct the multivariate effect model using the joint GEE model formulation (JGEE) . We applN independent pedigrees and each pedigree has ni subjects. We assume that the ni subjects have been sequenced in a genetic region of interest  that contains p variants. Let yik = T be the ni × 1 phenotype vector for the kth phenotype of the ith pedigree. Let yi =  be the (ni × K) × 1 response vector for the K phenotypes that we are interested in. Let xim = T be the ni × 1 vector for the mth covariate of the ith pedigree. Let xi =  be the ni × (q + 1) covariate matrix for the (q + 1) nongenetic covariates that we want to adjust for. Let αk = T be the (q + 1) × 1 vector of regression coefficients of the (q + 1) nongenetic covariates with the element αmk being the effect of the mth covariate on the kth trait. Let gi =  be the ni × p genetic matrix for p genetic variants in a target region of interest where gil = T is the ni × 1 vector for a genetic variant l . Let βk = T be the p × 1 vector of regression coefficients of the p genetic variants with the element βlk being the effect of the lth genetic variant on the kth trait.To describe the proposed multivariate trait association method based on the pedigree- or population-based structured data, we suppose that there are Xi = IK ⊗ xi be the (ni × K) × ((q + 1) × K) covariate matrix and Gi = IK ⊗ gi be the (ni × K) × (p × K) genotype matrix for the ith pedigree where IK is an identity matrix of dimension K × K and ⊗ stands for the Kronecker product. According to the generalized linear model [yijk is f(yijk) = exp[{yijkθijk − a(θijk) + b(yijk)}/ϕ] with two moments, μijk = E(yijk) = ∂a(θijk)/∂θijk and Var(yijk) = (∂μijk/∂θijk)ϕ, where ϕ is a scale parameter. Let θi = T be the (ni × K) × 1 vector with the components θik = T, k = 1, 2, ⋯, K and ηi = T be the (ni × K) × 1 vector with the components ηik = T = xiαk + giβk, k = 1, 2, ⋯, K for the kth trait of the ith pedigree.We let ar model , we assuK correlated traits and genetic variants, which is given as follows:g−1(•) is the inverse function of g(•) and is a response-specific link function [μi = T is the (ni × K) × 1 vector of the expected mean of the multivariate traits yi = T, α = T is the ((q + 1) × K) × 1 vector of regression coefficients of the (q + 1) nongenetic covariates for the K correlated traits, and β = T is the (p × K) × 1 vector of regression coefficients of the p genetic variants for the K correlated traits.Based on the joint GEE model formulation , we consfunction , μi = (μRni(φ) and RK(γ) be the ni × ni within-in cluster correlation matrix and the K × K multivariate-response cluster correlation matrix, which depend on a vector of parameters φ and γ, respectively. The (ni × K) × (ni × K) working (or approximate) covariance matrix of yi is given by [Ai = diag is a (ni × K) × (ni × K) block diagonal matrix with the components Aik = diag, k = 1, 2, ⋯, K being the ni × ni diagonal matrices. According to equation (Let given by . (2)Vi=AH be a p × p correlation matrix of genotype scores with element Hll′ for markers l and l′. Let ml denote the minor allele frequency (MAF) of the lth marker. Let ni × K) × 1) vector of the standard residuals with components Sik = T, k = 1, 2, ⋯, K, where Vi and μi. Here and henceforth, all estimates are calculated based on the null hypothesis of the genetic effects β equal to zero. All unknown parameters and the working within-in and multivariate-response cluster correlation matrices are estimated by the R package JGEE [We let age JGEE .(1) Homogeneous Kernel Statistic. We suppose that wl is a marker-specific weight of the lth variant and assume that the genetic effects on the K different phenotypes are homogeneous . Based on the JGEE model with the genotype as random variables considered, we propose the homogeneous quadratic (kernel) association statistic (HoK) as follows:Zl = ∑k=1KZlk, ,Aik, and ik = diag that is a ni × ni diagonal matrix for the kth phenotype of the ith pedigree. The null distribution of κHo asymptotically follows a mixture chi-square distribution ∑l=1pλlχl,12, where χl,12s are independent random variables following a chi-square distribution with one degree of freedom and  are nonzero eigenvalues of the null covariate matrix of Ωi is a ni × ni matrix of genetic correlations for all ni individuals in the ith pedigree, which has the same definition given by Schaid et al. [j and j′ in the ith pedigree is unknown, the elements of the genetic correlation Ωi can be estimated through genomic data [d et al.  and can d et al. . When thmic data , 45, andmic data (4)Ω^i=1(2) Heterogeneous Kernel Statistic. We assume that the genetic effects on the K different phenotypes are heterogeneous . The heterogeneous quadratic (kernel) association statistic (HeK) is defined bywlk is a marker-specific weight of the lth variant of the kth trait. The null distribution of κHe asymptotically follows a mixture chi-square distribution ∑l=1p × K) are nonzero eigenvalues of the null covariate matrix of p values of κHo and κHe are approximately calculated by Kuonen's saddlepoint method [κHo) and the HeK test (κHe) is shown in Appendix Theoretical t method  and obtaith pedigree. On the basis of the HoK test (κHo) and the HeK test (κHe) in equations , we propose the burden test (BT) as follows:We let quations  and 5)  g~iT=∑l=Then,The null distribution of BT asymptotically follows a chi-square distribution with one degree of freedom.pHo, pHe, and pBT denote the p values obtained by the HoK, HeK, and BT statistics. Based on the idea of the p value combination method through the Cauchy distribution [Let ribution –49, we p(1) Homogeneous Omnibus Test. Combining the pHo with the pBT, we construct the homogeneous omnibus test (HoO) as follows:FC−1 stands for the inverse cumulative distribution function of the standard Cauchy distribution.(2) Heterogeneous Omnibus Test. Combining the pHe with the pBT, we construct the heterogeneous omnibus test (HeO) as follows:OHo test and the OHe test asymptotically follow a standard Cauchy distribution [p values of the OHo test and the OHe test are calculated by the R package RNOmni [The null distributions of the ribution –49. The e RNOmni .The kernel statistic, the burden test, and the omnibus test are also applicable to the X chromosome. Additional technical information for extensions to the X chromosome is shown in Appendix p value SKAT statistic (mPK), and the minimum p value burden statistic (mPB) [We conduct the numerical simulation studies to assess the finite sample performance of the proposed methods and evaluate the comparison results with two existing methods, the minimum ic (mPB) . The twoic (mPB) . Based oic (mPB) , 43, 51,ic (mPB) . A 3 kb k for individual j in the ith family are generated from the multivariate linear model in equation T, Xi = I2 ⊗ xi, Gi = I2 ⊗ gi, α = T, β = T, and εi = T. Here, the elements xi0 = T of the covariance matrix xi =  is a 3 × 1 vector of all ones. The elements xi1 = T of xi are independently generated with an equal probability of being 0 or 1. The elements xi2 = T of xi are generated from a multivariate normal distribution with a mean of 0.5 and a covariance matrix with diagonal entries of 1 and all off-diagonal entries of 0.1. The regression coefficients of the covariate matrix xi for the kth correlated trait are given by αk = T = T and αk = T = T, respectively, for continuous traits and binary traits for k = 1, 2.In the heterogeneous population with nuclear family data considered, continuous and binary phenotypes for trait equation  with K =εi = T in equation ) with diagonal entries of 1 and all off-diagonal entries of 0.2 and a subject-across-response correlation matrix ) with diagonal entries of 0.3 and all off-diagonal entries of 0.1. Similarly, binary traits yi in equation ) and the same subject-across-response correlation matrix ) as the continuous traits yi in equation T, in equations T, are set by 0.095 × ∣log10(ml)∣ or 0.095 × log10(ml) corresponding to the risk or protective variant l, l = 1, 2, ⋯, p [β1 ≠ β2), the genetic effects β1 for the first traits yi1 are set as described above, while the genetic effects β2 for the second traits yi2 are set by zero. On the other hand, under the assumption that the genetic effects on the two different phenotypes are homogeneous , the genetic effects β1 and β2 for the first and second traits have the same settings as described above.For type I error simulations, the regression coefficients of genetic variants, quations  and 12)β =  with shape parameters λ1 > 0 and λ2 > 0 [wl of variant l on the phenotypes, we consider the unweighted marker-specific weight with wl = Beta = 1 and the weighted marker-specific weight with wl = Beta [We simulate 1,400 nuclear families with 800 nuclear families from the European samples and 600 nuclear families from African-American samples. The marker-specific weight d λ2 > 0 . To stud, 1, 25) . The empwl = Beta or wl = Beta for variant l for binary traits. On the other hand, the empirical type I error rates of the proposed methods for X chromosome analyses with binary traits are depicted in Table In summary, our simulation results show that the proposed multivariate trait association methods, HoK, HoO, HeK, HeO, and BT, have reasonable control of type I error rates for continuous traits or binary traits whether the marker is X chromosomal or autosomal. On the other hand, the existing methods, mPK and mPB, yield well-controlled type I error rates for the autosome analyses or the X chromosome analyses with continuous traits , regardlwl = Beta are higher than that with an unweighted marker-specific weight of wl = Beta = 1. The heterogeneous kernel statistic (HeK) has slightly greater empirical power rates than other methods, when the genetic effects on the different phenotypes are heterogeneous , and causal SNPs have positive effects or negative effects on phenotypes. On the other hand, the existing method, mPB, has bigger empirical power rates, when the genetic effects on the different phenotypes are heterogeneous , and all causal SNPs have a positive association on phenotypes. Moreover, the empirical power rates of the homogeneous omnibus test (HoO) are larger than that of the other six competing methods, when the genetic effects on the different phenotypes are homogeneous . Evidently, the seven competing methods have their respective advantages in identifying the association between genetic effects and multiple continuous traits for autosome analyses.Similar empirical power rates are obtained from the working within-cluster and multivariate-response correlation matrices of the proposed methods, HoK, HeK, and BT, considered to be unstructured. Hence, these empirical power rates are not shown in order to save space. On the other hand, the seven competing approaches display a similar performance in testing for the X chromosome analyses with continuous traits Appendix .β1 ≠ β2). On the other hand, the empirical power rates of the homogeneous omnibus test (HoO) are bigger than that of the other competing methods, when the genetic effects on the different phenotypes are homogeneous . As expected, in general, the heterogeneous kernel statistic (HeK) is more powerful than the homogeneous kernel statistic (HoK), when the genetic effects on the different phenotypes are heterogeneous . On the other hand, the homogeneous kernel statistic (HoK) is more powerful than the heterogeneous kernel statistic (HeK), when the genetic effects on the different phenotypes are homogeneous . In a word, the proposed methods, HoK, HoO, HeK, HeO, and BT, have their respective merits in examining the association between genetic effects and multiple binary traits for autosome analyses.Similarly, when the working within-cluster and multivariate-response correlation matrices of the proposed methods, HoK, HeK, and BT, are considered to be unstructured, the empirical power rates have similar results and thus they are omitted. On the other hand, the empirical power rates of the proposed methods for X chromosome analyses with binary traits are presented in Figure In summary, the seven competing methods, HoK, HoO, HeK, HeO, BT, mPK, and mPB, have their respective merits in diagnosing whether genetic effects are associated with multiple continuous traits for autosome analyses or the X chromosome analyses. Similarly, the proposed methods, HoK, HoO, HeK, HeO, and BT, have their respective advantages in examining whether there are associations between genetic effects and multiple binary traits for autosome analyses or the X chromosome analyses.To furthermore examine the performance of the proposed methods, additional simulation studies for continuous traits and binary traits are presented in Appendix wl = Beta under the alternative hypothesis for continuous data are 0.83 and 0.91 minutes, respectively, while that for binary data are 4.77 and 4.80 minutes, respectively. Therefore, in the current version, such a framework algorithm implementation is unsatisfactory for analyzing a large-scale high-dimensional data set in practice.However, we note that the proposed methods have a high computational cost, especially for binary data. Under our simulation setting and framework, we carry out a single simulated data set by using a computer based on one CPU core at 2.1 GHz. The average computational times of the homogeneous and heterogeneous tests with a weighted marker-specific weight In this investigation, we develop a retrospective framework for identifying the pleiotropic effects of genetic variants on multivariate traits by using collapsing and kernel methods with pedigree- or population-structured data. The proposed framework, corresponding to the burden test, the kernel test, and the omnibus test, provides a sound basis for genetic association analyses for autosomes and the X chromosome. The proposed multivariate trait association methods based on the JGEE model can flexibly accommodate continuous phenotypes or binary phenotypes and further can adjust for covariates.One critical advantage of the proposed methods is that the homogeneous kernel statistic (HoK), the heterogeneous kernel statistic (HeK), and the burden test (BT) retain all of the benefits of the retrospective tests proposed by Schaid et al.  who treaAnother important benefit of the proposed method is that the HoK test, the HeK test, and the BT test keep the benefits of the JGEE model that validly account for complex correlations between subjects within the cluster (within-cluster correlations) and between different phenotypes from the same subjects (multivariate-response correlations). Moreover, the proposed test statistics, HoK, HeK, and BT, based on the JGEE model can efficaciously account for covariate adjustment whether the phenotypes are continuous or binary.wl = Beta = 1 and an exchangeable structure of the working within-cluster and multivariate-response correlations are recommended for the practical data analysis if the data cannot sufficiently provide valid information for estimating the structures of the working within-cluster and multivariate-response correlations before the start of the data analysis. Moreover, the homogeneous kernel statistic (HoK) is more robust than the heterogeneous statistic (HeK) in controlling the empirical type I errors, because the null distribution of the HeK statistic asymptotically follows a mixture chi-square distribution with a larger degree of freedom, in comparison with the null distribution of the HoK statistic. However, the HeK statistic is more powerful than the HoK statistic when the genetic effects on the different phenotypes are heterogeneous.Our simulation studies show that an unweighted marker-specific weight On the other hand, our simulation results show that for the autosome analyses or the X chromosome analyses with continuous traits, the seven competing methods, HoK, HoO, HeK, HeO, BT, mPK, and mPB, show good performance with well-controlled type I errors, while the seven competing methods have their respective merits for identifying the association between the genetic effects and multiple continuous traits. In addition, our simulation results show that for the autosome analyses or the X chromosome analyses with binary traits, the proposed methods, HoK, HoO, HeK, HeO, and BT, can control empirical type I errors with lower correlations of phenotypes or with lower dimensions of phenotypes , while tThe proposed multivariate trait association methods have their limitations. First, these proposed methods cannot simultaneously include the continuous traits and binary traits in analysis. Thus, future studies are needed to extend the idea of the proposed multivariate trait association methods for simultaneously considering continuous traits and binary traits in analysis. Second, the multivariate trait association methods, based on higher correlations of phenotypes or higher dimensions of phenotypes, easily suffer from the problem of the inflated type I errors, especially when the binary traits are considered Appendix . Althoug


Function and bother are related but distinct aspects of health-related quality of life. The objective of this study was to compare quantitatively the relative impacts of function and bother in urinary, sexual, and bowel outcomes on health utility as a reflection of health-related quality of life in men with prostate cancer. Our analysis included participants in the Cancer of the Prostate Strategic Urologic Research Endeavor utility supplementary study, with a final cohort of 1617 men. Linear regression on the patients’ function and bother summary scores (0-100) from the University of California, Los Angeles Prostate Cancer Index was performed to predict bias-corrected health utilities. Urinary and sexual bother were associated with each health utility, and their coefficients were 3.7 and 20.8 times greater, respectively, than those of the corresponding function. To our knowledge, our study provides the first quantitative and direct comparison of the impacts of function vs bother on health utility. Health-related quality of life (HRQoL) is a critical determinant of satisfactory outcomes after prostate cancer (PCa) management and is affected differently by various treatments . BesidesStandardized patient-reported HRQoL questionnaires, such as the University of California, Los Angeles Prostate Cancer Index (UCLA-PCI), measure function and bother separately for each HRQoL domain. However, the extent to which the subscore for function or bother reflects subjective HRQoL remains unclear.The HRQoL score is needed to calculate a health utility based on prederived weights or formulae ,11. HealRecently, our group determined robust utilities for various outcomes among men participating in the Cancer of the Prostate Strategic Urologic Research Endeavor (CaPSURE) . In thisWe used data from 1617 patients in the CaPSURE utility supplementary study (CaPSURE-USS), which was a nested cross-sectional survey that measured utilities using the standard gamble method . The oriPatients’ function and bother were assessed using the UCLA-PCI standardized scores . Then the summarized score for each domain was calculated as an average value of all items in that domain. The primary outcome was the utility for each domain  status. We used bias-corrected utilities using 1-parameter weighting ,15,16. TWe used linear regressions to predict the utility associated with the summarized function and bother scores in each domain as simple models . Then we generated linear regression models using the scores of all individual UCLA-PCI questions to investigate detailed associations in a full model .2R, F statistics, root-mean-square error, and mean absolute error to evaluate the goodness of fit of models. The importance of individual variables was calculated as the relative contribution to variance explained (RCVE) (2Rfull − 2Rreduced)/ 2Rfull, where 2Rfull represents the share of explained variability by all the predictors in the model, whereas 2Rreduced represents the explained variability without the specific variable whose contribution is being evaluated. We performed subgroup analyses comparing the utilities predicted by the simple models with the actual utility values according to the initial treatments, disease status at the survey, and each functional status, respectively.To match scales between the standardized UCLA-PCI scores (0-100) and utilities (0-1), we divided the UCLA-PCI scores by 100. We used adjusted d (RCVE) . The forIn the regression models, urinary and sexual bother demonstrated a statistically significant association with each utility, whereas the corresponding summarized function scores were not associated . The simWith regard to urinary function, subjective urinary control (question 2 in the urinary domain) was found to have the highest association with urinary utility. Achieving intercourse  was most closely associated with sexual utility. All functional questions in the bowel domain  were correlated with bowel utility.All results for goodness of fit are shown in To our knowledge, our study is the first quantitative and direct comparison evaluating the relative impact of function vs bother outcomes on health utilities among men with PCa. We found that bother had a greater impact on urinary and, in particular, sexual HRQoL compared with function. This result is not entirely unexpected, because bother reflects how a patient perceives a given level of function. The high proportion of elderly men with baseline erectile dysfunction and decreased sexual activity may contribute to the weak correlation between sexual bother and function and their impacts on this health utility. Older men may also adapt more easily over time to sexual dysfunction than to urinary incontinence or bowel dysfunction . BecauseThe results regarding the bowel domain should be interpreted cautiously, because only 3.5% of men suffered from bowel issues. All results of goodness of fit were not high and low variance explained can be limitations of the models. Other limitations of our study include the restricted generalizability of our results because of the characteristics of the study population: 93.4% were white men living in the United States, and HRQoL may vary by race or ethnicity and/or geographic region. The majority of men in CaPSURE-USS were long-term survivors of PCa and relatively old at the time of the survey (mean age of 73 years). The findings should, therefore, be applied cautiously to men with short-term follow-up or to younger patients. The UCLA-PCI focuses on urinary incontinence rather than irritative or obstructive symptoms, so the results may not extend to these other subdomains and are more relevant to surgical than to radiation patients.Despite these limitations, our study helps us understand how HRQoL is influenced by patients’ functional status and recovery process. Our results suggest that urinary and sexual bother have greater impact on HRQoL than function per se in men with PCa.CaPSURE was funded until 2007 by TAP Pharmaceutical Products Inc. It is currently funded by US Department of Defense grant W81XWH-13-2-0074 and the University of California, San Francisco, Department of Urology. This study was also partly supported by a grant from the National R&D Program for Cancer Control, Ministry of Health and Welfare, Republic of Korea (HA17C0039).Role of the funder: None of the sponsors had any access to the data or any influence on or access to the analysis plan, the results, or the manuscript.Disclosures: All authors report no conflict of interest.pkaa044_Supplementary_DataClick here for additional data file.


We conclude that the covariance between random effects is a key parameter for estimation, especially when partitioning phenotypic variance by multi-omics layers.As a key variance partitioning tool, linear mixed models (LMMs) using genome-based restricted maximum likelihood (GREML) allow both fixed and random effects. Classic LMMs assume independence between random effects, which can be violated, causing bias. Here we introduce a generalized GREML, named CORE GREML, that explicitly estimates the covariance between random effects. Using extensive simulations, we show that CORE GREML outperforms the conventional GREML, providing variance and covariance estimates free from bias due to correlated random effects. Applying CORE GREML to UK Biobank data, we find, for example, that the transcriptome, imputed using genotype data, explains a significant proportion of phenotypic variance for height (0.15,  Linear mixed models have bias due to the assumed independence between random effects. Here, the authors describe a genome-based restricted maximum likelihood, CORE GREML, which estimates covariance between random effects. Application to UK Biobank data highlights this as an important parameter for multi-omics analyses of phenotypic variance. This is largely because GWASs typically examine SNP trait associations one at a time, generating a large number of tests across the genome for which the stringent Bonferroni correction is applied. To overcome this problem, a whole-genome approach that jointly considers all available SNPs has been introduced2, allowing estimation of the proportion of phenotypic variance explained by genome-wide SNPs, i.e., SNP-based heritability. Central to this approach is the use of linear mixed models3 (LMMs)—extensions of random-effects models or variance-component models4—which treat SNP effects as random.Genome-wide association studies (GWASs) have been incredibly successful in identifying genetic variants associated with complex traits. However, the proportion of phenotypic variance explained by genome-wide significant single nucleotide polymorphisms (SNPs) is far lower than the narrow-sense heritability estimate5. As multi-omics data become increasingly available6, variance partitioning using LMMs will become indispensable to uncover the relative contributions of multiple ‘omes’ to phenotypic variation. Alongside GREML, linkage disequilibrium score regression (LDSC) provides an alternative way for SNP-based heritability estimation and variance partitioning, using only GWAS summary statistics without the need to access individual genotypes8. LDSC or stratified LDSC also treats SNP effects as random8, and for the same set of individual-level genotype data, this approach generates similar estimates as GREML11.Using genome-based restricted maximum likelihood (GREML) for parameter estimation, LMMs are a key tool not only for SNP-based heritability of complex traits but also for variance partitioning in general. For example, when heritability is partitioned by functional annotation of SNPs using a LMM with multiple random effects, GREML estimation has provided important insights into the latent genetic architecture of complex traits4, GREML assumes independence between random effects when estimating variance components. However, it is questionable if this assumption is always valid, especially for genomic analyses of complex traits. For example, gene regulatory networks shared between functional categories may generate non-negligible correlations between effects of these categories on phenotypes12. In the context of phenotypic variance partitioning by multi-omics layers, effects of genetic variants and their expression levels on phenotypes are likely correlated15, as exemplified by overlaps between GWAS loci and expression quantitative trait loci17 (eQTL). Given these justifiable covariance terms in genomic analyses of complex traits, the naive assumption of independence between random effects held by GREML can lead to a biased partition of phenotypic variance and false inferences on the underlying architecture of complex traits.Following classic LMMsHere we introduce an alternative GREML, named CORE GREML (CORE for COvariance between Random Effects), which fits the Cholesky decomposition of kernel matrices in an LMM, to estimate the covariance between a given pair of random effects. Using extensive simulations, we show that CORE GREML outperforms GREML, providing estimates of variance and covariance components that are free from bias due to correlated random effects. We also apply CORE GREML to real data from the UK Biobank, to demonstrate its use for genomic partitioning analyses and for genome-transcriptome partitioning of phenotypic variance. We conclude that the covariance between random effects is a key parameter that needs to be estimated, especially for multi-omics analyses of phenotypic variance.n = 10,000) under settings where the covariance between random effects was zero, positive, and negative . To validate CORE GREML, we simulated 500 replicates of phenotypic data n = 0,000 und18  with zero covariance between effects of genomic regions on phenotypes , the CORE GREML vs. GREML comparison yielded significant results (at a significance threshold of 0.05) for 19 replicates out of 500, giving an estimated type I error rate of 0.038 for detecting covariance terms. Similarly, for data simulated using the genome-transcriptome model under the null setting (see Methods for the simulation model), the estimated type I error rate was 0.042. Thus, for both simulation scenarios, type I error rate was not inflated.In terms of parameter estimation, regardless of the simulation model and the parameter setting, CORE GREML consistently yielded unbiased estimates of all model parameters  and comparing CORE GREML estimation from fitting an estimation model that has the correct assumption about the genetic architecture, referred to as the “true model,” with that from fitting a “wrong model” that has an incorrect assumption about the genetic architecture.Incorrect assumptions in the estimation model about the genetic architecture of the trait can also bias variance-components estimation in the context of GREML19.We found that misspecification of genetic architecture in the estimation model in general biased CORE GREML estimation of variance components but not the covariance term for genome-transcriptome partitioning of phenotypic variance  model  affects the variance of SNP-specific effects on phenotypes, set at the recommended default −0.2519). We found that for all traits, the GCTA model had a better fit than the LDAK model, irrespective of estimation method . Given the above, unless specified otherwise, results presented in the main text below are GCTA-based; LDAK-based results are included in Supplementary Tables In light of the above results, to reduce the chance of misspecification of genetic architecture for real data analyses, we fitted two estimation models, the Genome-wide Complex Trait Analysis (GCTA) modely = g + ε, and a “G-T model” that decomposes phenotypic effects into the random effects of the genome and the imputed transcriptome and residuals, i.e., y = g + t + ε. We declared the presence of significant effects of the imputed transcriptome when the G-T model had a better fit than the G model using the likelihood ratio test with one degree of freedom. We found significant effects of the imputed transcriptome for all traits, except fluid intelligence . It is noted that the set of SNPs for estimating g0 were ones that remained after applying our quality control criteria, whereas the 1,316,391 SNPs for the transcriptome imputation (hence for estimating g1) were based on the transcriptome imputation protocol18. The two sets of SNPs overlap by 716,636. If the two-GRM model (y = g0 + g1 + ε) is no different from the genome-imputed transcriptome model (y = g0 + t + ε), their likelihoods would be very similar. Contrary to this, we found that y = g0 + t + ε had a much better fit than y = g0 + g1 + ε for all traits, except heel bone mineral density, years of education, and fluid intelligence , suggesting that the transcriptomic effects on phenotypes of the selected traits are genuine. Taken together, our results thus far indicate that the variance components of interest differ from zero for all traits with fluid intelligence being the exception for the genome-transcriptome partitioning of phenotypic variance. Importantly, these results established the basis for our subsequent covariance estimation.To validate the transcriptomic effects on phenotypes revealed by the G-T model, we performed a five-fold cross-validation, in which the phenotypic prediction accuracy of the G-T model was compared against that of the G model. For each trait, we randomly split the sample into a training set (~80%) and a validation set (~20%), and iterated this process five times in a manner such that validation sets did not overlap across iterations. To derive the prediction accuracy for the two models, we computed the Pearson’s correlation coefficient between the observed and predicted phenotypes of each trait in each iteration and averaged correlation estimates across five iterations. Figure p = 0.85 and 0.37 for height and sitting height, respectively). We subsequently used estimates from the reduced model for these two traits. In genome-transcriptome analyses, we found significant covariance between the random effects of the genome and those of the imputed transcriptome for height, sitting height, heel bone mineral density, and diastolic blood pressure .To show that covariance terms estimated by CORE GREML are genuine biological parameters, we validated the covariance between the random effects of the genome and the imputed transcriptome using the same five-fold cross-validation procedure as before . In this instance, the phenotypic prediction accuracy of CORE GREML was compared against that of GREML. We chose genome-transcriptome analyses for validation since significant covariance emerged from four traits in contrast to two traits in genomic partitioning analyses. Figure To show the impact of neglecting significant covariance terms, we compared variance-component estimates from GREML with those from CORE GREML. In both genomic partitioning and genome-transcriptome analyses, variance estimates by GREML for correlated random effects were larger than those by CORE GREML Figs.  and 3. A5. Notably, GREML estimates of these functions were larger than CORE GREML estimates whenever there was a significant covariance term  proportions of phenotypic variance explained by the genome  and by the imputed transcriptome  in a genomic partitioning analysis. In fact, the correlations between off-diagonal entries of kernel matrices used in our analyses are moderate to high  may be met in many cases. However, when phenotypic variance is further partitioned, e.g., by functional genomic region or omic layer, using a model with multiple random effects, the covariance terms between these random effects can be substantial, as we demonstrated using the genomic partitioning analyses and the genome-transcriptome partitioning analyses for complex traits. Unless these non-negligible covariance terms are properly accounted for, variance-components estimation would be biased, resulting in misleading inferences on the latent omic architecture of complex traits, as shown by simulation results. Therefore, we recommend that covariance terms between random effects need to be carefully checked and properly modeled for genomic analyses of complex traits.21, but they are specific to experimental studies and are not readily applicable to genome-wide analyses for human complex traits. To our knowledge, CORE GREML is the first of its kind in variance partitioning analyses, which correctly models the covariance between random effects.CORE GREML can serve as a useful tool for detecting and estimating covariance terms between random effects, as demonstrated using analyses of both simulated and real data. Prior to the proposal of CORE GREML, there have been several attempts to relax the assumption of independence between random effects15, they were based on a limited number of SNPs and genes. In contrast, the association estimates by CORE GREML were based on aggregated effects of genome-wide SNPs and those of all available gene expression levels jointly on phenotypes, thereby providing an overall picture of the proportion of phenotypic variance shared by the whole genome and the transcriptome.We demonstrated the use of CORE GREML in genomic partitioning analyses and genome-transcriptome partitioning of phenotypic variance, and found that significant covariance terms mostly emerged from the latter (for four traits out of ten). Although genome-transcriptome associations have been reported by numerous studies18. As a result, the imputed transcriptome would only have captured part of transcriptomic effects on phenotypes; hence, the phenotypic variance explained by the transcriptome would have been underestimated. However, our intention of using the imputed transcriptome is not to accurately estimate variance explained by the transcriptome; but as a proxy of the transcriptome to demonstrate the use of CORE GREML for genome-transcriptome partitioning analyses. When actual gene expression levels become available for future analyses, we expect an additional gain in explained phenotypic variance.Of note, our study used imputed, as opposed to measured, gene expression. As demonstrated by our cross-validation results, the imputed transcriptome already improves phenotypic prediction accuracy, hence it allows phenotype forecasting, e.g., for newborns, solely on the basis of genotype information. It is noted though, the gene expression imputation models used in our study on average explains 13.7% of gene expression variation22. Although the former is essentially the effects of imputed gene expression on phenotypes, the latter is the effects of common SNPs (see Eq. (3) in Yao et al.22). Despite the conceptual similarity, our study is different from Yao et al.22 in two key aspects. First, implied from the model in Yao et al.22, gene expression levels were based on SNPs at cis-eQTLs only; however, in our study gene expression levels were computed using genome-wide SNPs. This may explain that the estimated phenotypic variance due to imputed gene expression in their study tends to be smaller than that in ours for BMI, standing height, heel bone mineral density and years of education. Second, unlike CORE GREML used in our study, the model used by Yao et al.22 does not account for covariance between effects of the genome and the imputed transcriptome on phenotypes.Although the model with the random effects of the imputed transcriptome fit the phenotypic data much better than the reduced model without the imputed transcriptomic effects, the two models explained the same amount of total phenotypic variance Fig. . Notably23 or MAF bin25, where correlations between the random effects in the model are intuitive. Even for the simplest partitioning of phenotypic variance that separates genetic variance apart from residual variance, CORE GREML can be useful, if genetic effects and residuals are correlated due to confounding, associations or interactions between genetic and environmental effects26. One may also be interested to partition genome-wide SNPs into subsets by LD structure and MAF bin and specify a separate random-effect term for each subset in CORE GREML. However, this would inevitably result in many random-effect terms in the model. Given that CORE GREML is computationally intensive, estimation would eventually become infeasible for models with a large number of random effects . Further studies are required to develop computationally efficient algorithms for CORE GREML, e.g., using summary statistics. In addition, we only validated CORE GREML for quantitative traits in this study. Validation of our method for binary traits is required in future studies.Importantly, the proposed CORE GREML can be used to analyze and dissect the shared effects among omic layers, beyond the genome and the transcriptome, including proteome, metabolome and exposome, when multi-omics data become available. We anticipate, based on our genome-transcriptome analyses of complex traits, that covariance between the random effects of omic layers is a key parameter, such that CORE GREML will be an important tool for multi-omics partitioning analyses. Other potential applications of CORE GREML include phenotypic variance partitioning by chromosome19 and found the GCTA model in general had a better fit than the LDAK model for our traits. In the absence of the knowledge of the true genetic architecture, it is recommended to vary parameter settings of the LDAK model more systematically  and choose the best fitting model via likelihood comparison before applying CORE GREML.Finally, we showed, using simulations, that misspecification of LD and MAF dependent genetic architecture can cause substantial bias in variance-components estimation by CORE GREML, although covariance estimation seems robust. However, the likelihood of the true estimation model in general is much greater than a wrong model, suggesting that likelihood-based comparisons of models that assume different genetic architectures is a useful way to reduce the chance of mis-specifying genetic architecture. For demonstration, we fitted the GCTA model and the LDAK model with the recommended default settingIn this study, we introduce a generalized GREML, referred to as CORE GREML, which relaxes the assumption of independence between random effects held by classic mixed-effects models for variance-component analyses. Using both simulations and real data, we showed that in the presence of non-negligible covariance terms, CORE GREML improved genomic partitioning and multi-omics partitioning analyses by the conventional GREML. We conclude that the covariance between random effects for analysis of complex traits is a key parameter for estimation and, hence, recommend that covariance terms should be carefully checked and properly modeled.https://www.ukbiobank.ac.uk) for our analyses.We used data from the UK Biobank , can be written asA LMM can be written asg and ε, i.e., cor = 0. When relaxing this classic assumption, Eq. , the model can be written asgi and gj. However, this assumption can be violated if the effects of two categories on phenotypes are associated, e.g., through the same gene pathway. We relax this assumption and write the variance–covariance matrix of all observations, var(y), asAi is the genetic relationship kernel matrix constructed using SNPs from the ith functional category, Ai, and gi and gj. It is noted that the correlation term between genetic effects (gi) and residuals (ε) is not included and hence assumed to be zero in Eq. ,,4), is6\This CORE GREML approach can be easily extended to phenotypic variance partitioning using multi-omics data, e.g. genome-transcriptome analyses (see Genome-Transcriptome Partitioning Model section below).g, e). When there is non-negligible cor, the phenotypic variance should be written as For Eq. , the staith genetic component isFor Eq. , a gener31, the sampling variance of heritability for this example can be obtained as32 of CORE GREML.Using the Delta methodThe correlation between two random (genetic) effects . The computational requirements for fitting a model with two random effects using CORE GREML are outlined in Supplementary Table CORE GREML analyses were performed using open-access software, MTG2  < 0.6, an MAF < 0.01, a Hardy–Weinberg equilibrium p-value < 1e − 4, or a call rate < 0.95. We then selected HapMap phase III SNPs, which are known to yield reliable and robust estimates of SNP-based heritability35, for downstream analyses. We filtered individuals who had a genotype-missing rate > 0.05, were non-white British ancestry, or had the first or second ancestry principal components outside 6 SDs of the population mean. We also applied relatedness cut-off quality control to exclude one of any pair of individuals with a genomic relationship > 0.025. From the remaining individuals, we selected those who were included in both the first and second release of UK Biobank genotype data. We calculated the discordance rate of imputed genotypes between the two versions and excluded individuals with a discordance rate > 0.05. Eventually, genotypes of 1,131,002 SNPs from 91,472 individuals remained for data analysis.The UK Biobank  contains health-related data from ~500,000 participants aged between 40 and 69 years, who were recruited throughout the UK between 2006 and 2010https://nealelab.github.io/UKBB_ldsc/), which included standing height, sitting height, BMI, heel bone mineral density, fluid intelligence, weight, waist circumference, hip circumference, diastolic blood pressure, and years of education36. Heritability estimates for all selected traits were at least 20 times greater than their SEs, to ensure that they were significantly different from zero. We further verified SNP-based heritability of these traits using GREML and estimates are shown in Supplementary Fig. To preclude negligible heritability as a possibility for negative findings , we deliberately chose ten UK Biobank traits available to us with the largest heritability estimates by an independent open source  in three sequential steps as follows: (1) adjustment for age, sex, birth year, social economic status (by Townsend Deprivation Index), population structure , assessment center, and genotype batch using linear regression; (2) standardization; and (3) removal of data points outside ±3 SDs from the mean. The distributions of phenotypes of the ten traits are shown in Supplementary Fig. http://gusevlab.org/software/) as follows: (1) regulatory regions that consist of coding regions, untranslated regions and promotors; (2) DHSs; and (3) all other regions. We assigned each SNP into 1 of the 3 categories, resulting in 75,396 SNPs in the regulatory regions, 255,665 in the DHS, and 799,935 in all other regions. Prior to the assignment, genotype data were quality-controlled (see above for details).The genome was annotated using three pre-defined functional categories (18 (https://github.com/hakyimlab/PrediXcan), we imputed expression levels of 2028 to 9630 genes for 43 non-sex-specific tissues , which were trained using 2,496,846 SNPs of European individuals from the Genotype-Tissue Expression project37.Using PrediXcanThe phenotypic variance of each selected trait was partitioned using two separate random-effects models (see below for model description). The “genome-transcriptome” model partitions phenotypic variance into variation from the genome, the transcriptome and unknown sources , whereas the “genomic partitioning” model assumes phenotypic variation comes from the genome and residuals, and further partitions genetic variance by functional category of the genome. Three functional categories were under consideration, namely, regulatory regions , DHSs, and all other regions.Each partitioning model was fitted using the conventional method, i.e., GREML, and the proposed alternative, i.e., CORE GREML. Essentially, GREML sets all covariance terms between random effects to zero, whereas CORE GREML treats these terms as free parameters for estimation. To detect significant covariance terms, we performed likelihood ratio tests to determine whether the model fit by CORE GREML was better than that by GREML.y is a n × 1 vector of phenotype data, μ is the grand mean, g, t, and ε are the main genetic, transcriptomic, and residual effects, following g ~ N SNPs for n  individuals, and Qnxp contains standardized imputed expression of p  genes collapsed across 43 tissues for the n individuals. Essentially, entries of Anxn and Tnxn describe pairwise similarities between individuals based on their genotypes and imputed gene expression, respectively.Assuming that the phenotypes are pre-adjusted for fixed effects, the genome-transcriptome partitioning model can be expressed asσgt is the covariance between the effects of the genome and the transcriptome on phenotypes. Here we assume no correlation between residuals and genomic or transcriptomic effects.The variance–covariance matrix of phenotypic observations isy is a n × 1 vector of phenotype data that are decomposed into the grand mean μ, the genetic effects due to regulatory, N, positive and negative covariance between random effects in the variance partitioning model. Simulations were based on quality-controlled genotype data and imputed transcriptome data from a random sample of 10,000 UK Biobank individuals. The genotype data contained a total of 1,131,002 SNPs (see Genotype data above) and the imputed transcriptome contained imputed expressions of 227,664 genes collapsed cross 43 non-sex-specific tissues (see Gene expression imputation above).σgt varied across parameter settings  at a controlled rate of type I errors, by comparing the model fit of CORE GREML with that of GREML using likelihood ratio tests. Under all settings, we assessed if CORE GREML can produce unbiased estimates of variance and covariance components. To show the impact of neglecting genuine covariance terms, we also provided parameter estimates by GREML.38, we parameterized genetic architecture by MAF and LD, assuming the variance of SNP-specific effects on phenotypes, var(β), for any given SNP i, is proportional to its LD score, w, and MAF, f, expressed as α and γ control the extents to which w and f affect var(β). By altering values of α (either −1 or −0.25) and γ (either 0 or 1), we simulated phenotypic data under three different genetic architectures , each including three scenarios of the covariance between the random effects of the genome and those of the transcriptome, namely, null (σgt = 0), positive (σgt = 0.2), and negative (σgt = −0.2), as for the CORE GREML validation simulations . Each scenario had 500 replicates of simulated phenotypic data . For a given genetic architecture, we fitted two estimation models, both using CORE GREML, but one assuming values of α and γ the same as those of the simulation model  and the other always assuming α = −1, γ = 0 .To test the sensitivity of CORE GREML estimation to a wrong assumption about the genetic architecture in the estimation, we also simulated phenotypes under genetic architecture. Following past studiesmentclass2pt{minimFurther information on research design is available in the Supplementary InformationPeer Review FileReporting Summary