The last decade witnessed an unprecedented increase in the adoption of genomic selection (GS) and phenomics tools in plant breeding programs, especially in major cereal crops. GS has demonstrated the potential for selecting superior genotypes with high precision and accelerating the breeding cycle. Phenomics is a rapidly advancing domain to alleviate phenotyping bottlenecks and explores new large-scale phenotyping and data acquisition methods. In this review, we discuss the lesson learned from GS and phenomics in six self-pollinated crops, primarily focusing on rice, wheat, soybean, common bean, chickpea, and groundnut, and their implementation schemes are discussed after assessing their impact in the breeding programs. Here, the status of the adoption of genomics and phenomics is provided for those crops, with a complete GS overview. GS’s progress until 2020 is discussed in detail, and relevant information and links to the source codes are provided for implementing this technology into plant breeding programs, with most of the examples from wheat breeding programs. Detailed information about various phenotyping tools is provided to strengthen the field of phenomics for a plant breeder in the coming years. Finally, we highlight the benefits of merging genomic selection, phenomics, and machine and deep learning that have resulted in extraordinary results during recent years in wheat, rice, and soybean. Hence, there is a potential for adopting these technologies into crops like the common bean, chickpea, and groundnut. The adoption of phenomics and GS into different breeding programs will accelerate genetic gain that would create an impact on food security, realizing the need to feed an ever-growing population. Zea mays L.), rice (Oryzae sativa L.), and wheat (Triticum aestivum L.)  . Howeverivum L.) , 2013. Pivum L.) .Sub1 QTL for submergence tolerance in rice, Fhb1 QTL for providing tolerance to fusarium head blight in wheat, and QTL for providing resistance to cyst nematodes in soybean (Glycine max L.) , amplified fragment length polymorphism (AFLP), randomly amplified fragment polymorphic DNA (RAPD), simple sequence repeats (SSR), and single nucleotide polymorphism (SNP) have been developed and led to the identification of several quantitative trait loci (QTL) by linkage mapping in most crops . This wa max L.) .As the excitement about linkage mapping began to fade in the early 2000s, association mapping emerged as a new technique for studying marker-trait associations . AssociaBy the late 2000s, plant breeders realized that they needed a technique that can not only identify associated QTL, but provides enough information to improve complex quantitative traits, for which previous mapping techniques had failed. Bernardo  achievedTechniques like linkage and association mapping, marker-assisted selection (MAS), and GS need accurate phenotyping information for obtaining the desired results. GS requires phenotypic information for building models, and MAS requires phenotypic information for validating that a particular marker is associated with a trait . In a laSeveral ground-based and aerial sensing platforms are being used with multiple sensors for measuring various traits in plants at different growth stages accurately, rapidly, and precisely . The advPhaseolus vulgaris L.), chickpea (Cicer arietinum L.), and groundnut (Arachis hypogaea L). These six crops are chosen after considering the different rate of development during the last decade and importance in the human diet and crops were chosen separately from each cereal, legume and oilseed category; 2) offer an overview of GS and its implementation in cereal, legume, and oilseed breeding programs; 3) present developments in phenotyping platforms and imaging sensors for collecting phenotypic data; 4) discuss the status of below ground phenotyping techniques in plant breeding; and 5) discuss the merging of GS, machine learning, and phenomics information for increasing the genetic gain of breeding programs. This review is unique as it combines GS and phenomics in several important crops and will assist upcoming plant breeders understand the progress of this technology.The main objectives of this review are to 1) provide current status and overview about the advancements in genomics and phenomics for rice, wheat, soybean, common bean (https://www.fao.org/statistics/en/). The average productivity increased from 1.9 to 4.7 ton/ha in rice, 1.1–3.5 ton/ha in wheat, 1.1–2.8 ton/ha in soybean, 0.5–0.9 ton/ha in common bean, 0.6–1.0 ton/ha in chickpea, and 0.8–1.6 ton/ha in groundnut from 1961 to 2019 , wheat is one of the most widely studied crops at the genetic and cytogenetic level . Advancement of genomics started after 2010 in soybean with the genome sequence of cultivated soybean variety Williams 82 (Wm82) in the United States , collar rot (Sclerotium rolfsii), dry root rot (Rhizoctonia bataticola), botrytis grey mold (Botrytis cinerea), and fusarium wilt (Fusarium oxysporum) that reduce crop yield. Chickpea is a rich source of dietary protein, minerals, carbohydrates, and essential nutrients, thus has the potential for improving malnutrition problems in south Asia and sub-Saharan Africa, where it is mostly grown . Chickpea can produce 3.0–4.0 tons/ha, but currently it is restricted to ∼1 ton/ha due to limited work on biotic and abiotic stresses . Total ply grown . The lasly grown . The adoly grown . The devGroundnut or peanut is a nutritious oilseed and legume crop grown on 29.5 M ha in more than 100 countries globally, with a total productivity of 48.8 tons during 2019. Africa (55%) and Asia (40.3%) together have more than 95% of the groundnut cultivation area, account for 31.5 and 59.6% of the total production, respectively (FAO 2019). All parts of groundnut are a nutrition source for humans and animals. Groundnut plays an important role in fighting malnutrition as 80% of its seed consists of nutritious fats and proteins; furthermore, the crop can improve soil fertility and break the disease cycle when grown under rotation with cereal crops . PreviouAs mentioned in the introduction, GS is a technique for predicting GEBVs using training and testing populations . GS has Several factors affect the performance of GS models. They have been explored in multiple studies during the last decade, ranging from training population size, relatedness between training and testing population, cross-validation strategy, marker density, heritability of the trait, population structure, and prediction model . It is oSince GS uses genome-wide markers, proper genotyping is required. To date, several genotyping platforms like RFLP, AFLP, SSR, DArT, and SNP chips have been explored for GS; however, since 2012, with the emergence of the GBS platform, it has dominated all previous platforms due to the low cost, genome-wide coverage, and reduced sampling bias compared to SNP chips . It has 1) or (F2:3) for a rapid generation cycle with a short interval. This selection is used to predict the breeding values and helps the researchers select parents for new crosses or remove inferior performing lines earlier in the pipeline  and blast (Magnaporthe oryzae) are devastating diseases of rice observed worldwide, and accurate detection and management are the focus of several breeding programs. RGB and multispectral imaging sensors on UAVs are an affordable and user-friendly option for disease detection and rating  (Other root phenotyping approaches include positron emission tomography (PET), magnetic resonance imaging (MRI), thermal neutron tomography, and neutron radiography. MRI uses the absorption and re-emission of electromagnetic radiation from the nuclei to determine its root architecture and functional attributes . But MRIar (GPR) . But simar (GPR) .Developing crop varieties which remain productive on marginal soils and under water deficit is the main aim of several breeding programs, especially in Asia and Africa, owing to climate change . BreedinIn a recent study, two hundred wheat lines were screened for root dry matter, root diameter, seminal axis root length, root dry matter, and branching pattern in seedling growth over the hydroponic system for performing MTAs . From thAs discussed previously, GS aids in predicting GEBVs and in increasing genetic gain by reducing variety development time and cost per cycle and increasing selection accuracy. Phenomics allows generation of high-quality quantitative data and effectively characterizes large breeding populations . It has Most of the GS studies conducted so far use a single trait (univariate) statistical model to predict one trait at a time and do not benefit from the genetic correlation among two or more traits . HoweverOwing to the ability of phenomics to collect a large amount of data due to its high spatial and temporal variation, it sometimes creates the big data problem, where feature selection needs to be performed, and complex machine and deep learning models are needed to build the relationship between features and predictors. Several machine and deep learning models, such as random forests, ensemble-based methods, support vector machine, multilayer perceptron, convolutional neural network, and recurrent neural network, are often employed for analyzing phenomics data and predicting traits with GS models. The main interests for these models in plant breeding are useful due to their powerful ability to learn the complex/hidden non-linear relationship in the data to predict complex traits and usually result in higher prediction accuracy than a mixed linear GS model. We explored six important self-pollinated crops consumed by 90% of the world population. Most of the advancements in genomics and phenomics over the last decade have been observed in wheat and rice. The genome sequencing of other crops and the adoption of high throughput genotyping tools have paved the way for understanding various underlying genetic mechanisms. These crops can utilize phenomics in coming years after seeing the progress and benefits achieved in wheat and rice. Several GS models varying from traditional BLUP based model to machine/deep learning models have been explored for prediction. Furthermore, the inclusion of genotype by environment interaction in these models has delivered good prediction accuracy for predicting untested lines in new environments. All the GS models discussed in this study, including genotype and environment interaction, will assist the plant breeder in making improved selection decisions. Multi-trait GS models also indicate their success for predicting low heritable traits and will be explored in future years for prediction under multi-environment scenarios, with the inclusion of phenomics datasets, for understanding genotype by environment interactions.The yield trends of crops across different continents is very diverse, and it is evident that in Asia and Africa, use of advanced genomic and phenomic technologies can improve/enhance grain yield. Furthermore, public breeding programs play a predominant role in these crops. To translate the advantage of GS and phenomics in their programs, low-cost genotyping and phenotyping needs to be developed and used. In this context, easy to handle, reliable, and affordable low throughput platforms pave the way, and among such tools, RGB cameras make good candidates. Below ground phenotyping is tedious for a plant breeder and is being ignored by most programs. However, several field and lab-based root phenotyping tools were launched in the last decade and their potential is being realized. Further refinement and throughput will pave a new way to better understand root traits in field crops. This is extremely important for continuously increasing drought, salinity, aluminium, and heavy metal sensitivity to plants. As the total number of studies for phenotyping the roots traits increase, this will ultimately aid in predicting new genotypes using GS once enough data are collected for each crop under the different breeding programs.This review highlights the advantages of combining genomics and phenomics, especially in wheat and rice. There is a need to merge and adopt these two disciplines at a fast pace in other crops to increase their genetic gain. GS has been shown to increase genetic gain by increasing selection accuracy and intensity with reduction of cycle time, which can be further enhanced by using phenomics, and machine/deep learning models in the breeding programs due to big data sets. These tools could aid in screening large number of lines with less phenotyping cost and efforts, allowing better exploration of the genetic diversity of particular crops for various traits. Phenomics is assisting plant breeders in integrating physiological breeding in addition to using molecular and genetic tools for selection. Thus, future studies in breeding will focus on merging all these tools and domains to reach the required rate of genetic gain for grain yield.


While recent advancements in computation and modelling have improved the analysis of complex traits, our understanding of the genetic basis of the time at symptom onset remains limited. Here, we develop a Bayesian approach (BayesW) that provides probabilistic inference of the genetic architecture of age-at-onset phenotypes in a sampling scheme that facilitates biobank-scale time-to-event analyses. We show in extensive simulation work the benefits BayesW provides in terms of number of discoveries, model performance and genomic prediction. In the UK Biobank, we find many thousands of common genomic regions underlying the age-at-onset of high blood pressure (HBP), cardiac disease (CAD), and type-2 diabetes (T2D), and for the genetic basis of onset reflecting the underlying genetic liability to disease. Age-at-menopause and age-at-menarche are also highly polygenic, but with higher variance contributed by low frequency variants. Genomic prediction into the Estonian Biobank data shows that BayesW gives higher prediction accuracy than other approaches. Few genome-wide association studies have explored the genetic architecture of age-of-onset for traits and diseases. Here, the authors develop a Bayesian approach to improve prediction in timing-related phenotypes and perform age-of-onset analyses across complex traits in the UK Biobank. Understanding the environmental and genetic basis of the time at which symptoms first occur is critical for early screening programs and for gaining insight into disease development and progression, especially as the pathological processes of many age-related diseases may be triggered decades before the first symptoms appear. Evidence suggests that genome-wide analyses conducted with case-control phenotypes tend to have less power in comparison with their age-at-onset analysis counterparts3. Genetic predictors created from case-control studies have been shown to be predictive of age-at-diagnosis4, implying that early-onset is to a certain degree indicative of a higher underlying liability of disease. However, our understanding of the genetic architecture of reproductive timing, and the age at which symptoms first develop for common complex disorders, remains limited.Genome-wide association studies (GWAS) have greatly expanded our understanding of the genetic architecture of complex traits, but have largely focused on binary phenotypes and quantitative traits5 for each single nucleotide polymorphism (SNP) j ∈ {1, . . . , M} can be formulated as h0(t) is the baseline hazard at time t, hi(t) is the hazard for individual i, xij is the standardised jth SNP marker value, with βj the effect size of the jth SNP and M the total number of SNPs8. Recently there have been improvements in the computation times using some approximations for single-marker Cox PH regression9, however, this approach still yields marginal effect size estimates as the markers are not fitted simultaneously. Residual based approaches have also been widely used, which first regress the phenotype on covariates such as gender or age at entry in Cox PH model, and then use the residuals in a second regression on the SNP data, with martingale residuals i, δi is the failure indicator , ti, ti is the follow-up time for individual i, Zi is the vector of variables used in the first regression step and γ the vector of corresponding parameter estimates11. The martingale residual approach retains the linearity between the effect and the phenotype and given the model in the second step it can also be very fast. However, the failure time and censoring indicator are combined to one summary statistic, rather than including censoring information specifically via likelihood. Therefore, the martingale residual approach does not use the censoring information efficiently diminishing the power of this model. Rather than testing markers one-at-a-time, their effects can be estimated jointly in a mixed-effects Cox PH model, referred to as a frailty model, specified as β is the effect for one SNP being tested along with other fixed effects such as age or sex, b ~ N is the N-dimensional vector of random effects (N is the sample size), Σ: N × N is the genetic relationship matrix, σ2 is the variance of the genetic component, λ0(t) is the baseline hazard function and λi(t∣b) is the hazard for individual i. This idea has been long limited by computational resources and in the latest implementation (COXMEG)12 analyses are constrained to around ~10,000 individuals. For joint marker effect estimation, there is also the Cox-LASSO model13 which has been recently developed for genetic data in the R package snpnet15. Fully parametric alternatives are also the Sparse Bayesian Weibull regression (SBWR), which may outperform LASSO-based approaches16, but like other Bayesian methods such as SurvEMVS17 or a semi-parametric g-prior approach of Held et al.18 the ultrahigh dimensions of genetic data limit their application. Therefore, approaches that can efficiently handle both the complexity and scale of many millions of sequenced individuals with time-to-event outcomes have not been extensively developed, limiting our understanding and our ability to predict disease progression and the timing of symptom onset.Statistical modelling of time-to-event data is a highly active research area and is frequently applied to clinical and pharmacogenetic studies. Analogous to single marker regression in GWAS analyses, a Cox proportional hazards (PH) modelβj that represent the effect of each marker conditional on the effects of all the other markers accounting for genetic architecture; (3) provides a determination of the probability that each marker and genomic region is associated with a phenotype, alongside the proportion of phenotypic variation contributed by each, and (4) gives a posterior predictive distribution for each individual. Regardless of the phenotypic distribution, our suggested approach greatly improves genomic prediction for the timing of events for each individual and enables better insight behind the genetic architecture underlying time-to-event traits.Here, we take an alternative approach to obtain accurate inference in full-scale phenotype-genotype sequence data sets, by proposing a mixture of regressions model with variable selection, using different regularisation parameters for genetically motivated groups (see “Methods” section). Our suggested model fits all of the markers jointly in a Bayesian framework using the Weibull assumption for the phenotypes. We show that this approach: (1) allows for a contrasting the genetic architectures of age-at-onset phenotypes under this flexible prior formulation; (2) yields marker effect estimates M markers are split between Φ different groups. The groups can be for example formed based on marker-specific genomic annotations, MAF grouping, grouping based on LD score, etc. We assume for an individual i that the age-at-onset of a disease Yi has Weibull distribution, with a reparametrisation of the model to represent the mean and the variance of the logarithm of the phenotype asμ is the intercept, φ, βφ are the marker estimates for the corresponding group, zi are additional covariate values , δ is the additional covariate effect estimates and α is the Weibull shape parameter (see “Methods” section). For each group, we assume that βφ is distributed according to a mixture of Lφ Gaussian components. Each marker (from group φ) estimate βjj ∈ {1, . . . , M} is related to a corresponding indicator variable γj ∈ {0, . . . , Lφ} where Lφ is the number of mixture distributions. βj have zero values if and only if γj = 0. We assume that non-zero βj, where marker j belongs to group φ, that has been assigned to mixture component (γj = k ≥ 1) comes from a normal distribution with zero mean and variance φ and k is stored in Lφ + 1-dimensional vector πφ. Thus the mixture proportions, variance explained by the SNP markers, and mixture constants are all unique and independent across SNP marker groups.An overview of our model is as follows, suppose that βj. With T parallel workers (MPI tasks), Bulk Synchronous Parallel (BSP) Gibbs sampling can sample T marker effects when sequential Gibbs samples a single one, but BSP requires an extra synchronisation step of the tasks after each of them has processed u markers (see “Methods” section). After each worker has processed u markers, we synchronise the workers by transmitting the residual vector across workers. Given our assumption that the phenotype follows a Weibull distribution, we are using a numerical method, Adaptive Gauss-Hermite quadrature, for calculating the mixture membership probabilities for variable selection, and Adaptive Rejection Sampling (ARS) for estimating the marker effects. We implement these approaches to take full advantage of the sparsity of genomic data, converting computationally intensive calculations of exponents and dot products into a series of summations. We provide publicly available software (see “Code Availability”) that has the capacity to easily extend to a wider range of models (not just Weibull) than that described here. Our software enables the estimation of 2,975,268 SNP inclusion probabilities split between T = 96 workers, using 12 compute nodes and synchronisation rate of u = 10, mixture allocation and effect sizes in 371,878 individuals with an average of 49.7 s per iteration without groups model and 50.0 s per iteration with the groups model; using 151,472 individuals with T = 64, 8 compute nodes and u = 10 without groups we get an average of 24.8 and with groups, we get an average of 26.4 s per iteration. Here, we have chosen to run the chains for 10,000 iterations leading to execution times of 69 h  to 139 h . The run times ultimately depend upon compute cluster utilisation and the genetic architecture of the phenotype, as calls to the ARS procedure are linear with the number of markers. The calculations were done by using Helvetios cluster of EPFL (see Code availability).We develop a computational framework that overcomes previous limitations for the application of age-at-onset models to large-scale biobank studies. In the model likelihood, we account for right censoring, a situation where only the last known time without an event is recorded, with the event potentially taking place sometime in the future (see “Methods” section). Although we did not apply it in our final analysis, we also formulate the model to accommodate left truncation, a situation where individuals are not missing from the data at random, creating differences in the genetic composition of individuals across age groups (see “Methods” section). We implement a parallel sampling scheme for Eq.  that all13, Bayesian regression mixture model BayesR19 applied on martingale residuals and marginal single marker regression (OLS) applied on martingale residuals. First, we show that the previous statement holds even in the case of model misspecification  in the Supplementary Note of Supplementary Information), with differing θ value (see “Methods” section and Supplementary Note in Supplementary Information). In a simulation study of N = 5000 individuals and 50,000 uncorrelated markers with p = 500 randomly selected markers as causal variants, our BayesW model obtains higher out-of-sample prediction accuracy than a Cox-LASSO or a martingale residual approach used in several recent studies . Such phenotypic distributions are easier to discriminate meaning that for distributions where θ > 1 the same difference in genetic values leads to greater phenotypic distribution differences in Kullback-Leibler divergences compared to θ = 1. Our approach achieves better precision-recall as compared to these approaches  then due to the inflated genetic variance hyperparameter  (see “Methods” section). We show that (a) our model captures accurately the effect size distribution  as the metric to summarise the significance of genetic regions. PPWV shows the probability that a genetic region explains at least some fixed proportion of the genetic variance. We show below that false positives are controlled for under all generative models when conducting LD clumped based variable selection using a PPWV threshold of ≥0.9 . We show that the log-scale SNP heritability definition is valid under a Weibull assumption and across the range of theta when restricting the markers entering the model Fig. e. In addM = 2,975,268 SNPs for five traits: two reproductive phenotypes of age-at-menopause  and age-at-menarche  and three common complex diseases  of time-to-diagnosis of type-2-diabetes (T2D) , coronary-artery-disease (CAD)  and high blood pressure (HBP)  . For the group model, the four group-specific mixtures for each of the 20 groups were chosen to be 10 times larger  such that they would represent 0.01%, 0.1%, 1% or 10% of the group-specific genetic variances. Additional variables such as sex, UK Biobank assessment centre, genotype chip, and the leading 20 PCs of the SNP data (see “Methods” section) were used as fixed effects in the analysis. We conducted a series of convergence diagnostic analyses of the posterior distributions to ensure we obtained estimates from a converged set of chains  explain a far smaller part of the total genetic variance with age-at-menarche having almost no genetic variance  and only a small amount coming from that mixture for age-at-HBP diagnosis  and age-at-CAD .Under the assumption that the traits are Weibull distributed, this gives log-scale SNP (pseudo-)heritability estimates (see Supplementary Note of Supplementary Information) of 0.26  for age-at-menopause, 0.41  for age-at-menarche, 0.36  for age-at-diagnosis of HBP, 0.48  for age-at-diagnosis of CAD, and 0.52  for age-at-diagnosis of T2D. Both the model with and without groups reach similar conclusions in terms of partitioning markers between mixtures  and determined the genetic variance each of those regions explains. Then, we calculated the probability (PPWV) that each such region contributes at least 1/1000, 1/10,000 or 1/100,000 of the total genotypic variance, providing a probabilistic approach to assess the contribution of different genomic regions to time-to-event phenotypes. The smallest threshold was chosen to be 1/100,000 of the total genotypic variance corresponding to the smallest mixture component models were estimated with which also represents the magnitude of the smallest effect size we intend the model to capture. We find 291 LD clumped regions for age-at-menarche with ≥0.95 PPWV of 1/100,000, 176 regions for age-at-menopause, 441 regions for age-at-diagnosis of HBP, 67 regions for age-at-diagnosis of CAD, and 108 regions for age-at-diagnosis of T2D from our BayesW grouped mixture of the regression model Fig. a. Our gr28. Even though fastGWA is a frequentist and BayesW is a Bayesian method and the comparison between the two approaches is not comprehensive, we still use it as both methods should control for false discovery rate  using two metrics. As some of the Estonian Biobank time-to-event phenotypes are censored, we choose to calculate the R2 values between the predicted values and the martingale residuals from the Cox PH model where the true phenotypes are regressed on sex. In addition, we calculate Harrell’s C-statistic29 from the Cox PH model where the true phenotypic values are regressed on the predicted values and sex. BayesW outperforms Cox-LASSO for all phenotypes  compared to Cox-LASSO’s 0.593 (0.00455) for age-at-menopause and for age-at-menarche we get C-statistic of 0.598 (0.00290) with BayesW compared to Cox-LASSO’s 0.580 (0.00294). For the age-at-diagnosis traits, we obtain R2 values of 0.0047, 0.0236, and 0.0441 for BayesW and 0.0030, 0.0135 and 0.0271 with Cox-LASSO for CAD, T2D, and HBP, respectively. This shows that our BayesW model gives a higher prediction accuracy compared to the Cox-LASSO method, in-line with our simulation study results.We used the estimates obtained from the group-specific model to predict time-to-event in 22, we develop a group-specific BayesR approach and we use this to analyse the indicator variable  for HBP using the same data and a liability model to facilitate a direct comparison of the methods. For CAD and T2D, we use the results of the companion paper, where there were almost twice as many case observations  as it included those with the confirmed diagnosis but no age information and 8.4 million SNPs were analysed. For the prediction of age-at-diagnosis, we compared the R2 values between the predicted values and the martingale residuals from the Cox PH model and the Harrell’s C-statistic  and for T2D BayesR marginally outperforms BayesW . A similar ranking can be observed when using Harrell’s C-statistic for comparison  and area under PR curve  ; for predicting CAD or T2D despite the increase case sample size, BayesR only marginally outperforms BayesW . Therefore we get very similar prediction accuracies with both methods when predicting case-control phenotypes although the BayesW model was estimated using time-to-event phenotypes with fewer cases for T2D and CAD.We then compared the BayesW prediction results to those obtained from a case-control analysis. In a companion paperA finding of similar prediction accuracy is unsurprising given the striking concordance between the results of the two models when partitioning the genotype into 50 kb regions. We calculated  the mean proportion of genetic variance attributed to each 50kb region for the model using case-control phenotype (BayesR) and for the model using time-to-event phenotype (BayesW). Both models attribute similar amounts of genetic variance to the same 50 kb regions  regression models generally increase statistical power compared to case-control studies using logistic regression model3. Our results support this and we expect the benefits to become more evident as the number of cases accrues with accurate age-at-diagnosis information.Genome-wide association studies of time-to-event phenotypes are critical for gaining insights into the genetics of disease etiology and progression, but their application has been hampered by the computational and statistical challenges of the models, especially when the predictors are ultrahigh-dimensional. Our hybrid-parallel sampling scheme enables a fully Bayesian model to be applied to large-scale genomic data and we show improved genomic prediction over competing approaches, not only in the 5 that uses a non-parametric estimate for the baseline hazard and then estimates other effect sizes proportional to this hazard. Our BayesW model is also a proportional hazard model with the constraint that the baseline hazard follows a Weibull distribution and thus marker effect size estimates have a similar interpretation as those from a Cox PH model. Interestingly, the results from both our simulation study and real data analysis show that when quantifying the significance of the markers and estimating the marker effect sizes, it might not be pivotal to capture the baseline hazard with a non-parametric method. The simulations show that even in the misspecified cases BayesW performs better compared to the semi-parametric Cox model, demonstrating that using a parametric assumption might be more descriptive than simply using a Cox PH model from standard practice.A typical approach in the time-to-event analysis is the Cox PH model30, or on the log-frailty scale in Cox PH model31. Transforming the log-scale heritability to the original scale32, has then required approximations and the term of original scale heritability has not been easy to explain and use33. Here, using a similar idea of partitioning the total phenotypic variance into genetic and error variance components, we present an expression for SNP heritability on the log-time scale. We then show that there exists a natural correspondence between log-scale and original scale heritability, without the need for any approximations, with log-scale and original scale heritability giving similar estimates if the Weibull shape parameter tends to higher values. Therefore, under Weibull assumptions, we provide a definition of SNP-heritability for time-to-event traits for the GWAS-era.There has been a significant amount of work on the heritability of the time-to-event traits. For example, it has been suggested to define heritability in the Weibull frailty model on the log-time scaleThere are a number of key considerations and limitations. The assumption of a Weibull distribution for the traits considered here can induce bias in the hyperparameter estimates, although we have shown that this assumption yields accurate results in terms of prediction regardless of the phenotypic distribution. A third parameter could be introduced through the use of a generalised gamma distribution and this will be the focus of future work as it should allow for unbiased hyperparameter estimation irrespective of the trait distribution. In this study, we have used hard-coded genotypes to make the method computationally efficient which can result in reduced covariance between the imputed marker and the trait. However, we do not believe this to be a hindrance to our method or the application in this work as hard-coded genotypic values will likely be the norm with the upcoming release of whole-genome sequence data and our aim is to provide a time-to-event model that is capable of scaling to these data requirements. We apply our approach only to markers that are imputed in both the UK Biobank and the Estonian genome centre data and by selecting markers present in both populations we are favouring markers that impute well across human populations.In addition, despite allowing for left-truncation in the likelihood, we focus on presenting a series of baseline results before extending our inference to account for differences in sampling, semi-competing risks across different outcomes, genomic annotation enrichment, and sex-differences both the effect sizes and in the sampling of different time-to-event outcomes all of which require extensions to the modelling framework, which are also the focus of future work. Furthermore, we do not consider time-varying coefficients or time-varying covariates, which may improve inference as multiple measurements over time are collected in biobank studies. Nevertheless, this work represents the first step toward large-scale inference of the genomic basis of variation in the timing of common complex traits.Yi as the time-to-event for individual i, with Weibull distribution Yi ~ W, where a and bi are correspondingly the shape and scale parameters. The survival function isWe define Yi ~ W then K is the Euler-Mascheroni constant, K ≈ 0.57721. The parametrisation for the variance is only dependent on one parameter which we denote as α = a. As we are interested in modelling SNP effects β, covariates δ  and the average scale for time-to-event μ (intercept), it is possible to introduce them in the following way xi is the vector of scaled SNP marker values and zi is the vector of covariate values for an individual i and π = 3.14159. . . is a constant. The third and the fourth moment for We are interested in modelling the mean and the variance of the time-to-event. Unfortunately, the mean and the variance of Weibull are dependent as they share both parameters in their expressions. Moreover, as the expressions for mean and variance contain gamma functions it is rather difficult to dissect the mean and variance to be dependant only on one parameter. One possible solution is to use Yi with a series of covariates (SNP markers in this case) X using a mixture of the regression model, with γj as the mixture indicator, with γj = k if jth marker is included in the kth mixture component of the model, k ∈ {1, . . . , L}, and γj = 0 if it is not included into the model. The expected value of time-to-event logarithm is then a linear combination of the markers included in the model plus the effect of the covariates and the intercept (μ) as in Eq.  comes from a normal distribution with zero mean and variance As a baseline model, we propose to test the association of s in Eq.  and erros in Eq. . βj haveYi is correspondinglyn individuals is thendi is the failure indicator and d is the number of events at the end of the periods; ai is the time of left truncation. It is possible to use the model without left truncation. In order to do so, for every i, we will assume that ai = 0. Whenever ai = 0, we will naturally define The survival and density function for α be a gamma distribution with parameters α0, κ0βj be normal:ασ and βσδq (qth covariate) be normal with variance parameter μ be normal with variance parameter γj be multinomial:k are stored in L + 1-dimensional vector π with the prior for π a Dirichlet distributionI(⋅) is the indicator function and pL is the L + 1-dimensional vector with prior values. For the exact values of the prior specification see Data Analysis Details.Let the prior distribution of γk denotes the set of indices j for which γj = k. The conditional posterior distribution for π is Dirichlet distributionThe conditional posterior distribution for α, μ, βj and δq. However, the conditional posterior distributions are log-concave (see Supplementary Note), and thus the sampling for α, μ, βj and δq can be conducted using adaptive rejection sampling requiring only the log posteriors. Denoting βj− as all the β parameters, excluding βj, and δq− as all the δ parameters, excluding δq, these areUnfortunately, there is no simple form for the conditional posteriors of 34, where the spike part of the prior has a point mass of 0. SNP will be assigned to a mixture component by comparing the ratios of the marginal likelihood. For mixture selection for the jth SNP, we need to find the following marginal likelihood for every k. Suppose here that C0 > 0 is the factor for the 0th mixture (spike)D represents the observed data, Q is a positive constant that is not dependent on k andγj is :C is a positive constant that is not dependent on k. Denoting j in the component k can be calculated askjth variable is included in the kth mixture component.We intend to do variable selection and select mixture components by using the idea of spike and slab priorsIn general, it is not possible to find an analytic expression for the integrals presented in Eq. , thus soui is analogous with ai instead of yi. We introduce a reparameterisation with variable sWe start by rewriting the expression Eq.  as28\docfrom Eq. 30\documen in Eq. , the terC0 > 0, then the corresponding spike distribution is absolutely continuous. As we would like to use Dirac spike instead, we define the corresponding marginal likelihood by finding the limit of the expression in the process C0 → 0+.If the smallest mixture variance factor C0 in the limiting process so without the loss of generality we define C0 through an auxiliary positive integer variable l as f ≤ 1 for every possible combination of arguments, because in the data censoring or event occurs only after entering the study, we can write thatWe are only interested in from Eq.  we get tgk(s) as defined in Eq. (gk(s) and m is the number of quadrature points, tr is the roots of mth order Hermite polynomial and wr are corresponding weights35.It is possible to use Gauss-Hermite quadrature, however, it can happen that for adequate precision one has to use a large number of quadrature points leading to more calculations. Adaptive Gauss-Hermite quadrature can make the procedure more efficient. For any function d in Eq. , we can gk at this mode. As we assume that the effects sizes are distributed symmetrically around zero, we use Finding the posterior mode can be computationally cumbersome, calculating k > 0 the inclusion probabilities areγj = 0) the markerCombining the previous results we get a numerical solution for calculating the posterior inclusion probability. For every α, μ and βj, δq, we use Adaptive Rejection Sampling, initially outlined by Gilks and Wild36. The prerequisite of the method is the log-concavity of the sampled density function.To sample 37). Although the derivative-based method might result in a more accurate upper hull, thus leading to faster sampling, it would still require evaluating derivatives and thus we employ the derivative-free method.The idea of the method is to build an envelope around the log-density. The lower hull is constructed by evaluating the function at some pre-specified abscissae and connecting the evaluation results with linear functions resulting in a piece-wise linear lower hull. The upper hull can be constructed either by using tangents evaluated at the prespecified abscissae (Derivative based ARS) or by extending the linear functions obtained in the construction of the lower hull . In the Supplementary Note, we provide proof of the log-concavity of the functions sampled.In addition, the derivative-free method requires specifying the minimum and maximum value of the distribution, an assumption that is often incorrect. In practice, it poses no problems as we can simply set the required minima and maxima to be extreme enough so that the distribution is very unlikely to reach those values. To sample intercept T = 1 and the synchronisation rate u = 1 then Algorithm 2 reduces down to Algorithm 29.We summarise the serial sampling algorithm in Algorithm 29 along with the specification for the prior distributions and the initialisation of the model parameters. Algorithm 2 summarises the Bulk Synchronous Gibbs sampling for BayesW that extends Algorithm 29. If the number of workers μ, and a single Weibull shape parameter α, but now φ, βφ are the marker estimates for the corresponding group. Each k is stored in Lφ + 1-dimensional vector πφ and these mixture proportions γj = k ≥ 1) is a normal distribution with zero mean and variance φ and πφ and mixture proportions Here, we now assume that the SNP marker effects come from Φ of disjoint groups, with a reparametrisation of the model parameters to represent the mean of the logarithm of the phenotype asξij represents the jth SNP allele count  for the ith individual, and j−th SNP in our sample. In the regular setting we would like to use standardised count values  in the adaptive Gauss-Hermite quadrature can be expressed assj and j.There are three equations where we can apply sparsity. Firstly, Eq.  for the Secondly, we can use the knowledge about sparsity to simplify expression Eq. . More spsion Eq.  we can ri. We notice that it is sufficient to use three sums  that we denote as For all cases, after each update, we need to recalculate the difference ions Eq.  and  was set to be h2 = 0.5. Then, we generated phenotypes from the generalised gamma distribution (see Supplementary Note), retaining the mean and the variance on a logarithmic scale and thus fixing the heritability, while varying the θ parameter of the generalised gamma distribution between 0 and 2 . For these data sets, we also varied the censoring levels of 0%, 20% and 40% (see Supplementary Note). For each of the censoring and phenotypic distribution combinations, 25 replicate phenotypic data sets were created, giving a total of 375 data sets. The prior parameters for α and μ were set the same way as described in Data Analysis Details.We conducted simulations to analyse the performance of our model under model misspecification, where the phenotypic distribution does not conform to a Weibull distribution, and to different censoring levels in the data. We assessed (i) estimation of hyperparameters, (ii) false discovery rate, and (iii) prediction accuracy. We used 13, a martingale residuals approach with single-marker ordinary least squares (OLS) regression11, and martingale residuals with a Bayesian regression mixture model with a Dirac spike (BayesR)19. For each of the 25 simulation replicates, across the five generalised gamma θ parameters, we calculated the correlation between the simulated genetic values and a genetic predictor, created from the regression coefficients obtained from each approach, in an independent data set , with results shown in Fig. θ ∈ {0, 1, 2} and censoring levels 0%, 20% and 40%  levels across the 25 replicate simulations given fixed posterior inclusion probabilities  and we simulated the phenotypes from Weibull distribution, with 25 simulation replicates. We used this data to compare BayesW to the same other methods described above, by calculating the correlation of simulated genetic value and a genetic predictor in an independent data set . We present these results in Supplementary Fig. To test the impact of LD among the markers, we used UK Biobank chromosome 22 imputed genotype data . We ran 10 simulations with three different numbers of causal loci: 200, 2500 and 4000. The phenotypes were simulated from Weibull distribution with a fixed heritability of h2 = 0.5. All the models were executed with three variance components   and then assigning the effects to the index SNPs of randomly selected clumps.To validate properties of polygenicity, variance partitioning between mixtures and false discovery rate we used UK Biobank chromosome 1 imputed genotype data that was LD pruned with threshold p = 2, 500 randomly selected causal SNPs, with heritability h2 = 0.5. The phenotypes were simulated from Weibull distribution and models were run with three variance components . Models were run by varying the number of tasks  between 1, 4, 8, 16 and synchronisation rate (number of markers processed by each task until synchronisation) between 1, 5, 10, 20, 50 . To infer ancestry, 488,377 genotyped participants were projected onto the first two genotypic principal components (PC) in 2504 individuals of the 1000 Genomes project with known ancestries. Using the obtained PC loadings, we then assigned each participant to the closest population in the 1000 Genomes data: European, African, East-Asian, South-Asian or Admixed. As we wished to contrast the genetic basis of different phenotypes, we then removed closely related individuals as identified in the UK Biobank data release. While we expect that our model can accommodate relatedness similar to other mixed linear model approaches, we wished to compare phenotypes at markers that enter the model due to LD with underlying causal variants, and relatedness leads to the addition of markers within the model to capture the phenotypic covariance of closely related individuals.We restricted our discovery analysis of the UK Biobank to a sample of European-ancestry individuals  were selected those with minor allele frequency (MAF) 0.0025 and rs identifier, in the set of European-ancestry participants, providing a data set of 9,144,511 SNPs, short indels and large structural variants. From this, we took the overlap with the Estonian Biobank data to give a final set of 8,433,421 markers. From the UK Biobank European data set, samples were excluded if in the UK Biobank quality control procedures they (i) were identified as extreme heterozygosity or missing genotype outliers; (ii) had a genetically inferred gender that did not match the self-reported gender; (iii) were identified to have putative sex chromosome aneuploidy; (iv) were excluded from kinship inference. Information on individuals who had withdrawn their consent for their data to be used was also removed. These filters resulted in a dataset with 382,466 individuals. We then excluded markers of high LD by conducting LD pruning using a threshold of r2 = 0.9 for a 100kb window leaving us with a final set of 2,975,268 markers. This was done in order to decrease the number of markers that were in high LD and thus giving very little extra information but requiring more than two times the computational resources. Genotype quality control was conducted using plink version 1.938.We used the imputed autosomal genotype data of the UK Biobank provided as part of the data release. For each individual, we used the genotype probabilities to hard-call the genotypes for variants with an imputation quality score above 0.3. The hard-call-threshold was 0.1, setting the genotypes with probability ≤0.9 as missing. From the good quality markers (with missingness less than 5% and i we created a pair of last known time (logarithmed) without an event Yi and censoring indicator Ci . If the event had not happened for an individual, then the last time without having the event was defined as the last date of assessment centre visit minus date of birth .We then selected the recorded measures for the 382,466 to create the phenotypic data sets for age-at-menopause, age-at-menarche and age-at-diagnosis of HBP, T2D or CAD. For each individual N = 151,472 women of which 108, 120 had the event and 43, 352 had not had an event by the end of the follow-up. For time-to-menarche we used UKB field 2714 and we excluded all women who had no record for time-to-menarche which left us with a total of N = 200,493 women of which all had had the event. For the age of diagnosis of HBP we used the UKB field 2966 for and we left out individuals who had the HBP diagnosed but there was no information about the age of diagnosis (UKB field 6150) which left us with a total of N = 371,878 individuals of which 95,123 had the event and 276, 755 had not had an event by the end of the follow-up. For age of diagnosis of T2D, we used either the UKB field 2976 or field 20009 or the mean of those two if both were available. We excluded individuals who had indicated self-reported “type 1 diabetes” (code 1222) and had Type 1 Diabetes (ICD code E10) diagnosis; we also excluded individuals who did not have any recorded time for the diagnosis of T2D but had indicated secondary diagnosis (UKB fields 41202 and 41204) of “non-insulin-dependent diabetes mellitus” (ICD 10 code E11) or self-reported non-cancer illness (UKB field 20002) “type 2 diabetes” (code 1223) or “diabetes” (code 1220). That left us with a total of N = 372, 280 individuals of which 15, 813 had the event and 356, 467 had not had an event by the end of the follow-up. For the age of diagnosis of CAD, we used either the minimum of age at angina diagnosed and age heart attack diagnosed (UKB fields 3627 and 3894) or the minimum age indicated to have either two of diagnoses  in UKB field 20009 or the mean of those if both were available. We excluded individuals who did not have any information about the time of diagnosis but had the following primary or secondary diagnoses: ICD 10 codes I20, I21, I22, I23, I24 or I25; self-reported angina (code 1074) or self-reported heart attack/myocardial infarction (code 1075). That left us with a total of N = 360, 715 individuals of which 17, 452 had the event and 343, 263 had not had an event by the end of the follow-up.For age-at-menopause, we used UKB field 3581 to obtain the time if available. We excluded from the analysis (1) women who had reported having and later not having had menopause or vice versa, (2) women who said they had menopause but there are no record of the time of menopause (UKB field 2724), (3) women who have had a hysterectomy or the information about this is missing (UKB field 3591), 4) women whose menopause is before age 33 or after 65. This left us with a total of 39 commit version b8044f1, to account for the population stratification in a standard way. We did not include any covariates of age or year of birth because these are directly associated with our phenotypes.In the analysis we included covariates of sex, UK Biobank recruitment center, genotype batch and 20 first principal components of the LD clumped set of 1.2 million marker data set, calculated using flashPCA240. For the Estonian Biobank Data, 48,088 individuals were genotyped on Illumina Global Screening (GSA) , OmniExpress (N = 8102), CoreExome  and Hap370CNV  arrays. We selected only those from the GSA array and imputed the data set to an Estonian reference, created from the whole genome sequence data of 2244 participants41. From 11,130,313 markers with imputation quality score >0.3, we selected SNPs that overlapped with the UK Biobank LD pruned data set, resulting in a set of 2,975,268 markers. The phenotypic data was constructed similarly to the phenotypes based on the UK Biobank data for the NEst = 32,594 individuals genotyped on the GSA array. For time-to-event traits, if no event had happened then the time is considered censored and the last known age without the event was used, calculated as the last known date without event minus the date of birth. Because only the year of birth is known, birth date and month were imputed as July 1 for age calculations.The Estonian Biobank cohort is a volunteer-based sample of the Estonian resident adult population. The current number of participants-close to 52,000–represents a large proportion, 5%, of the Estonian adult population, making it ideally suited to population-based studiesFor age-at-menopause, we excluded women who had reported having menstruation stopped for other reasons which resulted in 6434 women who had had menopause and 12,934 women who had not had menopause. For age-at-menarche we excluded women who had not reported the age when the menstruation started which resulted in 18,134 women. For both age-at-menarche and age-at-menopause, if the event had occurred, self-reported age during that event was used.Initially, the cases of CAD, HBP or T2D were identified on the basis of the baseline data collected during the recruitment, where the disease information was either retrieved from medical records or self-reported by the participant. Then, the information was linked with additional health insurance information that provided additional information on prevalent cases. To construct the phenotypes for the time-to-diagnosis of CAD, HBP or T2D for the individuals with the corresponding diagnosis we used the age at the first appearance of the respective ICD 10 code that was also used for creating the UK Biobank phenotypes. If the self-reported data about the ICD 10 code has only the information about the year, the date and month were imputed as July 1 and if only the date is missing then the date was imputed as 15. Respective case-control phenotypes for CAD, HBP or T2D were defined 0 if the person had not had an event (censored) and 1 if the person had had an event and these binary indicators were adjusted for age and sex. For the T2D phenotype, we excluded individuals with a diagnosis of T1D. For CAD we resulted with 30,015 individuals without the diagnosis and 2579 individuals with a diagnosis, for HBP we resulted with 24,135 individuals without the diagnosis and 8459 individuals with a diagnosis and for T2D we resulted with 30,883 individuals without the diagnosis and 1457 individuals with a diagnosis.μ and δ the mean is chosen 0 and variance α we choose α0 = 0.01 and κ0 = 0.01; for φ in with groups model, we set parameters to be ασ = 1, βσ = 0.0001; for π and πφ the prior parameters is set to be a vector of ones. The model without groups was executed with mixture components 0.00001, 0.0001, 0.001, 0.01 , and the model with groups was executed with (group-specific) mixture components . Guided by our simulation study  themselves distributed over compute nodes. For the analyses we used 8 tasks per node; due to the differences in sample size we were using the different number of nodes to accommodate the data in memory: for time-to-menopause, we used 8 nodes, for time-to-menarche we used 10 nodes and for time-to-diagnosis of CAD, HBP or T2D we used 12 nodes. This resulted in splitting the markers between 64 workers for time-to-menopause, 80 workers for time-to-menarche and 96 workers for time-to-diagnosis of CAD, HBP or T2D. For the last case, the average number of markers assigned to one worker is 30,992. We chose to use a synchronisation rate of 10 meaning the synchronisation between all of the workers was done after sampling 10 markers in each of the workers. Both the choice of the maximum number of workers and the synchronisation rate are stringent options considering our simulation study results plotted in Supplementary Fig. 21. PPWV requires first setting a threshold of the proportion of the genetic variance explained. Then, based on the posterior distributions we calculated the probability that each region explained more than the specified threshold of the proportion of the genetic variance and this quantity is denoted as PPWV. The regions were defined via the LD clumping procedure  resulting in regions that have high inter-region correlations but low intra-region correlations. For these LD clumped regions we used thresholds of 1/100,000, 1/10,000 and 1/1000 of the total genetic variance. The smallest threshold for PPWV is 1/100,000 of the total genetic variance as this gives the same magnitude as the smallest mixture component (0.00001) used in the models. The smallest mixture component reflects the smallest effect size the model is intended to capture. The thresholds of 1/10,000 and 1/1000 of the total genetic variance are chosen 10 and 100 times greater than the smallest threshold to point out the regions with larger effect sizes. To check the significance of the gene-associated regions we used more stringent thresholds of 1/10,000 and 1/1000 of the total genetic variance as gene-associated regions can contain greatly more markers. Furthermore, to make gene-associated regions more comparable, we fixed an upper bound of 250 for the markers that can contribute to a gene-associated, markers exceeding the bound were randomly discarded.For testing region-based significance for BayesW, we used a Posterior Probability of the Window Variance (PPWV)28. Because there were no results for our definition of time-to-CAD or time-to-T2D we used time-to-angina and time-to-heart attack summary statistics for comparison with CAD and time-to-diabetes for comparison with T2D. We called an LD clumped region significant if the region contained at least one SNP with a p-value < 5 × 10−8. To do the prediction into the Estonian Biobank we only used the markers with p-value < 5 × 10−8. We did the predictions only for age-at-menarche and age-at-menopause since the number of significant markers for them is higher.To do the comparison in terms of discovered regions and prediction accuracy we used the summary statistics from the fastGWA method15 with UK Biobank data and then used the estimates to make predictions into Estonian Biobank. To make the two models comparable, we used exactly the same data sizes for estimating the models on the UK Biobank as were used with the BayesW. For all of the traits, we decided to use 95% of the sample size as the training data and the rest as the validation data. This was done in order to minimise the loss in power due to not using the entire sample and 5% of the sample gives a sufficiently large validation set. We ran the Cox-LASSO model using snpnet with 16 threads and allocating 250 GB of memory. This was sufficient to find the optimal hyperparameter for the traits of time-to-menopause  and time-to-CAD . However, for the other traits, the snpnet procedure ran out of memory and it was decided to use the results from the last available iteration . For the traits for which it was not possible to detect the optimal hyperparameter a sensitivity analysis was done by comparing with the previous iterations. Prediction accuracy was virtually the same between the last available iteration and some iterations before that suggesting that the last available iteration was already providing a hyperparameter close to the optimum.To do the comparison in terms of predictive accuracy with a competing method we also trained the Cox-LASSO method with R package snpnetXEst is NEst × M matrix of standardised Estonian genotypes (each column is standardised using the mean and the standard deviation of the Estonian data), M × I matrix containing the posterior distributions for M marker effect sizes across I iterations. To calculate the prediction into Estonia we used the BayesW model with groups using 3000 iterations which gave us posterior predictive distributions of the genetic values with 3000 iterations. To create the final predictor, we calculated the mean genetic value for each individual across 3000 iterations. We also created the predictor using the estimates from Cox-LASSO by multiplying the standardised Estonian genotype matrix with the vector of Cox-LASSO effect size estimates. We evaluated the performance of the two predictors by comparing them to the true phenotype value and calculating R2 and Harrell’s C-statistic29. Instead of using the exact phenotypes the martingale residuals from the Cox PH model where the true phenotype was regressed on the gender (if applicable) were used to calculate the R2. That enables calculating the R2 value using also the censored individuals. Harrell’s C-statistic was calculated from the Cox PH model where the true phenotype was regressed on the predictor and gender (if applicable).The prediction based on BayesW into Estonian Biobank 42.The BayesW calculations have been performed using the facilities of the Scientific IT and Application Support Center of EPFL and the Helvetios cluster. All of the post-analysis steps were conducted using R software (version 3.6.1)Serial algorithm for BayesW sampling from the posterior distribution Data: Matrix x of standardised genotypes, matrix with covariate data z, vector of last time without an event y, vector of failure indicators d, prior hyperparameters α0, κ0, ασ, βσ, pL, iterations I. K is the Euler-Mascheroni constant.Initial values: Initially, we exclude all the variables from the model, thus the initial γj = 0, ∀ j ∈ {1, . . . , M}. We set βj = 0, ∀ j and δq = 0, ∀ q. The initial value for α is chosen to be the suitably transformed variance of the log sample. The initial value for μ is the mean of the log sample. The M.Parameters for prior distributions: We set priors weakly informative. Otherwise, if available, prior information could be used. To get weakly informative priors, for α prior, we set parameters to be α0 = 0.01 and κ0 = 0.01; for ασ = 1, βσ = 0.0001; for μ prior, we set parameter δq we set parameter π, pL is a vector of ones.i: εi = yi − μ  1  Initialise for every for iteration ← 1 toIdo  2  εi ← εi + μold  3     Add the previous effect to the residual: μ using ARS;  4     Sample εi ← εi − μ  5     Subtract the new effect from the residual:   6     Shuffle foreach covariate q do  7       8       δq using ARS  9        Sample εi ← εi − ziqδq10       α using ARS;11     Sample 12     Shuffle 13     Calculate exponentiated residuals: foreach marker j do14     if then15     16         Calculate if then17     18         19         20         Calculate γj;21     Sample mixture indicator if γj > 0 then22     βj from the γjth conditional distribution using ARS;23         sample εi ← εi − xijβj; 24           25           if γj = 0 then26     βj = 0;27          set π;28  Sample 29  Sample Bulk Synchronous Parallel Gibbs sampling with BayesW. Data, parameter initialisation and prior values are set as in Algorithm 29.Input: Define T parallel workers (tasks) and synchronisation rate u. Each worker t ∈ {1, . . . , T} has its corresponding vector of marker effects βt , indicator values γt to update and set of T messages Δεt: N × 1, N is the sample size.  1 Initialise variables;for iteration ← 1 to I do  2 μ, δ, α (as in Algorithm 29);  3   Update foreach subset of size u do  4   εt ← 0, t ∈ {1, . . . , T};  5    Δfor t ← 1 to T in parallel do  6     foreach column j from a subset of size u of the columns assigned to worker t do  7     if then  8       ϵ;  9       Calculate if then10      11       12       13       Calculate γj;14       Sample mixture indicator if γj > 0 then15       βj from the γjth conditional distribution using ARS;16       sample if γj = 0 then17       βj = 0;18       set if then19       εti ← Δεti − xij20        Δu markers;21     Wait until all workers are finished processing their sets of 22     23     π, 24   Update Further information on research design is available in the Supplementary InformationPeer Review FileReporting Summary


PEVs) for genomic best linear unbiased prediction (GBLUP) models using the Algorithm for Proven and Young (APY), extend it to single-step GBLUP (ssGBLUP), and apply this algorithm for approximating the theoretical reliabilities for single- and multiple-trait models in ssGBLUP. The PEV with APY was calculated by block sparse inversion, efficiently exploiting the sparse structure of the inverse of the genomic relationship matrix with APY. Single-step GBLUP reliabilities were approximated by combining reliabilities with and without genomic information in terms of effective record contributions. Multi-trait reliabilities relied on single-trait results adjusted using the genetic and residual covariance matrices among traits. Tests involved two datasets provided by the American Angus Association. A small dataset (Data1) was used for comparing the approximated reliabilities with the reliabilities obtained by the inversion of the left-hand side of the mixed model equations. A large dataset (Data2) was used for evaluating the computational performance of the algorithm. Analyses with both datasets used single-trait and three-trait models. The number of animals in the pedigree ranged from 167,951 in Data1 to 10,213,401 in Data2, with 50,000 and 20,000 genotyped animals for single-trait and multiple-trait analysis, respectively, in Data1 and 335,325 in Data2. Correlations between estimated and exact reliabilities obtained by inversion ranged from 0.97 to 0.99, whereas the intercept and slope of the regression of the exact on the approximated reliabilities ranged from 0.00 to 0.04 and from 0.93 to 1.05, respectively. For the three-trait model with the largest dataset (Data2), the elapsed time for the reliability estimation was 11 min. The computational complexity of the proposed algorithm increased linearly with the number of genotyped animals and with the number of traits in the model. This algorithm can efficiently approximate the theoretical reliability of genomic estimated breeding values in ssGBLUP with APY for large numbers of genotyped animals at a low cost.The objectives of this study were to develop an efficient algorithm for calculating prediction error variances  of an animal measures its genetic merit. For calculating EBVs, pedigree and genomic information are jointly used in a procedure called single-step genomic best linear unbiased prediction (ssGBLUP). Genetic evaluations report each EBV with its reliability, which measures how accurate the breeding value estimation was. Calculating EBV with ssGBLUP for large datasets is computationally expensive; Therefore, the Algorithm for Proven and Young (APY) was developed to reduce its computational cost. However, the procedure for obtaining the reliabilities of EBV is still computationally unfeasible to apply. Thus, this study aimed to develop a new method for approximating reliabilities for ssGBLUP with APY for large datasets. We required this new method to be accurate and with fewer computational requirements than the estimation of breeding values by itself. The method that we develop consists of accumulating pedigree and genomic information in successive steps, allowing for computational efficiency. Using a dataset with more than 300,000 genotypes in a pedigree of 10,000,000 animals provided by the American Angus Association, we showed that our proposed method is accurate and computationally efficient, with a correlation of 0.98 between the approximated and target values running in less than 12 min. MMEs) are used for measuring the variation in the estimated breeding values (EBVs) from genetic evaluations; therefore, their calculation is necessary. Under the correct model specification, the reliabilities can be calculated as a function of the prediction error variances (PEVs), which are obtained from the diagonal of the inverse of the coefficient matrix of the MMEs (SNP) for genetic evaluations, such a structure was lost, and the existing algorithms were not useful.Reliabilities obtained from the inverse of the mixed model equations (the MMEs . The dirthe MMEs  or theirthe MMEs  by explothe MMEs . HoweverGBLUP) and marker effect models −1X′)W .Here, we assume that Read and store Approximate Calculate Overwrite Invert Overwrite i refers to the ith element of ith row of Overwrite Obtain PEVs from ERCs) such as in The method presented for single-trait APY-GBLUP can be used for approximating the reliabilities for a single-trait APY-ssGBLUP by using different procedures based on effective records contributions  in A1 using ERCs instead of Approximate pedigree reliabilities for genotyped animals without taking into account the information provided by non-genotyped animals by removing their contributions. This will be referred to as approximating reliabilities of Obtain final reliabilities for genotyped animals using formulas (18 to 27) from Back-solve the reliabilities for genotyped animals to get ERCs. A detailed procedure for back-solving the reliabilities to obtain ERCs is explained in the R0) covariances among traits, and the number of progenies for each trait. Then, for each animal, its adjusted reliability for the jth trait is equal to:For extending both single-trait APY-GBLUP and APY-ssGBLUP to multi-trait models, single-trait reliabilities were adjusted using the genetic and residual covariance matrices among traits following the method of Where,jth trait, and is Q a diagonal matrix with jth trait is non-missing and zero otherwise. It is worth noticing that the dimension of all the matrices in equation (6) is equal to the number of traits. Therefore, calculating equation (6) is not computationally demanding. For more details, we refer the reader to and D requires reading the data file only twice. Although D by accounting for all the cross-classified effects, which can be more robust. The inversion of MKL; Intel corporation), whereas the inversion of http://www.openmp.org). The most time-consuming steps from Algorithm A1 are 4 to 6. Letting For calculating the reliabilities of APY-GBLUP, only the nonzero elements of when n→∞ . Thus, fover 25k .For approximating the reliabilities under APY-ssGBLUP, the animals should be ordered such that the parents precede their progeny for obtaining ERCs . This waPWG). In Data1 used for multi-trait analysis (Data1_mt), 78,641 animals had phenotypes for at least one of the three traits: PWG, birth weight (BW), or weaning weight (WW), and the number of animals in the pedigree was 172,089. Because of limitation in the computation of the inverse of the left-hand side of the MMEs for multi-trait models with many genotypes, the total number of genotyped animals in Data1_mt had to be reduced to 20,000, of which 9,477 were noncore.The two datasets used in this study to test the approximated reliabilities of genomic EBV from APY-ssGBLUP were provided by the American Angus Association . A small dataset (Data1) was used for comparing the approximated reliabilities with the reliabilities obtained by the inversion of the left-hand side of the MMEs. A large dataset (Data2) was used for evaluating the computational performance of the algorithm. Data1 for single-trait analysis (Data1_st) consisted of 50,000 animals that had genotypes for 39,759 SNPs after quality control. From the genotyped animals, 10,523 were randomly selected as core based on the number of core animals proposed by The second dataset (Data2) had 335,325 genotyped animals, and the number of noncore animals was equal to 324,802. In single-trait analysis (Data2_st), 4,218,407 individuals had records for PWG, and the total number of animals in the pedigree was 10,213,401. The multi-trait analysis with Data2 (Data2_mt) used the same number of animals in the pedigree as in Data2_st, but 8,681,659 animals had phenotypes for at least one of the three traits. A summary of all datasets can be found in PreGSf90  was usedThe methods presented in the previous sections were programmed in Fortran 95 and compiled with the Intel Fortran Compiler version 15.0.3 with options -fpp and -O3. All the computations were performed on a Dell PowerEdge R740XD server with 1.5 TB of memory, 45 TB of disk, and two Intel Xeon Gold 6258R processors with 56 threads each; however, the number of threads for all the computations was limited to 32.The comparison between reliabilities obtained from inversion of the MME with required . When suAs can be noticed, the method developed in the present study is much faster than the current methods for approximating reliabilities. For example, Since the reliability estimation is done after estimating breeding values, An efficient method for calculating exact genomic reliabilities with APY skab353_suppl_Supplementary_MaterialsClick here for additional data file.


Recent studies have shown the ubiquity of pleiotropy for variants affecting human complex traits. These studies also show that rare variants tend to be less pleiotropic than common ones, suggesting that purifying natural selection acts against highly pleiotropic variants of large effect. Here, we investigate the mean frequency, effect size and recombination rate associated with pleiotropic variants, and focus particularly on whether highly pleiotropic variants are enriched in regions with putative strong background selection. We evaluate variants for 41 human traits using data from the NHGRI-EBI GWAS Catalog, as well as data from other three studies. Our results show that variants involving a higher degree of pleiotropy tend to be more common, have larger mean effect sizes, and contribute more to heritability than variants with a lower degree of pleiotropy. This is consistent with the fact that variants of large effect and frequency are more likely detected by GWAS. Using data from four different studies, we also show that more pleiotropic variants are enriched in genome regions with stronger background selection than less pleiotropic variants, suggesting that highly pleiotropic variants are subjected to strong purifying selection. From the above results, we hypothesized that a number of highly pleiotropic variants of low effect/frequency may pass undetected by GWAS.The online version contains supplementary material available at 10.1007/s00439-021-02308-w. The analyses of thousands of genetic variants obtained in the last decades by Genome-Wide Association Studies (GWAS) have provided a great advance in the knowledge of the understanding of genetic variation, particularly for human traits  was considered, and the corresponding gene or intergenic name associated with that SNP was assumed to be a potential causal locus. The contribution to heritability from each locus was calculated as h2 = 2β2q(1 − q) where β is the locus estimated effect and q its frequency. For the sake of robustness, only traits with a wide and well-known genetic background composed by at least 30 unique genes detected were considered. In addition, we restricted the traits analyzed to those represented by at least three different studies. More details of the procedure can be found in López-Cortegano and Caballero , 2, 3, etc. traits. The average homozygous effect size, minor allele frequency (MAF) and contribution to heritability from each locus, were obtained for each pleiotropy degree.B statistic attached to each genomic position of the genome represents the expected reduction in nucleotide diversity at a neutral site due to purifying selection at other sites  for the value of B on the degree of pleiotropy. Since, as mentioned above, the value of B depends both on the intensity of natural selection and the rate of recombination (RR), we also obtained the partial regression coefficients (b′) of B on the degree of pleiotropy and RR. These were obtained with the R command summary (lm(y ~ x1 + x2)), where y is the dependent variable (B) and x1 and x2 are the predictor variables . For genomic regions we averaged the B and RR values for all positions within each region. For genes, we averaged the corresponding values for all positions from the start to the end of the gene. Finally, for SNPs, the values of B for each SNP position were considered. All genomic regions, gene and SNP coordinates were fitted to the genome version GRCh37 (hg19), using the dbSNP database  for SNPs, and the RefSeq database  for genes. Recombination rates for each SNP, gene or genomic region fitted to the GRCh37 coordinates were obtained from the human genetic map , and the same was observed for the standard deviation of effect sizes . The MAF of variants gradually increased with the pleiotropy degree , and this, along with the increased effects sizes, accounted for a higher contribution to heritability for the most pleiotropic classes .The total number of pleiotropic loci found in our study was 629, which is a 23% of all loci analyzed Table S1). Gastrointestinal, skeletal and cardiovascular functional domains presented the highest proportions of pleiotropic loci when averaging traits , and the neurological/psychiatric domain, the lowest one (18%) . As expected, the higher the pleiotropy degree, the lower the number of variants found, with the highest degree being 12 . The mean effect size steadily increased with the pleiotropy degree 2).Fig. 2B statistic) is given in Fig. B on the degree of pleiotropy of b′ = 0.010, p = 0.06) and for the data of Pickrell et al. . However, for the two much larger datasets of Watanabe et al.  and Shikov et al. , the relationship was significantly negative and of similar magnitude. The results presented in Fig. B on the degree of pleiotropy of: b′ = 0.012 (p = 0.02), − 0.004 (p = 0.4), − 0.017 (p < 2 × 10−16), − 0.013 (p < 2 × 10−16), respectively. The results of Watanabe et al.  or genes or SNPs were considered instead . Analogously, the results of Shikov et al.  anIn agreement with the results of Chesmore et al. , we founAnother difference between our results and those of Chesmore et al.  refers tWe found an increase in minor allele frequency with the degree of pleiotropy Fig. c, in accB and the degree of pleiotropy , which was also found as highly pleiotropic by Chesmore et al.  as well as for individual SNPs . Moreover, for Shikov et al. Below is the link to the electronic supplementary material.