Mendelian randomization is the use of genetic variants as instruments to assess the existence of a causal relationship between a risk factor and an outcome. A Mendelian randomization analysis requires a set of genetic variants that are strongly associated with the risk factor and only associated with the outcome through their effect on the risk factor. We describe a novel variable selection algorithm for Mendelian randomization that can identify sets of genetic variants which are suitable in both these respects. Our algorithm is applicable in the context of two-sample summary-data Mendelian randomization and employs a recently proposed theoretical extension of the traditional Bayesian statistics framework, including a loss function to penalize genetic variants that exhibit pleiotropic effects. The algorithm offers robust inference through the use of model averaging, as we illustrate by running it on a range of simulation scenarios and comparing it against established pleiotropy-robust Mendelian randomization methods. In a real-data application, we study the effect of systolic and diastolic blood pressure on the risk of suffering from coronary heart disease (CHD). Based on a recent large-scale GWAS for blood pressure, we use 395 genetic variants for systolic and 391 variants for diastolic blood pressure. Both traits are shown to have significant risk-increasing effects on CHD risk. It applies the theory of instrumental variable analysis from the causal inference literature, using genetic variants associated with the risk factor as instruments. Mendelian randomization relies on the idea that, since genetic variants are randomly inherited and fixed at conception, they should be uncorrelated with potential confounders of the relationship between the risk factor and outcome and are therefore suitable to use as instruments. This approach has received much attention since the seminal paper of Davey Smith and Ebrahim,Formally, Mendelian randomization relies on three basic assumptions:There is no confounding in the association between the genetic variants and the outcome.The genetic variants are associated with the risk factor of interest.The genetic variants only influence the outcome via their association with the risk factor and not through alternative causal mechanisms.The three assumptions are shown in As mentioned earlier, Assumption 1 is usually justified on the basis of Mendelian inheritance: an individual’s genotype is randomly assorted at conception and not influenced by external confounding factors. To ensure the validity of Assumption 2 Mendelian randomization analyses utilize the results of large consortia meta-GWAS, which use sample sizes of tens or hundreds of thousands of individuals to identify genetic variants robustly associated with a trait. In particular, many recent Mendelian randomization studies have taken a two-sample approach in which genetic associations with the target risk factor and with the outcome are assessed in separate datasets, in order to leverage their respective  meta-GWAS. However, these GWAS results are rarely available as individual-level data. Usually, only a set of summary statistics, such as univariate SNP-trait associations and corresponding standard errors (SEs), are reported. As a result, a large number of recent Mendelian randomization investigations rely on summarized data.5 and median estimation,6 while several algorithms for pleiotropy-robust Mendelian randomization have been developed recently.15 Recent reviews and comparison of the various methods have also been conducted.17Assumption 3, often called the exclusion restriction or no-pleiotropy assumption, has received much attention in the recent literature. In practical applications, we typically do not know which genetic variants exhibit pleiotropic effects and there is need for methods to perform Mendelian randomization in the presence of pleiotropic variants. Traditional approaches include MR-Egger regression18 JAM was originally proposed for fine-mapping genetic regions. Similar to other recently proposed fine-mapping algorithms,20 JAM is designed to work with summary GWAS data. The algorithm performs variable selection to identify genetic variants robustly associated with the trait. Genetic correlations are taken into account by estimating them from a reference dataset such as 1000 Genomes or the UK Biobank. Variable selection is performed according to a Bayesian stochastic search algorithm, which can explore the complete space of causal configurations. Consequently, JAM is able to explore complex models with large numbers of variants, as recently demonstrated while fine-mapping dense genotype data for prostate cancer risk.21In this article, we add to the relevant literature by proposing a new method for variable selection and causal effect estimation in Mendelian randomization. Our method is derived as an extension of the JAM algorithm .22 in which the likelihood is augmented with a loss function before obtaining a Bayesian loss- posterior distribution. JAM’s likelihood is hence combined with a loss function that penalizes models containing variants with heterogeneous univariate causal effect estimates. Our algorithm performs variable selection and returns variant-specific posterior inclusion probabilities, which can be interpreted as probabilities of each variant being a valid instrument, and posterior model probabilities, which can subsequently be used to estimate the causal effect of interest by averaging across model-specific estimates. Uncertainty in which variants should be excluded on the basis of pleiotropy is reflected by averaging estimates over competing selections of instruments.We develop a novel model averaging variable selection algorithm for Mendelian randomization, which we call JAM-MR (JAM for Mendelian randomization). To do so, we modify JAM’s variable selection to downweight genetic variants which exhibit heterogeneous effects. We use the recently proposed framework of general Bayesian inference,19 and colocalization.23 The Bayesian paradigm allows us to incorporate prior information on the suitability of genetic variants as instruments into the analysis. It also allows us to model the uncertainty in genetic associations with the risk factor, which is often ignored by other approaches. JAM-MR also offers a natural framework for incorporating genetic correlations, when conducting Mendelian randomization with several genetic variants coming from the same gene region.Conceptually, the JAM-MR algorithm has a number of attractive features. The use of Bayesian stochastic search has already proven useful in other areas of statistical genetics, such as fine-mappingThe performance of our algorithm is illustrated in a range of simulated datasets, as well as in a real-data application where we investigate the causal effect of systolic and diastolic blood pressure on the risk of CHD. We instrument blood pressure using a recently published large-scale meta-GWAS, which combined results across more than one million individuals, and therefore base our Mendelian randomization analysis on larger sample sizes and more genetic variants compared with previous studies in the literature. Our results strengthen the claim for a risk-increasing causal relationship between blood pressure traits and CHD risk.2 |2.1 |2.1.1 |18 for a more detailed description.We start by providing a brief outline of the JAM algorithm. The reader is referred to the relevant paperP correlated genetic variants G1, … ,GP, identify which variants are causally and independently associated with a trait X of interest. The genetic variants are typically located in the same gene region. The main purpose of JAM is to perform variable selection for the genetic variants, given a set of GWAS summary statistics and an external dataset from which to estimate genetic correlations.The JAM algorithm is primarily a tool for fine-mapping densely genotyped DNA regions. The standard fine-mapping question is as follows: given i, let gi =  and xi be the allele counts and trait measurements, respectively, and denote G = (gij) the genetic matrix. We assume that trait values and allele counts per variant have been centered. Typically, the “individual-level data” xi,gi are not available in practice. Instead, we only have access to a set of univariate association estimates For individual 2.1.2 |P genetic variants were assumed to be associated with X, the JAM algorithm would model the trait asJAM uses linear regression to model the trait: if all 𝛾 ∈ {0,1}P be a vector of binary indicators, such that 𝛾j = 1 if variant Gj is included into the JAM model and 𝛾j = 0 if it is not. Moreover, let R𝛾 = {j ∈ {1, … ,P} ∶ 𝛾j = 1} be the set of genetic variants included in the model, and P𝛾 =|R𝛾|be the number of variants in R𝛾. In practice JAM implements variable selection, reflecting that, in many applications, only a subset of the variants should be used to model the trait. Let 𝛽𝛾 denote the subvector of 𝛽 containing only those entries 𝛽j for which 𝛾j = 1, G𝛾 be the N1 × P𝛾 submatrix of G containing only those columns j for which 𝛾j = 1 and g𝛾i be the ith row of G𝛾. Let 2.1.3 |𝛽𝛾, JAM uses a conjugate g-prior,The likelihood 𝜏 is a constant. By default, the algorithm sets 𝜏 = max, as has been previously recommended by various authors.25 The residual variance where aX, bX. Finally, JAM uses a Beta-Binomial prior on the space of all possible models,for fixed B denotes the Beta function. This is a binomial prior on the number of variants associated with X, with the prior proportion of causal variants treated as a hyperparameter and assigned its own Beta prior. By default, a𝜔 = 1, b𝜔 = P, which corresponds to an expectation that a single variant will be associated with the trait. The Beta-Binomial prior assigns the same prior probability to models of equal size, and the same prior inclusion probability to different variants. In applications where prior information on the associations on specific variants with the risk factor is available, this information can be incorporated into JAM by partitioning the variants into subgroups and assigning a different prior inclusion probability to variants in each subgroup.where 2.1.4 |A Bayesian posterior distribution over all the parameters  can be obtained according to the standard principles of Bayesian inference:Conditional on a particular combination of causal variants, posterior inference on the regression parameters where 2.1.5 |𝛽𝛾 and the residual variance An advantage of the linear regression setting is that it allows for fast and efficient variable selection. The regression coefficients wherep(𝛾∣x) ∝ p(x ∣𝛾)p(𝛾). The normalizing constant of that density can be difficult to compute, as it would involve a summation over all possible models. To avoid this, JAM implements a reversible-jump MCMC algorithm,26 which only requires the posterior to be evaluated up to a normalizing constant. JAM implements a standard reversible-jump algorithm with addition, deletion and swapping of genetic variants as possible moves. The stochastic search algorithm allows exploration of an unrestricted model space, without the need to set limits on the maximum number of causal variants. Consequently, JAM is able to efficiently explore complex causal configurations among large numbers of genetic variants. Posterior model probabilities can be estimated by the proportion of iterations JAM spends in each model.This leads to the marginal model posterior, 2.1.6 |xi,gi only through the quantities xTx, GTx and GTG. These quantities are sufficient summary statistics for linear regression. In the setting of genome-wide association studies, 18 by the univariate variant-trait association estimates X) measured in the GWAS for the trait X. This variance is sometimes reported by genetic association studies, but can be approximated using the SNP-trait associations, SEs and effect allele frequencies if it is not available.27 Finally, the matrix GTG models (N1 − 1 times) the genetic correlations between variants G1, … ,GP and can be approximated if a reference dataset, such as the 1000 Genomes dataset or the UK Biobank, is available. This yields the following summary-data approximation for S(𝛾):One of JAM’s main advantages is that it does not require access to individual-level data. Variable selection is implemented according to The model-specific marginal posteriors (7) and (8) can be approximated by summary GWAS data in a similar way.2.1.7 |GTx can be derived by mapping univariate log-odds ratios to the univariate effects that would have been estimated if the binary trait was modeled by linear regression. The same strategy is employed in other linear-based summary data frameworks, including LDPred,28 to which we refer readers for a detailed description of this mapping.The JAM algorithm assumes that the trait of interest is continuous, as is often the case for risk factors used in Mendelian randomization studies. JAM then relies on a linear modeling framework to relate univariate summary statistics to multivariate effects via the transformation 2.2 |2.2.1 |We now describe our extension to the standard JAM algorithm that facilitates pleiotropy adjustment and accurate causal effect estimation in Mendelian randomization studies, which we call JAM-MR.𝜃 according to a ratio formula,29For the JAM-MR algorithm, we work in the context of two-sample summary-data Mendelian randomization. This framework allows researchers to leverage the power and large sample sizes of large consortia GWAS studies that are already available for many important traits, without the requirement that both the risk factor and the outcome should be measured on the same GWAS. It requires access to two different genetic association studies, one from which to obtain the SNP-risk factor summary statistics G1, … ,GP used in the JAM-MR algorithm can be either correlated or independent. In fact, Mendelian randomization studies usually rely on independent variants, so we will mainly focus on that case.The genetic variants 2.2.2 |30In order to construct the JAM-MR algorithm, we first outline how a fully Bayesian algorithm for Mendelian randomization can be developed. Fully Bayesian algorithms for Mendelian randomization are not common in the literature, despite some recent developments.xi,yi,Gi1,Gi2 were available. The data can be modeled in terms of the genetic associations with the risk factor and outcome, 𝛽X,𝛽Y, and the corresponding residual variances, 𝛽Y = 𝜃𝛽X and model the causal effect parameter explicitly). The likelihood for the risk factor and outcome in such a model would have the form x,G1) and  come from two independent samples, the likelihood factorizes intoSuppose again that individual-level data For Bayesian inference, this likelihood needs to be combined with a set of prior distributions on the model parameters, The term in the brackets is the posterior distribution targeted by the JAM algorithm (6) while for the remaining terms some modeling assumptions about the SNP-outcome associations are necessary—for example, a normality assumption 2.2.3 |22 This new theoretical framework constitutes a generalization of the core Bayesian paradigm. For a dataset 𝜆 and prior distribution (𝜆), the standard Bayesian updating scheme, Instead of full Bayesian inference, the JAM-MR algorithm implements a simplified version of the above procedure. To do so, it utilizes a recently proposed framework for performing Bayesian inference using loss functions.w is a tuning parameter that determines the influence of the loss function relative to that of the prior. If where Sometimes this new framework is implemented using both a log-likelihood and a loss function,22which is equivalent to using 2.2.4 |𝛾 and not the full set of parameters.JAM-MR utilizes general Bayesian inference to perform variable selection for Mendelian randomization. Our new algorithm uses the Bayesian framework only for variable selection instead of performing full Bayesian inference. Its target parameter is therefore the model indicator w balances the impact of the pleiotropic loss on model selection relative to that of the JAM likelihood and the prior.JAM-MR uses an implementation of the general Bayesian framework that is similar to (15), using both a “likelihood part” and a loss function. This can be motivated by comparing (15) to the posterior distribution (13). In the “likelihood part,” we model genetic associations with the risk factor using 𝛾 can be specified as in JAM. A marginalization procedure similar to that used in JAM reduces the “likelihood part” to the marginal likelihood (9).Prior distributions for the SNP-risk factor association parameters 10 The intuition is that genetic variants which are valid instruments yield the same univariate causal effect estimates, up to some random variation. On the other hand, estimates based on pleiotropic variants can exhibit systematic differences, especially if the variants operate on different causal pathways towards the outcome, because the estimated causal effects depend on the strength and direction of the pleiotropic G-Y association. This suggests that our loss function should up weight modelswith homogeneous univariate causal effect estimates, as such models are likely to contain valid instruments, and downweight modelswithheterogeneousestimates,asatleastsomeofthegeneticvariantscontainedinthemarelikelytobepleiotropic.We now discuss how to specify the loss function. It is common in the Mendelian randomization literature to use some measure of heterogeneity between univariate causal effect estimates as a proxy for pleiotropic behavior.Consequently, in order to penalize pleiotropic models, we need a loss function that measures the heterogeneity of univariate estimates. A simple option isWhereR𝛾. The loss function (16) is the sum of squared differences of SNP-specific estimates is the inverse-variance-weighted (IVW) causal effect estimate based on variants in the set Note that the loss function (16) is not defined for the null model and models containing only one genetic variant. Such models carry no evidence as to whether the variant included is valid or pleiotropic. Therefore, we ignore these models and restrict JAM-MR to only consider models with at least two genetic variants. Since JAM-MR works with a single risk factor, our assumption that there are at least two genetic variants is similar to overidentification assumptions used in econometrics to test for instrument validity. In both cases, the assumption is needed because a single instrument would provide insufficient information to assess violations of the instrumental variable assumptions. However, JAM-MR and overidentification tests use the assumption in different ways. For example, in Sargan’s J test, the number of instruments minus the number of endogenous variables determines the degrees of freedom of the test’s chi-squared distribution. On the other hand, the process of fitting JAM-MR is not directly affected by the number of genetic variants, provided there are two or more variants.An alternative loss function can be obtained by weighting the individual causal effect estimates in (16) by the inverse of their squared SEs,10 to obtain a pleiotropy-robust exhaustive-search model averaging procedure for Mendelian randomization. In the simulations and the real-data application of this article, we have used the weighted loss function (19).This second loss function has the advantage that it also models uncertainty in the univariate ratio estimates y,x,g1,G2), the Bayesian loss-posterior for JAM-MR’s variable selection isCombining the general Bayesian inference framework (15), the JAM likelihood (9) and the loss function (19) and using the summary-level data S(𝛾) can be computed from (11). Note that this formula still depends on the tuning parameter w; the specification of a value for w will be discussed in P possible models. However, the loss-posterior can easily be evaluated up to a normalizing constant and hence it can be used as the target distribution for JAM-MR’s reversible-jump MCMC algorithm. This stochastic search procedure scales well to large datasets containing hundreds of genetic variants, is quite flexible and can explore large parts of the model space.where 15Since we are using heterogeneity as a proxy for pleiotropic behavior, JAM-MR implicitly makes a plurality assumption: the algorithm targets the largest set of genetic variants with homogeneous  univariate causal effect estimates, and it is assumed that this set corresponds to the valid instruments. Similar assumptions have been made by other methods.2.2.5 |The JAM-MR posterior probabilities can subsequently be used to obtain an overall estimate of the causal effect of interest, according to a model averaging procedure. The use of model averaging enables quantification of uncertainty related to the choice of instruments and allows many genetic variants to have small contributions to the overall causal effect estimate.For each model, the algorithm computes a model-specific causal effect estimate with variance𝛾 assigned positive posterior probability by the variable selection procedure. where the summation is over all models A straightforward choice for the model-specific estimates would be the IVW estimate (17). However, we have observed empirically that this choice often results in underestimation of the overall causal SE and a more elaborate modeling assumption may be needed. Because JAM-MR uses the heterogeneity of ratio estimates in order to model pleiotropic behavior, it will tend to favor sets of variants with homogeneous causal effect estimates and consequently underestimate the overall heterogeneity. This manifests as overly precise SEs, as estimated from the naive IVW formula. A relevant example is given in the 31 to use a multiplicative random effects model of the form 𝜙. The overdispersion parameter 𝜙 can be estimated using weighted linear regression.31 On the other hand, to model the potential downweighting of valid genetic variants with ratio estimates that may seem outlying due to random variation, we modify the random-effects model further and use a truncated normal distribution for the univariate causal effects:To better model this behavior and attenuate the overprecision, we modify the IVW estimates as follows. The standard IVW formula can be motivated as the maximum likelihood estimate obtained from the heteroskedastic linear regression model l, u denote the lower and upper truncation points. Using (23) we can obtain a model-specific likelihoodwhere N random variable. The likelihood (24) can then be maximized numerically to obtain a model-specific estimate l to the value R𝛾 that is smaller than all the ratio estimates of SNPs in 𝛾. If no such R𝛾 that is still larger than the ratio estimates of all SNPs in 𝛾  can be performed using gradient-based methods, since the derivatives of the likelihood (24) are straightforward to compute. To estimate the causal SE, we rely on frequentist asymptotic arguments and compute the sample Fisher information matrix,Optimization of (24) over . The maximum likelihood estimate for 𝜃 and its SE are then used as the model-specific estimates in (21), (22), from which to obtain the overall causal effect estimate.evaluated at the maximum likelihood estimates , plus an implementation for w0 = 0.In order to balance between these two extremes, JAM-MR implements a grid search to select the value of the tuning parameter. A range of candidate values is specified and the stochastic search is run for each value separately. Candidate values for the grid search should scale according to the sample size w, the algorithm then selects the one for which the smallest causal SE (22) is obtained. Intuitively, the collection of genetic variants that attains the smallest SE should correspond to the set of valid instruments, as both the inclusion of a pleiotropic variant and the removal of a valid SNP from that model will increase the causal SE. In our simulations, we have found this criterion to offer a reasonable compromise in the bias-variance trade-off that underpins the selection of a value for the tuning parameter.Among the candidate values for 3 |3.1 |We conducted a simulation study in order to assess the performance of the JAM-MR algorithm and compare it to established approaches for Mendelian randomization. The Mendelian randomization model that we used for the simulations wasGj on U,X,Y are denoted by 𝛼j,𝛽j,𝛿j, respectively. The parameters 𝛿j represent the direct effects of genetic instruments on the outcome. The parameters 𝛼j represent effects mediated by confounders, which we use to model violations of the InSIDE assumption (“instrument strength is independent of direct effects”). For genetic variants that are valid instruments, 𝛿j = 𝛼j = 0.Where N1 = N2 = 300 000 individuals and P = 400 independent genetic variants. These sample sizes are in line with recent Mendelian randomization analyses using composite risk factors. Of the 400 genetic variants, P1 = 120 (30%) were simulated as pleiotropic and P − P1 = 280 were simulated as valid instruments.Our simulations were conducted using the statistical software R. To inform the simulations, we used sample sizes and numbers of genetic variants similar to those in the real-data application presented in the following section. Specifically, we used a two-sample Mendelian randomization setting and generated two sets of individual-level data for gij were generated from a Binomial, with effect allele frequencies fj ∼ U. The genetic effects on the risk factor were simulated as 𝛽Xj ∼ 0.5 + N, which we empirically observed to yield a realistic pattern of genome-wide significant p-values, the majority of which were between 10−7 and 10−50. The error term variances were set equal, cX = cY = 1.Genetic variants 𝛼j, 𝛿j were set equal to zero for all the valid instruments. For invalid instruments, we generated three simulation scenarios. In the first scenario , we let 𝛼j = 0 and 𝛿j ∼ ±N, setting the sign of each 𝛿j to be positive or negative with equal probability. In the second scenario  we also set 𝛼j = 0 but the direct effect parameters 𝛿j were restricted to be positive, 𝛿j ∼ N. Finally, in the third simulation scenario , the SNP-confounder effects were also randomly generated, 𝛼j ∼ N. The direct effects 𝛿j were generated as in the second scenario. The three simulation scenarios are summarized in p-values for SNP-trait associations and univariate causal effect estimates for our simulations are provided in the The pleiotropic effect parameters 𝜃 = 0) and one with a positive (𝜃 = 0.3) causal effect. This resulted in a total of six simulation settings. Each setting was replicated 1000 times. In each replication, we generated the individual-level data and then performed univariate linear regression between each genetic variant and the risk factor and outcome to obtain GWAS summary statistics. Using these summary statistics, we then implemented JAM-MR as well as a range of already existing algorithms for Mendelian randomization. For comparison purposes, we also computed “oracle” IVW causal effect estimates using only the valid instruments. For more details on how JAM-MR and the competing methods were implemented in our simulation, we refer to the For each scenario, we implemented two sets of simulations, one with a null  R package R2BGLiMS. An outline of each of the competing methods and their implementation in R is provided in the 3.3 |𝜃 = 0.3 being a rather strong effect that most methods could easily identify and the fact that pleiotropic bias in our simulations was in the direction away from the null.The results of this simulation experiment are reported in 𝜃 = 0. This was the case even for the standard inverse-variance weighted method, which does not perform pleiotropy adjustments . The JAM-MR implementation attained SEs close to those obtained by the oracle IVW and had one of the smallest mean squared errors among the competing approaches. In terms of calibration of confidence intervals, JAM-MR exhibited a slight inflation of Type I error rates for 𝜃 = 0 and a moderate reduction in coverage for 𝜃 = 0.3. Most of the competing methods attained nominal Type I error rates for 𝜃 = 0, but the Lasso, MR-Presso and the contamination mixture performed worse than JAM-MR. The contamination mixture in particular exhibited significant inflation; this was due to the fact that the method’s default values for its tuning parameters were not suitable for modeling the simulated datasets. For 𝜃 = 0.3, all methods except the mode, MR-Raps and possibly IVW exhibited a reduction in coverage. The performance of most methods was slightly worse in this case compared with a null causal effect. We conjecture that this might have been due to mild effects of weak instrument bias. Instrument strength in our simulations was calibrated using heuristic arguments based on our real-data application, as discussed earlier. The value of the F statistic in our simulations was well above the threshold of 10 that indicates weak instrument bias, ranging between 50 and 100 in most scenarios. Nevertheless, it is possible that a small amount of bias might have remained, affecting causal effect estimates in the third significant digit and causing the mild undercoverage observed by many methods. Weak instrument bias is known to act towards the null in two-sample Mendelian randomization, which would explain why most methods performed better for 𝜃 = 0, and would also explain why undercoverage is observed even for the oracle IVW procedure.In the first simulation , all Mendelian randomization methods provided nearly unbiased estimates of the causal effect of interest, especially for In the second scenario , there were larger deviations in the performance of the various methods. Bias in causal effect estimates was observed for the majority of methods, with mode-based estimation, the contamination mixture and JAM-MR being the least biased algorithms on average. Among these three approaches, JAM-MR and the contamination mixture attained smaller mean squared errors. The SEs computed using JAM-MR were smaller than those of the mode and again close to those of the oracle estimator. On the other hand, the mode-based approach offered better calibration of Type I error rates and coverage of confidence intervals; in fact, the method was conservative and attained Type I error rates below the 5% threshold. Both JAM-MR and the contamination mixture exhibited an inflation in Type I error rates, although the issue was less pronounced for JAM-MR. A similar pattern of results was observed in simulation scenario 3 .16 MR-Egger also depends on the InSIDE assumption, therefore its performance was poor in scenario 3 where this assumption was violated. The remaining methods were subject to different degrees of bias in both directional pleiotropy scenarios with the Lasso approach being the least biased. This inaccuracy also gave rise to large Type I error rates.Among the other methods, MR-Egger returned fairly accurate causal effect estimates and well-calibrated confidence intervals in scenario 2, but had a significantly higher SE compared with all the other approaches. The lack of power associated with MR-Egger is well-known in the Mendelian randomization literature.The Type I error rates observed for some of the competing methods are rather extreme. These error rates were mainly due to biases in causal effect estimates. Our simulation was rather challenging, as it contained a large number of pleiotropic variants with similar univariate causal effect estimates and this was difficult for some of the methods to tackle. For example, the MR-Presso method relies on outlier removal and can be expected to struggle in simulations where many genetic variants exhibit similar pleiotropic effects. We should mention that many of the existing methods would perform quite well in different simulation scenarios, and some of them (such as the median and MR-Raps) come with theoretical guarantees of good asymptotic performance. Hence, their poor behavior here should not be taken as a general-purpose criticism towards these methods.w and for the grid search version of the algorithm. An explanation for this could be that our heuristic multiplicative truncated random effects procedure, though more efficient than standard IVW, is still unable to perfectly model the uncertainty in model-specific estimates. Alternatively, the undercoverage could be an indication that JAM-MR’s reversible-jump MCMC algorithm exhibits insufficient mixing. In each iteration, our algorithm proposes moves that modify the selected model by only a single SNP, and can find it difficult to move between two models containing completely different sets of SNPs, when such models have high posterior probability; focusing on one of the two models would result in overprecise estimates. Regardless of the reason, the below-nominal coverage of the algorithm can be a concern, but we believe that JAM-MR still possesses some utility as a sensitivity tool, by virtue of offering accurate causal effect estimation, as well as additional diagnostics for pleiotropy such as posterior inclusion probabilities for each variant and plots of causal effect estimates per value of w, as presented below.In comparison, JAM-MR offered causal effect estimates with low bias even in challenging simulation scenarios. On the other hand, the algorithm’s suboptimal calibration of confidence intervals is a concern and merits some discussion. Upon inspection, the algorithm’s estimated SEs are smaller than the empirical SD of estimates across multiple iterations; this happens both for fixed In the 𝜃 = 0.3. In addition, in the right-hand panel of w. The plot illustrates how the algorithm can provide insight into the pattern of pleiotropy in the data. Causal effect estimates for small values of w were much larger than those for moderate and large values, indicating that pleiotropic variants are present in the dataset and cause upwards bias if not accounted for. As we increase w towards the minimum-standard-error value, plotted in red, causal effect estimates become unbiased. However, for large values of w the estimates become more variable and eventually unstable, as we discuss in the In the left-hand panel of 𝜃 = 0.3. We used three JAM-MR runs with a small (w ≈ 0.03N1), a moderate (w ≈ 0.4N1) and a large (w ≈ 15N1) value for the tuning parameter. Pleiotropic variants are colored red. The plot illustrates that our heterogeneity penalization procedure makes accurate selection of the valid instruments, when properly tuned (center panel). When a small value of w is used (left panel), the algorithm may retain some pleiotropic SNPs in the analysis. On the other hand, when w is large (right panel), the algorithm will correctly downweight pleiotropic variants but will also downweight some of the valid SNPs.The JAM-MR algorithm returns posterior inclusion probabilities for each genetic variant, which can be plotted in a Manhattan plot. In In our simulations, we used a 51-point grid search in order to tune the JAM-MR algorithm. In practice, it may be unnecessary to use such a large grid - often, even a grid as small as 5 to 10 values will provide a reasonable fit. To verify that, we compared the causal effect estimates obtained from JAM-MR using a 51-point grid search to those obtained using a smaller grid with only nine points, nested within the 51-point grid. The resulting causal effect estimates were quite similar for all simulation scenarios. In particular, the mean absolute difference in causal effect estimates obtained by the two grid-search procedures across all our simulations was 0.0029, and the mean absolute difference in SE estimates was 0.00047.Overall, the JAM-MR algorithm was able to identify the causal effect of interest with reasonable accuracy in all our simulations, yielded SEs close to those obtained by the oracle IVW procedure, and fairly small root mean squared errors. This came at the expense of a slight inflation in its Type I error rates across most simulation scenarios. The most promising competing approaches were mode-based estimation and the contamination mixture. JAM-MR exhibited smaller mean squared errors than the mode, which produced better-calibrated confidence intervals in exchange. The contamination mixture yielded similar mean squared errors as JAM-MR but with a slightly larger Type I error rate inflation.4 |4.1 |32 and it is generally accepted that high blood pressure has an increasing effect on the risk of suffering from CHD, despite the fact that a previous Mendelian randomization analysis5 did not identify a causal relationship.We now illustrate the use of the JAM-MR algorithm in a real-data application. We conduct a Mendelian randomization analysis to assess the effect of blood pressure on the risk of CHD. This application has been studied in the past33 which identified hundreds of genetic variants associated with blood pressure. The authors meta-analyzed data from the International Consortium for Blood Pressure (ICBP), the UK BioBank, the US Million Veterans Project, and the Estonian Genome Center Biobank. In total, a sample of approximately 1 million individuals of European descent was analyzed. The study confirmed previously reported findings about 258 known and 92 reported but not validated genetic variants associated with blood pressure. It also identified a total of 535 novel associations.Our analysis is novel not in the question it aims to answer but in the data sources it uses. We utilized a recently published meta-GWAS study33 This resulted in a total of P1 = 395 genetic variants for systolic and P2 = 391 genetic variants for diastolic blood pressure; our analysis was therefore based on larger numbers of genetic variants than previous Mendelian randomization investigations.We used two blood pressure traits for our analysis, namely, systolic and diastolic blood pressure. For each trait, we used all 258 genetic variants with an already established relation to blood pressure, as well as any of the reported-but-not-validated and novel variants reported to be associated with that trait. Among the novel findings, we excluded from consideration variants which were associated with blood pressure in the “discovery” GWAS but not in the “replication” GWAS.33 We used estimates based on the ICBP dataset of N1 = 299 024 individuals, as this was the only dataset for which genetic associations were reported for all variants. Since the genetic associations with blood pressure were replicated in independent datasets,33 winner’s curse bias is unlikely to have had a serious effect in our analysis. For the selected variants, we obtained genetic associations with CHD risk from the CARDIoGRAMplusC4D Consortium,34 based on a sample of N2 = 184 305 individuals. The variants were mostly independent, as a result of LD pruning.33Genetic associations with systolic and diastolic blood pressure were obtained from the 4.2 |We implemented JAM-MR and the competing Mendelian randomization methods to identify the causal effect of systolic and diastolic blood pressure on CHD risk. The results are listed in e0.24 = 1.024 to e0.4 = 1.041. All methods were able to reject the null causal hypothesis at a 95% significance threshold, except the mode-based estimator whose SE was unusually large. JAM-MR estimates were very close to those reported by other methods. The minimum causal SE criterion suggested using an implementation with w ≈ 0.5N1, for which the corresponding JAM-MR log-odds ratio estimate was 0.039 ).The reported effects from the various methods confirm that increased blood pressure, both systolic and diastolic, has a deleterious effect on CHD risk. For systolic blood pressure, the causal effect reported by the various methods was in the region of 0.024 to 0.040, corresponding to an odds ratio of e0.067 = 1.069. These were attained for w ≈ 0.29N1. This difference is mainly due to JAM-MR downweighting genetic variants that are weak instruments; the estimate of 0.067 is close to what would be obtained by running the competing methods using only the 258 genetic variants with established associations with blood pressure. Once again, all methods were able to reject the null causal hypothesis at a 95% level, except the mode-based estimator.For diastolic blood pressure, most of the established Mendelian randomization methods reported causal effect estimates between 0.048 and 0.061, corresponding to odds ratios between 1.049 and 1.063. The JAM-MR estimates were slightly larger, yielding a log-odds ratio of 0.067 ) and an effect estimate of 4.3 |The two JAM-MR implementations assigned posterior inclusion probabilities larger than 0.5 to 142 genetic variants for systolic and 169 variants for diastolic blood pressure. Of the 253 variants for systolic blood pressure that were assigned posterior probability lower than 0.5, 116 had univariate causal effect estimates below the overall reported value of 0.037 and 137 had univariate estimates above that value. For diastolic blood pressure, there were 222 variants assigned posterior probabilitieslowerthan0.5,121ofwhichhadsmallerand101hadlargerunivariateratioestimatesthanthereportedvalue of 0.067. Not all genetic variants were downweighted due to pleiotropy; some were assigned low posterior probabilities because their association with the two blood pressure traits were not strong enough. Nevertheless, the downweighting of genetic variants both larger and smaller than the overall causal effect estimates in almost equal numbers suggests a fairly balanced pleiotropic pattern. In line with that observation, the reported estimates in w values. There is perhaps a slight downward bias for diastolic blood pressure with small w values, as we speculated by inspecting the results of To further explore the effect of pleiotropic variants on the estimated effect of blood pressure on CHD risk, we can use the sensitivity analysis plots of w become unstable, with both inaccurate causal effect estimates and increased SEs leading to the behavior observed in the sensitivity plot of w is not directional, therefore it was averaged out across the 1000 iterations and only the inflation in SEs could be observed. w is evident.As in the simulation study, JAM-MR estimates with unnecessarily large 5 This seeming inconsistency is due to the larger sample sizes and numbers of SNPs that we have used in this article. The previous analysis5 was based only on 29 genetic variants that were known to be associated with blood pressure at that point, and its power to detect a causal association was lower.The MR-Egger method rejects the null hypothesis of no causal association and indicates the existence of a causal effect on CHD risk for both blood pressure traits. MR-Egger was the method used to generate the null findings in previous work.The mode-based estimation method was implemented using the default function in the R package MendelianRan-domization. The method yielded much larger SEs than other algorithms. This was due to the presence of a few genetic variants with extreme outlying effects in our dataset, which were in turn the result of weak instrument bias. The presence of weak instruments can have adverse effects on the performance of the mode-based approach. In fact, the effects can be more severe than those reported in 5 |In this article, we have developed a new algorithm for causal effect estimation in Mendelian randomization when some of the candidate instruments are pleiotropic. Our algorithm uses Bayesian variable selection to identify sets of genetic variants with homogeneous causal effect estimates, and model averaging to obtain causal effect estimates and SEs. A wide range of simulation studies and a real-data application using data from a recent GWAS to study the effect of blood pressure on CHD risk were conducted to evaluate the effectiveness of the JAM-MR algorithm, focusing primarily on implementations in large-scale GWAS studies with hundreds of genetic variants that can explain a decent proportion of variation in the risk factor.w values.Compared with other approaches for Mendelian randomization with pleiotropic variants, the JAM-MR algorithm offers accurate causal effect estimation but is hindered by a slight reduction in coverage of confidence intervals. This issue makes it difficult to draw causal conclusions using JAM-MR alone, but we believe that the algorithm can still act as a sensitivity analysis tool for Mendelian randomization studies. To that end, the algorithm can facilitate the use of prior information on instrument validity; this point was not discussed extensively in the current paper but we intend to explore it in the future. JAM-MR also yields posterior inclusion probabilities for genetic variants, which can be interpreted as a measure of validity for each variant, and can be used to visualize the impact of pleiotropy in an application, by plotting causal effect estimates and confidence intervals for various JAM-MR also provides a natural framework for incorporating genetic correlations into a Mendelian randomization analysis and selecting the most relevant variants from a densely genotyped region. Common approaches for Mendelian randomization typically assume that genetic variants are independent therefore we have focused on simulations with independent variants in this article. In related work, we are investigating the advantages of utilizing JAM-MR’s variable selection compared with pruning and other approaches for Mendelian randomization with correlated instruments, in order to incorporate multiple correlated effects from regions which harbor complex genetic signals for the trait of interest.The most notable limitation of our approach is the algorithm’s imperfect calibration of SEs. The inflation of Type I error rates was not as severe as for some of the other Mendelian randomization approaches in many applications: Type I error rates usually ranged between 7% and 10%. If JAM-MR is to be used to assess the existence of a causal relation between a risk factor and an outcome of interest, a heuristic yet practical adjustment could be that researchers report 99% confidence intervals instead of the traditional 95% threshold, when using the algorithm.w that does not have to rely on a grid search. The time required to implement the JAM-MR algorithm is longer than that for other Mendelian randomization approaches: for the simulated datasets with P = 400 genetic variants, the running time was about 3 to 5 minutes per value of w. A grid search over many values takes longer, but can be implemented more efficiently if researchers have access to cluster computing facilities, as runs for different w values can be executed in parallel. Nevertheless, it would be desirable to obtain an accurate causal effect estimate based on a single w value.It would also be useful to devise an efficient automatic procedure for specifying the tuning parameter JAM-MR’s stochastic search procedure is quite flexible and can explore large parameter spaces of causal configurations; as a result, the algorithm is quite efficient when used with large numbers of genetic variants. The stochastic search targets the largest homogeneous set of genetic variants. If smaller sets of genetic variants with homogeneous ratio estimates exist, the stochastic search will sometimes identify them; in practice, this can be checked by inspecting the list of posterior model probabilities. Such sets may correspond to several pleiotropic variants acting on the same causal pathway to the outcome, and the biological interpretation of these sets can be an interesting question in applications. We note however that JAM-MR often requires a very large number of iterations in order to identify such sets. To improve its efficiency at identifying such sets, JAM-MR could benefit from a more elaborate implementation of the reversible-jump procedure, for example, using parallel tempering.In its current form, the JAM-MR algorithm is not fully Bayesian. Although it utilizes Bayesian variable selection to obtain posterior model probabilities and facilitate model averaging, the process of causal effect estimation for each model is conducted using classical IVW formulas. A fully Bayesian algorithm for Mendelian randomization would require further modeling assumptions for the causal effect parameter and would return a posterior sample for that parameter. This is another interesting potential extension of JAM-MR.Inconclusion, JAM-MRperformspleiotropy-robustcausaleffectestimationforMendelianrandomization.Inthisarticle, we have introduced the algorithm, evaluated its performance in simulation studies and used it to conduct a Mendelian randomization analysis of the effect of blood pressure on CHD risk, using a recent large-scale blood pressure meta-GWAS. We hope that JAM-MR will become a useful addition to the Mendelian randomization literature.https://github.com/pjnewcombe/R2BGLiMS.The JAM-MR algorithm has been implemented in R as part of the GitHub package R2BGLiMS, available at supinfo


Watermelon traits such as yield and other agronomic traits are highly environmentally sensitive and regulated by multiple genes; therefore, by understanding the genetic structure, the heritability and genetic influence of different traits can improve them. Five watermelon lines were crossed in a 5 × 5 full diallel parental design to estimate the genetic effect and heritability of fruit traits. Treatments were evaluated on the farm using a random complete block design. Analysis of the results showed a significant difference between genotypes, which was observed for all the studied traits at the probability level of 1%. Hayman's graphical method showed that the contribution of the nonadditive effects was more important than that of the additive effect to control most of the traits. Fruit maturation and pericarp thickness traits were regulated by incomplete dominance gene effects, and other traits were regulated by overdominance effects. The trait heritability varied between at least 0.013 and 0.352 for the fruit weight and fruit number, respectively. Results demonstrated that some traits can be modified based on the heterozygosity and production of hybrid variety methods, while the hybrid and selection in an advanced generation method can be suggested in watermelon breeding programs to breed other traits. Citrullus lanatus var. Lanatus) with 2n = 2x = 22 chromosomes [The family Cucurbitaceae has different species, and the only cultivated species of this genus is watermelon (omosomes . Watermeomosomes . Differeomosomes . Therefoomosomes –7. Agronomosomes .Cucurbitaceae). Then, the application of these results in other studies, for example, in the research by Wu et al. [β-cyclase (lcyb) protein sequences and watermelon flesh color, and in another study by Reddy et al. [Cucumis melo var. momordica).The first step in the plant breeding programs is to study genetic diversity, as was performed by many studies on the genetic diversity of watermelon. For example, Mohosina et al.  divided u et al. , expressy et al. , the corIn plant breeding, to achieve cultivars with desirable agronomic traits, knowledge of the genetic structure of the crossed parents is very important for the researcher to choose the appropriate breeding method. Knowledge about how the inheritance and type of action of genes controlling the traits is the basis of designing a suitable breeding method to achieve the goals of genetic breeding. One of the methods, by which genetic information can be obtained easily and in a relatively short time, is the method of diallel crosses; the principles and foundations of which were presented by Jenks, Hayman, and Griffin in the 1950s –21. AmonDiallel analysis has been used as a suitable and efficient method for quantitative trait genetic analysis to estimate the genetic structure and genetic effects of agronomic traits considered by breeders . Among tF1 hybrid P2 × P8 (Pusa Madhuras×Hara Madhu), as the superior hybrid. In addition, these studies show that inheritance and control of the studied traits are equally affected in both additive and nonadditive effects. However, in the study by Sapovadiya [Several studies based on the Griffing diallel method had been done on watermelon to estimate combining ability , 23–28. povadiya  based onAlthough diallel crosses have been studied to investigate the combing ability (GCA and SCA) of different traits of crops, few works have been done on watermelon, especially on how to inherit its quantitative traits in the form of diallel crosses. Also, the type of gene action  in controlling fruit traits in watermelon by the Hayman method has not been done much. Therefore, for better planning and selection of breeding methods according to the type of genes in controlling watermelon traits, a 5 × 5 diallel design was conducted. Therefore, the purpose of this study was to investigate the effect of genes, heritability, and control of fruit traits in watermelon in order to select appropriate breeding methods and take an effective step in producing high-yielding watermelon cultivars.Citrullus lanatus plants (Orzoeiyeh (P1), Hejrak (P2), Gerd (P3), Neyshabour (P4), and Yazd (P5)) from plant materials collected from different parts of Iran. Firstly, to select these five parents, 38 watermelon genotypes that were collected from different regions and gene banks were planted in a statistical design and different traits were measured and their diversity was estimated by cluster analysis. Finally, based on the greatest genetic distance in terms of different traits and cluster analysis results, popular cultivars were selected from each group and used in this study, and these five parents were the most popular cultivated genotypes in the population.This study was performed on five accessions of F1 families, the purity of the five parent accessions was achieved after several generations of self-pollination. Afterwards, from April to May 2017, seeds of 5 parental lines were sown for crossing all the parents together based on the diallel design; it produced 20 F1 hybrids with 5 inbreeding lines. A randomized full block design (RCBD) with three replicates was used in this study. Therefore, the F1 offspring and their parents were planted in the research farm based on this design from May to October 2017 at the Graduate University of Advanced Technology, Kerman. This research farm was located at the Graduate University of Advanced Technology with E 57° 17′ longitude and N 30° 1′ latitude and with the altitude of 2020 m above sea level. Climatic characteristics of this region were hot and dry climates. The average annual temperature was 16 degrees, average annual rainfall was 136 mm, and average humidity was 31% [Prior to crossing to obtain different  was 31% .Vr) of each array component and the covariance of the array's parent offspring (Wr). Scaling tests were performed to verify the fit of the additive-dominant model to the specified data using regression analysis and variance analysis for arrays (Wr + Vr and Wr − Vr) and t2 tests. Hayman's analysis method shows the additive and dominant effects related to the inheritance of the studied traits based on the analysis of variance and covariance, and by using this method [Vp: parents' variance; Wr: parents' and their offspring covariance in the rth row; Vr: variance of row r; mp: mean of parents; and mo: mean of offspring.After the analysis of variance, the data were tested using an additive dominance model, which required calculating the variance . If the line intersects the Wr axis in the positive part , there will be a reason for the effect of incomplete dominance of genes. Also, if this line intersects the Wr axis in the negative part, i.e., it is a < 0, therefore, h > d; it is a reason for the overdominance gene effect. When the regression line is tangential to the parabola plot, it is a sign of the lack of dominance and, consequently, the additive effect of genes. For different degrees of dominance, the dominant homozygous parent is always at the bottom and the recessive homozygous parent is always at the top of the regression line [Drawing the regression line ion line –19.E, and five different genetic components that have been defined by Jinks and Hayman and were estimated as follows [D: variations due to an additive effect; H1: the variation components due to the effect of gene dominance; H2: calculations to predict the proportion of positive and negative genes in parents; F: mean of Fr values over arrays, where Fr is the covariance of additive and dominance effects in a single array and F is positive where dominant genes are more frequent than recessive; E: expected environmental component of variation; and h: dominance effects .The total phenotypic variance was divided into its environmentally induced part  follows –19: D: vThe relationship between variances, covariances, and genetic parameters is as follows:F1 generation results were analyzed according to Hayman's method, and graphical analysis was performed using it [D), nonadditive variance (H1 and H2), and covariance of additive effects with dominance (F) were also estimated by the proposed Hayman's regression method [Parental and using it . Geneticn method , 19. TheH2/4H1 is the proportion of genes with positive and negative effects in parents. h2/H2 denotes the number of gene groups/genes, which control the character and exhibit dominance.The ch trait , 19. H2/F1 generation, the broad and narrow sense heritability values were calculated for each trait from the following relationship [In the tionship , 19:F1) or hb2 isThe broad sense heritability (F1) or hn2 isThe narrow sense heritability (Wr + Vr and the parent mean), and positive and negative effects on the parent ratio were calculated based on Hayman's proposed methods [Wr on Vr was used for epistasis and the hypothesis of Hayman's method test [https://data.cimmyt.org/dataset.xhtml?persistentId=hdl:11529/10548045.Also, other indices and parameters such as the dominant and recessive gene ratio, direction of dominance gene effects , c, and d , the Wr − Vr test, and the genetic components of variances and other statistical indices, viz., D, H1, H2, h2, E, and F, for the studied traits are shown in Furthermore, the preliminary test results of the Jinks-Hayman model for the slope of the regression line a” effect in controlling all traits, except the fruit weight per plant, flesh weight, and skin thickness (b” effect (except the fruit number per plant). Based on the significance of the “b1” effect, it was found that there was a difference between the parents and progeny for all the traits which showed the directional dominance for traits, except for the fruit number per plant, pH, and sugar content. Although, the study by Badami et al. [The results of Hayman's variance analysis  showed thickness . Moreovehickness  due to ti et al.  based onb2” effect , and the “d” effect showed simple reverse effects and was significant for the studied traits, except for the fruit number per plant and pH traits .The dominant and recessive gene frequency in parents was not the same for all the traits, and the significance of the “” effect  determin” effect  was deteH traits . TherefoD showed that additive effects were involved in controlling traits. This parameter was significant for the fruit number per plant, fruit weight per plant, fruit length, flesh weight, and sugar content traits, showing the role of additive effects in controlling them. The significance of H1 and H2 parameters also demonstrated the role of dominance effects in controlling the traits. Considering the significance of these parameters for the fruit maturity period, fruit weight per plant, fruit length, flesh weight, skin thickness, pH, and sugar content traits, the role of the dominance effect in the inheritance of them was determined.Estimates of statistical indices and genetic components for the studied traits are presented in D, H1, and H2 parameters showed the role of simultaneous additive and nonadditive effects in controlling traits that were significant for fruit weight per plant, fruit length, flesh weight, skin thickness, pH, and sugar content traits. But the other traits were only controlled by nonadditive effects. Studies by other researchers have reported similar and different results for the type of genes controlling these traits [The significance of e traits , 26, 27,H2/4H1 ratio, which shows an estimate of the ratio of dominant genes with increasing (positive) to decreasing (negative) effects, in the best case, i.e., in the symmetry of gene frequencies in parents, will be 0.25. Otherwise, it indicates the asymmetry of the dominant positive and negative alleles of the genes controlling the relevant traits in the parents. In other words, the farther this ratio is from 0.25, asymmetric and nonuniform distributions of positive and negative dominant alleles will show more in parents. These ratios were 0.165, 0.19, 0.23, 0.24, 0.22, 0.20, 0.21, and 0.16 for the fruit number per plant, fruit maturity period, fruit weight per plant, fruit length, flesh weight, skin thickness, pH, and sugar content, respectively, indicating that the dominant additive and subtractive genes in the parents were different for all the traits. The overdominance effects played a role in controlling all the traits, except the fruit number per plant, which was due to the point that the mean degree of dominance The r parameter, which shows the correlation between parents and Wr + Vr, indicates the behavior of dominant alleles. If it is negative, it indicates that the dominant alleles are subtractive and these alleles decrease the trait. Also, if it is positive, it indicates an additive effect of dominant alleles and these alleles increase the trait. In this study, the r parameter or correlation coefficient , these parents cannot be used to increase this trait in breeding programs. According to the results of this experiment, which showed that genetic control of these traits was under the influence of the overdominance gene effects, breeding methods based on hybridization and heterosis phenomenon should be used to improve these traits in breeding programs. Due to the different patterns of gene expression from one environment to another, different strategies should be used to modify these traits in different environmental conditions.Also, the regression line  Figures , meaningThe role of overdominance and incomplete dominance effects in controlling studied traits was identified in this study. According to the results, the genetic control of the traits was different; therefore, a specific breeding strategy should be used for each trait. In most of the traits studied, genes with the overdominance effects had a dominant role in controlling the traits. In addition, the estimation of private heritability of traits indicated that most of the studied traits were of moderate private heritability and nonadditive effects played a greater role in controlling them. Hence, the direct selection breeding method would not be very successful in improving the genetic value of the population for these traits, but using the heterosis phenomenon and the crossing the parents to produce hybrids or direct selection of lines in the last generations could improve these traits.


Combining the results of metabolome-genome wide association studies using the 10,488,742 imputed SNPs, 103, 160, 83, 43, and 109 candidate genes were identified as significantly associated with HCW, REA, AFAT, LMY, and CMAR , respectively. By applying functional enrichment analyses for candidate genes of each trait, 26, 24, 26, 24, and 28 significant cellular and molecular functions were predicted for HCW, REA, AFAT, LMY, and CMAR, respectively. Among the five topmost significantly enriched biological functions for carcass merit traits, molecular transport and small molecule biochemistry were two top biological functions associated with all carcass merit traits. Lipid metabolism was the most significant biological function for LMY and CMAR and it was also the second and fourth highest biological function for REA and HCW, respectively. Candidate genes and enriched biological functions identified by the integrative analyses of metabolites with phenotypic traits and DNA variants could help interpret the results of previous genome-wide association studies for carcass merit traits. Our integrative study also revealed additional potential novel genes associated with these economically important traits. Therefore, our study improves understanding of the molecular and biological functions/processes that influence carcass merit traits, which could help develop strategies to enhance genomic prediction of carcass merit traits with incorporation of metabolomic data. Similarly, this information could guide management practices, such as nutritional interventions, with the purpose of boosting specific carcass merit traits.Improvement of carcass merit traits is a priority for the beef industry. Discovering DNA variants and genes associated with variation in these traits and understanding biological functions/processes underlying their associations are of paramount importance for more effective genetic improvement of carcass merit traits in beef cattle. This study integrates 10,488,742 imputed whole genome DNA variants, 31 plasma metabolites, and animal phenotypes to identify genes and biological functions/processes that are associated with carcass merit traits including hot carcass weight (HCW), rib eye area (REA), average backfat thickness (AFAT), lean meat yield (LMY), and carcass marbling score (CMAR) in a population of 493 crossbred beef cattle. Regression analyses were performed to identify plasma metabolites associated with the carcass merit traits, and the results showed that 4 , 6 (creatinine,  Detecting more candidate genes and functional or causal DNA variants through genome-wide association studies (GWAS) and understanding the biological background of the relationship between the genome and phenome could help improve the accuracy of genomic selection for complex traits including carcass merit traits10.Carcass merit traits, including hot carcass weight (HCW), rib eye area (REA), average backfat thickness (AFAT), lean meat yield (LMY), and carcass marbling score (CMAR), are economically important traits in beef cattle since they directly influence the meat product yield and quality grade, and therefore profitability. For example, sufficient marbling is important for beef tenderness, juiciness and flavor, so the degree of marbling in beef is the primary factor determining quality grade of the meat. However, carcass merit traits are expressed late in life and the measurement of these traits for individual live animals is relatively expensive via ultrasound technologies. In many cases evaluation occurs post mortem, thereby eliminating breeding stock with superior breeding values for the traits. The development of genomic prediction provides an opportunity to assess genetic merit of animals as early as birth14. Among the omics-based intermediate phenotypes, metabolites have been reported to be associated with carcass merit traits of livestock17 and their variation is influenced by genetic effects19. Therefore, we hypothesize that combining metabolomic data into GWAS of whole genome DNA variants could help detect key candidate genes and functional or causal DNA variants associated with carcass merit traits.As more omics-based intermediate phenotypes, based on gene expression, protein and metabolite analysis, become available, integrating multi-omics data to further elucidate genetic influence of complex traits holds great promiseIn this study, the data of 5 carcass merit traits  and 31 plasma metabolites were collected from a beef cattle population consisting of 493 crossbred bulls, heifers, and steers. Our objective was to identify significant single nucleotide polymorphisms (SNPs), candidate genes and biological functions associated with carcass merit traits through integration of carcass merit traits, plasma metabolites and whole genome sequence variants. Linear regression models were first used to identify metabolites associated with carcass merit traits. Metabolome-genome wide association studies (mGWAS) were then performed with 10,488,742 imputed whole genome SNPs to identify significant SNPs for the trait associated metabolites. Candidate genes were mapped based on significant SNPs and gene functional enrichment analyses were subsequently performed on candidate genes of each trait to predict biological functions/processes associated with carcass merit traits in beef cattle.P-values less than 0.05, 3 , 3 , 1 (methanol), 1 (l-lactic acid), and 3  metabolites were identified as associated with HCW, REA, AFAT, LMY, and CMAR, respectively. However, some metabolites with a P-value from 0.05 to 0.1 explained more than 1% of the phenotypic variance of the associated carcass merit traits, thus, a relatively relaxed threshold of P-value < 0.1 was chosen to include more metabolites that may be potentially associated with carcass merit traits. For HCW, at P-values less than 0.1, 3-hydroxybutyric acid, acetic acid, citric acid, and choline were the associated metabolites, accounting for 1.92%, 1.69%, 1.48%, and 1.09% of the phenotypic variance in HCW, respectively. Creatinine, l-glutamine, succinic acid, pyruvic acid, l-lactic acid, and 3-hydroxybutyric acid were significantly associated with REA, and these six metabolites accounted for 1.87%, 1.79%, 1.60%, 1.56%, 1.04%, and 0.80% of phenotypic variance, respectively. AFAT was associated with fumaric acid, methanol, d-glucose, and glycerol, and these four metabolites explained 2.40%, 1.71%, 1.67%, and 1.39% of the phenotypic variance, respectively. l-lactic acid and creatinine were associated with LMY and accounted for 2.71% and 1.42% of phenotypic variance, respectively. Five metabolites, including succinic acid, fumaric acid, lysine, glycine, and choline, were associated with CMAR and each respectively accounted 1.76%, 1.36%, 1.22%, 1.20%, and 1.05% of the phenotypic variance in CMAR, respectively. Most of these metabolites were mainly associated with a single trait. However, a few metabolites were associated with more than one trait. For example, 3-hydroxybutyric acid was associated with both HCW and REA. l-lactic acid and creatinine were both associated with REA and LMY. Choline was associated with HCW and CMAR, and fumaric acid was associated with both AFAT and CMAR  are presented in Table Genomic inflation factors for all association analyses ranged from 0.95 to 1.01  were associated with HCW, REA, LMY, and CMAR.Through integrating the metabolite and carcass merit trait regression analyses and the mGWAS results, a total of 103, 160, 83, 43, and 109 candidate genes were found to be associated with HCW, REA, AFAT, LMY, and CMAR, respectively  associated with HCW, REA, AFAT, LMY, and CMAR, respectively  were involved in concentration of corticosterone and lipid, and quantity of steroid and steroid hormone  were involved in multiple metabolic processes related to fatty acid and lipid metabolism  were involved in carbohydrate metabolic processes such as carbohydrate biosynthesis  associated with CMAR were involved in both lipid and carbohydrate metabolism.Of the identified candidate genes for HCW, REA, AFAT, LMY, and CMAR, 99, 149, 78, 42, and 102 genes were respectively mapped to the IPA database for functional enrichment analyses. Briefly, 26, 24, 26, 24, and 28 cellular and molecular functions were identified as significantly  to identify metabolites potentially associated with carcass merit traits, therefore, validation in independent beef cattle populations or further studies considering a wider range of metabolites is warranted. It is also worthwhile to analyze metabolites on samples collected at different developmental stages to see whether and how the associations between the metabolites and the carcass traits may change. Furthermore, we observed that a majority of the significant metabolites were only associated with one trait. However, some metabolites in the current study were associated with two traits, indicating potential biological relationships between these traits. For example, in this study, we observed that 3-hydroxybutyric acid was associated with both HCW and REA, and beef cattle with high HCW and REA had lower concentration of 3-hydroxybutyric acid, indicating that animals with high HCW and REA may have better carbohydrate metabolism. Additionally, 3-hydroxybutyrate is the main representative of ketone bodies and one important function of ketone bodies is to provide acetoacetyl-CoA and acetyl-CoA for the synthesis of cholesterol, fatty acids, and complex lipids20. Thus, a lower concentration of 3-hydroxybutyric acid may lead to reduced lipid synthesis in animals with high HCW and REA. Interestingly, creatinine, the final catabolite of muscle energy metabolism21, was positively associated with both REA and LMY in the current study  were associated with the concentration of creatinine which is a metabolite associated with REA and LMY. These results indicated that these genes may be associated with the synthesis or degradation of creatinine in animals and thereby influence the related traits. Similarly, HLTF, GYG1, RYR2, RBM47, and APBB2 were reported to be associated with REA and CMAR by Wang et al.24. Our results showed these genes were associated with succinic acid which was negatively associated with both REA and CMAR. Both examples represent one of situations that genes may influence different traits by regulating the same metabolites, and the mechanism of how these genetic variants affect the concentration of metabolites still needs more studies. According to our results, we would like to highlight that some genes could affect the same carcass merit traits by influencing different metabolites. For instance, AMN was associated with both 3-hydroxybutyric acid and citric acid, and both metabolites were identified as associated with HCW. Therefore, information obtained via analyzing metabolites could improve the understanding of genetic effects on these phenotypes. In our companion paper by Li et al.26, similar findings were also observed. These two studies indicate that metabolites play important roles in the variation of both feed efficiency and carcass merit traits. Integration of metabolomic and genomic data could help identify functional or causal SNPs or genes, and interpret the biological meaning of the candidate genes identified in GWAS. In addition, these two studies investigated the associations between different omics levels, which could shed light on the interrelationship between different omics layers and potential molecular mechanisms underlying these traits. Therefore, our findings have broadened our knowledge on the genetic and molecular mechanisms of these traits. Based on what we have learned from these two studies, we recommend applying such multi-omics analyses to study other important traits in beef cattle.Generally, identification of SNPs and genes associated with carcass merit traits mainly relies on association studies between DNA variants and the traits. For example, Wang et al.27 for the prediction of three traditional milk traits in dairy cows. Melzer et al.27 applied regression methods to identify important milk metabolites and then those SNPs with significant genetic effects on important metabolites were identified and used to predict milk traits. Compared with the classical approach that uses all SNPs  in prediction, this metabolite approach could achieve similar prediction precision with less than 1% of the total amount of SNPs. Fontanesi28 suggested integration of metabolomic data would be useful if the heritability of a trait is low, if a trait is hard to be precisely and directly measured on the animals, or if the prediction accuracy was limited by the small number of phenotyped animals in the training populations. Since carcass merit traits are expressed at later stages of animal production and are usually measured by sacrificing potential breeding stock, these traits are more suitable for DNA marker based genomic prediction, and incorporating metabolomic data into the genomic prediction has the potential to enhance the genomic prediction accuracy. In addition to metabolites, the information carried by other omics data, such as RNA and protein, also helps to prioritize SNPs associated with complex traits, and can further contribute to improving genomic prediction accuracy of these traits. For example, Fang et al29 applied an extended genomic best linear unbiased prediction (GBLUP) model called genomic feature BLUP (GFBLUP) that included a separate random effect for the joint action of SNPs within genomic features which were obtained from RNA differential expression analyses. Compared to GBLUP, the accuracy of genomic prediction for mastitis and milk production traits with GFBLUP was marginally improved (3.2 to 3.9%) in within-breed prediction but significantly increased (164.4%) in across-breed prediction. Theoretically, the genomic features could be defined from various sources of biological knowledge  and the GFBLUP model could be applied to other complex traits. Therefore, it would be of interest to investigate the accuracy of prediction for carcass merit traits with and without utilizing multi-omics data.In addition to adding more information to known associations, incorporating metabolomic data can help us identify additional novel associations as metabolites represent a level close to the final phenotypes . In the current study, some additional candidate genes were reported to be associated with carcass merit traits. Therefore, we expect that including the candidate gene SNPs in the DNA marker panel or increasing the weight applied to such SNPs could either improve accuracy of genomic prediction of the traits or decrease the DNA marker density used in genomic prediction while retaining accuracy. A preliminary attempt of this latter option was done by Melzer et al.31, which indicated those overlapping top biological functions potentially have important biological meaning for the carcass merit traits in beef cattle. Since our carcass data were collected from animals that were finished for meat production, genes involved in these overlapping top biological functions, such as lipid metabolism and carbohydrate metabolism, likely play a more important role in determining the carcass merit traits. Therefore, the identification of top and other enriched biological functions and their corresponding genes will not only improve our understanding of the underlying biology but also help prioritize candidate genes and related putative causal SNPs for future studies. Additionally, construction of gene networks for biological functions could help us elucidate complicated connections among genes and disentangle the potential relationships among genes, biological functions and phenotypes. For example, molecular transport was identified as a top enriched biological function associated with all carcass merit traits and its network of HCW as an example showed that some of the associated genes were involved in concentration of lipid and corticosterone, and quantity of steroid and steroid hormones  spectroscopy as described by Li et al.18. Briefly, blood samples were thawed at room temperature and centrifuged at 10,000 rpm for 10 min to separate the plasma. Plasma was then filtered through 3 kDa molecular weight cut-off filters  to exclude macromolecules, including lipids and proteins. After filtration samples with a low volume were diluted with high-performance liquid chromatography (HPLC) water to 570 μl to ensure an adequate volume for NMR acquisition. Samples were further prepared in a 5 mm NMR tube  that contained a total of 700 μl including 570 μl filtered serum, 60 μl DSS and 70 μl D2O. Spectra were acquired on a 500 MHz VNMRS spectrometer equipped with a 5 mm cold probe . Metabolites were identified and quantified with a targeted profiling approach using the Profiler and Library Manager modules in the same software which contained 304 total metabolites as references. Each spectrum was reviewed by a separate analyst and a final review was performed on all of the spectra before exporting concentration results. Concentration measurements were adjusted to report metabolite concentrations (µM). In total, 33 metabolites were initially identified and quantified using NMR. However, two of them were excluded due to missing values, resulting in 31 metabolites for further analyses.All animals in this study were cared for according to the guidelines of the Canadian Council on Animal Care (2009) and the experimental procedures were approved by the University of Alberta Animal Care and Use Committee (AUP00000777). In total, 493 bulls (n = 93), heifers (n = 125) or steers (n = 275) from different herds including Charolais (n = 73), Hereford-Angus crosses (n = 191), and Beefbooster composite breed  were used. Among these animals, 277 animals from two herds had implants, while 216 animals from the other three herds had no implant. The effect of the factor of implant was examined using statistical analysis, and its effect has been captured under the herd variable through subsequent phenotypic value adjustment. Animals were born between 2002 to 2011 and initially measured for feed intake using the GrowSafe system  at the feedlot test station under multiple projects, which were described previously45. In summary, hot carcass weight (HCW) in kg was obtained by summing up the weight of each side of the carcass that was split during dressing, about 45 min post-mortem. Average backfat thickness (AFAT) in mm, rib eye area (REA) in square centimeters, and carcass marbling score (CMAR) at the grading site between the 12th and 13th ribs was assessed by trained personnel. Carcass marbling score was measured as a continuous variable from 100 (trace marbling or less) to 499 (abundant or more marbling) to reflect the amount of fat deposit interspersed between the muscle fibers  of the longissimus thoracis. Lean meat yield (LMY) was calculated as LMY, % = 57.96 +  −  −  as an estimate of saleable meat in the carcass37.In order to collect carcass data, animals were then slaughtered after the feedlot tests at either a commercial plant or the Lacombe Research and Development Centre (LRDC) abattoir when a majority of them reached > 8 mm backfat as predicted from ultrasound measurement. The processes of carcass data collection were previously described40 using Beagle 5.1 software46. Briefly, we first imputed from the 50 K SNPs to the AffyHD panel of 444,558 SNPs with 4,247 animals of mixed beef breeds in the reference population. We then imputed from the imputed AffyHD panel to the whole genome sequence variants with the reference population of 3,093 Bos taurus animals from the 1000 Bull Genomes Project47 (run 7). Finally, 53,258,178 variants (SNPs and indels) on 29 autosomes were obtained after the imputation with average imputation accuracy of 0.97 across the DNA variants with a standard deviation (SD) of 0.08, which was assessed through a five-fold cross-validation as described in our previous studies40. For each SNP, post-imputation quality control was then performed to filter out the imputed variant genotypes if one of the following conditions was met: (1): SNPs on 29 autosomes that had an imputation accuracy < 0.95; (2): minor allele frequency < 0.05; (3) SNPs failed to pass the Hardy–Weinberg equilibrium test . A total of 10,488,742 SNPs remained for subsequent analyses after the quality control.DNA was extracted from the blood samples using DNeasy Blood & Tissue Kit  and then genotyped using the Illumina BovineSNP50 v2 BeadChip . For the SNP imputation, a step-wise procedure was applied as described in our previous studies49. The value of K = 6 was chosen because it had the smallest cross-validation error and yielded the most accurate breed composition prediction based on prior knowledge of breed composition on a subset of animals. Residuals of metabolomic and phenotypic data beyond 3 standard deviations from the mean of residuals were considered as outliers and excluded from further analyses. The descriptive statistics of phenotypic data on carcass merit traits and metabolites are shown in Table P-value < 0.1.Phenotypic values of carcass merit traits and metabolites were adjusted for factors including animal type , birth year, herd, feedlot pen, animal age at slaughter, and breeding composition using linear regression models. The breed composition of each animal was predicted based on their 50 K SNPs using ADMIXTURE software to account for population stratificationmlma) as implemented in GCTA software50. The linear mixed model can be described as follows:51, which is essentially the same as the second VanRaden’s formulation52. The SNP allele substitution effect was estimated and the significance test of the SNP allele substitution effect was conducted via a generalized least square F-test as implemented in the GCTA package. The SNPs with P-value < 1 × 10–5 were considered to be significantly associated with the metabolite according to the recommendation of The Wellcome Trust Case Control Consortium53. The phenotypic variance of the metabolite explained by each significant SNP was calculated by For the metabolome-genome wide association studies (mGWAS), the adjusted values of metabolites that were significantly associated with the carcass merit traits were the response variable in the single SNP-based mixed linear model association (54.A 140-kbp window (70-kbp upstream and 70-kbp downstream) of each significant SNP was used to map candidate genes based on ARS-UCD 1.2 bovine genome assembly from the Ensembl BioMart database (accessed in February 2021). The 70-kbp was the chromosomal length within which a high linkage disequilibrium phase correlation (www.Ingenuity.com). Significantly enriched molecular and cellular biological functions and gene networks  for each carcass merit trait were inferred and gene-sub-biological function/process interactions within the most significant molecular and cellular functions were predicted in the IPA.The Entrez gene IDs were used as gene identifiers and small nucleolar RNA and microRNA were excluded from gene functional enrichment analyses. Bovine genes were changed to known human orthologous genes from Ensembl, whereas for those genes without human orthologs their bovine gene IDs were maintained in the gene list. Then candidate genes of all metabolites associated with the carcass merit traits  as identified in the regression analyses between carcass merit traits and metabolites were combined and imported into the Ingenuity Pathway Analysis software (accessed in February 2021) (IPA; Supplementary Information 1.Supplementary Information 2.Supplementary Information 3.Supplementary Information 4.


The genetic basis of phenotypic traits is highly variable and usually divided into mono-, oligo- and polygenic inheritance classes. Relatively few traits are known to be monogenic or oligogeneic. The majority of traits are considered to have a polygenic background. To what extent there are mixtures between these classes is unknown. The rapid advancement of genomic techniques makes it possible to directly map large amounts of genomic markers (GWAS) and predict unknown phenotypes (GWP). Most of the multi-marker methods for GWAS and GWP falls into one of two regularization frameworks. The first framework is based on We have developed a proximal operator algorithm based on the recent LAVA regularization method that jointly performs https://github.com/patwa67/LAVAADMM.This study presents the LAVA ADMM that is capable of joint modelling of monogenic major genetic effects and polygenic minor genetic effects which can be used for both genome-wide assoiciation and prediction purposes. The statistical evaluations based on both simulated and real pig data set shows that the LAVA ADMM has better prediction properies than the LASSO, RR and EN. Julia code for the LAVA ADMM is available at: The online version contains supplementary material available at 10.1186/s12859-021-04436-6. Mendelian, or classical, genetics is the study of traits that is controlled by a single locus. A mutation in a single gene can cause a disease, or another phenotypic alteration, that is inherited according to Mendel’s principles. Those traits are also referred to as monogenic . In humap) than observations (n). Moreover, the predictors are often substantially correlated with each other. Joint modeling of regression coefficients through standard multiple regression is not feasible in these situations. For example, when In many situations, the genomic data will be wide, i.e. there will be many more predictor variables  converged faster for the methods with one regularization parameter (i.e. RR and LASSO) when using the upper confidence bound (UCB) acquisition function, and for the methods with two regularization parameters (i.e. EN and LAVA) the mutual information (MI) acquisition function worked better. The lower and upper bounds of mentclass2pt{minimmentclass2pt{minimmentclass2pt{minimc and d were computed as the difference between the regression coefficients of upper homozygote genotype 2 and the lower homozygote genotype 0 for each SNP. Most of the additive effects are captured by the c  vector of length n, X is a predictor (input) matrix of size b is a regression coefficient (parameter) vector of length p and e is a residual (error) vector of length length n. Regularization provides a tool to put constraints on the regression coefficients, and a general optimization model can be formulated asf(b) is a loss function and g(b) is a penalty function. Ridge regression  in in\documoration.  recommeny and X multiplied by 0.0001. All analyses were performed with a Lenovo ThinkPad laptop with Intel Core i5-8265U 16GB RAM and Windows 10.The LAVA ADMM algorithm was implemented in Julia 1.5  using thThe simulated data encompass 3226 individuals organised in a 5 generation pedigree originally created for the QTLMAS2010 work-shop . 20 indiThe continuous quantitative trait is controlled by 9 major QTLs at fixed positions, including two pairs of epistatic genes, 3 maternally imprinted genes and two additive major genes with phenotypic effects of − 3 and 3. The additive genes are positioned at SNP indices 4354 and 5327, whereas the major epistatic locus is at SNP 931. Moreover, 28 minor QTLs, randomly dispersed on chromosome 1–4, have their additive effects sampled from a truncated normal distribution and their effects vary between − 1.98 and 1.93. The QTLs are enclosed by 19 to 47 polymorphic SNPs located within 1 Mb distance from the QTLs. A total of 364 SNPs exhibit moderate to high linkage disequilibrium (LD) with the QTLs. Hence, the trait can be considered to be an example of oligogenic inheritance because it is controlled by both a few major QTLs and a larger number of minor QTLs. However, the true number and positions of the minor QTLs are unknown due to the random sampling of these QTL effects.In addition, one dominance locus was positioned at SNP number 9212 by allocating an effect of 5.00 to the heterozygote and a value of 5.01 to the upper homozygote. Furthermore, one over-dominance locus was placed at SNP 9404 by assigning an effect of 5.00 to the heterozygote, and an effect of − 0.01 to the lower homozygote and 0.01 to the upper homozygote. Lastly, by assigning a value of − 5.00 to the heterozygote, an effect of − 0.01 to the lower homozygote and 0.01 to the upper homozygote, one under-dominance locus was created at SNP id 9602. The effects of these new dominance QTLs were added to the original phenotype values. SNPs with minor allele frequency (MAF) less than 0.01 was discarded which ended up in 9723 markers. These SNPs were transformed into one-hot encoding which means one indicator variable for each genotype. Hence, the final number of genomic markers was 29169. Generation 1 to 4  were used as training data and generation 5  acted as test data.In order to evaluate the methods on a typical real data set, we used a public pig dataset containing 3534 individuals with high-density genotypes, phenotypes, and estimated breeding values for five anonymous traits . GenotypMost of the genotyped animals were measured for five purebred traits (phenotypes in a single nucleus line). Heritabilities ranged from 0.07 to 0.62. For this study, we chose the trait that had a heritability of 0.38. The phenotypic data points were adjusted for environmental factors and rescaled by correcting for the overall mean. By discarding individuals with missing phenotype data a final number of 3141 individuals was obtained.Additional file 1.Supplementary 1. Additive genetic effects from the c of the simulated QTLMAS2010 dataAdditional file 2.Supplementary 2. Additive genetic effects from the d of the simulated QTLMAS2010 data.Additional file 3.Supplementary 3. Dominance genetic effects from the c of the simulated QTLMAS2010 data.Additional file 4.Supplementary 4. Dominance genetic effects from the d of the simulated QTLMAS2010 data.Additional file 5.Supplementary 5. Additive genetic effects from the c of the Cleveland pig data.Additional file 6.Supplementary 6. Additive genetic effects from the d of the Cleveland pig data.Additional file 7.Supplementary 7. Dominance genetic effects from the c of the Cleveland pig data.Additional file 8.Supplementary 8. Dominance genetic effects from the d of the Cleveland pig data.