We predict that ~8 M individuals will be required to explain the full heritability of cortical surface area by genome‐wide significant SNPs, whereas sample sizes over 20 M will be required to explain the full heritability of depression. In conclusion, our findings are consistent with brain structure satisfying the higher power criterion of endophenotypes.Human brain structure traits have been hypothesized to be broad endophenotypes for neuropsychiatric disorders, implying that brain structure traits are comparatively “closer to the underlying biology.” Genome‐wide association studies from large sample sizes allow for the comparison of common variant genetic architectures between traits to test the evidence supporting this claim. Endophenotypes, compared to neuropsychiatric disorders, are hypothesized to have less polygenicity, with greater effect size of each susceptible SNP, requiring smaller sample sizes to discover them. Here, we compare polygenicity and discoverability of brain structure traits, neuropsychiatric disorders, and other traits  to directly test this hypothesis. We found reduced polygenicity (FDR = 0.01) and increased discoverability (FDR = 3.68 × 10 Brain structures have been hypothesized to be broad endophenotypes for neuropsychiatric disorders, and are expected to be comparatively “closer to the underlying biology”. We directly evaluate this hypothesis by estimating and comparing polygenicity and discoverability of cortical and subcortical brain structures, neuropsychiatric disorders, and other brain‐related traits using genome‐wide association study summary statistics. Second, mechanistic insight, because those variants associated with an endophenotype also influence risk for neuropsychiatric disorders, endophenotype associations are informative about the mechanisms leading to risk for neuropsychiatric disorders. Genome‐wide association studies (GWAS) have identified common genetic variants associated with many traits, including brain structure : the overall amount of trait variance explained by genetics; (b) polygenicity (πc): the proportion of susceptibility SNPs (sSNPs), LD‐independent loci associated with a trait that are not necessarily genome‐wide significant, relative to the total number of LD‐independent SNPs in the genome (M); and (c) discoverability (σ): the distribution of effect sizes of sSNPs on a trait. Higher polygenicity of a trait indicates more sSNPs that are associated with that trait , thickness (n = 35)  . From the Psychiatric Genomics Consortium (PGC) (https://www.med.unc.edu/pgc/download-results/), summary statistics for three psychiatric disorders  (European population) . Summary statistics for Autism Spectrum Disorder (ASD) were generated in our previous study . Summary statistics for cognitive function  http://web.pasteur‐lille.fr/en/recherche/u744/igap/igap_download.php, respectively. Summary statistics for anthropometric measurements (height and body mass index [BMI]) . Summary statistics for three categories of depression from the UKBB population were obtained from https://datashare.is.ed.ac.uk/bitstream/handle/10283/3083/mdd_broad_probable_icd.zip. Summary statistics of cortical structures for UKBB was obtained from UK Biobank Brain Imaging webpage (https://www.fmrib.ox.ac.uk/ukbiobank/)  . For case–control studies, effective sample sizes were further estimated by 4/(1/cases +1/controls)  implemented in GENESIS, SNPs with a low effective sample size (<0.67 × 90th percentile of sample size), or very large effect size (Z2 > 80) were removed. This function also removed SNPs within the major histocompatibility complex region. Only those SNPs in HapMap3  . GENESIS implements two models , which assumes that the distribution of effects for non‐null SNPs follows either a single normal distribution or mixture of two normal distributions . Variance parameters for the M3 model were estimated using output from the M2 model as recommended in the GENESIS documentation. To select the best fit model, we used the modified Bayesian information criterion (BIC), also implemented in GENESIS  from M3 was less than 5 or if the BIC for M2 was less than M3.  were internally generated in genesis.We ran the 2.4c). The total number of sSNPs was estimated byc is the proportion of sSNPs obtained by genesis and M is the number of SNPs in the reference panel . The number of sSNPs in the cluster with larger variance component for M3 was estimated by multiplying the proportion of sSNPs in that cluster  for each trait, which is the polygenicity (π2 is variance estimated by GENESIS.In order to compare effect size distributions across traits (discoverability), regardless of whether the traits were modeled with M2 or M3, we selected one quantity from the distribution: the 50th percentile of ranked sSNPs absolute effect size. In other words, the predicted effect size of an sSNP where half of all sSNPs have larger effect size in absolute value. In order to estimate this quantity, we used the distribution and quantile functions in R‐3.5.0. For phenotypes with M2, we usedx” via grid search such that21 is the variance in cluster 1, σ22 is variance in cluster 2 with smaller effect sizes, and s = seq.For phenotypes with M3, we found the smallest value “SE for each parameter  output from GENESIS, and plugging in these interval endpoints into the formula above  if the lower bound of proportion of sSNPs in cluster 1 was <0, we set its value to 0 (meaning that the sSNPs were considered to belong to cluster 2); and (c) if the upper bound of prop was >1, we set its value to 1.Then, 95% CI for each parameter were calculated by adding and subtracting 1.96 times the Equation  or 3) b bSE for 2.5projection function in GENESIS. We defined the sample size needed to achieve complete heritability as the sample size required for the GVpercentage to pass 99%. We computed this prediction with the best fit model for each phenotype. If the GVpercentage did not pass 99% at a sample size of 200,000,000, we showed the GVpercentage achieved at that sample size.We estimated the predicted heritability explained by genome‐wide significance (GWS) SNPs (GVpercentage) with a given sample size from 50,000 to 200,000,000  by applying the 2.6SE of each parameter estimate. After SEs were calculated, the heterogeneity (I2 statistic) between groups of traits was calculated based on a fixed effect model implemented in metagen function in Meta package (v4.12‐0) , we excluded cuneus thickness and cuneus surface area to avoid potential biases in comparisons  (v1.0.0) , the similarity between MRI segmentations from two scans of the same individual for subjects that passed visual inspection, were obtained from  and the total number of sSNPs as (πc × M); (b) discoverability quantified as the variance of effect sizes for non‐null sSNPs (σ12 and σ22); and (c) predictions of heritability explained by GWS  sSNPs in future sample sizes. We then compared these genetic architecture parameters across traits.We first obtained GWAS summary statistics of various traits including cortical  showed somewhat different proportions of the best fit model , though this difference was not significant  , there are a number of outlier sSNPs (p < 10−10) implying that these traits could be fit to more complex models  or one set of null SNPs and two sets of sSNPs at different levels of effect size (M3) for each trait  . We then compared polygenicity from the 34 cortical surface area traits to the 34 matched cortical thickness traits  Figure . Current) Figure , but theI2 = 64%; p = .02) in estimates from neuropsychiatric disorders, we also tested differences at the level of individual disorders. We found increased polygenicity in depression (FDR = 0.01) and schizophrenia (FDR = 0.03), while the remaining neuropsychiatric disorders have no significant differences  compared to cortical surface area. On the other hand, neurodegenerative disorders showed the opposite directionality, . This may reflect that those two neurodegenerative disorders have strong signals  that were removed by the quality control preprocessing within GENESIS. Overall, these results are consistent with predictions of cortical structure as satisfying the higher power criterion of an endophenotype when compared to a subset of neuropsychiatric disorders.Next, we tested differences in polygenicity for global cortical structure traits  compared to brain‐relevant and anthropometric traits. Consistent with previous findings . However, we did not observe statistically different effect sizes in cortical surface area compared to subcortical volumes (FDR = 0.05)  Figure .neuropsychiatric disorders = 4.29 × 10−9; FDRaddiction relevant traits = 4.70 × 10−18; FDRcognition = 1.15 × 10−11; FDRanthropometric measurements = 1.14 × 10−3; FDRneurodegenerative disorders = 0.014). There was also heterogeneity of discoverability estimates across neuropsychiatric disorders, and the observed significance between cortical structure and neuropsychiatric disorders were driven mainly by depression, as the significance decreased when removing depression from the group of traits (FDR without depression = 0.33) . In summary, we found evidence that sSNPs for brain structure have stronger effect size compared to aggregated estimates across multiple neuropsychiatric disorders, largely driven by the low effect sizes observed in depression. We also observed that brain structure traits have stronger effect sizes when compared to cognition or addiction relevant traits.We also statistically compared the effect size distribution of genetic variants across brain‐relevant traits including neuropsychiatric disorders, neurodegenerative disorders, addiction related traits, cognitive function, and anthropometric measurements Figure . The het) Figure . This wa3.4r] = −.65, p = 4.59 × 10−7; Supplementary Figure Previous studies have shown that increasing polygenicity is associated with decreased discoverability , or cognitive traits (FDR = 7.67 × 10−24)  and negative correlation between TRC and polygenicity  . This indicates that measurement error does influence observed genetic architecture, but only when the most poorly segmented regions are included.Genetic architecture may be influenced by measurement error, where increased measurement error may lead to decreased effect sizes and increased polygenicity. We therefore assessed the impact of reliability of MRI segmentations on polygenicity and discoverability using TRC obtained from a previous study  Kraft, . To exam3.6Finally, we predicted the number of subjects needed in future GWAS to identify all of the common variant loci (sSNPs) associated with a trait. In other words, we estimated the sample size needed to achieve 99% heritability explained by genome‐wide significant SNPs. Figure . We pred4Here, we directly tested one proposed property of brain structure traits as endophenotypes: higher power of genetic discovery. We evaluated this by estimating the polygenicity and discoverability (effect size distribution) across multiple cortical and subcortical brain structure traits and compared these to the same measures from neuropsychiatric disorders, neurodegenerative disorders, cognitive, addiction relevant, and brain related traits. We found that cortical structure traits have reduced polygenicity and increased discoverability compared to aggregated estimates across multiple neuropsychiatric disorders. This is consistent with brain structure satisfying the higher power criterion of endophenotypes.Our results have both practical and theoretical implications. Practically, brain structure traits have higher power than neuropsychiatric disorders so smaller sample sizes will be required to achieve equivalent gains in genetic discovery. The costs of phenotype acquisition for brain MRI are still high, but nevertheless large biobanks and integration of genotype data with electronic medical records are now developing that will allow GWAS of brain structure in sample sizes of hundreds of thousands in the near future , so future exploration of genetic architecture in increased sample sizes will help solidify the findings of decreased polygenicity and increased discoverability of brain structure traits relative to neuropsychiatric disorders and cognitive traits. Second, polygenicity and discoverability are related to measurement error, whereby lower discoverability and increased polygenicity are observed in very poorly segmented brain structures like frontal and temporal pole. So, differences in genetic architecture estimates may reflect our ability to accurately measure the phenotypes. Third, although there are several software packages that can be applied for estimating polygenicity and discoverability (Holland et al., We should interpret our results in light of some limitations. First, we found that increased sample size is associated with decreased estimated effect size distributions. Future brain structure GWAS in larger sample sizes may therefore lead to decreased discoverability estimates, but nevertheless we expect that those estimates will likely be contained within the CIs shown here, if the assumptions for the estimation procedure in GENESIS have been met. The sample sizes were smaller in cortical/subcortical structures compared to behavioral traits (e.g., mean value of sample size is 32,512 for cortical surface area 193,640 for behavioral traits, Welch's 5Overall, our results use estimates of genetic architecture to test long‐standing hypotheses about brain structure traits as endophenotypes for neuropsychiatric disorders.The authors declare no conflict of interests.Appendix S1. Supporting Information.Click here for additional data file.Appendix S2. Supporting Information.Click here for additional data file.


The application of polygenic scores has transformed our ability to investigate whether and how genetic and environmental factors jointly contribute to the variation of complex traits. Modelling the complex interplay between genes and environment, however, raises serious methodological challenges. Here we illustrate the largely unrecognised impact of gene-environment dependencies on the identification of the effects of genes and their variation across environments. We show that controlling for heritable covariates in regression models that include polygenic scores as independent variables introduces endogenous selection bias when one or more of these covariates depends on unmeasured factors that also affect the outcome. This results in the problem of conditioning on a collider, which in turn leads to spurious associations and effect sizes. Using graphical and simulation methods we demonstrate that the degree of bias depends on the strength of the gene-covariate correlation and of hidden heterogeneity linking covariates with outcomes, regardless of whether the main analytic focus is mediation, confounding, or gene × covariate (commonly gene × environment) interactions. We offer potential solutions, highlighting the importance of causal inference. We also urge further caution when fitting and interpreting models with polygenic scores and non-exogenous environments or phenotypes and demonstrate how spurious associations are likely to arise, advancing our understanding of such results. Yet, disentangling the relative importance of polygenic scores and environmental covariates is difficult. Various methodological concerns have been raised, including but not limited to, the power and predictive accuracy of polygenic scores4 and the non-exogenous nature of environmental exposures and their consequences7. Moreover, genes and environments do not operate independently, necessitating greater scrutiny of both conventional models and new methods addressing gene-environment-trait correlations9. Here we address further methodological problems arising from gene-environment correlations that have gone largely unrecognised yet make identification of causal effects and the accurate estimation of associations even more challenging.The importance of understanding the joint contributions of genetic and environmental variation underlying complex traits is widely recognised. The rise of polygenic scores has resulted in a surge of studies investigating the mediating and moderating roles of environments along with genetic confoundingendogenous selection bias in a regression analysis. Heritable covariates in regression models with polygenic scores are endogenous variables, and this can give rise to the problem of conditioning on a collider. Collider bias is an important statistical problem that destabilises regression models and it can arise for a variety of reasons, including sample selection and attrition10. We demonstrate that collider bias is likely to occur not only in genetic association studies but also in other analyses where polygenic scores are included, regardless of whether the main focus is mediation, confounding, or gene × environment (G × E) interaction.We illustrate how the presence of genetic predispositions associated with exposures to an environment (or phenotype in cases of genetic confounding) introduces 7) and studies using polygenic scores as instrumental variables (e.g. Conley and Zhang11). We expand this focus and show that if both genetic and environmental covariates are included in a statistical model, gene-environment correlations may lead to spurious estimates and effect sizes. Understanding the mechanisms that generate these dependencies is crucial for how we may solve the issue. We emphasise that researchers must grasp the conceptual differences between passive, evocative, and active gene-environment correlations12 and the potential sources of endogeneity of environmental covariates in order to obtain results that are not biased due to conditioning on a collider.Moreover, the issue we describe here is linked to a growing body of literature showing the heritability of environments, known as gene-environment correlation (rGE) and pleiotropy. To date, discussion of the methodological implications of these findings has focussed on the implications for G × E interaction studies  and an outcome (sex).The notion of In what follows we consider a situation in which genes and environment are correlated (for reasons discussed below) and the environmental variable(s) is affected by variables not measured in the study but which also affect the outcome. We then discuss the consequences of the resulting collider bias for both additive and G × E interaction models.15). Other examples include studies that examine genetic risk and lifestyle in relation to stroke and cardiovascular mortality . Another example is where income and labour market outcomes are predicted by educational measures  along with an educational attainment polygenic score . Or the study of the variation of exam scores in relation to school types and polygenic prediction of education .The first type of model we consider is the rather straightforward design where polygenic scores and environmental covariates (or phenotypes if they are used as covariates) are jointly included as a set of predictors for an outcome of interest. Such models are intended to reveal whether genetic influences confound associations between environments and outcomes or whether environments are mediators of the link between genetic variants and phenotypes. Examples include linking health disparities with socio-economic outcomes such as the relationship between attention-deficit hyperactivity disorder (ADHD), its polygenic prediction, and IQ on educational outcomes among teenagers . It could also be linked to school type since parents adjust their educational choices for children depending on their child’s characteristics which are partially due to genetic differences, reflecting evocative rGE 25.Dependencies could also arise when gene-environment correlation (rGE) plays a role. rGE occurs when exposure to an environment depends on heritable inclinations26 into horizontal, vertical, and spurious pleiotropy. Horizontal pleiotropy occurs when genetic variants are either linked to multiple traits, either directly or through a series of intermediate traits; vertical pleiotropy arises if we expect causation among a set of traits; and, spurious pleiotropy is the result of linkage disequilibrium (LD), misclassification, or other biases. The most relevant aspect for our argument is that any type of pleiotropy could cause gene-environment dependency. As noted in the literature27, pleiotropy is not unusual among heritable traits. Rather, the null hypothesis is that pleiotropy exists unless proven otherwise.It is also important to consider the role of pleiotropy as a cause of gene-environment dependencies. In general, pleiotropy refers to situations when one gene influences multiple traits or two traits share genetic variants. There are different mechanisms involved described by Van Rheenen et al.rg)27. To illustrate this point, we can consider studies on cardiovascular mortality and incidents of stroke which investigate the joint importance of polygenic risk scores and lifestyles17. Conceptualisation of lifestyle includes, but is not limited to, smoking and BMI, which both have moderate genetic correlations with heart attacks . Therefore, we expect non-exogenous environments to vary depending on the values of polygenic scores. Such dependencies, when accompanied by hidden heterogeneity that links environments with outcomes, will result in endogenous selection bias, which we describe now.One way to assess whether consistent pleiotropy, at the level of observed genetic variance, exists is via genetic correlations , U, linking E and Y biases not only their association but also the estimate of the direct effect of G on Y. This is driven by the fact that E is now a collider since it is a product of both G and U, as indicated by the arrows from U to E and G in Fig. We could simply omit the collider, E, from the model in Fig. To further illustrate this bias, we conducted a series of simulations of the simple linear model from Fig. We also develop a mathematical expression of the bias for linear models, which we provide in the Supplementary Information. However, it is important to note that even though we consider simple linear models in our simulations and derivations, the bias would also arise in other types of regressions, e.g., logistic regression or Cox models. Since the graphical approach we follow does not require any parametric assumptions, the functional form describing the relationship between variables is not relevant.17 and incidents of stroke16 mentioned previously, the focus was to model the risk of strokes or cardiovascular mortality (Y) according to genetic risks (G) and lifestyle profiles (E), employing Cox proportional hazards models. A covariate such as a lifestyle may not be exogenous—individuals choose to follow healthy or less favourable behaviours based on various observed and unobserved factors. Hence, we expect not only G-E dependency, at least due to pleiotropy, but also the presence of unobserved confounders, U. Here, potential factors for U include, but are not limited to, childhood poverty, social or geographically-related deprivation, and socio-economic position33. This makes the estimation of G–Y association subject to collider bias since we introduced the path that goes through this set of unobserved confounders, U, into the statistical model. While one of the desires of these studies is to show that a healthy lifestyle potentially attenuates genetic risks, observed attenuations should be treated with caution.It is also possible to exemplify the nature of inflation of the G-Y association by considering examples from the existing literature. Using the example of studies on cardiovascular mortality19, which we can analyse in a more straightforward manner because of the properties of linear models. Here the researchers regressed a polygenic score of educational attainment (G) on earnings (Y) and found that greater values of the genetic score were associated with greater income. What is relevant to our argument is that after the inclusion of educational controls (E) into the model, we observe a 59.5% decrease of the polygenic score coefficient (Table 5 Panel A in Papageorge and Thom19). In addition, the authors regress the educational attainment polygenic score (G) and years of schooling (E) on standardised job tasks (Y) to explain the trends they found in the models of earnings. If we take models on nonroutine analytic and nonroutine interactive tasks , we also see that the inclusion of educational controls results in about a 70% reduction in polygenic score coefficients. It is likely that such a change of G-Y association in both analyses is largely attributed to the extended set of educational controls, which includes both parental and respondent educational attainment. However, respondent’s own education in these models is an endogenous variable; hence, the decrease of polygenic score coefficients is likely to be due to collider bias at least to some extent. If we consider a moderate strength of association between the genetic score and respondent’s years of schooling along with additional assumptions about unobserved confounders linking educational attainment and the type of job tasks, we can show that around 15–20% of the polygenic score coefficient decrease is plausibly due to collider bias, following the derivations we include in the Supplementary Information.Another instance is the study from Papageorge and Thom19), we see that the baseline coefficient of the educational attainment polygenic score is 0.185. This reflects a model without any environmental and phenotypic covariates. In the model with educational controls, which are respondent’s years of schooling and parental education, the polygenic score coefficient drops to 0.055. Since the dependent variable is standardised, we can assess the relative importance of collider bias under additional assumptions. We assume that the coefficient of the correlation between polygenic score and respondent’s years of schooling is 0.300, and that there is an unobserved confounder U, which is positively correlated with both years of schooling and job task with coefficients of 0.250 . These are all plausible and rather modest suggestions if we look at the correlation matrix from Table 6 in Papageorge and Thom19. Following the derivations provided in the Supplementary Information, the inflation bias would be 0.021 under these assumptions, which explains a 16% downward change of the polygenic score coefficient. We provide detailed calculations for this case in the Supplementary Information.In particular, if we consider the model with nonroutine interactive job tasks as the dependent variable , our estimate of the GxE term would be spurious due to collider bias. We provide an illustration of such a case in Figure The inflation of the GxE interaction term is not observed under the scenario where confounder, U, does not interact with a covariate, E, and polygenic score, G. This setting is considered in Bun and Harrison44, ancestry2, and so forth. If a correlation arises as a result of pleiotropy, active or evocative selection, the assumption of non-causal association would be violated and require another set of solutions to avoid collider bias. The latter is relevant when a phenotypic variable is used as a covariate along with its polygenic prediction since the association would be at least partially causal.To address this issue, it is important to understand the nature of the correlation between polygenic scores and environment (or phenotype if it is used as a covariate)—whether a correlation is conditional and observed because of omitted confounders between G and E and/or a correlation reflects causal dependency. The former would necessitate controlling for the confounding factor: this could be parental characteristics (passive rGE)6. Briefly, techniques such as regression discontinuity and difference-in-difference designs, instrumental variables and quasi-natural experiments will produce unbiased results for both additive and gene-environment interaction models, conditional on certain assumptions being met. As an illustrative case, we considered the instrumental variable solution and include it in our code, which is available online. However, any technique that would disentangle the exogenous variation in an environmental covariate would produce estimates which are not distorted by collider bias; hence, a choice of a particular model would depend on the case in question and a set of assumptions that are unique for each method.To obtain unbiased estimates, we need to apply causal inference methods that seek to exploit the exogenous variation in an environmental covariate. A comprehensive discussion of methods available for researchers and applicable to the context of this paper is provided by Fletcher and Conley46. Since the type of collider we described here is the product of both observed and unobserved factors, calculation of bias magnitude would rely on additional assumptions about the structure of the error correlation between the environmental covariate and the outcome. It is possible to assess the strength of gene-covariate association by either directly measuring their correlation in the data or by taking into account genetic correlations for the trait in question. The influence of unobserved variables on the bias, however, makes it impossible to provide a definitive estimate of the issue. In the Supplementary Information, we show the sources of bias in coefficient estimates and error in R-squared. In line with the results of our simulation analyses, mathematical expressions also demonstrate that the strength of collider bias is positively associated with the strength of rGE and unobserved confounders. The use of sensitivity analyses is a valuable tool in showing how robust conclusions are to different degrees of unobserved confounding and, thus, of collider bias47.There are also existing ways to assess the magnitude of the bias for the general collider casesWe have discussed methodological considerations arising due to heritable environments (or phenotypes that are included into models as covariates) that have not yet been previously recognised. We demonstrated that the inclusion of environments that are products of polygenic scores may introduce endogenous selection bias through conditioning on a collider, leading to spurious associations. Particularly, we showed that the degree of bias depends on the strength of the gene-covariate correlation and of the omitted variable(s) linking the covariate and outcome. We also showed that the portion of explained variance is overestimated proportionally. We proposed some solutions that exploit the strength of causal inference methods: these are likely to be important not only for obtaining reliable results but also in the interpretation of existing studies.48 The graphical approach is a transparent method to demonstrate the biases and it does not require any parametric assumptions which is an advantage. We also provide equations and detailed derivations for the bias under the assumptions of linear regression in Supplementary Information to compliment graphical models. In the schematic diagrams (or directed acyclic graphs) presented in the paper, arrows indicate associated paths, dark purple circles represent observed variables, grey circles represent unobserved variables (which are confounders in our case), and collider variables are indicated by squares.We use a graphical approach to demonstrate the methodological problems. For the theory behind graphical models, see Pearl et al.https://github.com/eva-akimova/collider-simulations.git (https://doi.org/10.5281/zenodo.4184672), to reproduce the figures presented in the paper. Simulations were conducted in R49 using dplyr50, broom51, purrr52, mvtnorm53, ggplot254, cowplot55, tidyr56, AER57, forcats58 packages. First, we simulated scenarios of OLS regression for additive models. We allowed the G-E correlation to vary from 0 indicating no heritability to 0.5 indicating a highly heritable covariate. We considered three scenarios where U is a modest, moderate, or strong confounder of the E-Y association . We also considered the presence of direct effects from G to Y and E to Y which coefficients are both positive and 0.6. Second, for the gene-environment interaction models, we simulated the same settings and added the GxE coefficient at a fixed value of 0.1 for all scenarios.To further illustrate this bias, we conducted a series of simulations of the simple linear models. All simulations are detailed in Supplementary Information. All analyses code is publicly available at: Supplementary Information includes the detailed derivations for the bias under the assumption of linear relationships that are modelled using regression analysis and detailed information on simulations.Supplementary Information.


Considering all SNPs simultaneously, 0.69 of variance among the sires’ transmitting ability was explained. There were 24 markers with coefficients of determination for yearling weight gain >0.04. One marker, BTA-75458-no-rs on chromosome 5, explained ≈6% of the variance among the estimated breeding values for yearling weight gain. ARS-BFGL-NGS-39379 had the fifth largest coefficient of determination in the current study and was also found to have a significant effect on weight at an age of 13–14 months in a previous study on Holsteins. Significant genomic effects on yearling weight gain were mainly associated with milk production quantitative trait loci, specifically with kappa casein metabolism.Yearling weight gain in male and female Israeli Holstein calves, defined as 365 × ((weight − 35)/age at weight) + 35, was analyzed from 814,729 records on 368,255 animals from 740 herds recorded between 1994 and 2021. The variance components were calculated based on valid records from 2008 through 2017 for each sex separately and both sexes jointly by a single-trait individual animal model analysis, which accounted for repeat records on animals. The analysis model also included the square root, linear, and quadratic effects of age at weight. Heritability and repeatability were 0.35 and 0.71 in the analysis of both sexes and similar in the single sex analyses. The regression of yearling weight gain on birth date in the complete data set was −0.96 kg/year. The complete data set was also analyzed by the same model as the variance component analysis, including both sexes and accounting for differing variance components for each sex. The genetic trend for yearling weight gain, including both sexes, was 1.02 kg/year. Genetic evaluations for yearling weight gain was positively correlated with genetic evaluations for milk, fat, protein production, and cow survival but negatively correlated with female fertility. Yearling weight gain was also correlated with the direct effect on dystocia, and increased yearling weight gain resulted in greater frequency of dystocia. Of the 1749 Israeli Holstein bulls genotyped with reliabilities >50%, 1445 had genetic evaluations. As genotyping of these bulls was performed using several single nucleotide polymorhphism (SNP) chip platforms, we included only those markers that were genotyped in >90% of the tested cohort. A total of 40,498 SNPs were retained. More than 400 markers had significant effects after permutation and correction for multiple testing ( Numerous studies have considered the economic consequences of animal size for dairy cows (reviewed by ). Most sBrothersone et al.  wrote inSeveral studies have recently performed genome-wide association studies on calf weights for both dairy and beef cattle, but most of these analyzes were based on several thousand genotyped cows, e.g., . ConsideThe objectives of this study were to estimate genetic and environmental parameters of yearling growth rate for both male and female Israeli Holstein calves, to compute genetic evaluations for this trait based on the individual animal model, to estimate phenotypic and genetic trends, and to perform a genome-wide association study (GWAS) based on the genetic evaluations of bulls for yearling growth rate with genotypes. Finally, genomic locations with significant effects were compared to the effects found for these locations in previous studies and the effects associated with these markers on other economic traits in cattle.w = weight in kg at age a in days. This formula assumes a birth weight of 35 kg and includes birth weight in the calculation of YG. This value was found be the mean birth weight for Israeli Holsteins [The data were 898,014 weight records of Israeli Holsteins collected at commercial farms from January 1994 through February 2021. Records of calves resulting from multiple births were deleted. In addition, records of calves with unknown sire or dam, and weights prior to age 150 days or after 500 days were deleted because an adjustment of weight to the age of one year would not be reliable at these ages. Weller and Ezra  estimateijklT = the lth YG record of calf j from herd k; iG = the ith genetic group effect for animals with unknown parents; jA = the additive genetic effect of calf j; jP = the permanent environmental effect of calf j; kH = the effect of herd-year-season (HYS) k; 0a.5, a, and a2 are the square root, linear, and quadratic effects of age; and ijkle = the random residual effect. The square root, linear, and quadratic effects of age were all significant (p < 0.001) in [A variance component divided by the sum of the A, P, and e variance components. Repeatability was defined as the sum of the A and P variance components divided by the sum of the A, P, and e variance components. Data set 2 was analyzed including calves of both sexes and of males and females separately. Prior to analysis of both sexes, the records of females were multiplied by the square root of the ratio of the male and female additive genetic variance components from the individual sex analyses to bring the records of both sexes to equal genetic variances.A subset of this data, consisting of records recorded from 1 January 2008 through 31 December 2017, was used to estimate REML variance components for YG, corrected for age at weight. This was denoted “data set 2”, and details are also given in Genetic evaluations for YG were computed for all animals included in data set 1 by the same model as data set 2, except that genetic groups for animals with missing parents were defined by sex of animal, birth year, and which parents were unknown. Although the Israeli dairy cattle population is 99% Holstein, a small fraction of cows was also mated to other bulls, and additional groups were determined by breed of sire for breeds other than Holstein. A total of 64 genetic groups were defined. As in the analysis of data set 2; G, HYS, and age effects were fixed; the other effects were random; and separate HYSs were defined for male and female calves. Although this precluded the need to include a sex-of-calf effect, it does not correct for the fact that variance components were also different by sex. To correct for this, we applied the procedure of Weller and Ezra .A and P variance components as derived from the REML analysis were calculated relative to the residual variance component.For male calves, P and residual variances for females are changed by this ratio.Records of female calves were multiplied by the square root of the ratio of the genetic variances between male and female calves. Thus, the additive genetic variance component is now equal for both sexes, and the P variances for each sex. For males, the diagonal elements are augmented by the ratio of the P and residual effects. For females, the diagonal elements are augmented by (fP × mA)/(fA × mR), where fP = the P variance for females, mA = A variance for males, fA = A variance for females, and mR = residual variance for males.The mixed model equations are then constructed with different fR × mA)/(fA × mR), where fR = the residual variance for females and the other terms are as defined previously. All records of females are then multiplied by the inverse of this ratio in the mixed model equations.Although the residual variance for males is absorbed from the mixed model equations, the residual variance for females is not. The corrected residual variance for records of females is then computed as:  of calves born since 1 January 1992, on their birth dates. Phenotypic trends for YG were computed based on the weight records closest to 365 days as the regression of YG on the birth dates of animals with records born since 1992. The reliabilities of the EBV were estimated using the algorithm of Misztal and Wiggans , as corrThe correlations were computed among the single sex evaluations of male ancestors derived from the analysis of data set 2 and the combined sex evaluations from data set 1, for bulls with reliabilities >0.9 in the analysis of data set 1. The correlations were also computed between sire EBV with reliabilities >0.5 for YG; EBV for all traits included in the Israeli breeding index, PD16; maternal and direct effects on calving traits; and 17 conformation traits. Genetic evaluations for milk, fat, protein, somatic cell score (SCS), fertility, and persistency were derived by the multitrait animal model as described . Genetic−6 if the substitution effect obtained from the actual data was greater than the permutation effects. To assess the variance explained by all SNPs, we used GCTA-GREML software [The genomic analysis included all Israeli Holstein bulls with genotypes born from 1991 with reliabilities >0.5 from the animal model analysis of data set 1. Of the 1749 Israeli Holstein bulls genotyped, 1445 had genetic evaluations with reliabilities >0.5. As genotyping of these bulls was performed using several SNP chip platforms, we included only those markers that were genotyped in >90% of the tested cohort. A total of 40,498 SNPs were retained. GWAS were computed as described in ,14. The software . The genhttps://www.animalgenome.org/cgi-bin/QTLdb/BT/download?file=gffUMD3.1, accessed on 21 March 2021), and the gtf file with the annotated bovine genome from Ensembl . Both files are based on the bovine reference genome assembly UMD 3.1, corresponding to this study’s markers coordinates. The GALLO package [We obtained the annotation file containing the quantitative trait locus (QTL) database for cattle from the Animal QTLdb . By application of the GCTA-GREML software [The Manhattan plot for the GWAS results for YG is given in software , consideSCO2 (Synthesis of Cytochrome C Oxidase 2). This gene is highly expressed in human and cattle adipose tissues, and downregulation of this gene is associated with fat gain and increased insulin resistance [All markers with coefficients of determination >0.04 for YG are presented in sistance .We investigated the association between the YG significant markers and previously reported cattle QTL. The results are summarized in As most previous studies show, heritability for YG is within the range of 0.25 to 0.4 ,20. Stanhttps://www.animalgenome.org/cgi-bin/QTLdb/BT/index, accessed on 21 March 2021), although the vast majority have not been confirmed on independent studies. Among those associations that have been confirmed by several independent studies, the causative polymorphism has been determined in only a few cases [In 2008, Maher  summarizew cases .Compared to other economic traits that have been analyzed by GWAS in commercial animal populations, YG in dairy cattle is somewhat unique in that this trait has relatively high heritability but has not been under intensive selection in dairy cattle.SCO2. SCO2 is one of the mitochondrial COX assembly factors. A previous study showed that reduction in SCO2 activity leads to increased fat mass, adipogenesis, and insulin resistance [SCO2 regulates its activity and might contribute to the phenotypic variation in the YG.Although 24 markers with coefficients of determination >0.04 were found, some of these are closely linked and most likely have detected the same causative polymorphism. With respect to independent confirmation of our results, we consider only studies that analyzed growth traits on the major dairy breeds, including Holsteins. None of the 24 markers listed in sistance . It is, GWAS often provides multiple QTLs that harbor numerous genes. It is therefore challenging to infer the actual genes and polymorphism that contribute to the phenotypic variation. To address this issue, we performed an enrichment analysis for the genes spanning the marker positions. If the actual genes involved in the tested traits are part of the same pathways, we expect them to be represented among the QTLs genes more than by chance. This analysis revealed significant enrichment of the FGFR pathway . The FGFAlthough YG is positively correlated with all three milk production traits and longevity, YG is also genetically correlated with mature cow size and smaller cows require less feed for maintenance. Therefore, a case can be made for application of some selection pressure to reduce YG considering that YG has a positive genetic trend and has economically unfavorable genetic correlations with female fertility and the direct effect of dystocia.pnominal < 1 × 10−8) after correction for multiple testing and 24 markers had coefficients of determination >0.04. Considering all SNPs simultaneously, 0.69 of variance among the sires’ transmitting ability was explained. The growth rate QTLs are mainly co-associated with milk production QTLs, specifically with kappa casein metabolism. ARS-BFGL-NGS-39379 had the fifth-largest coefficient of determination in the current study and was also found to have a significant effect on weight at age 13–14 months in a previous study on Holsteins. Negative selection on YG as part of a properly weighted selection index may be justified.YG of male and female calves is highly correlated genetically; thus, records from both sexes can be combined into a joint genetic analysis. The genetic trend for YG in the Israeli Holstein population was 1 kg/year. YG is positively correlated with milk production traits but economically negatively correlated with fertility and the direct effect of dystocia. In the genome wide association study, >400 markers were significant (


Paeonia rockii), a unique ornamental and edible oil woody species in the genus Paeonia, a better understanding of the genetic composition of these complex traits related to flowers and fruits is needed. Therefore, we investigated the genetic diversity and population structure of 160 P. rockii accessions and conducted single-marker association analysis for 19 quantitative flower and fruit traits using 81 EST-SSR markers. The results showed that the population had a high phenotypic diversity  and a high level of genetic diversity . These accessions were divided into three subgroups by STRUCTURE analysis and a neighbor-joining tree. Furthermore, we also found a low level of linkage disequilibrium between these EST-SSRs and, by single-marker association analysis, identified 134 significant associations, including four flower traits with 11 EST-SSRs and 10 fruit traits with 32 EST-SSRs. Finally, based on the sequence alignment of the associated markers, P280, PS2, PS12, PS27, PS118, PS131, and PS145 may be considered potential loci to increase the yield of flare tree peony. These results laid the foundation for further analysis of the genetic structure of some key traits in P. rockii and had an obvious potential application value in marker-assisted selection breeding.Allelic variation in floral quantitative traits, including the elements of flowers and fruits, is caused by extremely complex regulatory processes. In the genetic improvement of flare tree peony ( Paeonia rockii; FTP) is one of the most representative species in the Paeonia section Moutan that is native to China (Paeonia suffruticosa) in China  from core germplasm resources of FTP and 81 EST-SSRs were used for single-marker association analysis to explore the allelic effects of floral and fruit trait variation. The results of this study laid a foundation for further identifying the key trait linkage loci of FTP, which is of great value to the genetic improvement of related traits.A total of 160 accessions (each containing three clones), representing the core germplasm resources of FTP that had been demonstrated and described by Seven flower quantitative traits, namely, flower diameter (FD), petal length (PL), petal width (PW), flare length (FL), flare width (FW), petal number (PN), and carpel number (CN), in the 160 accessions were measured at full bloom using digital calipers  in May 2019. In addition, a total of 12 fruit quantitative traits, namely, the number of carpels with seeds (NCWS), multiple fruit fresh weight (MFFW), single fruit length (SFL), single fruit width (SFW), single fruit height (SFH), single fruit pericarp thickness (SFPT), multiple fruit seed number (MFSN), multiple fruit seed fresh weight (MFSFW), individual fruit number (IFN), individual seed number (ISN), individual fruit fresh weight (IFFW), and individual seed fresh weight (ISFW), were measured using digital calipers and electronic balances in September 2019. All measurements were carried out, as described in The total genomic DNA of 160 accessions was extracted from fresh young leaves using a DNAsecure plant kit , following the manufacturer’s instructions. The quality of the extracted DNA was determined by electrophoresis on 2% agarose gels and visualization using a UnicoUV-visible Spectrophotometer . Then, the DNA were diluted with deionized water to 20–30 ng/μl and stored in a freezer at –20°C.2O, 1 μl of 20–25 ng/μl genomic DNA, and 0.5 μl of 10 μmol/L each of forward and reverse primers, and the procedure was performed as described by The polymorphism of 140 previously developed SSR markers  was evalW test and for homogeneity of variance with Levene’s test, and the nonnormal data were logarithmically transformed. In addition, Benjamini-Hochberg (BH) FDR correction was used to correct the values of p for Pearson’s correlation analysis.The statistics software SPSS 18.0 , number of effective alleles (NE), Shannon’s information index (I), observed heterozygosity (Ho), expected heterozygosity (He), inbreeding coefficients (FIS), and Nei’s genetic diversity (GD). The polymorphism information content (PIC) of each locus was calculated using the Microsatellite Toolkit. GENEPOP 4.2  was detected by STRUCTURE 2.3.4 through an admixture model  and ΔK  and ΔK . For 10 analyses . Then, tDISTRUCT . The matr2), which was calculated by using TASSEL 2.0.1. r2 = 0.1 was taken as the critical value to determine whether two loci had LD (r2 > 0.1: LD). Then, a mixed linear model (MLM) of TASSEL 2.0.1 was used to incorporate SSR data, phenotypic traits, Q matrix, and kinship matrix for association analysis. Q was the matrix of the optimal K value obtained through Structure Harvester. The kinship matrix was calculated by SPAGeDi 1.2  correction under QVALUE in R (d) to additive (a) effects was used to assess the gene effects of the significant loci obtained by association analysis. The boundaries of additive effect, partial to full dominance and overdominance, were |d/a| ≤ 0.50, 0.50 < |d/a| < 1.25, and |d/a| > 1.25, respectively. The calculation formulas of additive (a) effects and dominance (d) were as follows: 2a = |GBB − Gbb|; d = GBb − 0.5  was used to find the complete ORFs of the associated marker sequences and translate the ORF sequences into protein sequences . Then, tp < 0.05), of which 65 were very significant correlations . The NE ranged from 1.006 to 8.516, with an average of 2.603. The I varied from 0.021 to 2.357 (mean 1.026). The mean values of HO and HE were 0.501 and 0.524, respectively. The mean FIS of 81 SSRs was −0.439, among which 63 pairs were negative. In addition, the PIC values ranged from 0.006 to 0.871, with an average of 0.476. After Bonferroni multiple comparisons, 43 SSR sites deviated significantly from HWE. Therefore, these deviated sites were removed in the subsequent analysis and 38 SSR sites were ultimately used for subsequent population structure analysis and association mapping.In total, 81 polymorphic EST-SSRs were used to evaluate the diversity of 160 accessions . Then, 4K value increased, LnP(D) progressively increased overall  slowed down. Moreover, the ΔK value corresponding to the maximum K value was the population structure; therefore, the 160 accessions could be divided into three subgroups. The outputs of K = 3 were visualized through CLUMPP and DISTRUCT, and all samples showed a wide range of mixed lineages. Then, we visualized the outputs from K = 2–5 and found that each sample was highly heterozygous. The Q matrix output of the three subgroups can be used for structure-based association analysis. The phylogenetic tree divided these samples into three major clades  of the loci sites had significant LD. In all 698 locus pairs (r2 < 0.1), 17.62, 30.23, and 50.14% of the locus pairs displayed linkage equilibrium at p < 0.05, p < 0.01, and p < 0.001, respectively. Therefore, the overall LD level of 38 EST-SSRs in the 160 accessions was low, and most loci were in linkage equilibrium with each other. However, there were also several loci with significant LD levels between them, such as marker P235-PS50 .The LD levels of 38 EST-SSRs in the 160 accessions were evaluated . Based op < 0.01. Then, FDR test was carried out on these 21 significantly associated combinations (Q < 0.05), and 17 significantly associated combinations of four flower traits and 11 EST-SSRs were ultimately identified, with an explainable rate of 2.23–26.34%  single-marker associations were detected by the MLM model, and a total of 21 (7.89%) significant associations between four floral traits and 14 EST-SSRs were detected under the condition of The number of significantly associated combinations for each trait ranged from two (FD) to eight (FL). There were two markers that were significantly related to FD and CN, and the highest interpretation rates were found for locus PS19, which were 26.34 and 25.38%, respectively. There were eight EST-SSRs that were significantly related to FL, with interpretation rates of 5.03–12.53% (mean 10.15%), and the highest explanatory rate was locus PS91. Among them, there were one and seven combinations with additive effects and overdominance, respectively. There were five EST-SSRs that were significantly related to FW, with explanatory rates ranging from 6.72 to 14.68% (mean 9.43%), and the highest interpretation rate was locus PS43. No significant associations with PL, PW, and PN were detected in this study. Additionally, five EST-SSRs were markedly associated with more than one trait. For instance, P26 was significantly related to FD, FL, and CN, while P235, P281, and PS24 were significantly associated with FL and FW.p < 0.01. Then, an FDR test was carried out on these 118 significantly associated combinations (Q < 0.05), and 117 significantly associated combinations of 10 fruit traits and 32 EST-SSRs were ultimately identified, with explanatory rates of 0.18–43.40%  single-marker associations were detected by the MLM model, and a total of 118 (25.88%) significant associations between 10 fruit traits and 32 EST-SSRs were detected under the condition of The number of significantly associated combinations for each trait ranged from 1 (IFN) to 25 (MFSN). Of these, IFN had one significant association; NCWS had two significant associations; SFW had three significant associations; ISN had eight significant associations; SFL had 12 significant associations; IFFW had 13 significant associations; MFFW and IFSW had 15 significant associations each; MFSFW had 23 significant associations; and MFSN had 25 significant associations. We also found that in ISN, ISFW, and MFSFW, the interpretation rates of P280 were the highest, while in MFFW and SFL, the interpretation rates of P318 were the highest. No locus significantly associated with SFPT and SFH was detected in this study. The phenomenon in which one marker was significantly associated with multiple traits was more common in this part of the study. In the correlation analysis of fruit traits, ISN, IFFW, and ISFW were all significantly correlated with each other. In the association analysis results, six markers were significantly associated with these three traits at the same time, namely, loci P150, P242, P280, PS12, PS59, and PS91. Similarly, loci P242, PS12, PS131, PS145, PS24, PS27, PS50, PS59, PS85, and PS91 were significantly associated with MFSN, MFFW, and MFSFW.via the National Center for Biotechnology Information (NCBI), the gene sequences of P280, PS12, and PS145 were related to fruit and seed development in other species. The explanatory rate of P280 with MFSFW was the highest, at 43.4%, which was predicted to be WRINKLED Like 1 (WRIL1) belonging to the basal ANT subgroup of the APETALA2/ETHYLENE-RESPONSIVE ELEMENT-BINDING FACTOR (AP2/ERF) family AINTEGUMENTA (ANT) group. This gene, which has a typical AP2 domain, has a similarity of 44.79% with AtWRIL1 in Arabidopsis, and its main function is now known to regulate lipid biosynthesis , GATA8  and the kinship matrix , ERF3 (PS27), and WRKY22 (PS131) may form a network to participate in nitrogen metabolism. In addition, WRKY22 was also involved in leaf senescence. Relevant studies have shown that the combination of related genes can delay the senescence process of plants and significantly increase yield , indicating that individuals with P26 heterozygosity sites may produce flowers of larger diameter. For fruit traits, ISFW was considered a yield indicator, with 15 markers significantly associated with it. Among them, 11 markers, namely, P242, PS59, PS91, P280, PS24, PS131, PS64, PS145, P150, PS118, and PS27, showed the same pattern of gene action (overdominance), while the effects of P280 (d/a = 1.4674) and PS64 (d/a = 4.0449) were positive, showing that individuals with P280 and PS64 heterozygous loci might generate heavier seeds. In summary, marker combinations based on the gene effect value of associated loci can be used for the early selection of target traits.The analysis of the genetic regulatory relationships between different significant association sites of the same quantitative trait was helpful for breeding with significant association combinations. For example, in floral traits, FD was an important ornamental character that determined its ornamental value to a large extent. P26 was significantly associated with FD and showed an overdominant effect (The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found in the article/XG and FC: conceptualization and validation. XG: formal analysis, data curation, writing – original draft preparation, and visualization. XG, XC, and XT: investigation. XC and XT: resources. XG, CH, FC, YZ, XC, and XT: writing – review and editing. FC: supervision and project administration. All authors contributed to the article and approved the submitted version.XC and XT are employees of Beijing Guose Peony Technology Co. Ltd., Beijing, China.The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


Yield is a complex trait and reliant on various related variables called yield components. In this study, the five most important yield component traits in soybean were measured using a panel of 250 genotypes grown in four environments. These traits were the number of nodes per plant (NP), number of non-reproductive nodes per plant (NRNP), number of reproductive nodes per plant (RNP), number of pods per plant (PP), and the ratio of number of pods to number of nodes per plant (P/N). These data were used for predicting the total soybean seed yield using the Multilayer Perceptron (MLP), Radial Basis Function (RBF), and Random Forest (RF), machine learning (ML) algorithms, individually and collectively through an ensemble method based on bagging strategy (E-B). The RBF algorithm with highest Coefficient of Determination (R2) value of 0.81 and the lowest Mean Absolute Errors (MAE) and Root Mean Square Error (RMSE) values of 148.61 kg.ha-1, and 185.31 kg.ha-1, respectively, was the most accurate algorithm and, therefore, selected as the metaClassifier for the E-B algorithm. Using the E-B algorithm, we were able to increase the prediction accuracy by improving the values of R2, MAE, and RMSE by 0.1, 0.24 kg.ha-1, and 0.96 kg.ha-1, respectively. Furthermore, for the first time in this study, we allied the E-B with the genetic algorithm (GA) to model the optimum values of yield components in an ideotype genotype in which the yield is maximized. The results revealed a better understanding of the relationships between soybean yield and its components, which can be used for selecting parental lines and designing promising crosses for developing cultivars with improved genetic yield potential.Improving genetic yield potential in major food grade crops such as soybean  Glycine max L. Merrill) is the world’s most widely grown leguminous crop and an important oil and protein source for food and feed [Soybean  algorithms, as one of the reliable and efficient computational approaches, were successfully implemented in different fields of study, such as traffic crash frequency modeling , 23, envOne of the recent agriculture trends is the use of ML algorithms for analyzing big data , 30. EmeOne of the major impediments of using individual ML algorithms is the high probability of overfitting in single predictive algorithms . To overIn soybean cultivar development programs, an optimum selection among important yield components can significantly improve the yield genetic gain. Therefore, the implementation of optimization methods in this field is of particular interest. Optimization algorithms for improving important traits are becoming more and more attractive in plant science , 53. GenAlthough the successful uses of ML ensemble methods have been reported in different agriculture-related fields , 56, 57,2.Two hundred and fifty soybean genotypes were grown under field conditions at two locations: Palmyra  and Ridgetown , in Ontario, Canada in 2018–2019. The population used in this study was selected from the core germplasms of soybean breeding programs at the University of Guelph, Ridgetown campus that have been created over 35 years and used for cultivar development and genetic studies. The experiment was conducted using randomized complete block designs (RCBD) with two replications in four environments (two locations × two years), consisting of 2000 phenotypic plots in total. Each plot consisted of five rows, each 4.2 m long and 40 cm spacing between each row. The seeding rates used in this study were 50–57 seeds per m-1) of each plot was measured by harvesting three middle rows and adjusting to a 13% moisture level. Soybean yield components, including NP, NRNP, RNP, and PP, were hand-measured by randomly selected ten plants from each phenotypic plot for each genotype. Also, the PP to NP ratio (P/N) for each genotype was calculated using the following equation:PP indicates the number of pods per plant, and NP indicates the total number of nodes per plant.Seed yield  as one of the spatial error control methods to reduce and minimize the possible error in the field data , 59. Then number of independent observations [MLP, RBF, and RF are the most commonly used ML algorithms with distinct functions and abilities that are used for predicting yield in plant crop species , 48, 62.rvations . The finrvations . All thervations  was usedFor obtimizing the values of NP, PP, RNP, NRNP, and P/N in a theoretical maximized yield ideotype genotype, the best ML algorithm was linked to the genetic algorithm (GA). In order to have the best implement of GA, some parameters such as mutation rate, crossover fraction, and the number of chromosomes should be determined. For that, optimum values of the mentioned parameters are estimated by the trial and error methods. In brief, a chromosome is known as a set of variables that define as a possible solution to the problem. In this study, all the possible combinations of the yield component traits were considered as different chromosomes to form the maximum soybean yield production. All proposed chromosomes are constructed from the initial population, which is the first step in the GA optimization process . CrossovY′ stands for predicted value, Y is the field measured value, and n is the number of observations for a given genotype.The original dataset consists of 250 observations was randomly divided into training and validation sets based on the five k-fold cross-validation method  with tengoodness of fit between observed and the predicted values were performed using the coefficient of determination , ggvis , and ggp[ggplot2  packagest2 [MASS  package -1 with a mean and standard deviation of 4.22 and 0.57 ton ha-1, respectively.Based on the field data analyses, the average yield of 250 soybean genotypes was estimated to be between 2.58 to 5.71 ton har = 0.71) was found to be the strongest followed by its correlation with NP (r = 0.68), RNP (r = 0.67), and P/N (r = 0.64). The negative correlation between soybean seed yield and NRNP was estimated as r = -0.29  and the lowest values for MAE (148.61 kg.ha-1) and RMSE (185.31 kg.ha-1) ; however, they had different values for MAE and RMSE. In comparison with the MLP algorithm, RF had the lower MAE and RMSE with 156.28 kg.ha-1 and 194.75 kg.ha-1, respectively (-1), and RMSE (211.57 kg.ha-1) among the ML algorithms. In addition to individual evaluations of the three ML algorithms, an ensemble learning was also developed, which outperformed all the individual machine learning algorithms, attaining an R2 value of 0.82. The E-B method had the acceptable RMSE and MAE with a value of 184.35 kg.ha-1 and 148.37 kg.ha-1, respectively  . The R2 ectively . MLP hadectively . In gene-1 can be achieved in an ideotype soybean genotype in which the values of NP, NRNP, RNP, PP, and P/N are 17.32, 3.07, 14.25, 48.98, and 2.83, respectively.The aim of the current study, not only was to predict soybean seed yield using yield components, but also to estimate the optimum values of these traits, i.e., NP, PP, RNP, NRNP, and P/N, to maximize the final yield production in a given genotype. The results of the optimization process using GA, as the single objective evolutionary optimization algorithm, are presented in One of the objectives of this study was to investigate the potential use of soybean yield components such as NP, PP, RNP, NRNP, and P/N for predicting the final seed yield production. Many studies reported the reliance of the final yield production on the performance of several yield-related traits , 77–80. 2, RMSE, and MAE values [Conventional statistical methods such as ANOVA and simple regression methods are typically recommended for small datasets with limited dimensions , 87. HowE values –94. In tE values . HoweverE values , 97. TheE values , 98. ThiE values .2 values, the MAE and RMSE values were lower in RF. MLP, as one of the neural network algorithms, is highly susceptible to possible instabilities/noises caused by non-heritable factors. The MAE and RMSE values of this algorithm were the highest among all the tested ML algorithms that may indicate the sensitivity of the algorithm to noises. As a result, using this algorithm may not recommend for analyzing phenotypic data that are largely affected by environmental factors. RF with the R2, MAE, and RMSE values of 0.80, 156.28 kg.ha-1, and 194.75 kg.ha-1, respectively, was selected as the second-best ML algorithm for predicting soybean seed yield in this study. Although the difference between RF and RBF in terms of R2 values was not statistically significant (data are not shown), they were statistically different for their MAE and RMSE values. The performance of the RF algorithm relies on processing large dimensional data based on generalized error estimation [Although RF and MLP had the same Rtimation , 101. Altimation , and thitimation , 103.In addition to individual comparison of the three tested ML algorithms, we developed a bagging ensemble model by combining the results of RBF, MLP, and RF in this study. Since the RBF had the highest performance in predicting the soybean seed yield, this algorithm was chosen as the metaClassifier for developing the E-B algorithm. Using the E-B model, a slight improvement was obtained in predicting the total soybean seed yield from its component traits. Diversity and sufficiency are two of the most important principles in selecting base learners for an ensemble model . It meanper se. This strategy can improve genetic yield potential in varieties by setting up selection criteria on yield components and making the selection more efficient [Selecting high-yielding lines has always been one of the major goals in plant breeding programs that can be performed using either direct or indirect selection approaches . Selectifficient . In ordefficient , 113. Affficient , plant pfficient , and remfficient .Efficient breeding approaches for improving the genetic potential of complex traits such as yield in soybean can be developed based upon secondary traits that govern the final performance of the complex traits. Thus, in order to increase the genetic yield potential in soybean, breeders can select soybean genotypes with improved yield component traits. However, measuring yield components in a large soybean breeding program is time-consuming and labor-intensive, which limit the implication of this approach in cultivar development programs. The main objective of this study was to evaluate the potential use of yield component traits for estimating final seed yield in soybean using different ML and E-B algorithms, which in turn can be used by breeders for selecting parental lines and designing promising crosses for developing cultivars with improved genetic yield potential. The results of the current study showed that RBF is a reliable solo ML algorithm for predicting the soybean seed yield from its component traits. However, an E-B algorithm that was built by combining the three ML and using RBF as its metaClassifier outperformed all individual ML algorithms and, therefore, it is recommended for predicting the soybean seed yield exploiting yield component traits. In the current study for the first time, we coupled E-B algorithm with GA in order to estimate optimum values of yield component traits in a theoretical genotype in which the yield is maximized using the real field data. The results seem to be promising but are recommended to be evaluated in new and independent breeding populations before using in cultivar development programs for selecting high-yielding potential genotypes. This information can be also used, in combined with molecular marker technology, for developing genomic-based toolkits that can be used for selecting genotypes with improved genetic yield potential at early generations.S1 Table(DOCX)Click here for additional data file.


South Korea dairy farmers are less interested in improving milk production-related traits for various reasons, including the milk production quota system. Consequently, they have become more interested in improving traits associated with longevity. Therefore, the purpose of this study was to analyze the culling patterns and survival rates of Korean Holstein cows and evaluate genetic characteristics related to parity and longevity of each lactation by using test data collected in South Korea. This study presents evidence that individuals’ genetic backgrounds differ with respect to differences in the individuals’ survival periods. Three lactation periods in each parity were split according to lactation days: 0–90 days (1), 91–299 days (2), and 300 days to the next calving (3). The heritability of the first, second, and third parities were 0.020, 0.028, and 0.039, respectively. In all parities, the heritability in late-lactation was higher than that in early- and mid-lactations. Most genetic correlations for survival in the first parity were higher than those in the second and third parities. The genetic correlation in the same lactation in the first and second parities were lower than those in the intervals in the second and third parities. The results of this study may serve as a basis for developing a more accurate model for evaluating longevity traits in South Korea.Longevity is closely related to the survival rate of dairy cattle and refers to the period during which the cow has economic value, from first calving to culling. The purpose of this study was to analyze the culling patterns and survival rates of Korean Holstein cows and evaluate genetic characteristics related to parity and longevity of each lactation by using the test day milk yield collected in South Korea. The performance data of the dairy cattle were collected from 2004 to 2019 by the Nonghyup Dairy Cattle Improvement Center. The collected 1,702,304 records were used as pedigree data through the Korea Animal Improvement Association. The lactation period was divided into early-lactation , mid-lactation , and late-lactation . The heritability of longevity for the first, second, and third parity was 0.020, 0.028, and 0.039, respectively. In all parities, the heritability in late-lactation was higher than that in early- and mid-lactation. Most genetic correlations for survival in the first parity were higher than those in the second and third parities. The results of this study may serve as a basis for developing a more accurate model for evaluating longevity traits in South Korea. The purpose of the Dairy Cattle Improvement Project of South Korea is to increase the income of farmers by improving their productivity and enhance the competitiveness of sire in South Korea. Domestic dairy farmers are less interested in improving milk production-related traits for various reasons, including the milk production quota system. Consequently, they have become more interested in improving traits associated with longevity, which have been introduced in regions that have made the most advancements in dairy technology such as North America.Longevity is closely related to the survival rate of dairy cattle and refers to the period during which the cow has economic value, from first calving to culling. In general, longevity is a desirable trait affecting dairy farmers’ profitability ,2. The mIn the dairy industry, it has also been reported that genetic evaluation for longevity has been conducted in all advanced countries . LongeviCalculating the survival probability for each parity and lactation is possible in South Korea using the milking and culling records collected as test records. The survival lifetime can be explained by the different reasons for culling collected from dairy farmers this way, and the reasons for culling vary by parity and lactation in days . TherefoThe performance data related to milk production and udder health of dairy cattle were collected from 2004 to 2019 by the Nonghyup Dairy Cattle Improvement Center. The 1,702,304 records collected through the Korea Animal Improvement Association were used as pedigree data.The records of 190,773 cows were used for estimating genetic parameters after removing censored data, cases with unknown parent records, and data with less than five contemporary groups (herd-year). The genetic evaluation was using 298,975 records, including censored  and uncensored  data among the test records collected for breeding value estimation. The uncensored data refers to cases with known culling time. The censored data refer to the data that cannot measure the exact productive life because it is unknown if an individual has survived, or the exact date of culling is unknown. The detailed reasons for culling and records are in The lactation period was divided into early-lactation , mid-lactation , and late-lactation .When an individual survived in the subsequent lactation period from the current lactation period, the individual was coded as 1. If an individual did not move to the next lactation period because of culling or death, it was coded as 0. Moreover, for late-lactation cases (more than 300 milking days), if a test record existed for the subsequent parity, it was set to 1. If not, it was set to 0 .Y is the survival of each lactation (1 or 0), HY (herd-year) is the effect of the contemporary group, Season with four levels  is the calving season effect, and FCa is the first calving age (<25 months of age: 0 and ≥26 months of age: 1), a is a random genetic effect, and e is a residual effect.The following animal model was used for estimating genetic parameters.Although the contemporary group used for the genetic evaluation model of production traits and conformation traits is usually set to herd-year-season (HYS) and analyzed, this study used HY and season separately because the Korean dairy cattle population was too small to use HYS. A previous study also showed that the contemporary group of using HYS as a fixed effect and that of using HY and season as fixed effects had the best fit, respectively .The multiple trait model that evaluates nine traits simultaneously demands a computer with high computation power. Therefore, because of computing power limitations, the parameter estimation was conducted by converting the nine traits  into 83 combinations of three traits. Afterward, the results were combined. The genetic parameters used in this study were estimated using the WOMBAT Program .The genetic and phenotypic parameters of economic life by parity  were calculated using the following equations.When making a genetic variance–covariance matrix using the diagonal mean of the genetic variances estimated from 83 combinations and the off-diagonal using the mean genetic correlation, it does not form a positive definite matrix. Therefore, this study generated a genetic variance matrix for estimating the breeding value through the following process for producing a positive definite matrix from a genetic variance–covariance matrix .(1) Genetic variance–covariance matrix was calculated using the mean of genetic parameters and mean correlation coefficients estimated from the initial variance component (3) If an eigenvalue of (4) The genetic variance matrix to be used for the genetic capacity evaluation was calculated as follows in this study.The off-diagonal matrix of In this study, the BLUPF90 Program  of the BThe following equation was used to estimate the direct herd life (DHL) for the longevity trait by combining the breeding values for each parity and lactation while considering the mean survival rate of daughters for the sire estimated above sI=mean i , and i until the culling of the individual.And the cumulative survival rate of the trait In this study, the calving interval was calculated by setting the 1st–2nd calving as 426 days, the 2nd–3rd calving as 426 days, and the 3rd–4th calving as 432 days.This study used the result of the national genetic evaluation published by the National Institute of Animal Science(NIAS) in April 2020 to compare breeding values between the longevity traits estimated above, milk production traits , those between 25 conformation traits (stature (ST), chest width (CW), body depth (BD), angularity (ANG), rump angle (RA), rump width (RW), rear leg side view (RLSV), foot angle (FA), fore udder attachment (FUA), rear udder height (RUH), rear udder width (RUW), udder suspensory (US), udder depth (UD), front teat placement (FTP), teat length (TL), overall conformation score (OCS), loin strength (LS), udder texture (UT), rear teat placement (RTP), heel depth (HD), bone quality (BQ), rear leg rear view (RLRV), height at front end (HFE), locomotion (Loc), and body condition score), and those between Udder Composite Index, Feet and Leg Composite, Milk, Fat, Protein composite, and Korean Type Production Index. This study used 383 sires with at least 50 daughters from DHL evaluation and 90% or higher reliability in EBV (adj-305 milk yield and OCS) for the accuracy of analysis .https://www.vit.de/, accessed on 22 May 2020) reported that the culling rate was 17–24%, 31–35%, and 44–49% during early-, mid-, and late-lactation period, respectively. The culling rate in Korean Holstein was slightly different with 8–13%, 19–29% and 56–71% during early-, mid-, and late-lactation periods, respectively  which were significantly affected by the survival period are presented in ectively .9. Heise  reported9. Heise ,11. ConsThe multiple trait model is a method to utilize genetic and environmental correlations between traits by analyzing two or more traits simultaneously. The multiple trait model including traits with low heritability is useful when the difference between genetic and environmental correlations is high  or when the heritability of one trait is considerably higher than that of another trait . In thisIt is not easy to directly compare the genetic parameters of longevity between countries because the lactation period is set differently between countries. Therefore, if the heritability and genetic correlations by parity are unified, it will be possible to compare countries directly and the results are shown in When cows survived in all three parities, heritability was estimated as 0.085, and it was 0.020, 0.028, and 0.039 in the first, second, and third parities, respectively. The heritability of this study was lower than that of Boettcher, et al. , and HeiThe correlation between the DHL and the breeding value for longevity of each parity ranged between 0.76 and 0.96, showing a highly positive correlation. In all parities, the breeding value correlation between late lactation and DHL was always 0.9 or higher, which is a strong correlation .Sewalem, Miglior, Kistemaker, Sullivan, Huapaya and Van Doormaal  reportedIn this study, the correlation between early-lactation and mid-lactation in the first parity, early-lactation in the second parity and mid-lactation in the third parity was 0.9 or lower, which was different to the results of the previous studies. However, it could be explained by the differences in the culling rates in South Korea between early-, mid-, and late-lactation periods. It could also be because the heritability of early- and mid-lactation (0.003–0.014) was different to that of late-lactation (0.022–0.028). The analysis results of this study indicated that DHL included survival rates for all lactation periods. Moreover, it may be possible to compare different evaluation traits using these results.In addition, this study analyzed the correlation of DHL with four traits related to milk production evaluated in South Korea, the breeding value of conformation traits, and four indices using these traits’ breeding value A–C. The The results of Heise, Liu, Stock, Rensing, Reinhardt and Simianer  showed tThe breeding value correlation between DHL and conformation traits is shown in The correlation between ST and breeding value by lactation was negative. Like milk production-related traits, it showed the highest breeding value correlation in early and mid-lactation of the first parity. However, the correlation between these traits was negative. The breeding value correlations of lactation with CW, BD, AN, and RW were negative (−0.44 to −0.05). CW and BD showed a moderate negative correlation (−0.43 to −0.34) in all parities. ANG also showed a negative correlation in all periods. In particular, correlations in late-lactation showed relatively stronger negative relationships. Similar to ANG, RW also showed a relatively high negative correlation in late-lactation of each parity. FA showed a strong breeding value correlation with early-lactation in most parities.Breeding value correlations of traits related to udders mostly showed moderate positive correlations. In FUA, early-lactation in all parities was the highest with 0.10, 0.17, and 0.14, respectively, and the breeding value in mid-lactation had the strongest correlation. RUH, US, and UD showed positive correlations ranging from 0.22 to 0.52. In particular, UD was 0.36–0.52, showing a moderate correlation. In addition, correlation tended to be stronger in later lactation and later parity.In particular, the results of this study revealed that the breeding value correlations with TL, OCS, and LS, which South Korean dairy farmers were keen on understanding, were not high (−0.10 to 0.13), unlike the expectations previous to the study. LS showed a negative correlation in the range of −0.12 to 0.07.Moreover, UT was 0.28 (L1.1), 0.26 (L1.2), 0.14 (L2.1), 0.20 (L2.2), and 0.10 (L3.1) in early- and late- lactation of each parity. In late-lactation, the correlations of the first parity, second parity, and third parity were low with −0.01, −0.01, and 0.03, respectively. In RTP, which farmers paid much attention to during milking, there were positive correlations in all periods, but they were weak (0.04–0.12). HFE showed a negative correlation (−0.35 to −0.29) in all parities and lactation periods.This study calculated the breeding value correlations between longevity-related DHL by applying UDC, FLC, MFP, and KTPI estimated by combining the breeding value estimated from Korea Genetic Capacity Evaluation . As shown above, along with the results showing relatively stronger positive correlations with udder-related traits, the udder index calculated by combining udder traits showed intermediate correlations in the range of 0.30–0.44. In addition, as parity increased, the correlation between the breeding value and the udder index tended to increase from 0.29 to 0.44. FLC also showed a positive correlation. In the case of MFP, relatively high breeding value correlations were found in early- and mid-lactation of the first parity, similar to milk production traits. Lastly, KTPI, composited using milk production traits and body shape-related traits, showed positive correlations (0.25–0.36) in early- and mid-lactation of all parities with a small culling rate.The breeding value correlation between DHL and conformation-related traits revealed that the correlations with RUH, US, and UD among traits-related to udder were relatively high (0.25– 0.52). In particular, the correlation with UD was the highest at 0.52. It appeared to be related to the heritability of udder-related traits associated with milk production after calving to the subsequent breeding. It is possible to improve the longevity of the individuals by improving these traits.In contrast, the results of this study showed that longevity had a negative correlation with traits directly related to body size . Specht, et al.  reportedVacek, et al.  reportedDHL showed positive correlations (0.14–0.42) with UDC, FLC, MFP, and KTPI, calculated by combining these traits. It is possible to, directly and indirectly, select the longevity of Korean dairy cattle using these indices.In addition, this study analyzed both censored and uncensored data. The results of this study will provide baseline information for estimating life expectancy solely by using censored data.This study presented evidence that individuals’ genetic backgrounds differed with respect to differences in the individuals’ survival period. It provides the basis for developing a more accurate genetic evaluation model for traits related to longevity that will coherently explain the genetic correlation structure. It will also be possible to provide information on the economic aspects to Korean dairy farmers by comparing breeding value between milk production-related traits, conformation-related traits, and reproduction-related traits.DHL and each lactation period showed a strong correlation. Milk production traits, udder-related traits, and composite indices  had positive correlations. The results indicate that these traits are closely related to the longevity and survival of cows. It is believed that it will be possible to select a sire based on these traits.The results of this study may serve as the basis for developing a more accurate model for evaluating longevity traits in South Korea.