Obsessive–compulsive disorder (OCD) is a heritable childhood-onset psychiatric disorder that may represent the extreme of obsessive–compulsive (OC) traits that are widespread in the general population. We report the heritability of the Toronto Obsessive–Compulsive Scale (TOCS), a new measure designed to assess the complete range of OC traits in youth. We also examined the dimensional nature of the TOCS and the degree to which genetic effects are unique or shared between dimensions. OC traits were measured using the TOCS in 16,718 youth (6–18 years) at a science museum. We conducted a factor analysis to identify OC trait dimensions. We used univariate and multivariate twin models to estimate the heritability of OC trait dimensions in a subset of twins (220 pairs). Six OC dimensions were identified: Cleaning/Contamination, Symmetry/Ordering, Rumination, Superstition, Counting/Checking, and Hoarding. The TOCS total score (74%) and each OC dimension was heritable (30–77%). Hoarding was not highly correlated with other OC dimensions, but did share genetic effects. Shared genetics accounted for most of the shared variance among dimensions, whereas unique environment accounted for the majority of dimension-specific variance. One exception was Hoarding, which had considerable unique genetic factors. A latent trait did not account for the shared variance between dimensions. In conclusion, OC traits and individual OC dimensions were heritable, although the degree of shared and dimension-specific etiological factors varied by dimension. The TOCS may be informative for genetic research of OC traits in youth. Genetic research of OC traits should consider both OC dimension and total trait scores. OCD is familial and heritable5. In children and adolescents, genetic effects account for most of the variance (heritability estimates of 45–65%)6 and the stability of symptoms across development7, with common environment only playing a role in early adolescence9. However, there are few replicated genetic risk variants for OCD. Gene discovery has been hampered by relatively small samples, which international consortia are working to overcome10.The role of genetics in the etiology of obsessive–compulsive disorder (OCD) is well established11. Clinical samples are slow and expensive to collect. Alternatively, if OCD represents an extreme of obsessive–compulsive (OC) traits widely distributed in the population, then genetic research could focus on quantitative OC trait measures. A quantitative trait-based measure assesses the full range of OC traits  to capture all their variance, which could boost power for genetic studies12, especially in general population samples.Exclusive reliance on OCD diagnosis and clinic samples may be another reason for slow progress. Diagnoses are useful in clinical practice, but in genetic research, could obscure phenotypic and genetic heterogeneity, hide variation in symptom severity among cases, and miss sub-threshold OCD casesExisting OCD scales can generate quantitative scores by summing the number of symptoms exhibited by an individual. However, symptom counts typically generate J-shaped distributions in population samples because most people have few or no OC symptoms. These kind of scores are suboptimal for quantitative analyses.14, indicating considerable phenotypic heterogeneity. Twin studies in adults indicate that each of these dimensions is heritable with genetic influences that are shared among dimensions and unique to each dimension16, suggesting that phenotypic heterogeneity reflects genetic heterogeneity. In youth, the heritability of OC traits and dimensions is unclear17, particularly the degree to which individual dimensions are mediated by unique and shared genetic influences. Hoarding is considered a distinct dimension in adults and youth and is classified as its own disorder in the Diagnostic and Statistical Manual of Mental Disorders, 5th edition (DSM-518). Twin studies in adults suggest that hoarding shares genetic influences with other OC dimensions20. In youth, it is not known whether hoarding shares genetic risk with other OC dimensions. The present study aimed to uncover the degree of shared and unique genetic effects on OC dimensions in youth.Another reason for slow progress in genetic discovery might be the apparent phenotypic heterogeneity in OCD. In adults and youth, OC symptoms generally cluster into four dimensions: symmetry, forbidden thoughts/checking, cleaning, and hoarding21) to measure widely distributed OC traits in youth and to capture OC dimensions. Each item on the TOCS queries whether a given OC symptom occurs far less often (lower extreme), an average amount of time, or far more often (upper extreme) in the child than in typically developing peers. Questionnaires of this type are designed to generate scores that are widely and more normally distributed in the general population compared with the J-shaped distributions found with typical symptom-based measures  of youth to identify OC dimensions. We used structural equation modeling in 220 twin pairs to examine the heritability of the individual dimensions, their co-heritability and test the fit of the independent and common pathway models. Current evidence indicates that the best fitting model is unclear in adults16 and unknown in youth. The expectation was that if a common pathway model fit best, then shared etiological factors were mediated by an underlying latent trait, whereas if an independent pathway model fit best, then shared etiological factors influenced dimensions directly. If the variance shared by multiple dimensions is captured by a global latent trait, research designs should focus on that latent trait. Conversely, if unique genetic factors influence each OC trait dimension directly, research designs should focus on the individual OC dimensions.We tested whether (1) the TOCS captured OC dimensions, (2) if these dimensions were heritable as well as co-heritable, (3) if these dimensions had shared and unique etiological factors, and (4) whether shared variance between the dimensions was best explained by a common pathway  or independent pathway model. To address these questions, we factor analyzed the TOCS in a population-based sample .Our sample included 16,718 participants with complete data ; 50.5% male) from the 17,263 youth (6–18 years of age) recruited at the Ontario Science Centre, a local science museum in Toronto, Canada. Informed consent, and assent where applicable, was obtained and the protocol was approved by The Hospital for Sick Children Research Ethics Board. We collected behavioral information about the participants from themselves if they were thought to be capable of self-reporting (18.2%) or from their parents (81.8%). Participants provided a saliva DNA sample using 2 mL Orageneσ = 0.94) and to discriminate between an OCD measure  Spearman correlation = 0.5) and an ADHD measure , Spearman’s correlation = 0.02)21. Because the TOCS total score  was significantly associated with age, gender and respondent (p < 0.05), we created standardized TOCS z scores. Total scores were modeled using linear regression controlling for age and gender, for parent- and self-respondents separately and residual scores were obtained. Participants were divided into 30 groups according to respondent (parent- or self-report), gender, and integer age. Parent-report groups included integer ages from 6 to 15 and self-report groups included integer ages from 13 to 17. Standardized scores corresponding to the empirical percentile of each individual were assigned within each of the 30 groups separately. We compared the heritability of the TOCS total score to an established measure of OC symptoms: the CBCL-OCS23. Each of the eight CBCL-OCS items were scored on a scale of 0–2 , and were summed to generate a total score (range: 0–16).A computerized, English questionnaire covered demographics, medical history, and two measures of OC traits and symptoms. The TOCS had 21 items scored on a scale of − 3 to + 3 . The TOCS has been shown to have high internal consistency , a panel consisting of 15 autosomal, codominant, unlinked loci, and the sex-determining marker, amelogenin amplified in a single PCR26. We classified twin pairs as monozygotic (MZ) if all 16 markers were identical between the pair; otherwise, they were classified as dizygotic (DZ)26. We had four sets of DZ triplets. We randomly selected two siblings from each triplet to be a DZ twin pair and excluded the other sibling. Our final twin sample included 60 MZ twin pairs  and 160 DZ twin pairs . The mean age of the twins was 10.5 years (SD 2.6) and no individuals had a reported diagnosis of OCD.We estimated heritability from 220 twin pairs. Their zygosity was initially determined by a twin questionnaire adapted from Cohen et al.Exploratory factor analysis with principal components using varimax rotation was conducted in SAS 9.3 to examine the underlying dimensional structure of the TOCS. We also conducted promax rotation because of the expected correlation of TOCS items. Phenotypic correlations between the OC trait dimensions were examined using Pearson’s correlations with IBM SPSS Statistics 21.0. We assigned items to factors when the factor loading was > 0.7 and had factor loadings < 0.4 for all other factors. We considered scree plots, cumulative variance explained and interpretability when selecting the number of factors.n = 3)) in OpenMx27. For analyses using standardized z scores, age, sex, and respondent (parent or self) covariates were not included in the models because these factors were incorporated during z score calculation (see above). Saturated model fit was conducted to test the assumption of equality of means and variances between the MZ and DZ twins28. The goodness of fit parameters used to compare twin models were the likelihood-ratio chi-square statistic (χ2) and Akaike’s information criterion (AIC).Intraclass correlations (ICCs) for each trait and across traits were examined within MZ and DZ twins. The heritability of total OC traits and each OC trait dimension was estimated by structural equation modeling with age, sex, and respondent included as covariates using full information maximum likelihood  to the nested sub-models .We decomposed the total variance of the CBCL-OCS, TOCS total scores, and each of the TOCS OC trait dimensions identified in our factor analysis into genetic and environmental factors. Genetic variance could be attributable to additive effects (A), and/or dominance (non-additive) effects (D). Environmental variance was partitioned into common environmental (C) influences, which are shared by family members, and unique environmental (E) factors, which include measurement error. The within-pair additive genetic correlation (A) was set at 1 for MZ twins and 0.5 for DZ twins, whereas the dominance genetic correlation (D) was fixed at 1 for MZ twins and 0.25 for DZ twinsn = 54).To examine sex differences in heritability, we observed intra-pair correlations by zygosity and sex. Our sample only had 33 DZ female twin pairs, and because the opposite-sex DZ twin correlations (0.40) were generally similar to the DZ same-sex twin correlations  we did not further test sex differences in heritability. We could not examine differences in heritability by respondent because there were very few self-reporting twins  and environmental (C and E) variance components that are correlated. A correlation between the A variance components of two OC trait dimensions was interpreted as an indication of a shared genetic influence . A correlation between the C or E variance components was interpreted as an indication of overlapping environmental influences . A genetic or environmental correlation of zero between two OC trait dimensions would indicate independence in the variance components explaining the variation in those two dimensions.30. In the common pathway model, the co-variance of the OC trait dimensions is accounted for by a single latent phenotype influenced by shared additive genetic (Ac), common environment (Cc), and unique environment (Ec) factors. The model estimates dimension-specific genetic (As), common environment (Cs), and unique environment factors (Es). The independent pathway model accounts for co-variance of the dimensions by estimating Ac, Cc, and Ec factors that directly influence each dimension  and dimension-specific variance is accounted by estimating As, Cs, and Es factors for each dimension. The best fitting model was selected using the AIC.To understand how A, C, and E factors influence the co-variance between all trait dimensions, we compared the correlated factor model (which examines correlations between each pair of dimensions) to the common and independent pathway models 21, are shown in Supplemental Fig. We selected a six-factor structure as it minimized cross-loading between factors while including as many items as possible, accounted for the most variance (75%), and produced the same factor structure for both parent- and self-reported data (an asset for use in population-based samples). Two items, “experiences unwanted upsetting thoughts or images” (referred to as ‘upsetting’) and “spends time checking and rechecking homework” (referred to as “homework”), factored separately from the other items and were excluded from the final factor model. Upon re-examination of these two items, we considered the ‘upsetting” item too general, capturing a broad, non-specific trait, and the “homework” item, intended to capture a checking compulsion, too specific. The six factors were as follows: Cleaning/Contamination, Symmetry/Ordering, Superstition, Rumination, Counting/Checking, and Hoarding Table . Distribr = 0.70). Hoarding was less correlated with the other five dimensions (r = 0.31–0.52) and the TOCS total score  and for Symmetry/Ordering and Rumination (0.86). Unique environmental influences also accounted for significant co-variance between OC trait dimensions. The Cleaning/Contamination and the Symmetry/Ordering dimensions showed highest E correlations (0.53). The lowest unique environmental correlation was for the Cleaning/Contamination and Hoarding dimension (0.04).df = 2604, p = 0.02) and independent pathway  models to the correlated factor model  for the OC trait dimensions. The independent pathway model was the most parsimonious. Model fit was unchanged by removing the Hoarding dimension (data not shown). As shown in Fig. We compared the fit of the ACE common pathway  from previous studies34.Overall, there was considerable convergence in the factor structure of the TOCS and existing OCD scales37. The TOCS also had higher heritability estimates than the CBCL-OCS, an established heritable OC measure9, supporting the utility of the TOCS in genetic research. Each OC trait dimension was heritable, a finding that converges with previous studies of youth and adults39. Environmental factors contributed significantly to only the Cleaning/Contamination dimension, suggesting a distinct etiological mechanism. This finding is consistent with a previous study in an adolescent population-based sample that also reported low additive genetic effects for this dimension17. The effect of common environment on Cleaning/Contamination may result from family values, education or parental modeling.The estimated heritability of the TOCS total score was 74%, which was higher than estimates for OC symptoms from previous twin studies using existing OC measures15 reported dimension-specific genetic effects for most OC dimensions. However, both ours and this previous study reported that unique environment accounted for most of the variance specific to each dimension (Es), suggesting that environment and potentially error (which is not separable in ACE models) are the biggest contributors to making dimensions different. One implication is that genetics may play a larger role in what makes dimensions similar, whereas unique environment may play a bigger role in what makes dimensions different.Phenotypic heterogeneity in the TOCS, demonstrated by six OC dimensions, also reflected some etiological heterogeneity. Genetic factors contributed considerably to all OC dimensions, although less so for Cleaning/Contamination. All OC dimensions were also co-heritable indicating that they share some genetic influences. However, the degree to which genetic effects influenced the variance shared by dimensions (Ac) rather than the dimensions individually (As) varied by dimension. For many OC dimensions, shared effects (Ac) accounted for genetic influences with the other dimensions, suggesting that similar genetic factors played an important role across phenotypically separate, but correlated, OC dimensions. By contrast, what made the OC dimensions different was accounted for mostly by unique environment (Es) rather than genetic factors (As). A notable exception was Hoarding that had considerable genetic effects that were shared with the other dimensions (Ac) but also had considerable Hoarding-specific genetic influences (As). A similar pattern was observed for Superstition. Unlike our study, a previous study of female adults16. An important difference seems to be the number of OC dimensions captured by the scale they used—an independent model fits best when there were more dimensions15, whereas a common model fit best when there were fewer dimensions16. Our finding that shared etiological factors contribute to OC dimensions in youth without being mediated by a latent trait suggests that simply measuring overall OC traits will not uncover the full spectrum of genetic influences on OC dimensions and that OC traits are heterogeneous.The finding that an independent model best explained the shared variance between dimensions highlights both shared and unique etiological influences. If a common pathway model had fit best, shared etiology of the OC dimensions would have been attributable to a latent trait . If that were true, it would be necessary to calculate a latent trait for genetic studies rather than studying individual dimensions. OC dimensions can be highly correlated and still fit an independent model. Both independent and common models demonstrate that the co-variance between dimensions can be accounted for in part by shared genetic factors. However, in the independent model, those shared genetic factors influence each dimension directly, whereas in the common model these shared genetic factors are so closely related that they influence each dimension through a common factor. Results from previous studies in adults on the fit of the common and independent pathway for OC traits are mixed18. In our study, Hoarding did not correlate as well phenotypically with the other dimensions or the TOCS total score but was genetically correlated with all other OC trait dimensions. Hoarding also differed from the other dimensions in that genetic influences contributed significantly to both shared and dimension-specific variance. Excluding Hoarding from the independent pathway model did not affect model fit, suggesting that Hoarding was not obscuring a latent OC trait that accounted for the other OC dimensions. A study of adult twins also showed that hoarding had both specific and unique genetic effects and that an independent model fit the data best with or without the hoarding dimension15. In another previous adult twin study, hoarding and total OC symptoms were not highly genetically correlated but did share additive genetic effects20. Probable hoarding disorder and OC symptoms also shared genetic effects in another sample of adult twins19. Classifying hoarding as a distinct condition is useful in the clinic and may help in identifying hoarding-specific mechanisms. However, our results and others indicate hoarding shares considerable genetic risk with other OC trait dimensions. Disorders may share their genetic etiology even when phenotypically and clinically distinct40.In the DSM-5, hoarding is considered both a possible symptom of OCD and a symptom of a distinct hoarding disorderWe conclude that the TOCS is useful for studying both an overall OC trait and individual dimensions. This measure identified several heritable OC dimensions similar to those in previous studies. OC dimensions were correlated to a significant extent, but appeared to have different etiological mechanisms. Even less phenotypically correlated dimensions shared genetic risk. We are currently working on a genome-wide association study of OC traits in this sample to uncover whether genetic variants are associated similarly across OC dimensions. Specifically, we will test if Hoarding and Cleaning/Contamination are differentially associated with genetic variants. To uncover the genetics of OC traits and OCD, OC trait dimensions should be considered both individually and together.Supplemental Material


Researchers have increasingly employed family-based or longitudinal study designs to survey the roles of the human microbiota on diverse host traits of interest . Such study designs are useful to properly control for potential confounders or the sensitive changes in microbial composition and host traits. However, downstream data analysis is challenging because the measurements within clusters  tend to be correlated so that statistical methods based on the independence assumption cannot be used. For the correlated microbiome studies, a distance-based kernel association test based on the linear mixed model, namely, correlated sequence kernel association test (cSKAT), has recently been introduced. cSKAT models the microbial community using an ecological distance , and then tests its association with a host trait. Similar to prior distance-based kernel association tests , the use of ecological distances gives a high power to cSKAT. However, cSKAT is limited to handling Gaussian traits  and a single chosen distance measure at a time. The power of cSKAT differs a lot by which distance measure is used. However, choosing an optimal distance measure is challenging because of the unknown nature of the true association. Here, we introduce a distance-based kernel association test based on the generalized linear mixed model (GLMM), namely, GLMM-MiRKAT, to handle diverse types of traits, such as Gaussian , Binomial  or Poisson  traits. We further propose a data-driven adaptive test of GLMM-MiRKAT, namely, aGLMM-MiRKAT, so as to avoid the need to choose the optimal distance measure. Our extensive simulations demonstrate that aGLMM-MiRKAT is robustly powerful while correctly controlling type I error rates. We apply aGLMM-MiRKAT to real familial and longitudinal microbiome data, where we discover significant disparity in microbial community composition by BMI status and the frequency of antibiotic use. In summary, aGLMM-MiRKAT is a useful analytical tool with its broad applicability to diverse types of traits, robust power and valid statistical inference. The recent surge in next-generation sequencing technologies has dramatically advanced the human microbiome studies by enabling generic characterization of the microbes in the human body  In Materials and Methods, we address methodological details. (2) In Simulation, we address extensive simulation experiments. (3) In Real data applications, we apply aGLMM-MiRKAT to real familial and longitudinal microbiome data sets, where we test the association of the microbial community composition with BMI and the frequency of antibiotic use, while making interesting testing attempts and interpretations. (4) In Discussion, we finish with discussion and concluding remarks.yij denote a host trait of interest  for the j-th measurement in the i-th cluster , zijk denote the abundance level of the k-th OTU among p OTUs in the microbial community , and xijl denote a covariate among q covariates  that we want to adjust for . We also let N denote the total number of measurements  is a canonical link function  and μij = E(yij). α = xij = i is the random effect for the pre-specified sij to account for the within-cluster correlation in responses , yij are independent with a diagonal variance-covariance matrix sij = 1, υi is the random intercept which is assumed to follow a normal distribution N is the random intercept and slope which are assumed to follow normal distributions υi1 ~ N is a function which features the microbiome effect.where H0: h(zij) = 0  and, notably, with different specifications for h(zij), we can characterize different association patterns between microbial composition and a host trait. One may specify h(zij) as a fixed effect using a linear or non-linear function for the OTUs. For example, we can specify h(zij) = β, where φ(·) is an element-wise transformation  function and β = p OTUs, and then test H0: β = 0 using a p-degrees of freedom test. However, because of the high-dimensional nature of the data  and, for example, the resulting issue of low-rank matrices, testing H0: β = 0 with fixed effects might be challenging or even impossible. Therefore, we apply the kernel trick  = zij = p × 1 vector for the p OTUs and ωij's are coefficients; as such, h(·) lies in a reproducing kernel Hilbert space spanned by κ. Then, via the connection between kernel machine regression and mixed effect models  with matrix forms for all the measurements across all the clusters (Equation 2).To address details on the kernel matrix μ = N × 1 vector, α = q+1) × 1 vector, X = N × (q+1) matrix, γ =  is an N × 1 vector, and δN × 1 vector. Again, δ is assumed to follow a distribution with mean zero and variance-covariance matrix τK. We further assume that the two random effects γ and δ are independent as in  externally models ecologically meaningful pairwise similarities (correlation) in microbial composition among all the measurements across all the clusters, where the block-diagonals , , ,  model the within-cluster similarities while the off-diagonals model the between-cluster similarities. The extent of OTU abundance and phylogenetic information is properly modulated by different ecological distance measures . Following Lin, , the var to Lin, .y*= y-diag is the variance function. This test statistic (Equation 4) is the penalized quasi-likelihood estimating equation in Breslow and Clayton . Hence, we use a reduced-form statistic (Equation 5).where  Clayton  and the  Clayton  under th Clayton . Similar Clayton , while ap-value from multiple item-by-item GLMM-MiRKAT analyses (Equation 6).The testing performance depends on the choice of distance measure , where Γ = {Jaccard dissimilarity, Bray-Curtis dissimilarity, Unweighted UniFrac distance, Generalized UniFrac distance (θ = 0.5), Weighted UniFrac distance}. Obviously, we do not report the genuine minimum p-value  as it is. Instead, TaGLMMMiKAT (Equation 6) is the test statistic of aGLMM-MiRKAT, and we estimate the p-value for aGLMM-MiRKAT (PaGLMMMiKAT) using a permutation approach . Our extensive simulations reveal that aGLMM-MiRKAT maintains high power throughout all surveyed association scenarios, while the item-by-item GLMM-MiRKAT analyses are limitedly powerful only for some association scenarios. Further details are addressed in the Simulation section.where p-values for the item-by-item GLMM-MiRKAT tests and aGLMM-MiRKAT using a permutation approach. Our permutation approach is semi-parametric as we fit the null model g], we permute (1) the whole clusters (only the exchangeable clusters which have the same number of measurements) and (2) the measurements within each cluster, simultaneously. For the random slope model  . It is n (kgm2)] .p-value: < 0.001), Bray-Curtis dissimilarity , unweighted UniFrac distance  or generalized UniFrac distance (θ = 0.5)  estimates significant association between BMI and microbial composition, while GLMM-MiRKAT using weighted UniFrac distance  does not. This matches with our visual inspection of the smallest separation for the weighted UniFrac distance .We fitted GLMM-MiRKAT with random intercepts for BMI in continuous scale (Gaussian traits) adjusting for age. GLMM-MiRKAT using Jaccard dissimilarity , Bray-Curtis dissimilarity , unweighted UniFrac distance , generalized UniFrac distance (θ = 0.5) , weighted UniFrac distance ] or adaptive  analysis. This power loss, of course, is related to the reduced sample size in the selected comparison. This may also indicate that BMI in continuous scale is better informed than BMI in binary scale, which matches with our simulation result, where the Gaussian models are more powerful than the Binomial models  adjusting for age, comparing the normal and obese populations . However, we could not find any significant association by any item-by-item . We also found the significant association for aGLMM-MiRKAT .We fitted GLMM-MiRKAT with random intercepts for the number of antibiotic use (Poisson traits)  of antibiotic use) adjusting for gender. We found significant association between the number of antibiotic use and microbial composition by all the item-by-item analysis [i.e., Jaccard dissimilarity  microbiome studies. GLMM-MiRKAT can relate microbial community composition with any type of host traits that are distributed as an exponential family distribution. Thus, GLMM-MiRKAT can be regarded as an extension of cSKAT  within-cluster correlation structure. Therefore, for the random intercept model, our permutation approach works in any study design with either balanced or unbalanced numbers of measurements per cluster. However, for random intercept model, we permute only the whole exchangeable clusters. Therefore, for the random slope model, our permutation approach is limited to the balanced study design with a sufficient number of whole exchangeable clusters. In practice, the random intercept model has been more widely used for many prior tests . However, without loss of generality, the methods can be applied to any other microbial communities, such as the kingdom of yeasts, fungi or viruses, or the lower level microbial assemblages  (Koh et al., HK, NZ, and YL developed the method. HK performed the simulation experiments and real data analyses, and developed the software package. NZ, XZ, and JC contributed to simulations and real data analyses. HK and NZ wrote the manuscript. All authors read and approved the final manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Sparse Cholesky factorIzation LMM (Sci-LMM), a modeling framework for studying population-scale family trees that combines techniques from the animal and plant breeding literature and from human genetics literature. The proposed framework can construct a matrix of relationships between trillions of pairs of individuals and fit the corresponding LMM in several hours. We demonstrate the capabilities of Sci-LMM via simulation studies and by estimating the heritability of longevity and of reproductive fitness (quantified via number of children) in a large pedigree spanning millions of individuals and over five centuries of human history. Sci-LMM provides a unified framework for investigating the epidemiological history of human populations via genealogical records.The rapid digitization of genealogical and medical records enables the assembly of extremely large pedigree records spanning millions of individuals and trillions of pairs of relatives. Such pedigrees provide the opportunity to investigate the sociological and epidemiological history of human populations in scales much larger than previously possible. Linear mixed models (LMMs) are routinely used to analyze extremely large animal and plant pedigrees for the purposes of selective breeding. However, LMMs have not been previously applied to analyze population-scale human family trees. Here, we present  The advent of online genealogy services allows the assembly of population-scale family trees, spanning millions of individuals and centuries of human history. Such datasets enable answering genetic epidemiology questions on unprecedented scales. Here we present Sci-LMM, a pedigree analysis framework that combines techniques from animal and plant breeding research and from human genetics research for large-scale pedigree analysis. We apply Sci-LMM to analyze population-scale human genealogical records, spanning trillions of relationships. We have made both Sci-LMM and an anonymized dataset of millions of individuals freely available to download, making the analysis of population-scale human family trees widely accessible to the research community. Together, these resources allow researchers to investigate genetic and epidemiological questions on an unprecedented scale. Genealogical records can reflect social and cultural structures, and record the flow of genetic material throughout history. In recent years, very large pedigree records have come into existence, owing to collaborative digitization of large genealogical records ,2 and toTraditional human pedigree studies collect a large number of independent families which are analyzed separately and then meta-analyzed. However, this approach is not suitable for population-scale pedigrees, because such pedigrees cannot be decomposed into mutually exclusive families . Hence, Pedigree studies often employ LMMs to decompose the phenotypic variation among individuals into variance components such as genetic effects and shared environment . LMMs haLMMs and their extensions have recently gained considerable popularity in human genetics studies for the purposes of estimating heritability –32 and gIn recent years, animal and human studies have been using different techniques to scale LMMs to datasets with millions of individuals. Animal studies typically fit large-scale LMMs via restricted maximum likelihood (REML) , by explHuman genotyping studies do not give rise to sparse data structures. Instead, human studies have managed to scale LMMs to large datasets via two approaches. The first approach applies REML, either via supercomputers with thousands of CPUs and terabytes of memory , or by aThe second approach to scale LMMs uses the method of moments rather than REML, by solving a set of second moment matching equations –53. SuchHere we present Sci-LMM, a statistical framework for analyzing population-size pedigrees that combines techniques from animal and human genetic studies. Sci-LMM uses sparse data structures as is common in animal studies, and supports both moment and REML estimators. The moment estimator is based on a common technique called Haseman-Elston (HE) regression ,66 , using a large cohort spanning millions of genealogical records and several centuries of human history. We estimate that both traits have a substantial heritable component, with an estimated 22.1% heritability for longevity and 34.4% for reproductive fitness. Sci-LMM enables analysis of large pedigree records that was not previously possible.n individuals with observed phenotypes y1,…,yn, and covariates vectors C1,…,Cn, and consider a set of n×n covariance matrices M1,…,Md, where i and j according to the kth covariance structure, up to a scaling constant. We assume that the vector y = T follows a multivariate normal distribution:Consider a sample of C = T is an n×c matrix of covariates (including an intercept), β is a c×1 vector of fixed effects, Σ is the covariance matrix of the vector y, th variance component, and I is the identity matrix. The parameters to estimate are the fixed effects β and the variance components Here, c is the number of covariates.The restricted log-likelihood given by :lR any product of the form Σ−1v for some vector v is computed using L and two triangular solvers (forward and backward substitution); and (4) the gradient of Our solution combines several ideas: (1) we maximize Σ−1(y−Cβ) and log|Σ|. The first term can be computed exactly via either conjugate gradient iterations or by explicitly computing the Cholesky factorization of Σ and then applying forward and back substitution. The second term can be computed via the Cholesky factorization of Σ. The Cholesky factorization can be computed efficiently via the CHOLMOD routines TY can also be computed efficiently in a similar manner.Typically, the fixed effects nditions . Afterwaq≜TY, S = T, we have:By following the notation of  and denoq and S efficiently as follows:By applying a few matrix manipulations, we can compute S−1var(q)S−1, where var(q) is given by:The sampling variance of the estimators is given by Exactly, via: Approximately, via: where y′ is sampled from This quantity can be computed in two ways:y′ vectors and approximating the right hand side. It can be substantially faster than the exact approach (because it circumvents expensive matrix-matrix multiplications) and obtain excellent accuracy. The Sci-LMM software uses the approximate approach by sampling ~100 random y′ vectors.The approximate approach uses Monte-Carlo approximations, by randomly sampling HE regression provides a simple technique for excluding specific pairs of individuals (e.g. spouses) from the analysis without excluding the individuals themselves. This can be useful when trying to limit confounding due to factors such as assortative mating. Excluded pairs can be omitted by zeroing the covariance matrix entries of corresponding pairs. Importantly, this technique cannot be used in REML, because the resulting covariance matrices may not be positive definite. We note that another potential approach to capture environmental risk factors is including shared effects with a suitable incidence matrix , but thiHE regression is a convenient theoretical framework to analyze the factors affecting estimation accuracy. HE regression can be considered as a special form of linear regression, where off-diagonal entries of covariance matrices serve as explanatory variables. Hence, good accuracy is obtained when measured and unmeasured explanatory variables are uncorrelated with other explanatory variables .Specifically, obtaining accurate estimates requires (1) that the off-diagonal entries of the LMM covariance matrices are uncorrelated with each other; and (2) that they are uncorrelated with covariance due to unmeasured environmental factors. While the first requirement can be easily tested, the second one requires making strong assumptions about the structure of environmental covariance. For example, if latent environmental factors are shared between spouses but not between parents and children, we may wish to exclude spouses from the analysis. Unfortunately, we not know the structure of environmental covariance for the traits studied in this work, and we leave its investigation for future work.aij, is the probability that a randomly selected allele in an autosomal locus was originated from the same chromosome of a shared ancestor between individuals i and j [The IBD kinship coefficient of two individuals, denoted as  i and j ,86, and fi is the inbreeding coefficient, defined as half of the IBD coefficient of the parents of individual i [rij is the coefficient of relationships, defined as:Here, vidual i , and rijA is a least common ancestor of individual i and j in the pedigree graph ; the summation is performed over every path connecting individuals i,j in the pedigree graph, culminating at some ancestor A, such that the path does not contain the same individual twice; and |path| is the path length.The quantities in the above equation are defined as follows: L and H of its decomposition A = LHLT, where L is a lower triangular matrix such that Lij contains the fraction of genome shared between individuals i and her ancestor j, H is diagonal, and the matrices are ordered such that ancestors precede their descendants 2 for two-loci epistasis, 3 for three-loci epistasis and so on [Epistatic covariance encodes the assumption that variants interact multiplicatively to affect a given phenotype, and is proportional to the exponent of the corresponding IBD coefficient, i.e.,  of the IBD matrix as fixed effects, using sparse matrix routines . The incWe generated pedigrees mimicking real family patterns in the United States, partially based on publications by the United States Census Bureau ,93. We iIn each generation we generated households, where every household includes either one individual or two individuals with different genders, and every individual can belong to zero, one or multiple households. The number of households in each generation was 62.5% of the number of individuals in that generation. 68% of the households included pairs of individuals, and the rest included a single individual. Every individual in every generation (except for the top one) was born to parents from a randomly selected household from the previous generation  or from two generations in the past .After generating all individuals, we omitted randomly selected edges until obtaining the desired sparsity factor, up to 10% error. We then created corresponding IBD, dominance and epistasis matrices.Mk from U and scaling them such that they sum to 1.0; (2) Generating 5 binary and 5 normally distributed covariates; and (3) generating fixed effects from n is the sample size.Finally, we generated phenotypes using The parameters differentiating the various experiments are: (1) cohort size ; (2) sparsity factor ; and (3) the subset of matrices used. We generated 10 different datasets for every unique combination of settings, except for matrices with 2M individuals, for which we generated a single pedigree with ten different phenotype vectors due to runtime considerations.All experiments were conducted using a Linux workstation with a 24-cores 2GHz Xeon E5 processor and 256Gb of RAM.To evaluate the capabilities of Sci-LMM, we generated large synthetic pedigrees spanning 20 to 40 generations and various family structures, under a wide variety of settings. The pedigrees included 50,000–2,000,000 individuals, amounting to trillions of pairs of relatives. A subset of the individuals in each generation consists of children of individuals from either the previous generation, or from two generations in the past. To simulate patterns observed in real datasets, the simulations also included consanguinity, half-siblings, and individuals with less than two recorded parents (Methods).In each simulation we generated a normally distributed phenotype, using a covariance matrix with additive, epistatic and dominance effects and ten binary covariates. Unless otherwise stated, the sparsity factor (the fraction of non-zero entries in each matrix) was 0.001. Ten different datasets were generated for each combination of sample size and sparsity factor.In all settings, Sci-LMM yielded empirically unbiased estimates of the variance components, using both REML and HE regression. As expected, estimation accuracy increased with sample size, though the estimators became slightly less accurate when increasing the number of variance components, . SpecifiA comparison of the REML and the HE results shows that that HE was slightly more accurate in the presence of <100,000 individuals , and REM11 pairs of possible relatives and a sparsity factor of 0.001.We evaluated the time Sci-LMM requires to construct covariance matrices. The covariance matrices computation is dominated by the IBD matrix construction, because the other matrices can be computed trivially given this matrix (Methods). The IBD matrix construction scaled linearly with the number of non-zero entries in the matrix . For exaWe next investigated the runtime for variance component estimation using REML and HE. REML estimation for samples with 500,000 individuals (representing 250 million covariance entries) required less than 24 hours , whereasfitNullModel function of the GENESIS package [lmekin function of the coxme package [We also compared Sci-LMM with several REML software packages –72. We c package  and the  package , both ofFinally, we compared Sci-LMM to WOMBAT in the presence of only an additive covariance matrix. WOMBAT is more computationally efficient in this setting because it uses a mixed model equations (MME) solver . MME solWe used Sci-LMM to estimate the heritability of longevity and reproductive fitness, based on large-scale pedigree records obtained from the Geni genealogical website  . The dataset included 9.7 million pairs of individuals with a kinship coefficient corresponding to a > = 20-degree relationship . Sci-LMMNext, we used Sci-LMM to estimate the heritability of longevity with covariates encoding gender, year of birth (yob), yob raised to second and third power, and the top 10 principal components of the IBD matrix, and with covariance matrices encoding IBD and pairwise epistasis (Methods). A dominance matrix was not included because the analysis included a relatively small number of full-sibs or double first cousins, rendering this matrix almost equivalent to the identity matrix .The Sci-LMM REML estimates were: IBD: 22.1% (s.e. 0.8%); pair-wise epistasis: 0.001% (s.e. 1.7%); environmental effects: 74.4% (s.e. 1.0%) . The HE We next estimated the heritability of reproductive fitness, quantified by number of children. To limit confounding due to non-genetic factors, we applied stringent filtering of individuals. We removed individuals with less than two children records, because the family records of such individuals are more likely to be incomplete. We additionally removed individuals with a shared household (spouses and children) and individuals who are first- or second-degree relatives of another individual in the data set. The filtered dataset included ~45,000 individuals. We used the same covariates as before, applied a Box-cox transformation to induce normality for number of children, and excluded epistatic interactions from this reduced dataset, because they led to large standard errors.The REML and HE estimated heritability of reproductive fitness were 28.4% (s.e. 0.5%) and 34.4% (s.e. 1.2%), respectively. These results indicate a substantial genetic component for reproductive fitness, in line with previous work . HoweverFinally, we performed a series of experiments to examine the robustness of our longevity analysis to potential confounding factors . First, We have described a statistical framework for analysis of large pedigree records spanning millions of individuals. Our framework includes methodologies for constructing large sparse matrices given raw pedigree data, and methodologies for LMM analysis with random effects described by these matrices. Taken together, the proposed solution enables an end-to-end analysis of population scale human family trees.In this work we focused on partitioning phenotypic variance into additive genetics, epistasis and dominance. However, the LMM framework is flexible and can be extended in various directions. For example, sparse LMMs are often used to model transmissible phenomena –104, whiWe evaluated two methods for variance components estimation: REML and HE regression. REML is more accurate than HE and provides a likelihood-based solution, which can be used for model comparison and hypothesis testing. HE estimates are less accurate but can be more robust to modeling violation. Importantly, HE regression can mitigate confounding due to environmental factors by zeroing selected entries in the covariance matrices, which may be especially suitable for studying human genealogical records (Methods). Hence, the two methods are complementary in terms of their strengths and weaknesses. In practice, we found that it is difficult to scale REML to datasets with more than 500,000 individuals with a sparsity factor of 0.001. Our recommendation is to use REML when it is feasible and all model assumptions hold, and HE regression otherwise. We note that REML estimation can be substantially faster when not fitting epistatic interactions by using a mixed model equations approach , which iOur work demonstrates the technical feasibility of studying population scale human family trees. However, the analysis of human genealogical records is challenging due to imperfect data and the difficulty of controlling for confounding factors. Potential issues include non-paternity, cryptic relatedness, missing or false genealogical records, genetic nurture ,106, envIn this work we focused on analyzing large pedigree records with no measured genotypes. In recent years, the advent of biobank-sized datasets allows analyzing population-scale genotyped cohorts. The two study types are complementary because biobanks cannot be used to investigate longevity, traits with a late age of onset, or epidemiological and sociological questions on historical scales. We anticipate that cohorts combining both types of data will become increasingly common. For example, we and other online genealogy platforms allow users to upload their genetic information and link it with their genealogical profile. Such combined datasets have been extensively explored in the animal breeding literature ,108–112.S1 Text(PDF)Click here for additional data file.S1 TableBoth tools used only 1 CPU thread, and WOMBAT was executed with the—meuwissen flag. The estimated values were essentially the same for both tools in all cases. WOMBAT and Sci-LMM REML both crashed in the presence of pedigrees with ≥500,000 individuals.(PDF)Click here for additional data file.S2 TableThe epistasis estimates were very close to zero in all cases and are omitted for clarity.(PDF)Click here for additional data file.S1 FigNodes represent individuals, and edges represent parent-child relations. Only red individuals have full information records (TIF)Click here for additional data file.S2 Fig(TIF)Click here for additional data file.


In the pig production industry, artificial insemination (AI) plays an important role in enlarging the beneficial impact of elite boars. Understanding the genetic architecture and detecting genetic markers associated with semen traits can help in improving genetic selection for such traits and accelerate genetic progress. In this study, we utilized a weighted single-step genome-wide association study (wssGWAS) procedure to detect genetic regions and further candidate genes associated with semen traits in a Duroc boar population. Overall, the full pedigree consists of 5284 pigs (12 generations), of which 2693 boars have semen data  and 1733 pigs were genotyped with 50 K single nucleotide polymorphism (SNP) array.TDRD5, QSOX1, BLK, TIMP3, THRA, CSF3, and ZPBP1; Sperm motility, PPP2R2B, NEK2, NDRG, ADAM7, SKP2, and RNASET2; Sperm progressive motility, SH2B1, BLK, LAMB1, VPS4A, SPAG9, LCN2, and DNM1; Total morphological abnormalities, GHR, SELENOP, SLC16A5, SLC9A3R1, and DNAI2.Results show that the most significant genetic regions (0.4 Mb windows) explained approximately 2%~ 6% of the total genetic variances for the studied traits. Totally, the identified significant windows  explained 28.29, 35.31, 41.98, and 20.60% of genetic variances (not phenotypic variance) for number of sperm cells, sperm motility, sperm progressive motility, and total morphological abnormalities, respectively. Several genes that have been previously reported to be associated with mammal spermiogenesis, testes functioning, and male fertility were detected and treated as candidate genes for the traits of interest: Number of sperm cells, In conclusion, candidate genes associated with Duroc boars’ semen traits, including the number of sperm cells, sperm motility, sperm progressive motility, and total morphological abnormalities, were identified using wssGWAS. KEGG and GO enrichment analysis indicate that the identified candidate genes were enriched in biological processes and functional terms may be involved into spermiogenesis, testes functioning, and male fertility. In the pig production industry, artificial insemination (AI) plays an important role in enlarging the beneficial impact of elite boars. Sperm quality, affected by genetic and environmental factors simultaneously and with moderate to low heritability , 2, is eH , Zwindow and gwindow represent genotypes and marker effects of SNPs located in the window, respectively).With the fast development of sequencing technology and commercially availability of dense marker panels, researchers can precisely identify quantitative trait loci (QTL) by searching for association between genetic markers and phenotypic records, which is known as genome-wide association study (GWAS) . Though riances, , 8) as rIn several previous studies related to boar semen traits, associated genetic markers or candidate genes have been reported for both sperm quantity (i.e. number of sperm cells per ejaculation) and quality  traits, –13. HoweIn this study, we identified genomic regions associated with semen traits in a Duroc boar population via wssGWAS . In ordeFigure positive regulation of myeloid cell differentiation (GO:0045639), steroid hormone mediated signaling pathway (GO:0043401), granulocyte colony-stimulating factor receptor binding (GO:0005130), and enzyme regulator activity (GO:0030234). The enriched KEGG pathway is involved in thyroid hormone signaling (ssc04919). For sperm motility, three GO terms were enriched for the candidate genes, while no KEGG pathway was targeted. The GO terms play roles in cellular calcium ion homeostasis (GO:0006874), protein phosphorylation (GO:0006468), and protein kinase activity (GO:0004672). For progressive motility, seven GO terms and two KEGG pathways were enriched for the identified candidate genes. These GO terms and KEGG pathways are involved in nucleotide metabolic and biosynthesis processes. Three GO terms were enriched for the total morphological abnormalities, while no KEGG pathway was targeted. These GO terms are related to transition between slow and fast fiber (GO:0014886), sensory perception of sound (GO:0007605), endochondral ossification (GO:0001958), cullin family protein binding (GO:0097602), RNA polymerase II transcription factor activity, and sequence-specific DNA binding (GO:0000981).Table DNAI2 was previously reported to be associated with sperm motility of large white boars [In this study, we conducted association study for boar semen traits in a large Duroc population. To our knowledge, this is the first time that genome-wide association study results are reported in such a large Duroc boar population. Based on the literature (cited above), some of the identified candidate genes have been previously reported to be involved in mammal spermiogenesis, testes functioning, and male fertility. Among the candidate genes identified in this study, te boars , while nte boars , where wte boars . Therefote boars  by assigSus scrofa chromosome), 3, 5, 9, 10, 12, 14, 15, 16, and 17 were identified. Sixty-five genes were annotated in these genomic regions according to the National Center for Biotechnology Information (NCBI)  and quiescin Q6/sulfhydryl oxidase 1 (QSOX1), located in the region of 121.15–121.95 Mb on SSC9, were considered as candidate gene for the number of sperm cells , among which TIMP3 was identified in the region of 11.71–12.51 Mb on SSC5, were reported to exist in human seminal plasma as complexes and may play roles in sperm function [thyroid hormone receptor alpha (THRA), identified in the region of 22.19–22.56 Mb on SSC12, is responsible for sertoli cell development, sperm production, and testis weight [Colony stimulating factor 3 (CSF3), another gene located in the region of 22.19–22.56 Mb on SSC12, is involved in reactive oxygen species (ROS) of sperm cells and associated with abnormalities in sperm histone transition [Zona pellucida binding protein 1 (ZPBP1), a spermatid and spermatozoon protein, is responsible for sperm morphology. Lin et al., (2007) reported that male mice lacking ZPBP1 were sterile, with abnormal round-headed sperm morphology and no forward sperm motility.For number of sperm cells, 18 relevant QTL regions located on SSC2 (SSC: function . The thys weight  by regulphosphatase 2 regulatory subunit B beta gene (PPP2R2B), located in the region of 147.71–148.51 Mb on SSC2, was indicated to be associated with biology of sperm in a Co-Expression Network Analysis conducted in Bos taurus and Sus scrofa [NIMA related kinase 2 (NEK2) gene, located in the region of 131.55–132.35 Mb on SSC9 and showed biased expression in reproductive tissues (testis of human and ovary of swine), several studies have demonstrated its role in the process of meiotic cell cycle and the first meiotic division in mouse spermatocytes [Ndrp3, a member of the N-myc downstream regulated gene family (NDRG), is enhanced expressed specifically in germ cells and involved in regulation of the male meiosis in mouse [ADAM7), which located in the region of 8.49–8.86 Mb on SSC14, is associated with epididymosomes, integrated into sperm plasma membrane, and related to fertility and sperm integrity [SKP2) was identified in the region of 21.48–21.86 Mb on SSC16. The Skp2 protein is the receptor subunit of an SCF-type ubiquitin ligase and is a major regulator of the progression of cells into S phase of the cell cycle, Skp2-deficient mice shown reduction in the number of mature gametes and thus decreased fertility [Skp2 was reported to be involved in the manipulation of boar sperm production [RNASET2) gene, located in the region of 1.77–2.16 Mb on SSC1, have been reported to be involved in sperm motility impairment [AKAP4 in human sperm tail and subsequently reduced sperm motility by suppressing PKA/PI3K/calcium signaling pathways [For sperm motility, 20 relevant QTL regions located on SSC1, 2, 3, 4, 7, 8, 9, 12, 13, 14, 15, 16, and 17 were identified. Fifty-nine genes were annotated in these genomic regions  was identified. Previous study in mice have shown that expression of Sh2b1 on the mRNA level in obesity mice was decreased, which resulted in significant reduction in male fertility and exacerbated reproductive toxicity and germ cell mutagenicity [BLK), also identified for the number of sperm cells, is an Src-related tyrosine kinases in the events leading to proper sperm formation, was reported to potentially involved in sperm function and integrity [LAMB1) gene, which directs the expression of beta-galactosidase during development of the mouse testis and ovary [VPS4A) gene, located in the region of 17.44–17.8 Mb on SSC6, was previously reported to be involved in human sperm acrosome reaction [SPAG9) gene, which codes an acrosomal molecule that plays role in acrosome reaction and sperm-egg binding [LCN2) gene, located in the region of 268.37–268.73 Mb on SSC1, is involved in sperm maturation and fertilization [DNM1), another gene located in the same region on SSC1, acts as a key mediator of sperm acrosome formation and function, is essential for mammalian spermatogenesis [For progressive motility, 24 relevant QTL regions located on SSC1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, and 17 were identified  gene, which shows significant association with embryos survival rate and involved in fertilization rate of dairy cattle [SELENOP), another gene located in the same region, is abundantly expressed in the testis and affected the sperm quality of rats [SLC16A5) gene, for which a protein-coding variation is associated with cisplatin-induced ototoxic effects and involved in human germ cell testicular cancer [SLC9A3R1), another gene located in the same region, is important for mouse sperm capacitation through functional interactions with SLC26A3, SLC26A6, and CFTR [DNAI2) gene, of which mutations may resulted in defects of sperm flagella in human [For total morphological abnormalities, 16 relevant QTL regions located on SSC1, 3, 4, 8, 9, 10, 11, 12, 13, 15, 16, and 17 were identified  in the literature provided evidence of role of the identified genes in semen traits. The identified candidate genes may provide knowledge for developing genomic evaluation models by assigning higher weights to genetic markers located in these regions.6 mL− 1). For the traits of total morphological abnormalities, number of sperm cells, and motility, 143,113 individual ejaculates (53 ejaculates per boar on average) of 2693 boars were used for the association study . SNPs with unknown positions (9405 SNPs) and that unable to map to Sscrofa11.1 (1842 SNPs) were removed from the dataset. Moreover, SNPs on sex chromosomes (2469 SNPs) were excluded to avoid sex bias in constructing genomic relationship matrix and calculating genetic breeding values (1231 boars and 392 sows were included in the genotype file and used for constructing the genomic relationship matrix). Autosome SNPs were filtered via plink [p-value ≥10−6 (2680 SNPs removed). After quality control, 1623 pigs, among which 1231 had semen evaluation data, and 28,289 SNPs remained for further analysis. Missing genotypes were finally imputed using Beagle software (version 4.1) [Total DNA of 1733 pigs were extracted (semen of boars and ear tissue samples of sows) using the genome extraction kits produced by Wuhan NanoMagBio Technology Co., Ltd. Those pigs were genotyped using the GGP 50 k SNP array , which contains 50,703 SNPs. Overall, 1627 out of the genotyped pigs were related to boars with phenotypic records through pedigree and used for further analysis. SNPs, for which the Sscrofa10.2 physical positions were provided by the SNP array producer, were remapped to Sscrofa11.1 using the genome remapping procedure available in NCBI (ia plink  using thion 4.1) , 49.The weighted single step GWAS procedure , 50 was y denoted the response; X, Z, and W denoted the design matrices; b was fixed effects ; Age and Intv denoted the age of the boars at months when ejaculating and collection interval (days), respectively. I denoted the identity matrix. The inverse of H matrix, a blend of pedigree and genetic marker derived matrices [A denoted the pedigree derived relationship matrix; A22 was a sub matrix of A corresponding to the genotyped individuals; Gω = 0.9G + 0.1A22, these weights were used to scale the genomic information to be compatible with the pedigree information and to control bias [Z was a matrix of genotypes ), D denoted a diagonal matrix contained the SNP weights, pi denoted MAF of the ith SNP, and m denoted the number of SNPs.The following repeatability model was used in this study.matrices , 8, was rol bias , 52; \dop matrix , where ZG were calculated in an iterative way used by Wang et al., [Dt)( = λZDt)( = λZDt + 1)  procedur et al., , 50. An The procedure was run for three iterations, as used in Wang et al.,  and MarqThe iteration steps mentioned above were run with BLUF90 software families  in a iteQTL regions were selected according to genetic variance of chromosome windows. Windows explained higher than 1% genetic variances were selected as candidate QTL regions, within which candidate genes were searched. The threshold of 1% was set based on literature , 57–59 ahttp://www.ncbi.nlm.nih.gov) were searched in the selected QTL regions. Kyoto Encyclopedia of Genes and Genomes (KEGG) [https://david.ncifcrf.gov/) [Candidate genes on “National Center for Biotechnology Information”   and Genes (KEGG)  enrichmerf.gov/) .Additional file 1: Table S1. Genomic regions explained more than 1% of genetic variance for the number of sperm cells.Additional file 2: Table S2. Genomic regions explained more than 1% of genetic variance for sperm motility.Additional file 3: Table S3. Genomic regions explained more than 1% of genetic variance for progressive motility.Additional file 4: Table S4. Genomic regions explained more than 1% of genetic variance for total morphological abnormalities.