Finally, our meta-analysis method is used in a small  study of interactions in myocardial infarction.A complex disease has, by definition, multiple genetic causes. In theory, these causes could be identified individually, but their identification will likely benefit from informed use of anticipated interactions between causes. In addition, characterizing and understanding interactions must be considered key to revealing the etiology of any complex disease. Large-scale collaborative efforts are now paving the way for comprehensive studies of interaction. As a consequence, there is a need for methods with a computational efficiency sufficient for modern data sets as well as for improvements of statistical accuracy and power. Another issue is that, currently, the relation between different methods for interaction inference is in many cases not transparent, complicating the comparison and interpretation of results between different interaction studies. In this paper we present computationally efficient tests of interaction for the complete family of generalized linear models (GLMs). The tests can be applied for inference of single or multiple interaction parameters, but we show, by simulation, that jointly testing the full set of interaction parameters yields superior power and control of false positive rate. Based on these tests we also describe how to combine results from multiple independent studies of interaction in a meta-analysis. We investigate the impact of several assumptions commonly made when modeling interactions. We also show that, across the important class of models with a full set of interaction parameters, jointly testing the interaction parameters yields identical results. Further, we apply our method to genetic data for cardiovascular disease. This allowed us to identify a putative interaction involved in Lp(a) plasma levels between two ‘tag’ variants in the LPA locus ( Interaction between organic molecules forms the basis of all biological systems. The availability of high-throughput genotyping and sequencing platforms enables us to cost-effectively genotype a large number of individuals. For sufficiently large datasets it is possible to reconstruct the genetic dependencies that underlie complex traits and diseases. However, there is a need for efficient statistical methodologies that can tackle the large sample size and computational resources required to study interaction. In this work we provide theory that reduces the required computational resources, and enable multiple research groups to effectively combine their results. Large data sets are vital to counter low statistical power due to low allele frequency, small effect sizes, and multiple testing. This has driven the GWAS field towards more collaborative efforts as well as meta-analyses. Fortunately, there exists a standardized statistical methodology that allows for reliability and comparability between different studies. In contrast, in association studies aiming at identifying interactions, or epistasis, there are multiple competing methodologies with unclear relationships. As a consequence, collaborative GWAS efforts have almost exclusively focused on single variant associations.Conceptually, interactions in association studies are generated when multiple genetic variants affect the dynamic, non-linear and inter-connected networks that underlie complex traits . CandidaFrom a statistical point of view it is non-trivial to define interaction  and it cMultiple tests for interaction have been proposed for case-control data. However, these tests typically depend on strong assumptions about the main effects, marginal effects or LD to reduce the computational complexity , 19, 20.A major complication in interpreting interactions is that they are inferred relative to a link function. This function determines the parameter subspace that belongs to the null model and is, in practice, unknown. Consequently, mis-specification of the link function causes an inflated error rate that increases with sample size, which cannot be resolved by replication in a separate cohort. Here we address this issue by testing interactions using a family of link functions. Specifically we use two families of link functions that has been proposed previously , 23. We We implement these new tests in a GLM-based analysis tool for both case-control and quantitative data. We investigate the impact of different parameterizations on both the false positive rate and the statistical power. We finally apply our Wald tests in two genome-wide interaction analyses. Firstly, we study a continuous phenotype, Lp(a), in the PROCARDIS cohort. Secondly, we perform a meta-analysis of myocardial infarction by combining results from the PROCARDIS cohort and the Myocardial Infarction Genetics Consortium cohort.In this subsection, we introduce the theory of GLMs and their application to genetic data. We then describe how a so-called parameterization can be used to incorporate genetic assumptions about a single variant. Finally, we describe how a multi-variant parameterization can be constructed from several single variant parameterizations.i, Yi is a measured phenotype and xi is a vector of predictor variables. The observed phenotype yi is modeled by its expected value E[Yi ∣ Xi] = μi along with a distribution from the exponential family that captures the stochastic variation around μi. We write yi ∼ f(μi) and refer to f as the dispersion distribution. The expected value μi is in turn related to the linear predictorψ(xi)β by g(μi) = ψ(xi)β, where g is the link function, ψ is an encoding of the predictor variables, and β is a vector of parameters. A combination of ψ and β is called a parameterization. The parameters are generally estimated according to the maximum likelihood principle by applying the iteratively reweighted least squares algorithm. The commonly used linear regression is a special case of GLMs, obtained by using the identity link function (g(μi) = μi) and the Normal dispersion distribution. Moreover, logistic regression is a GLM with the logit link function  = phβ. Here h can be the genotype of either a single variant or multiple variants. Furthermore, we can define a design matrix P, so that ph are the rows of P; the complete relationship between the mean levels of the genotypes and the parameterization can then expressed g(μ) = Pβ.As our focus here is interactions, we restrict the predictor variables saturated model, otherwise it is called unsaturated. Additionally, we will call a model full if the number of unique design vectors equals the number of possible genotypes, otherwise it is called constrained. A design matrix that recodes genotypes into dominant and recessive states is an example of a constrained model.In general, we will call any model where the number of parameters equals the number of unique design vectors a Single-variant models. We first introduce a notation for the parameterization for a single variant and, in the next section, we extend this notation to multiple variants. We let β = T, where α is referred to as the reference level, and β1 and β2 are deviations from the reference . For example, if ph =  then g(μh) = α + 2β1. For the single variant case, let A be the most frequent haploid genotype in the population considered, and a be the less frequent. We have three possible diploid genotypes {AA, Aa, aa} that are enumerated by h ∈ {0, 1, 2}. We now describe the parameterizations corresponding to the most common single variant genetic models.aturatedfullparameterizationsG—Genotypic. Let β1 be the mean difference in phenotype between the reference and the heterozygote, and β2 be the mean difference in phenotype between the reference and the minor homozygote. Then the genotypic can be expressed as follows:AD—Additive/deviation. Letting β1 denote the additive component, and β2 denote the deviation from additivity, this can be expressed as follows:SaturatedconstrainedparameterizationsR—Recessive. This model assumes that the effect allele has a recessive effect:D—Dominant. This model assumes that the effect allele has a dominant effect:H—Heterozygote. This model assumes an heterozygote advantage effect:SnsaturatedfullparameterizationsA—Additive. This model assumes that only additive effects are present:UG alternatively could be perceived as either a Heterozygote/deviation, HD, or a Recessive/deviation, RD.We note that From single to multi-variant models. We will now show how a multi-variant parameterization with relevant interaction parameters can be obtained from a set of single variant parameterizations. This provides a intuitive way to construct higher order interaction parameterizations. Thereafter, we provide two examples of the construction of bi-variant models (see dels see .Pi)( ⊗ βk−1)( (i) the intercept has been replaced by 1 in all parameter vectors β(⋅) on the right hand side of ii) the first element of the resulting vector β on the left hand side of iii) any product between two different parameters is replaced by a new parameter that represents interaction. The number of factors in each element represents the order of interaction and factors of order 1 are the main effects. The matrix P and the parameters β are now related to the mean values byA multi-variant parameterization can be constructed from single variant parameterizations using the Kronecker-product (⊗) . Given aMi and Mj, of those enumerated above in Section Introduction to generalized linear models in genetics. We denote the resulting bi-variant model Mi × Mj. We detail the construction of G × G and AD × AD in We will below mainly focus on bi-variant models resulting from applying Eqs This concludes our description of the GLM components used to model genotype-phenotype association. Using this uniform GLM framework, we formulate, in the next section, efficient tests for interaction applicable to very general families of GLMs, allowing any number of variants and a wide variety of phenotypes and link functions. Moreover, in Section Relating different GLMs, we use this framework in a comparative analysis of major GLM families with respect to differences in parameterization, link function and dispersion distribution.In this section, we first derive closed-form estimates of the parameters of a GLM and the corresponding covariance matrix for any saturated parameterization. We then build on this to design an computationally efficient test for interaction. A complete discussion and proofs can be found in A general Wald test for saturated models. We first describe how to estimate the parameters. Let N be a diagonal matrix in which the diagonal element Nhh is the number of individuals with the genotype corresponding to h. Let t be a vector in which element th contains the sum of the phenotypic values of individuals with the genotype corresponding to h. Maximum likelihood estimates of β is then obtained byP exists because P is a full rank square matrix when the parameterization is saturated , so that the dominant time complexity is linear in the number of samples ated see . The timvh = Var, which can be calculated from the specific dispersion distribution that is used, μh), nh is the number of samples with genotype h, and ϕ is the dispersion parameter of the GLM . The score test requires estimation only under the null hypothesis, the Wald test requires estimation under the alternative hypothesis, and the standard likelihood ratio test under both. For saturated GLMs, gression .β under the null model of no interaction, for which no closed form expression exists. Hence, in this case, we need to resort to the slower iteratively reweighted least squares (IRLS) algorithm. The average time complexity for estimation and testing based on the IRLS algorithm is A(n) is the expected number of iterations until convergence. Since the time complexity for computing the actual test statistics is similar to that of the Wald test, these two tests will have a theoretical complexity that is dominated by a factor that is linear in n. The practical difference is shown in A × A model with linear or logistic link function.The score test and the LRT additionally requires estimation of Unsaturated models. For an unsaturated model the design matrix P is no longer full rank, and for parameter estimation one usually must resort to an iterative algorithm. We develop a two-step closed-form estimation, in which we first transform the model to a convenient saturated model , by extending P and β with corresponding columns and parameters, respectively. We estimate all parameters in this model, and then back-transform them to the unsaturated model.Pβ be the parameterization of the unsaturated model, and PSβS be the parameterization of a corresponding saturated model. For convenience of notation let βS = Sβ. Given an estimate of the covariance matrix CS of the saturated parameters we can estimate β and the corresponding covariance matrix C byCS and mean Sβ. Maximum likelihood estimation of β in this model leads to the equations above . We show in Let The time complexity involves estimating the saturated model, transforming these estimates into estimates for the unsaturated model, and computing the final Wald test statistic. This time complexity is bounded by the estimation of the saturated model, so the total time complexity is still Fixed effect meta-analysis of multivariate Wald tests. We now describe how to combine association results, based on the Wald test, from multiple studies to perform a meta-analysis. Let i and χ2-distribution with four degrees of freedom. Of note, each covariance matrix is commonly small and the time for computing the combined covariance matrix is shorter than the time for estimating each individual covariance matrix.A practical problem for large-scale meta-analysis is that the result files from the interaction analysis of each study will be very large, typically in the order of 1 Terabyte. We suggest one possible solution to this issue. The analysis can be split into two stages. In the first stage, each study reports all variant pairs below some p-value threshold. A reduced set of candidate variant pairs are then created by taking the union of the significant pairs over all studies. In the second stage, all studies perform a second analysis of the reduced set of variant pairs. Finally, meta-analysis is performed on the result from the second stage. This effectively limits the storage space required, but may miss variant pairs with intermediate effects in all studies.In this section, we demonstrate some aspects of how interaction models can be related to each other within the GLM framework.Linear reparameterizations of saturated models are equivalent. We first show that a certain class of parameterizations are equivalent with respect to the Wald test, providing some insight to when two parameterizations are identical in terms of the inference of epistasis.δ and δ′, respectively, are linear transformations of each other, i.e. δ′ = Bδ, then the corresponding Wald tests are equivalent,G × G and AD × AD parameterizations is equivalent. In general, any joint test of interaction for saturated reparameterizations, will have identical results. However, this is not true if parameterizations are non-linear transformations of each other, i.e. if they use different link functions.We start by demonstrating an important property of the Wald test; if two parameterizations with interaction parameters Of note, the degrees of freedom can be reduced if some of the genotypes contains very few samples. We can then avoid estimating the interaction parameters for these cells and adjust the degrees of freedom accordingly. This could, however, cause the GLMs to be non-equivalent in terms of the joint likelihood, and an interaction in one parameterization may not be reflected in another.Relating parameter estimates. We show above that two saturated full GLM reparameterizations are equivalent in terms of the joint Wald test. However, different saturated full parameterizations may emphasize different interaction components, which potentially reflect different biological mechanisms. We now investigate this further.Pβ and Qβ′ isβ are a function solely of the interaction parameters of β′. This is in contrast to the main effect parameters of β, which may depend on both main effect and interaction parameters of β′. The relationship between the interaction parameters of G × G and AD × AD is,G × G: δG×G = {0, 0, 0, 1}, but not in AD × AD: δAD×AD = {0.25, 0.5, 0.5, 0.25}. Conversely, a single additive-additive interaction is apparent in AD × AD: δAD×AD = {1, 0, 0, 0}, but not in G × G: δG×G = {1, 2, 2, 4}. We note that, for certain questions, exploring multiple parameterizations would be beneficial to allow the formulation of most-parsimonious constrained parameterizations, or to provide hypotheses on possible biological mechanisms for inferred interactions.The relationship between two saturated full parameterizations Pβ, to another parameterization, Qβ′, we find that the interaction parameters of β are no longer guaranteed to be a function of the β′ interaction parameters alone, but can depend on its main effects as well. More generally, constrained/non-saturated models reduce the number of interaction parameters  of saturated full parameterizations, either by constraining some parameters to be zero or to be functions of other parameters; some examples of the relation between the G × G parameterization and selected constrained or unsaturated parameterizations are given in When performing the corresponding investigation of parameter relations of a non-saturated full parameterization, μ, changes with the genotype. The choice among major classes of link function is further discussed, below, in Section Relating different GLMs and we will here focus on understanding the effect of a small change in the link function has on the parameters β. If the link function g(μ) is perturbed to g(μ) + ϵ (while keeping the design matrix P constant), we have the following approximate relationship between the parameters in the two models,xf(x) denotes the gradient of f(x) with respect to x. This means that the incorrect link function will introduce a bias in the parameter estimates, and the magnitude of that bias will depend on the rate of change of the inverse of the link function, as well as on the design matrix . However, for a given data set, the dispersion distribution is seldom varied because different dispersion distributions are often tightly connected to specific types of phenotypes  parameter θ, there exists a link function g(μ) such that θ = μ, then g(μ) is called the canonical link function of f . It also contains the identity link as a special case λ = 1. The second family, the Pregibon family, was suggested for studies of link functions for Binomial outcomes which contains the logit-link (λ = 0) as a special case [The first one is a symmetric version of the Box-Cox family of power transforms that lies between the ial case . These fa posteriori measure of link appropriateness.Evaluation of link function invariance is a computer-intensive approach as it requires re-analyses while stepping through different values of λ to identify any critical link function. In practice, we therefore apply this test on significant pairs, only, as an G × G joint, AD × AD joint, G × G separate, A × A, and R/D × R/D tests (described below). We then apply the G × G joint test to two biological data sets to demonstrate the efficacy and plausibility of our method. In the following sections, we will (unless it is clear from context) discriminate between a generative model referring to the model from which data is generated and a test model, which is used in the statistical test. For details on data generation, please see In the following sections we investigate how the false positive rate and statistical power is affected by different assumptions on the genetic effects. In both cases, we evaluate the following parameterizations: the We will now introduce five different strategies for testing interactions. We will use these strategies together with the Wald test described in Section Fast estimation and testing of interaction in generalized linear models, above, in our investigations of statistical power and false positive rate.G × G and AD × AD  to  to allow for easy comparison with G). A parameterization that features multiple interaction parameters can be tested either jointly or separately. Let the interaction parameter vector be denoted by δ = {δ11, δ21, δ12, δ22, }. A joint test evaluates the hypothesis that all interaction parameters are zero δ = 0 and a separate test evaluates the hypothesis that one or more interaction parameters are zero ∪h{δh = 0}; notice for the separate test that, while each test has a lower degree of freedom, the multiple testing burden will increase by a factor of 4. In our first two strategies, the interaction parameters are tested by a joint test and we will refer to these tests as G × G joint and AD × AD joint, respectively, while the third strategy is based on a separate test and the G × G parameterization, which we refer to as G × G separate.The first three are based on two different saturated parameterizations:  AD × AD . The G ×ussed by  . This model is often writtenδ11 instead of four as in the previous models. We refer to this test as the A × A test.For the next strategy, we introduce a non-saturated parameterization that assumes that the genotypic effect is completely additive in the number of minor alleles, and thereby ignores possible deviations. This corresponds to the Kronecker product of two additive-encoded uni-variant GLMs ; we also tested whether linkage disequilibrium (LD) affects the FPR. The results in A × A and R/D × R/D tests was inflated by the presence of dominant and recessive inheritance patterns. A second source of errors is LD, and the A × A test had a strongly inflated error rate when data was generated from R × A and R × D under both dispersion distributions. Moreover, also the R/D × R/D parameterization had an inflated error rate for all generative models under LD. The error rate was highest for data generated from R × A and a Binomial dispersion distribution. In general, the G × G joint, G × G separate and AD × AD joint tests are safe to use in the presence of LD, whereas the other tests must be treated with caution.We performed two experiments to investigate how potential model misspecifications affect the false positive rate (FPR) for the interaction tests described previously. In the first experiment we generated synthetic data from (null) models with no interaction, but where some assumptions of the evaluated tests fail , whereas the A × A test has on average 20% lower power. There is, as expected, no difference between the G × G joint and AD × AD joint tests. Despite the increased number of parameters of the joint tests these results holds also for lower sample sizes, see In the second experiment, we investigate the statistical power for the same tests over a large number of randomly sampled generative models. The results in G × G joint test in two different association analyses aimed at the quantitative outcome blood concentration of Lp(a) lipo-protein particles [G × G and AD × AD parameterizations are equivalent; our choice to use the G × G is arbitrary, but perhaps it is more easy to directly relate to the actual genotypes.We applied the articles  and a cacanonical link function in the respective analysis. The significant interactions from the discovery are then re-analyzed including the full test for link function invariance. The full link function invariance test is also applied in the replication analysis.The link family test was used to test for link function invariance. As this test is quite computer-intensive, we apply the following strategy for each association analysis: We first perform the large-scale discovery analysis using the We performed a classic discovery-replication analysis using quantitative data for circulating Lp(a). Lp(a) is an LDL-like lipoprotein particle also containing apolipoprotein , circulaL, the set of variants in approximate linkage equilibrium (≈1010 pairs), resulted in no variant pair passing the very severe Bonferroni-corrected significance level (p < 10−12).A full chip-wide analysis, testing all possible pairs from a priori selection procedure for single variant interaction candidates The vGWAS analysis is claimed to be independent of, and not biasing ensuing interaction analysis; in fact a theoretical proof is provided [V. We used a liberal p-value threshold of 10−4. The final interaction analysis was performed on all pairs between V and L. A single variant pair (rs13209686-rs1884480) was significant from this analysis   approachprovided . In a vGprovided ). Variani ∈  of the significant variant pair from the discovery analysis, we defined the set Fi of all variants in close proximity of i, and then analyzed all pairs of variants from F1 and F2. The top three pairs in the region are shown in p = 4.7378e − 07); the value of λ is consistent between discovery and replication and indicates that link should increase faster than linearly. Lastly, we performed a fixed-effect meta-analysis of the PROCARDIS and SCARF-SHEEP results for the rs3103353-rs9458157 pair. This resulted in a substantially stronger signal . Functional annotation analysis using HaploReg [SLC22A2 [AGPAT4. Both variants are situated in regions with histone or methylation patterns associated with enhancer activity—in adipose tissue and in blood or cardiovascular tissue, respectively. Gene Ontology analysis [SLC22A2 is a membrane transporter of organic cations, including histamine, choline and dopamine, while AGPAT4 is involved in metabolic processes related to triglyceride and phospholipid metabolism.Both rs3103353 and rs9458157 are located in the proximity of the previously associated LPA locus , but areHaploReg . Typically for complex diseases, such as CAD (and MI), very large sample sizes are needed for GWAS. Major collaborative efforts aimed at genetic analyses of CAD are therefore being performed, typically in large-scale meta-analyses driven by international consortia. As a proof-of-concept, we performed a two-cohort meta-analysis of gene-gene interactions associated to myocardial infarction (MI).9 pairs) using a fixed effect meta-analysis, as described above in Section Fast estimation and testing of interaction in generalized linear models. No variant pair was declared significant using the Bonferroni correction (p < 5.72 ⋅ 10−12), the top five variant pairs  are shown in The first cohort was obtained by augmenting the PROCARDIS MI case-control data with control data from the WTCCC  . Our simulations show that, under certain circumstances, incorrect parameterization in unsaturated or constrained GLMs may lead to incorrect interaction inferences. Specifically, when the data includes LD, tests using the constrained AGPAT2, involved in triglyceride metabolism, and SLC22A2, a transmembrane transporter, which previously has been indicated as involved in lipoprotein metabolism [We demonstrate the applicability of our method on experimental data aimed at Lp(a). Levels of circulating Lp(a) have an inverse association with coronary heart and are predominantly regulated by genetic variation in proximity of the LPA gene . Althougtabolism . Howeverp ≤ 5.72 ⋅ 10−12.We demonstrated the applicability of our meta-analysis method by analyzing two MI case-control cohorts. The genetic architecture of MI is known to be complex and single variant analysis has so far explained only a small fraction of its heritability. This motivated us to investigate the potential impact of interactions on MI. Our meta-analysis did not result in any interactions of genome-wide significance, which require passing the severe Bonferroni-corrected threshold canonical link function in the large-scale discovery phase, and performed the test for link function invariance only on those interactions that passed the discovery phase.The link function, a.k.a. the “scale”, is a crucial component of GLMs that relates the linear predictors to the phenotype mean. Because the link function used in a test can have a considerable impact on the resulting FPR, it is of interest whether the interaction is invariant of the scale, and a few tests for this have been devised. We applied the link family test, which evaluates invariance across a family of link functions , 23. ThiAlvarez-Castro and Carlborg  derived a posteriori to the initial analysis, fits a standard GLM including relevant covariates for the identified interactions and checks that the results still hold (this approach was used in the Lp(a) replication analysis). If this causes too many pairs to be identified, an alternative that is commonly applied in univariate meta-analyses is to first regress out relevant covariates and then model the residuals. Because residuals are expected to be normally distributed for all GLM models this can be generally applied; however, some interpretation is lost.Our test is currently limited to discrete genotype data, i.e., directly genotyped or hard-called imputed data . Meta-anG × G joint, AD × AD joint, G × G separate, A × A and R/D × R/D tests.We first considered two scenarios that may produce false positives: submodels in which one or more assumptions fails to hold, and tests where the incorrect link function is used. In both experiments, we evaluated the following five different tests: the D (LD) and we used two cases, high linkage (LD = 0) and low linkage (LD = 0.8). The genotypes were generated according to the following probability distributionq is the minor allele frequency of the first variant, and r is the minor allele frequency of the second variant. The parameter D = LD ⋅ Dmin where Dmin = min(1 − r), q(1 − r), (1 − q)r). The minor allele frequency was set to q = r = 0.3 and the sample size to 4,000. For each genotype, continuous phenotypes were then generated using six different null GLMs, obtained as the combinations of (i) a linear predictor from either of the A × A, R × A and R × D parameterizations, (ii) the identity link function, and (iii) either a Binomial and normal dispersal distribution . LD was measured by Lewontin’s A × A linear predictor, the identity link function and a normal dispersion distribution to generate phenotype from each genotype; we varied the size of the second variant’s main effect between −0.4 and 0.4 and the sample size over 2000, 3000 and 4000 .In the large power experiment, we focused on different The PROCARDIS study was approved by the Regional Ethics Review Board at Karolinska Institutet, Stockholm in Sweden  and by the Institutional Review Boards of the Mario Negri Institute, Milano in Italy, the University of Munster, Munster, in Germany, and the University of Oxford, Oxford, United Kingdom. The PROCARDIS study was supplemented with controls from the WTCCC study, UK. The WTCCC was approved by the relevant research ethics committees. All study participants provided their written informed consent to participate in the study, which was conducted in accordance with the Helsinki Declaration.The SCARF and SHEEP studies were approved by the local ethics committees at Karolinska Hospital, Stockholm , and Karolinska Insitutet, Stockholm , respectively.The MIGEN data was accessed from dbGAP (access.nr. phs000294.v1.p1) ; all parThe PROCARDIS multicentre study was designed to investigate early onset CAD. Cases with a documented CAD event before the age of 66 years were collected from Sweden, the UK, Germany and Italy. The full PROCARDIS cohort comprise 8410 cases and 5188 matched controls free from CAD. Here we have used a subset of the PROCARDIS cohort previously genotyped with the Illumina Human1M Quad chip and the Illumina Human610K chip . We onlyThe SCARF  and SHEEphs000294.v1.p1. This data contains 3,068 cases and 2,957 controls.The MIGEN cohort is a case-control study aimed at investigating the genetic basis of MI . The cohG × G joint test. In the discovery phase, a canonical link function and the appropriate dispersion distribution was used in an initial analysis. The significant pairs was then tested using the same G × G joint test, but using the link family test to step through a family of link functions; the latter approach was also used in the replication phase. For the continuous Lp(a) phenotype, the Normal dispersion distribution was used first with the canonical identity link and then with the symmetric Box-Cox family. For the binary MI case-control data, the Binomial dispersion distribution was used first with the logit link and then with Pregibon link family.We performed multiple experiments on biological data: exhaustive interaction scan in Lp(a), vGWAS scan in Lp(a), and a exhaustive meta analysis of CAD. All interaction tests was performed using the r2 < 0.5 . This resulted in 180,947 variants, and we tested for interaction between all possible pairs of these.The Lp(a) phenotype was log-transformed and outliers were removed in both PROCARDIS and SCARF-SHEEP. For the exhaustive interaction scan we applied the following QC filters: minor allele frequency 0.05, genotyping rate 0.1. We then removed variants in linkage disequilibrium using the Plink “–indep-pairwise” option, using a window size of 100, step size of 5 and a LD threshold of L. We then tested for variance heterogeneity of each variant with a Brown-Forsythe test [−4 is called V. We constructed all pairs of one variant from V and one variant from L (1.26 ⋅ 107 pairs) and tested these for interaction.For the vGWAS scan, we applied the same QC filters as above; the resulting set of SNPs is called the test . The set9 pairs).In the meta-analysis, we only considered variants that was genotyped both in MIGEN and PROCARDIS, this resulted in 132,181 variants. We applied the following QC filters: minor allele frequency > 0.05, genotyping rate > 0.1. We only tested pairs for which all genotypes had at least five samples in both cases and controls  due to population stratification. In all cases genomic inflation was low see  Figs. Thhttps://github.com/mfranberg/besiq.A c++ implementation of the software can be accessed at S1 TextThe text contains four sections: derivation of closed-form estimation of saturated full parameterizations, proof of that the fast estimation of unsaturated models is consistent, description of orthogonality for GLMs, and derivation of the fixed effect meta-analysis for Wald statistics.(PDF)Click here for additional data file.S2 Text(PDF)Click here for additional data file.S1 FigThe left and right subplots are for binary and continuous phenotypes respectively. The x-axis is the total sample size, for the binary phenotype this value is cases plus controls. The y-axis is the average time required to compute the p-value per variant pair on a log-scale. The colors represent the algorithm used to compute the test statistics, “Plink” refer to the –epistasis-test in the Plink 1.90 Beta software, “Wald” refer to the Wald test, and “IRLS” refer to the likelihood ratio test implemented using the iteratively reweighed least squares algorithm.(TIFF)Click here for additional data file.S2 FigThe x-axis is the number of minor alleles of the first variant. The y-axis is the expected value of the phenotype. The colors correspond to the number of minor alleles of the second variant.(TIFF)Click here for additional data file.S3 Figx-axis shows a threshold, t, for power to detect an interaction among 1012 variant pairs, and the corresponding y-axis shows the fraction of generative models, for which the analysis have a power greater than or equal to t. The rows correspond to the sample size. The columns correspond to the minor allele frequency of both variants in the pair. The line for the AD × AD joint test is often obscured by the line for the G × G joint test.For each plot, the (TIFF)Click here for additional data file.S4 FigThe x-axis is the expected log p-values and the y-axis is the observed log p-values. There is little deviation from the diagonal line and genomic inflation is low (λ = 1.102).(TIFF)Click here for additional data file.S5 FigThe x-axis is the expected log p-values and the y-axis is the observed log p-values. There is little deviation from the diagonal line and genomic inflation is low (λ = 1.100).(TIFF)Click here for additional data file.S6 FigThe x-axis is the expected log p-values and the y-axis is the observed log p-values. There is substantial deviation from the diagonal line; however, genomic inflation is low (λ = 1.109).(TIFF)Click here for additional data file.S7 FigThe x-axis is the expected log p-values and the y-axis is the observed log p-values. There is substantial deviation from the diagonal line; however, genomic inflation is low (λ = 1.047).(TIFF)Click here for additional data file.S8 FigThe x-axis is the total heritability of the generated variants. The y-axis is the estimated family-wise error rate. The colors correspond to the number of causal variants. The dashed line is the 0.05 FWER threshold. The error bar of each point is the 95% confidence interval of the corresponding FWER estimate. Each subplot corresponds to 4 different pruning strategies: no pruning, only ld-pruning, only vGWAS pruning, and both ld-pruning and vGWAS pruning.(TIFF)Click here for additional data file.S1 Tableϵ using parameterization matrix P. Here diag(x) denotes the diagonal matrix with the vector x as the diagonal.The first column is the name of the link function. The second column is the form of the link function. The third column is the bias when the link function is misspecified by a small perturbation (PDF)Click here for additional data file.S2 TableThe first column is the type of phenotype. The second column is the name of the dispersal distribution commonly used for the corresponding phenotype type. The third column is the canonical link function.(PDF)Click here for additional data file.S3 TableThe abbreviation ‘Enh’ stands for Enhancer and ‘Pro’ for Promoter, see the Roadmap epigenomics project for further details.(PDF)Click here for additional data file.S4 TableGene symbols for the closest genes (Tables 2 and 4) were used as search terms on the gene ontology website. Searches were limited to GO terms relating to Homo Sapiens and biological processes.(XLSX)Click here for additional data file.S5 Tableσ is the variance of the Normal dispersion distribution).The first column is the model, the second the LD of the model, the rest of the columns are the model parameters used in the simulation Click here for additional data file.S6 Tableσ is the variance of the Normal dispersion distribution).The first column is the effect size used on the x-axis in the plot. The rest of the columns are the normal model parameters Click here for additional data file.S7 Tableσ is the variance of the Normal dispersion distribution).The first column is the model, the second column is the effect size used on the x-axis of the corresponding plot, the rest of the columns are the parameters used in the simulation Click here for additional data file.S8 TableThe first column is the variance of the normal distribution, the second column the intercept, column four to eight are the main effects. The last column denotes the sum of the interaction effect sizes, these were used to generate all possible interaction models where the mean of the non-zero effect sizes equaled this number.(PDF)Click here for additional data file.S9 TableFor each variant pair, the p-value and sample size are given for the meta-analysis and the analyses of individual the MIGEN and the PROCARDIS cohorts.(PDF)Click here for additional data file.S10 Tablep < 5 ⋅ 10−8) variant v, this table shows the likelihood ratio (LR), the p-value (p) and the sample size (N) for the test of association of the rs3103353-rs9458157 interaction pair, when adjusting for v.For each genome-wide significant ((XLSX)Click here for additional data file.


Recent studies have indicated that quantitative autistic traits (QATs) of parents reflect inherited liabilities that may index background genetic risk for clinical autism spectrum disorder (ASD) in their offspring. Moreover, preferential mating for QATs has been observed as a potential factor in concentrating autistic liabilities in some families across generations. Heretofore, intergenerational studies of QATs have focused almost exclusively on Caucasian populations—the present study explored these phenomena in a well-characterized Hispanic population.The present study examined QAT scores in siblings and parents of 83 Hispanic probands meeting research diagnostic criteria for ASD, and 64 non-ASD controls, using the Social Responsiveness Scale-2 (SRS-2). Ancestry of the probands was characterized by genotype, using information from 541,929 single nucleotide polymorphic markers.In families of Hispanic children with an ASD diagnosis, the pattern of quantitative trait correlations observed between ASD-affected children and their first-degree relatives (ICCs on the order of 0.20), between unaffected first-degree relatives in ASD-affected families , and between spouses (mother/father ICC = 0.48) were in keeping with the influence of transmitted background genetic risk and strong preferential mating for variation in quantitative autistic trait burden. Results from analysis of ancestry-informative genetic markers among probands in this sample were consistent with that from other Hispanic populations.Quantitative autistic traits represent measurable indices of inherited liability to ASD in Hispanic families. The accumulation of autistic traits occurs within generations, between spouses, and across generations, among Hispanic families affected by ASD. The occurrence of preferential mating for QATs—the magnitude of which may vary across cultures—constitutes a mechanism by which background genetic liability for ASD can accumulate in a given family in successive generations. Recent studies have indicated that a substantial proportion of inherited liability to autism spectrum disorders (ASD) can be attributed to common, inherited variants –4. Even between generations through heritable genetic factors.These associations underscore the role of background genetic factors in risk for ASD. Quantitative trait scales such as the Social Responsiveness Scale-2 (SRS-2) , and thePrevious studies have also demonstrated evidence of assortative mating for sociality through spousal correlations in social responsiveness. Three previous studies involving both ASD-affected and ASD-unaffected families have reported significant spousal correlation for SRS-2 scores ranging from 0.26 to 0.39 . PrefereStudies of preferential mating for QAT have predominantly examined Caucasian populations, and it is possible that this phenomenon varies across cultures. Given the implications for accumulated liability in successive generations, and the fact that the influence of specific inherited liabilities for many complex genetic conditions have been shown to differ by ancestral origin –15, crosn = 85) and typical controls (n = 66). ASD diagnoses were ascertained from UM affiliated child/adolescent psychiatric clinics. The typical controls were families recruited from South Florida, including the University of Miami student community. All families contained at least one English-speaking parent, one parent of self-reported Hispanic ancestry, and one sibling. In 79.6 % of families, both parents were Hispanic. Five of the ASD families were known multiple-incidence autism families. One affected sibling in the family was designated as the proband/subject case to maintain independence.The study population was comprised of 151 Hispanic families recruited from the University of Miami Department of Psychology (UM), categorized into two groups by index subject: ASD diagnosis  and 83 Spanish-translation SRS-2 forms . Three different forms were administered based on age of the subject . We also differentiated the sample by informant . SRS-2 scores were obtained from student’s teachers for 50 subject cases (45 w/ASD diagnoses) and 43 of their siblings. Eleven preverbal individuals were evaluated with a video-referenced rating of reciprocal social behavior [The Social Responsiveness Scale-2 (SRS-2) is an extensively validated , 18 quanN = 61, SD = 26.6); non-verbal ASD subjects had a mean SRS-2 score of 104.8 —this difference did not reach statistical significance (p = 0.20).Although IQ data were not available for the sample, previous research has demonstrated that SRS-2 scores differ minimally as a function of IQ among verbal subjects , 19, 20.N = 200 ASD group, N = 363 typically developing group; matched for age and gender), the SRS-2 exhibited high discriminant validity between ASD and typical controls for both total score and subscale scores. The ASD sample mean for SRS-2 total raw score (102.2) was in keeping with means reported in previous US studies involving exclusively English-speaking informants [Fombonne et al. (2012) recently published findings on the psychometric properties of the Spanish version of the SRS-2 from a case-control study of ASD in Mexico . In thisOn average, respondents completing the Spanish version of the scale in this sample reported slightly higher SRS-2 scores than those completing the English version. This difference reached statistical significance difference for ASD-diagnosed children and their parents but not for siblings of ASD probands or for non-ASD families Table .Table 1CSixty nine ASD probands with sufficient DNA quantity and quality were genotyped using Illumina HumanCoreExome Chips  according to standard genotyping protocols. A total of 542,584 SNPs were genotyped. Quality filtering was performed, checking for low efficiency samples and SNPs. 656 SNPs with a call rate <90 % were removed and all samples had a >98 % call rate. No duplicate samples were identified and all individuals were confirmed as unrelated. After quality control screening there were 541,929 SNPs in 69 individuals.To contextualize the ancestry-informative genotypic data from the 69 ASD subjects, we included data from several publically available datasets: 165 CEU samples, 167 YRI samples, and 77 MEX samples were obtained from the HapMap project  represenPrincipal component analysis (PCA) was used for genomic data in order to elucidate the patterns of population structure in the 69 ASD subjects. PCA was performed in EIGENSOFT  jointly n = 5) for separate analysis.We analyzed SRS-2 distributions independently for ASD-diagnosed subjects, non-ASD controls, their siblings, and biological parents in order to characterize variations in distributions of SRS-2 scores by diagnostic status, language of the test, and informant. Intraclass correlations were calculated for associations between first-degree relatives separately by diagnostic group—ASD families and non-ASD families. For the children, the use of parent-report SRS-2 data optimized statistical power. There were insufficient multiplex families . Finally, we examined whether concordant elevation in parental scores of ASD subjects occurred more frequently than (a) would be predicted by chance; and (b) would occur among parents of non-ASD subjects. We did so by dividing the raw spouse-report SRS-2 data into four quartiles based on parental severity of SRS-2 scores as presented in Fig. p = 0.074; Fig. p < 0.001; Fig. In keeping with prior studies in non-Hispanic samples , SRS-2 sp < .0001), and 0.60 for spousal pairs in non-ASD families   were observed among ASD families than among non-ASD families , and the former substantially exceeded the proportion expected by chance, which was 6.25 %. These results were unchanged when restricting the analysis to data from one or the other specific language versions of the SRS-2 (Spanish vs. English).Trait correlations for specific pairings of first-degree relatives in both ASD and non-ASD families were generally moderate-to-strong, as shown in Fig. 01) Fig. , both ret scores >70. Although diagnostic information about the higher scoring individuals (predominantly fathers) is not available, these distributions are in keeping with prior studies in both the general population and in clinical family studies [Five out of 125 (4 %) parents in the control group and 22 out of 140 parents of ASD probands (15 %) had SRS  studies . Concern studies .r = −.259, p > 0.05; Table r = 0.072, p > 0.5). There was a small positive correlation between YRI ancestry and SRS-2 score .Figure This study confirmed substantial correlations in quantitative autistic trait (QAT) scores between first-degree relatives within and across generations, in a Hispanic cohort, and provides new validation data for such measurements in both English and Spanish versions of the assessment used, the Social Responsiveness Scale-2 (SRS-2). In an epidemiologic twin sample, we previously  showed tclinical-ASD [Furthermore, the data obtained in this cohort provide cross-cultural evidence of pronounced effects of variation in social responsiveness on mate selection, and confirms prior observations that concordant elevation in QAT scores of parents is associated not only with elevated QAT scores of offspring but with increased offspring risk for ical-ASD , 28. Indical-ASD , 0.26 [1ical-ASD , and 0.3ical-ASD , among c2), where r is the phenotypic correlation for spouses and h2 the heritability of the trait. This generates the expectation that full siblings should be somewhat less similar to each other than to their parents, which is what was observed among ASD families in this sample [In general, higher spousal correlations for an inherited trait operate to increase within-family aggregation of the traits over the course of successive generations . Under as sample .An ICC value of 0.7 is the upper limit of what could be expected on the basis of inheritance alone for first-degree relations, and this would require 100 % inheritance of the trait. Therefore, the observation of very high trait correlations in this range suggests that despite the high heritability of SRS scores reported in family  and intesubstantially lower than those between unaffected first-degree relatives in ASD-affected families—which we have previously reported [combination of background genetic liability and superimposed de novo mutations, which exert pathological shifts against a genetic background [Furthermore, the observation that trait correlations between ASD probands and their first-degree relatives are reported , 26—is ickground . The effckground . The resckground , 32.sub clinical elevation in quantitative autistic trait burden may be evolutionarily adaptive in certain contexts, though not in the extreme [Evidence in other species shows that assortative mating over the span of several generations can have a rapid and significant evolutionary impact on the demographics of a population . The ass extreme . For exa extreme . More ge extreme .Individuals of Hispanic identity are exceptionally diverse in ancestry and genetic origin compared to many other populations , and ourFinally, the Spanish language test yielded somewhat higher average SRS-2 scores for children with ASD and their parents. The complex history of Hispanic ancestry and colonialism is a potentially important factor when relating ancestry to biological traits and may contribute to correlations between language and genetic ancestry . The comThe present study provides strong evidence of familial aggregation of quantitative autistic traits (QAT) between generations (from parents to offspring) and within generations (between siblings) among Hispanic families. These traits represent measurable indices of inherited liability to ASD that may relate to common genetic variants. We also observed strong evidence of assortative mating for social behavioral characteristics indexed by the Social Responsiveness Scale, and this reflects a mechanism by which ASD traits might accumulate within families in successive generations. The degree to which this occurs may vary across cultures, and measurement of QATs in prospective parents may represent important indices of risk for the occurrence of ASD in their offspring.


To test our method we used public ENCODE whole genome DNase-I sequencing data, from a single sample, to estimate the functional potentials of the subset of 10,552 noncoding heterozygous single-nucleotide polymorphisms (SNPs) that were also present in the Genetic Analysis Workshop 19 (GAW19) family-based data set. We then built two covariance kernels, one nonweighted and one weighted by the functional potentials, and conducted kernel-based variance component association analyses against the 20,527 transcript expression phenotypes in the GAW19 family-based data set. We found signals of potential cis-regulatory effects, that surpassed the Bonferroni significance threshold, for ten transcripts. Stepwise removal of the cis-located SNPs from the weighted kernel lead to the disappearance of the association signal from our top transcript hit. We found compelling evidence of allele-specific cis-regulation for four transcripts using both kernels, and our results agree with previous research that suggests the involvement of specific cis-located variants in the regulation of their neighboring gene.We present a novel approach to detect potential  It has ng sites .cis-acting regulatory loci. Our approach differs from previous works [We have been investigating a systematic approach that uses DHSs to determine if noncoding single-nucleotide variation changes the local allele-specific chromatin accessibility, something that would indicate a potential regulatory role for a variant . We haveus works , 9 in thWe used single-nucleotide polymorphism (SNP) dosages from 959 genotyped individuals, transcript expression levels from 647 of those individuals, and the genealogies  that were provided as part of the Genetic Analysis Workshop 19 (GAW19) family-based data set . In addiWe compiled a reference panel of heterozygous SNP sites from the genotype calls from the high-coverage/high-quality WGS of NA12878. This independent genotypes source allowed us to analyze heterozygous loci where, because of either low coverage or complete allele-specific accessibility, only 1 allele is represented in the DNase-seq reads.We defined our chromatin accessibility measure to be equal to the DNase-seq read depth of each allele at a heterozygous locus. Based on our previous experience  the DNasA departure from the expectation of an equal chromatin accessibility measurement of the two alleles at a locus within a DHS is what we refer to as the locus functional potential (FP). We implemented the FP statistic as a likelihood ratio–based test that contrasts the observed allele read depths with their expected depth at known heterozygous loci within DHSs . A signiTo test our approach we used the real expression phenotypes from approximately 20,000 transcripts provided in the GAW19 family data set . In addiWe also used the sex, age, their interactions, and the smoking status at the first visit as covariates in all models. The first two principal components   of those heterozygous SNPs but only 10,618 (22 %) of them were present in the GAW19 dosages. Of the 10,618 heterozygous-in-NA12878 SNPs with a FP estimation that were present in GAW19, 66 (0.62 %) were monomorphic in the GAW19 dosages and were therefore discarded from further analysis. The remaining 10552 SNPs with FP estimates were used for the construction of our weighted and nonweighted covariance kernels.p values of the estimated effects  we found significant evidence of potential cis-regulatory effects for ten transcripts . While most of the trait heritability was explained by the covariance kernel, a substantial amount (between 14 and 28 %) still remained. Table We then analyzed the 20,527 transcript expression phenotypes in the GAW19 family data set using both the weighted and the nonweighted covariance kernels. After a genome-wide Bonferroni correction (−logcis-acting, regulatory elements. To that end we used a combined approach that integrates functional information, in the form of allele-specific chromatin accessibility measurements at DHSs, gene expression phenotypes, and a variance component model that estimates the proportion of a trait’s variance as a result of a localized relationship kernel.The objective of this study was to investigate the prioritization of SNPs based on their potential as functional, We constructed nonweighted and weighted covariance kernels, using the 10,552 SNPs with an available FP estimate, and obtained the proportion of variance in the levels of transcript expression that could be explained by them in the family data set. We identified a clear signal for eight transcripts when using the nonweighted kernel, and for two additional transcripts when using the weighted kernel (see Table cis-elements, but could also be the result of, for example, linkage disequilibrium with SNPs in closer proximity to the transcript.Some of our results are difficult to interpret because of the distance between the transcript location and the closest SNPs to it in our kernels. For transcripts GI_12056480-A and GI_15451941-S our results might indicate the presence of long-acting cis-acting variants. Particularly for the GI_23097237-S, GI_42544126-I, GI_4506738-S, and GI_41393558-I transcripts, corresponding to the CHST13, SF1, RP56KB2, and KIF1B genes, respectively. The SNPs with FP estimates that we incorporated in our covariance kernel near these genes are all located either within the gene or within the promoter region of the gene (see Table SF1 gene led to the degradation of the signal from the GI_42544126-I transcript (see Table cis-acting effect of the variants in the transcript expression. Furthermore, previous research provides additional compelling evidence for the implication of rs11718493 in the allele-specific methylation of CpGs and the regulation of CHST13 [KIF1B expression [However, close examination of the annotations of the significant transcripts in our results shows suggestive evidence of potential f CHST13 , 16, a ccis-regulated by the SNPs we identified in them.Our kernel-based variance component test was able to prioritize noncoding variation from whole-genome sequencing data based on their potential to regulate gene expression. An allele-specific chromatin accessibility measure was used as both a biologically meaningful filter for the selection of the variants and the weight of each variant in the covariance kernel. We observed compelling evidence to support the idea that four genes might be


Building an efficient reference population for genomic selection is an issue when the recorded population is small and phenotypes are poorly informed, which is often the case in sheep breeding programs. Using stochastic simulation, we evaluated a genomic design based on a reference population with medium-density genotypes [around 45 K single nucleotide polymorphisms (SNPs)] of dams that were imputed from very low-density genotypes (≤ 1000 SNPs).A population under selection for a maternal trait was simulated using real genotypes. Genetic gains realized from classical selection and genomic selection designs were compared. Genomic selection scenarios that differed in reference population structure (whether or not dams were included in the reference) and genotype quality (medium-density or imputed to medium-density from very low-density) were evaluated.The genomic design increased genetic gain by 26% when the reference population was based on sire medium-density genotypes and by 54% when the reference population included both sire and dam medium-density genotypes. When medium-density genotypes of male candidates and dams were replaced by imputed genotypes from very low-density SNP genotypes (1000 SNPs), the increase in gain was 22% for the sire reference population and 42% for the sire and dam reference population. The rate of increase in inbreeding was lower (from − 20 to − 34%) for the genomic design than for the classical design regardless of the genomic scenario.We show that very low-density genotypes of male candidates and dams combined with an imputation process result in a substantial increase in genetic gain for small sheep breeding programs. Although technical and economic interests of genomic breeding programs in sheep have been positively assessed by various authors –5, only In sheep, the profitability at the nucleus level remains a critical factor for the design of breeding programs . ImplemeVery low-density (VLD) genotypes in association with imputation techniques would reduce the genotyping costs of a genomic design. VLD panels based on a few hundred SNPs are available at lower cost than low-density genotypes (~ 15 K) and are mainly used for parentage assignment –20. SincFocusing on a breeding program applied to a small population of purebred sheep in which both AI and natural mating sires are used, the objectives of this study were to quantify the impact of (1) increasing the reference population size with female genotypes and (2) imputing genotypes of male candidates and females from very low- to medium-density SNP panels. The presence of all sires and grand-sires of dams and male candidates in the reference population for imputation is expected to limit the detrimental effect of using a VLD panel on imputation accuracy. Various designs were compared, including a classical selection scheme based on progeny testing of AI sires as a baseline scenario, and several genomic selection schemes. Five scenarios were assessed for genomic selection by varying the reference population component (with or without dams), and genotype information .2) between SNPs was equal to 0.13 at 50 kb, 0.07 at 200 kb and 0.05 at 1000 kb. These values of r2 were within the range of estimated r2 and Ne reported by Kijas et al. . Parents were randomly mated and replacements randomly selected during 20 reproductive cycles to establish a founder population. Ten years of selection for a maternal trait were then simulated by applying a classical progeny test design. At this time Time = 10), the average LD  from progeny-tested sires of the dairy Lacaune sheep breed were used to initialize population genomes . The genotypes were cleaned  and only autosomal SNPs with a known position on the genome were retained: 47,706 SNPs were used for the genome simulations. Using genealogical information, phased chromosomes were obtained from Fimpute . At Timeλ the chromosome length expressed in Morgan. Positions of the crossovers were allocated following a uniform distribution along the chromosome and a parental strand was randomly chosen  to start the haplotype reconstruction of the gamete. Mutations were simulated with a mutation rate of 1 × 10−6 per SNP per meiosis. At Time 0, founders were randomly selected according to a demographic structure  among individuals that were created in the last seven cycles, aged from 1 to 7 years old at Time 0.Accuracy of genomic predictions depends on the relationships between individuals of the reference population and candidate animals , 38. To VA = ∑ i=1nqtl2pi(1 − pi)qi2 [pi denotes the allele frequency of the first allele for QTL i and qi its effect. The QTL effects were rescaled to make the additive variance VA equal to 1. The maternal permanent effects, pe, were drawn from a normal distribution with mean 0 and variance rep = 0.5 and h2 = 0.25 denote the repeatability and the heritability of the trait. The residual effects were drawn from a normal distribution with mean 0 and variance kth phenotype yjk of a female j was simulated as yjk = TBVj + pej + ejk with TBVj the true breeding value depending on the effects of all QTL and the female genotype.One thousand quantitative trait loci (QTL) underlying the maternal trait under selection were randomly selected among SNPs with a minor allele frequency higher or equal to 0.05. Following Hayes and Goddard , QTL effpi)qi2 , where pThe estimation of breeding values was based on an animal best linear unbiased prediction (BLUP) model  for the nvld evenly spaced SNPs , the genome was divided into nvld windows. Each window was subdivided into three parts and a SNP was randomly selected from candidate SNPs located in the central part. If no candidate was available, a SNP located within the window or in adjacent windows was selected. The MD panel  included all SNPs except those selected as QTL.SNPs with a minor allele frequency higher or equal to 0.25 and not selected as a QTL were considered as candidates for inclusion in the VLD panel. To select Around 5000 females divided into 10 flocks were recorded each year for the trait under selection. According to Hill , and negAbout 600 male candidates were preselected among newborn males (~ 3500) based on the parental mean EBV in the classical design and on the parental mean GEBV in the genomic design. In practice, all candidates were born from proven AI sires  and genomic AI sires (genomic design) given their genetic superiority. About 10% of male candidates died within their first year. At 1 year of age, 50% of the male candidates were culled for non-genetic factors. In the classical design, the ten candidates with the highest parental mean EBV (out of 270 candidates) were selected and mated across flocks by AI to 1100 females  for progeny-testing. Two cycles later, the five AI sires with the highest EBV (including progeny records) were selected as proven AI sires and used at most for four cycles (from 4 to 7 years old). At each cycle, 1400 females were mated to proven AI sires. In the genomic design, MD or imputed MD genotypes of candidates were available. The ten candidates with the highest GEBV were selected as AI males and mated across flocks to 1100 females. Genomic AI sires were used at most for another cycle and mated to 1400 females. To mimic the variation in the AI dose production per male, the inseminated dams were randomly allocated to one AI sire with a maximum number of AI per male equal to 140 for 2-year old sires and to 250 for older sires. About 50 naturally-mated male replacements were selected among 260 candidates not selected as AI sires, based on their parental mean EBV in the classical design or on their GEBV in the genomic design. These naturally-mated sires were used at most for four cycles (from 2 to 5 years old).Five genomic scenarios Table  were assAt each cycle and for each scenario, the average TBV and inbreeding coefficient of females in their first parity were computed. The annual genetic gain and annual rate of inbreeding were estimated as the regression slopes of the average TBV and average inbreeding coefficient of first parity females on time between cycles 10 and 25. The means and standard deviations presented are based on 50 replicates. At time 25, we computed the average Pearson correlation between the TBV and the (G)EBV for dams and male candidates as well as the average concordance rate of imputed genotypes  of dam and male imputed genotypes.Table Annual genetic gains indicated in Tables The rates of inbreeding shown in Tables The EBV accuracies shown in Tables The concordance rates of imputation reported in Tables a for 500 SNPs, while the gain for 250 SNPs was close to, but still significantly higher (0.175Table 2 = 0.10 instead of 0.25) and a higher efficiency of the classical design (proportion selected of 0.33 instead of 0.5 to select proven AI sires) were also assessed. Based on ten replicates (results not shown), differences in genetic gain were slightly smaller for a low heritability or a higher efficiency of the classical design, but were of a similar magnitude to the results obtained with the classical design, GSscd and GSs_Icd in this study.In this study, our reference scenario the classical design produced a rate of genetic gain of 0.162ch breed , a breedThe rate of increase in inbreeding was lower in genomic scenarios for three main reasons: first, the average contribution of natural mating sires was identical across designs whereas the average contribution of AI sires was smaller in the genomic design ; second, the selection intensity applied on parent average genetic value to select male candidates in the genomic design was lower than that to select AI and natural mating sires in the classical design; and third, genomic information was expected to reduce the relative importance of pedigree information in breeding value estimation through improved estimation of the Mendelian sampling term . The latP = 0.068) in agreement with results obtained in sheep by Moghaddar et al. [Compared to the classical design, a genomic design based on a reference population of related sires resulted in an increase in genetic gain of 26.3% per year when candidates were genotyped with the MD panel (GSsc). This increase is inferior to the increase in gain of 50% calculated by Shumbusho et al.  for the r et al.  for a siSires of male candidates were selected among young males in the genomic design and progeny-tested males in the classical design. Although the accuracy of GEBV of young males was lower than that of progeny-tested males (0.71—results not shown), higher selection intensities and a shorter average generation interval resulted in a higher genetic gain. Regardless of the design, male candidates were preselected based on parent average genetic values. The genomic AI sires were selected on their own genomic value before 2 years of age with a proportion selected of 1/27 versus 1/2 at 4 years old after progeny-testing for AI sires in the classical design. Since genomic AI sires were used for a maximum of 2 years, the average generation interval was reduced (3 years instead of 4). The natural sires were used and selected at the same age with a proportion selected close to 1/5 but on their own genomic value in the genomic design or on parent average genetic value in the classical design.The comparison between scenarios with (GSscd) or without females (GSsc) included in the reference population shows that adding dam genotypes, along with their phenotypes, resulted in a doubling of the increase in genetic gain of the genomic design. Genotyping dams led to an increase in the average accuracy of GEBV for both dams 14%) and male candidates (34%). The effect of including cow genotypes in the reference population on accuracy and bias of genomic prediction has been widely reported in dairy cattle , 49–55. % and malP < 0.001). Comparing GSs_Icd based on 250 SNPs with the classical design, the differences in accuracy of GEBV for dams and male candidates were small (+ 0.02) but highly significant (P < 0.001) for dams and moderate for male candidates (P = 0.0565).When male candidates and dams were genotyped with a 1000-SNP panel and imputed genotypes were included for genomic evaluation (GSs_Icd), the additional genetic gain was reduced but it remained higher (+ 41%) than that obtained with the classical design. With a concordance rate of imputed animals of about 96%, the accuracy of GEBV of male candidates and dams decreased compared to that in GSscd (respectively − 11.3 and − 4.6%) but remained higher than accuracies obtained with a reference population based on sires (respectively + 15.9% and + 8.4% pts). Compared to the classical design, the increase in accuracy with a 1000-SNP panel was + 41% but this decreased to + 8% with a 250-SNP panel. Lower imputation accuracies obtained with lower density SNP panels and larger numbers of genotypes discarded from the genomic evaluation due to parent-progeny mismatches (results not shown), resulted in no increase in GEBV accuracy of dams when they were genotyped with the 250- or 500-SNP panel: both panels gave the same accuracies as the GSsc scenario in which dams were not included in the reference population. For male candidates, the accuracy of GEBV was lower for 500 (− 40.0%) and 250 (− 65.8%) SNPs than the accuracy obtained with 1000 SNPs but the increases in gain for genomic scenarios based on 250 and 500 SNPs, compared to the classical design, were highly significant  were removed in GSs_Icd_pop, due to mismatches between parents and progeny (results not shown). In GSs_Icd_pop, the annual genetic gain remained higher (+ 9.5%) than in the classical design but was substantially lower than in GSs_Icd (+ 45%). The contribution to the analysis of co-segregation between VLD and MD SNPs has already been reported in previous studies which highlight the beneficial effect of including relatives in the reference population for imputation –31, 34, Higher gain was obtained with a genomic design when including dam medium-density genotypes in the reference population. The economic value of such a scenario should be assessed over the whole population, including both the nucleus and commercial levels. However, as discussed for Australian sheep and beef cattle industries by van der Werf and Banks , for an Plieschke et al.  noted thUsing a stochastic model, we compared classical and genomic selection designs. Five genomic scenarios were assessed by varying the structure of the reference population  and the genotyping panels that were used . Compared to the classical design with progeny testing, genomic scenarios generated more genetic gain and limited the increase in inbreeding. This superiority was based on higher selection differentials that are applied to male candidates to select sires for both AI and natural mating. The combined use of very low-density genotypes for male candidates and dams together with imputation resulted in lower genetic gain than scenarios based on medium-density genotypes. However, the increase in gain was substantial compared to that in the classical design. Using very low-density SNP panels might be more profitable at the nucleus level given its lower cost relative to medium- or low-density SNP panels and multipurpose use (parentage assignment and genomic selection).


Number of functional teats is an important trait in commercial swine production. As litter size increases, the number of teats must also increase to supply nutrition to all piglets. Therefore, a genome-wide association analysis was conducted to identify genomic regions that affect this trait in a commercial swine population. Genotypic data from the Illumina Porcine SNP60v1 BeadChip were available for 2951 animals with total teat number (TTN) records. A subset of these animals (n = 1828) had number of teats on each side recorded. From this information, the following traits were derived: number of teats on the left (LTN) and right side (RTN), maximum number of teats on a side (MAX), difference between LTN and RTN (L − R) and absolute value of L − R (DIF). Bayes C option of GENSEL (version 4.61) and 1-Mb windows were implemented. Identified regions that explained more than 1.5% of the genomic variation were tested in a larger group of animals (n = 5453) to estimate additive genetic effects.VRTN mutant allele increased teat count by 0.35 (TTN), 0.16 (LTN and RTN) and 0.19 (MAX). 15, 18, 13 and 18 one-Mb windows were detected that explained more than 1.0% of the genomic variation for TTN, LTN, RTN, and MAX, respectively. These regions cumulatively accounted for over 50% of the genomic variation of LTN, RTN and MAX, but only 30% of that of TTN. Sus scrofa chromosome SSC10:52 Mb was associated with all four count traits, while SSC10:60 and SSC14:54 Mb were associated with three count traits. Thirty-three SNPs accounted for nearly 39% of the additive genetic variation in the validation dataset. No effect of piglet sex or percentage of males in litter was detected, but birth weight was positively correlated with TTN.Marker heritabilities were highest for TTN (0.233), intermediate for individual side counts (0.088 to 0.115) and virtually nil for difference traits (0.002 for L − R and 0.006 for DIF). Each copy of the Teat number is a heritable trait and use of genetic markers would expedite selection progress. Exploiting genetic variation associated with teat counts on each side would enhance selection focused on total teat counts. These results confirm QTL on SSC4, seven and ten and identify a novel QTL on SSC14.The online version of this article (doi:10.1186/s12711-016-0282-1) contains supplementary material, which is available to authorized users. The competition for teats leads to increased pre-weaning mortality due to crushing and starvation . Therefo2 swine populations and detected the largest QTL on either Sus scrofa chromosome SSC1 or 7 , whVRTN alleles . Additive genetic variance was estimated in this dataset both with and without fitted SNPs to determine the percentage of genetic variation represented by the effects of SNPs. Candidate genes were selected by manually inspecting genes that were positioned within QTL regions based on the UCSC Genome Browser Gateway (www.genome.ucsc.edu/cgi-bin/hgGateway) using Sscrofa 10.2 genome build.The SNP that explained the highest proportion of genomic variation for each 1-Mb window explaining more than 1.5% of the genomic variation for any count trait  was identified  and DIF (n = 6) were detected; however, these results are likely meaningless since the estimated genomic variation was nearly zero Table . A regioVRTN) that were located in 32 unique 1-Mb windows were selected for the validation phase. Three SNPs were removed because they created multi-colinearity among the regression coefficients due to high linkage disequilibrium. All three SNPs were within a 1-Mb window represented by multiple SNPs, which were associated with different phenotypes. In total, the SNPs accounted for ~39% of the additive genetic variation estimated for this dataset. The effect of VRTN was 0.335 for TTN. The magnitudes of the estimated additive effects for eight SNPs were between 0.10 and 0.14, 12 SNPs had additive effects that ranged from 0.05 to 0.10 and 12 SNPs had estimated effects that were <0.05 (Table Thirty-six SNPs (including the SNP in How the development of the mammary gland has evolved is an interesting issue. It was observed by Aristotle more than 2000 years ago (as referenced by Diamond ) and shoVRTN and number of teats. This association was also found by Ding et al. [ceramide kinase (CERK) which produces ceramide-1-phosphate and has a role in cell proliferation and migration [CELSR1, a regulator of planar cell polarity [COL1A1 gene. Mutations in COL1A1 cause osteogenesis imperfecta leading to reduced bone mass and increased fracture. Two other potential candidate genes are the homeobox proteins DLX3 and DLX4 at 26.2 Mb on SSC12 but, to our knowledge, no role of these genes in mammary gland development has been described. DLX3 induces the degradation of p63 [We detected several interesting genomic regions that affect number of teats in commercial-type pigs. Foremost, was the confirmation of the association between SNPs in g et al.  and Duijg et al. . While tg et al. . Howeverigration  and CELSpolarity . The regn of p63  a transcn of p63 . In spit2 populations [MPP7 and FRMD4A. FRMD4A resides within a copy number variation (CNV) region [FRMD4A and MPP7 regulate the polarization of epithelial cells [PLXDC2 as a possible candidate gene, which encodes a transmembrane receptor for the neurotrophic factor PEDF [The region with the most consistent and largest effects on all teat count traits (SSC10:52) was also found by Duijvesteijn et al.  in Largeulations –35. This) region  and bothal cells , 38. Theal cells  who suggtor PEDF . Hirookator PEDF  and Rodrtor PEDF  reportedTRPS1, which encodes a transcriptional repressor that regulates epithelial-mesenchymal transition [The region on SSC4:25 Mb explained a high percentage of the genomic variation for LTN and had an additive effect on TTN of 0.103 in the validation population. Guo et al.  and Tortansition  is requiansition .T-box 1 transcription factor gene (54.93 Mb). T-box transcription factors are critically important for normal tissue and organ development in the embryo. GWAS results for this region revealed a broad peak that spans several Mb. Several of the SNPs tested in the validation phase were in high linkage disequilibrium which made it difficult to directly pinpoint which SNP had the largest effect. Thus, multiple causative genes/variants are possible.The region on SSC14:51–55 Mb has not been associated with teat number in pigs based on QTLdb and no obvious positional candidate genes were identified. This region had a large effect on RTN and MAX and the association with TTN was validated in the larger population. The SNP with the greatest estimated effect on TTN in the validation data was located at 55.00 Mb on SSC15 only 70 kb from the MEGF6 and MEGF11) and the SSC14:41 Mb region contains two genes that are involved in the modulation of the NOTCH signaling pathway (DTX1 and RITA1), these genes have not been shown to affect mammary gland development.Other novel QTL regions with estimated additive effects exceeding 0.10 were on SSC1:181, SSC8:37, SSC13:146, SSC14:29 and SSC14:41 Mb. Unfortunately, identification of obvious candidate genes was unsuccessful. Although the SSC1:181 Mb region includes two genes for multiple epidermal growth factors  were identified to have CNV segregating within the studied population . AdditioVRTN gene that were observed on vertebra number and teat count [NR6A1 gene on SSC1 [NR6A1 is only observed in crosses between Asian and European breeds, we were not able to evaluate this result. In spite of these two co-localizations of vertebra number and teat count in the proximity of VRTN and NR6A1 as well as the speculation mentioned in Duijvesteijn et al. [WNT7B as candidate gene. However, there are clearly additional genetic factors that affect teat count since the USMARC Meishan population averages a 2.6 greater TTN [The concept that additional mammary glands arise from somite division in the developing embryo is supported by the effects of the at count , 17, 23 n et al. , none ofater TTN  while haFinal teat number is likely a composite trait for which the underlying genetic model begins with somite division, followed either by proliferation of the mammary buds and/or regression of milk buds which results in teat number at birth. If genetic variation exists for all segments of the mammary gland development, then more progress may be possible if selection is applied to the component traits. This was the hypothesis on which was based the study of the maximum number of teats on one side since this would be the best estimate of an animal’s true genetic potential for the initial phase of somite proliferation. Measures of difference (L − R and DIF) may reflect regression of mammary buds. Based on the lack of genomic variation detected for L − R and DIF, these traits appear to be controlled by non-genetic factors. These factors must have a role during gestation, but the estimated effects of litter variation were zero for both L − R and DIF in the model development phase of this study. Similarly, Borchers et al.  and FernFew studies have actually evaluated left and right teat count values in pigs. While approximately 60% of pigs have the same number of teats on each side , 10 and Studies of embryonic development in mice have shown that mammary glands of male fetuses regress between day 13.5 and 15.5 of gestation due to circulating androgens, such that after birth male pups do not possess mammary gland tissue . Based oBorchers et al.  reportedEstimated heritabilities indicate that selection would be successful for increased teat number. This was documented in sheep by Alexander Graham Bell  who obseEstimated heritabilities for teat number often range from 0.2 to 0.4 . The estIf final teat number is the result of the number of mammary buds initially developed and a maintenance (or regression) component, which acts randomly relative to the side of the developing organism, then MAX would be a better indicator of the initial number of mammary buds than RTN or LTN. L − R or DIF could predict the maintenance or regression component; however, genetic variation for these traits was nil, thus selection based on these measures would likely be ineffective. Among the traits analyzed here, TTN is the trait with the highest heritability and the most phenotypic variation and since it is the most important trait for swine production, selective pressure to increase lactational capacity in commercial sows should focus on TTN. Based on the association between birth weight and teat number, this would result in a serendipitous increase of piglet birth weight if this association is due to genetics.VRTN on teat number as well as the QTL located on SSC4:25 and SSC10:52 Mb, and we identified an important novel region on SSC14:51–55 Mb, which needs to be further studied. The most heritable trait that also possessed the most phenotypic variation was TTN. Therefore, without SNPs, single-trait traditional selection for TTN would yield the greatest gains; however, a selection index including TTN and MAX might slightly improve selection response. In this study, factors such as sex of the pig or of other pigs in the litter had no effect on TTN and there was no common environmental effect associated with fluctuating asymmetry for teat counts. The observed correlation between birth weight and teat number was unexpected and deserves further investigation. While it is clear that teat number can be increased in pigs, whether an increase in number of teats will result in increased total milk production still needs to be addressed.Selection to increase the number of teats is possible in pigs and use of genetic markers should expedite progress. Since individual side counts of teats is not commonly recorded, exploitation of loci that independently control RTN or LTN would be most effective with genetic markers. These results validate the effect of