Statistical genetics shows that the success of both genetic association studies and genomic prediction methods is positively associated with the heritability of the trait used in the analysis. Identifying highly heritable components of a complex disease can thus enhance genetic studies of the disease. Existing heritable component analysis methods use data from related individuals to compute linearly-combined traits to maximize heritability. Recent advances in acquiring genome-wide markers have enhanced heritability estimation using genotypic data from apparently unrelated individuals, which is referred to as the chip heritability. Novel statistical models are thus needed to identify disease components (subtypes) with high chip heritability.We propose an optimization approach to identify highly heritable components of a complex disease as a function of multiple clinical variables. The heritability of the components is estimated directly from unrelated individuals using their genome-wide single nucleotide polymorphisms. The proposed approach can also model the fixed effects due to covariates, such as age and race, so that the derived traits have high chip heritability after correcting for fixed effects. A new sequential quadratic programming algorithm is developed to efficiently solve the proposed optimization problem.The proposed algorithm was validated both in simulations and the analysis of a real-world dataset that was aggregated from genetic studies of cocaine, opoid, and alcohol dependence. Simulation studies demonstrated that the proposed approach could identify the hypothesized component from multiple synthesized features. A case study on cocaine dependence (CD) identified a quantitative trait that achieved chip heritability of 0.86 estimated using a cross-validation process. This quantitative trait corresponded to the likelihood of an individual's membership in a CD subtype. Clinical analysis showed that the subtype enclosed individuals who reported heavy use of cocaine but few withdrawal symptoms.Extensive experiments on both synthetic and real-world data demonstrate the effectiveness of the proposed approach as a means to find meaningful disease components with high chip heritability. Identifying genetic variation that underlies complex diseases has important implications in medicine. To date, genome-wide association studies (GWAS) have had limited success in dissecting the genetic etiology of complex diseases. For instance, very few associations identified for substance use disorders at a genome-wide significant level have been replicated -3. ComplThe ability to translate genotype information into a quantitative prediction of disease phenotypes is important for precision medicine . Genomich2 is defined by the percentage of phenotypic variance that is due to additive genetic effects  with L1 of size n × p and L2 of size n × (n − p). The two sub-matrices satisfy ỹ follows the following multivariate Gaussian distribution given the multivariate Gaussian assumption of y:The main idea of REML for estimating the variance components is to first eliminate the fixed effect due to covariates from the observed values of We have ỹ can be decomposed into:Then, the log likelihood of 2 is not a function of the fixed-effect parameter β. The two variance components, i.e.,l2, and there is no additional information in ℓ1 for estimating the variance components. Once β can be obtained as:Where ℓℓ2 is calculated as (after removing constant):The second log likelihood component It has been shown in an early work  that whe2 yields:Substituting these equations into the calculation of ℓβ can be obtained by:where y, C and Z, y is computed using the resultant optimal Given data on ,σe2;y˜2). The chiy is not known beforehand but needs to be derived from a set of known clinical variables. Let Xn×d be the data matrix of d clinical variables x for the same n subjects as in Z. A trait y is defined by a linear function of y = ┬x wwhere w is the vector of combination coefficients. Correspondingly, the trait values y = Xw. Unlike the heritability estimation process that finds the best values of y, the inverse problem searches for the best w so to form a trait y that maximizes the likelihood, , by substituting the β value, When where ThenCombining the objective function and the constraint together, the proposed optimization problem is formulated as:X. If overfitting occurs, the optimal w of Eq.(9) may correspond to a trait that has high heritability on the data that is used to train the linear model, but when the model is applied to a new sample, the trait has low heritability. In order to prevent overfitting and identify a trait with high heritability that can generalize, we incorporate a regularizer R(w) in our formulation (9). The optimization problem becomes:According to statistical learning theory , only maλ is a hyper-parameter and needs to be tuned, and λ can either be chosen by users according to domain knowledge or determined using a crossvalidation process as done in our experiments. According to learning theory T , H = , and then we have Xw = γH. By the change of variables, Problem (11) can be equivalently re-written as:Due to the use of the f denotes the objective function, g's denote the constraints, and e = 2d + 1, indicating the number of constraints in that group. It is straightforward to show that Eq.(12) is equivalent to Eq.(11) in the sense that at optimality w = u − v =γ(1 : d) γ− (d + 1 : 2d). When Eq.(12) reaches optimality, at least one of the two components i uand i vat ny i-th position of the two vectors will be 0. Otherwise, by setting i ≥ viu, or i < viu, we obtain a better solution withu, v). Therefore, at optimality, where γ can be calculated as:Eq.(12) is not a convex problem because of the quadratic equality constraint. However, it can be efficiently solved using a sequential quadratic programming (SQP) algorithm  because α be the Lagrange multipliers, the Lagrangian function of this problem can be written as:Let γ is computed as:and the Hessian of the Lagrangian function with respect to t-th iteration, we have the iterates γt and αt, and we first solve the following quadratic program to find the moving direction for γ and α,We iteratively search for the optimal solution to Eq.(12). In the γ, along which the objective of Problem (12) can be decreased. Letα is calculated ass and update γ and α as follows:The optimal solution ribed in  to find We summarize the proposed algorithm in Algorithm 1. It has been proved that a SQP based algorithm can converge to a local minimizer lem (12) .Algorithm 1 A sequential quadratic programming approach to solving Problem (11)We validated the proposed approach in both simulations and the analysis of a real-world data set that was aggregated from multiple genetic studies of cocaine dependence (CD).Semi-Structured Assessment for Drug Dependence and Alcoholism (SSADDA) dataset aggregated from genetic studies of drug dependence to evaluate the proposed algorithm. The SSADDA subjects were recruited from multiple sites, including the University of Connecticut Health Center, Yale University School of Medicine, the University of Pennsylvania School of Medicine, McLean Hospital and the Medical University of South Carolina. All subjects participated using procedures approved by the institutional review board at each participating site. There were 6,621 subjects genotyped with a total of 1,140,420 SNPs genome-wide. Among the subjects, 2,674 were stratified into the African American population using STRUCTURE software v2.3 , used in our experiments. Then, the fifth feature was directly computed as y1) with h2 of 0.8. Hence, if our approach worked, it should at least find this linear combination if there was no any other one that gave even higher h2. Note that the heritability of the fifth feature had to depend on the empirical estimation, but given how it was created, there were genetic effects in this feature.We then created 45 other features that had no genetic effects, and added a certain number of these features to the original 5 features to create 5 other datasets. Hence, there were in total 6 datasets for 1,752 subjects with 5, 10, 20, 30, 40 and 50 features. We used this set of data  in training to retrieve the combination models. Then we repeated the above procedure to create another independent set of data  to validate the resultant models.Setting 2: This setting assumed that the two covariates, sex and age, had fixed effects to the features and the final trait. We generated 5 features by adding fixed effects to the 5 useful features created in Setting 1. Because fixed effects do not change h2 of a trait, we computed a composite trait y2 using the same pre-specified weight vector w that was used in Setting 1. We then created 45 other features with only covariate effects using the procedure described early on. Five other datasets were generated consisting of 10, 20, 30, 40 and 50 features. Note that the optimal weight vector for these datasets should have zero entries for all features except the first 5 features that were synthesized. Similarly, a discovery suite of the six datasets and another suite of them were synthesized using the same procedure for training and validation, respectively.h2 of the features created in the synthetic datasets using GCTA software. The four features synthesized with a pre-specified h2 = 0.2 had empirical chip h2 values 0.2 ± 0.01 in these datasets. The chip h2 estimate of the fifth feature was 0.57 in the discovery set and 0.6 in the validation set. Because fixed effects do not affect trait heritability, the five relevant features and the final quantitative traits had the same empirical chip h2 in Settings 1 and 2. The features simulated with no genetic effects had h2 estimates that ranged from 0 to 0.05, and most of these features had estimates less than 10−5.We estimated the chip We first validated the proposed approach in a variety of experiments with the synthetic data. Then we applied our approach to the real-life cocaine use data to identify important components or subtypes of the disease defined by linear combinations of clinical features. Such a combination can be used to define a disease subtype because it produces a quantitative trait for each individual, which amounts to the membership likelihood of the individual in a subtype. Because the actual causal variants were known for synthetic data, we calculated the GRM of the individuals directly using the causal variants. In the case study for CD, because the real causal variants were unknown, we followed the commonly-used procedure in the literature on chip heritability estimation  and compTuning of the hyperparameter: For both the simulation and the CD case study, we performed 10 times three-fold cross validation (CV) to help determine a proper value of λ. At each fold of the CV, a linear model was derived by running the proposed method on 2/3 of the data in the dataset, and then tested on the remaining 1/3 of the data. The cross validated h2 of the derived trait was estimated using only subjects in the remaining 1/3 of the data which was not used to train the model. We ran the same CV process for each pre-specified choice of λ (the choices we used are reported in the results section) and chose the λ value that gave a trait of the highest cross validated h2 for each experimental setting.Evaluation metrics: We reported and investigated the CV performance  for each λ choice in each experiment. Once the best value of λ was chosen through the cross validation for a dataset, we applied the proposed approach with the best λ to the entire data in the dataset to derive a quantitative trait. The chip heritability of this trait was estimated using the separately-synthesized validation datasets in simulations and by another cross validation process in the case study. In other words, in simulations, we estimated the valiation chip h2 using the trait values computed by the linear model on the newly-synthesized validation samples. In the CD case study, we computed the trait h2 using SNPs of 2/3 of the subjects randomly sampled from the dataset and repeated the random sampling 10 times to report the averaged h2 value. We named this process the evaluation CV. Moreover, besides the heritability as a major evaluation metric, we also measured the effectiveness of our approach by comparing the derived trait models and the linear model implanted in the simulated data. We calculated the squared difference between the learned weights w, i.e., Comparison: The validated chip h2 of our derived traits was compared with that of all quantitative features in the data in both simulations and the CD case study. In each of the experiments, our derived trait was also compared with the commonly-used disease phenotype, often referred to as a symptom count, which was the quantitative trait created by equal weighted aggregation of all features in the data. Given that no prior method existed to identify heritable disease components using the genome-wide SNPs, on the real-life cocaine use data, we compared our approach with a recently published method [h2 of the two approaches using the evaluation CV. Note that the prior pedigree-based approach was actually given a favor because it used the superset of 2,674 subjects  to derive the trait in comparison with our approach that used only the 1,752 unrelated individuals.d method  that aimd method  to derivClinical interpretation: It is very important to understand the clinical implications of the quantitative trait  derived by our approach from the aggregated CD study data. From prior work [ior work ,31, we iλ values ranging from 0 to 0.04 with step size 0.002 for use in the cross-validation tuning process. The validation or test h2 for each λ choice was plotted for each of the six datasets in Figure 1 (for Setting 1 where data were generated without covariate effects) and Figure 2 (for Setting 2 where data were generated with covariate effects). The mean, median, and the standard deviations of the test h2 values in the cross validation were plotted for each tested λ. These two figures show that our approach could identify components (quantitative traits) with test h2 estimate of ~ 0.8, which was the heritability of the implanted heritable component (the simulated true model), for all datasets even with many irrelevant features in some of the datasets for both settings. This result demonstrates that our approach identified highly heritable disease components and could successfully correct for fixed covariate effects.We pre-specified 21 different λ was too small, especially when the number of irrelevant features grew to 25, 35 and 45. The larger the number of irrelevant features in the data, the more severe was the overfitting seen when λ was small. On the other hand, when λ was too large, underfitting could occur, so the test h2 showed a peak in most of the plotted curves. The best choice of λ for the six datasets  were 0, 0, 0, 0.002, 0.004 and 0.004 for Setting 1, and 0, 0, 0, 0,002, 0.004 and 0.006 for Setting 2.In addition, both Figures 1 and 2 show that there was overfitting of the learned models to the training data when λ was set to the optimal value for each of the six datasets, and we re-ran the algorithm to generate the final quantitative trait from each dataset. We then compared the chip h2 between these derived traits and the commonly used disease traits, such as individual features and the equally weighted aggregation of individual features. The results are shown in Figure 3 where all h2 values were estimated using only the validation datasets. Results from both settings are shown. For all the datasets and for both settings, our approach could recover the quantitative traits of h2 close to 0.8. When the number of irrelevant features in the data increased, the h2 values of the derived traits decreased as expected. Typically, when more irrelevant features were included in the data, the learning problem became more challenging. Because covariates do not affect h2 estimate when their effect is properly corrected during the estimation, we did not differentiate the two settings when discussing and comparing the h2 values of individual features and the aggregated traits. Recall that in our simulation, the highest chip h2 of individual features was 0.6 in validation. Figure 3 also included this most heritable feature for comparison. Among the traits derived by equally weighted aggregation, the one developed from the 5-feature dataset reached the highest h2 (= 0.66). As expected, chip h2 of these traits decreased along with the increasing number of irrelevant features. The trait developed from the 50-feature dataset had the lowest h2 (= 0.28). These results demonstrate that our approach could identify quantitative traits that are more heritable  than those commonly used. The squared difference (error) between learned weights and the optimal (implanted) weights: SE(w) together with the squared error between the derived traits and simulated traits: SE(y) are presented in Figure 4. The results from both settings are provided. We observed that clearly when the number of irrelevant features increased, the noisier data made the learning problem more difficult, and SE(y) and SE(w) increased in both of the settings.Then λ values in the simulations. In all expriments, we used age, sex and the first three principal components of the GRM as covariates. The test h2s of all traits derived for each λ choice in the CV process are plotted in Figure 5. It shows that there was overfitting when λ was too small as well. The trait derived with λ = 0.004 achieved the highest cross-validated h2 on average. We thus derived a model by running our approach with λ = 0.004 and all the 1,752 subjects in the data. We examined this model and the resultant quantitative trait as discussed in the proposed analyses section.In this study, we used the same pre-specified −5), thus its impact on the resultant trait was minimal. Variables with significant coefficients in the model are the seven cocaine criteria: F3 - using cocaine in larger amounts or over longer period than intended, F2 - withdrawal from cocaine, and F5 - great amount of time spent in activities necessary to obtain, use or recover from the effects of cocaine, had the highest impact on the trait. Both F3 and F5 had negative weights, which indicated that positive response to these two variables would lower the score or value of this trait. In contrast, F2 had a positive coefficient, which indicated that a positive response to this variable would increase the score. Variables F4 - persistent desire or unsuccessful efforts to cut down or control cocaine use, and F1 - tolerance to cocaine, had limited impact on the trait.The weights that each variable received in the model are shown in Figure 6. Of the 13 clinical variables, five (F8 - F12) received a zero coefficient, and were completely ruled out from the model. Variable F13 had an coefficient close to 0 (< 10h2 estimate of the trait derived by the proposed approach is 0.87 (with a standard error of 0.13). For comparison, we ran the approach proposed in [λ. For fair comparison, we first ran 10 times cross validation to choose a proper value for λ. With this λ, we developed a trait using the entire African American sample set. We then estimated the trait h2 using the exact same setting  as for the traits derived by the proposed approach. We estimated the h2 for all six continuous variables in the data using the same setting. All of the h2 values are plotted in Figure 7 together with the trait derived by the proposed approach. The figure clearly shows that the trait derived by the proposed approach had the highest h2 among all compared traits, and was significantly higher than that of the trait derived using the prior approach [h2 close to zero when estimated using our data. These results demonstrate the effectiveness of our approach in identifying disease components with high chip heritability from complex multivariate phenotypes. It is worth noting that the trait with the second highest h2 estimate was the one derived using the prior approach [The cross validated posed in , which iapproach . Note thapproach . This imapproach ) can ideFigure 8 shows the distribution of the trait values  of the subjects. It shows that based on the scores, the samples can be partitioned into four subgroups. There were 250 subjects  in Group 1, which had a mean score of -2.22. Group 2 consisted of 323 subjects and comprised 18.44% of the entire sample set. Its mean score was -0.8. Group 3 was the largest one and consisted of 821 subjects (46.86% of the sample). The mean score of this group was -0.2. Group 4 was the smallest, comprised of 237 individuals (13.53% of the sample), with a mean score of 1.22.To understand the clinical implications of the derived trait, we characterized the four groups using 11 important clinical variables, including the 7 CD criteria (F1-F7), the total number of CD criteria endorsed (F8), age of first cocaine use (F9), age when first diagnosed with DSM-IV CD (F11), and the transition time in years from first cocaine use to first DSM-IV CD diagnosis (F13). The results are summarized in Table 1. Only 5.6% of the subjects in Group 1 had experienced cocaine withdrawal symptoms (F2), despite the face that this group contained heavy cocaine users (as shown by other variables). Most of the subjects (99.6%) in this group reported using cocaine in larger amounts or over a longer period than indended (F3). Moreover, this group had the highest percentage of subjects (96%) who spent a great amount of time in activities related to cocaine (F5). Group 2 had the lowest percentage of subjects (23.6%) with tolerance to cocaine (F1), but the highest percentage of subjects (92.92%) who used cocaine despite knowledge that problems were likely caused by cocaine (F7). Subjects in Group 3 endorsed all of the seven CD criteria with similar percentages (87.04% was the lowest and 90.60% was the highest among the seven criteria). Group 4 had the lowest percentage of subjects who endorsed F3, F5 and F7. Group 4 also had the highest percentage of subjects with persistent desire or unsuccessful attempts to cut down their cocaine use (F4). Group 1 and Group 2 had a similar transition time from first cocaine use to first CD diagnosis (F13): 8.01 years for Group 1 and 8.07 years for Group 2. These were significantly shorter than the transition time in Groups 3 and 4, which were 12.15 years and 15.19 years, respectively.y = Xw. A quadratic optimization problem was formulated, in which optimal w was sought to optimize the log likelihood for estimating variance components in REML. In this formulation, variance components are set to their ideal values with the additive genetic variance component h2 than commonly-used disease phenotypes.We developed an approach to identify composite traits from multivariate phenotypes that are highly heritable, as estimated using genome-wide SNPs. The trait we derived is in the form of a linear combination of variables related to the phenotype, that is In this paper, the pairwise genetic relationship among subjects was estimated from genome-wide SNPs. However, it can also be estimated from SNPs restricted to a specific region, such as on a particular chromosome or in genes related to a pathway, to explore the genetic architecture of a trait. When SNPs within a specific region are used, the trait resulting from the proposed approach will achieve the maximized genetic variance component corresponding to this region. In an application, such as substance dependence, there are known pathways involved, so it may be of utility to determine whether there is a composite trait, the variance of which can be largely explained by the variants within the pathways. This will be a future application of our approach.The authors Jiangwen Sun and Jinbo Bi declare that they have no competing interests. Henry R Kranzler has served as a consultant or advisory board member for the following companies: Alkermes, Lilly, Lundbeck, Otsuka, and Pfizer. He is a member of the Alcohol Clinical Trials Group of the American Society of Clinical Psychopharmacology, which is supported by Abbvie, Alkermes, Ethypharm, Lilly, Lundbeck, and Pfizer.JB, HRK and JS designed the overall study of finding heritable disease subtypes. JS and JB designed the new algorithm discussed in this paper. JS implemented the algorithm and performed the data analyses. HRK provided the aggregated drug dependence data and helped in interpreting the results. JS and JB wrote the first draft of this manuscript.


We demonstrate the flexibility of identity-by-descent (IBD) graphs for genotype imputation and testing relationships between genotype and phenotype. We analyzed chromosome 3 and the first replicate of simulated diastolic blood pressure. IBD graphs were obtained from complete pedigrees and full multipoint marker analysis, facilitating subsequent linkage and other analyses. For rare alleles, pedigree-based imputation using these IBD graphs had a higher call rate than did population-based imputation. Combining the two approaches improved call rates for common alleles. We found it advantageous to incorporate known, rather than estimated, pedigree relationships when testing for association. Replacing missing data with imputed alleles improved association signals as well. Analyses were performed with knowledge of the underlying model. These estimates are computationally intense: for example, 727 cpu minutes for family 10 (83 members) on an Intel L5427 Xeon 2.50-Gz processor. However, the resulting sampled IBD graphs can be quickly reused for several types of analysis, including genotype imputation in pedigrees [Patterns of identity-by-descent (IBD) sharing within and across pedigrees are fundamental for the understanding of genetic variation, including its distribution, origin, and relationship to phenotype. Recent analytical and computational advances have allowed us to estimate the distribution of patterns of IBD sharing in large and complex pedigrees using the program gl_auto in the MORGAN v3.1 package (http://compgen.rutgers.edu/maps), excluding the 116 loci missing values. Kosambi positions were converted to Haldane positions to suit assumptions made by the Lander-Green algorithm [We analyzed GAW18 marker data for chromosome 3. We did not use the GAW18 sequence data because it included imputed variants, although our methods would work for sequence data as well. We obtained genetic map positions (cM) for the genome-wide association studies (GWAS) markers from the Rutgers sex-averaged interpolated positions of dbSNP Build 134 , only samples from family 10 (experiment F10), and the other families without family 10 (experiment FO). BEAGLE's 3,621 scaffold markers were chosen to be common (MAF >0.3) and evenly spaced (at least 0.05 cM apart). As with GIGI, we called both alleles of a missing genotype if Pr(genotype) greater than 0.8 and otherwise called one allele if Pr greater than 0.9.For comparison, we also used BEAGLE , which uWe imputed genotypes on family 10. We masked most genotypes in a subset of subjects for evaluation of imputation metrics  greater than 0.99 or to call one allele if Pr greater than 0.995, thus only calling alleles if essentially forced by the pedigree data. For loci with uncalled genotypes, we then used results from BEAGLE F10 + FO with call thresholds Pr(genotype) greater than 0.8 and Pr greater than 0.9. For loci with one uncalled allele by GIGI, we accepted the BEAGLE genotype if it included the single allele called by GIGI.We computed lod scores for family 10 and all cryptically related families at a subset of 44 positions from the IBD graphs, yielding a spacing of approximately 1 lod score per 5 cM. To obtain multipoint lod scores, we (a) used the program IBDgraph ,14 to idWe tested three parametric models. Model 1 is a quantitative trait locus (QTL) model with parameters defined by the single-nucleotide polymorphism (SNP) with the biggest contribution to the simulated trait variance. Because this SNP explains only 0.0229% of the simulated trait variance, model 1 tests whether we can detect a locus with a small effect size if it is modeled perfectly. Model 2 is a QTL that is the weighted average for all functional SNPs within the gene bearing the "biggest" SNP. The result is a common allele with small effect sizes and is an attempt to model the cumulative effects of several functional variants within a single gene. Model 3 is a perfectly penetrant additive locus, where affectation status indicates the subject carries the risk allele at the biggest SNP, and tests whether we could detect the SNP locus if it perfectly explained the trait variance. We compare results from the same IBD graphs with a typical variance components (VCs) lod score, as implemented by SOLAR .p-value, identified from a half genome scan [We used VC analyses to investigate association with the trait of candidate covariate SNPs while accounting for correlations among related subjects. We analyzed family 10 alone, as well as all 7 families jointly. Each of the top 5 SNPs, ranked by ome scan , was tesome scan  estimateWe also performed VC analyses with SOLAR with various combinations of imputed and observed genotype data within family 10 to evaluate the usefulness of imputed genotype data. For these analyses, we used the weighted average of genotype probabilities obtained from GIGI to provide an expected dose of the minor allele, given the observed data.only an outside reference sample for BEAGLE was ineffective for imputation of rare alleles that may be family specific, and (d) BEAGLE was able to call a small fraction (<20%) of rare alleles that were not observed in the genotyped subjects in family 10, but only when BEAGLE had access to reference samples from other families.Both GIGI and BEAGLE F10 + FO) achieved similarly high overall call rate  and accuracy  on the masked subjects in design 1. As demonstrated elsewhere 0 + FO ac, accuracNot surprisingly, both GIGI and BEAGLE called fewer genotypes starting with the design 2 data than the more complete design 1 , with high accuracy . Accuracy for rare variants was 99.5% for GIGI and 99.0% for BEAGLE, and accuracy at the most common variants was 99.9% for GIGI and 97.1% for BEAGLE. Combining GIGI and BEAGLE boosted both call rate (89.5%) and accuracy (98.6%) over the use of either alone, but only for more common variants. GIGI + BEAGLE continued to have high accuracy across MAF, with an accuracy of 99.3% for rare alleles and 98.6% for the most common alleles. Figure max = 5.36), this locus does not explain enough phenotypic variation within this family to provide measurable evidence of linkage (lodmax <0.5 for models 1 and 2). VC lod score analyses [Although enough copies of the risk allele segregate within the family to generate a linkage signal if the risk allele was indeed causal , although this was not true of the other simulated replicates. In contrast, family-based association testing with a mixed model was still able to detect association with the functional variants. We found that using the known pedigree structure in SOLAR provided similar but slightly stronger evidence for association than EMMAX, which treats subjects as unrelated but accounts for relatedness through an empirical covariance matrix. Finally, use of observed genotype data provides a stronger association signal than imputed data, although the difference between the two sets of The authors declare that they have no competing interests.All authors participated in study design and analysis. EMB and EMW drafted the manuscript, and all authors edited and approved the final manuscript.


Pseudomonas syringae pv. actinidiae is a serious disease of kiwifruit in New Zealand and other kiwifruit-producing countries. Data from a progeny test trial was available to identify parents with high breeding values for resistance. We successfully applied the GLIMMIX procedure for this purpose. Heritability for resistance was moderate, and we identified two parents and their family as having high potential for Psa resistance breeding. There are several potential pitfalls when using GLMMs with binary data and these are briefly discussed.Linear Mixed models (LMMs) that incorporate genetic and spatial covariance structures have been used for many years to estimate genetic parameters and to predict breeding values in animal and plant breeding. Although the theoretical aspects for extending LMM to generalised linear mixed models (GLMMs) have been around for some time, suitable software has been developed only within the last decade or so. The GLIMMIX procedure in SAS® is becoming popular for fitting GLMMs in various disciplines. Applications of GLMMs to genetic analysis have been limited, probably because of the complexity of the models used. This is particularly so for Proc GLIMMIX because, unlike ASReml software, it is not specifically tailored for analysis of breeding data and some pre-procedure coding is necessary. Binary data that fits the GLMM framework is commonly encountered in breeding experiments, such as when evaluating individuals for resistance by observing the presence or absence of disease. Bacterial canker (Psa) caused by  Pseudomonas syringae pv. actinidiae (Psa) is a pathogenic bacterium of kiwifruit (Actinidia spp.). The virulent form Psa-V is now well established in New Zealand following its first detection there in November 2010 . The disease has been very damaging to the New Zealand kiwifruit industry: the cost in net present value is expected to be between 310–410 million NZ$ over five years  A. chinensis have been reportedly showing higher resistance to Psa than diploid (2x) populations  of genotype random effects on continuous traits by fitting linear mixed models (LMMs) to familial data is well established. Statistical models with complex variance structures that account for pedigree as well as spatial trends within a field layout have been extensively applied to such data to assess if the trait of interest has a significant genetic component and is heritable  of McCullagh and Nelder  extendedA. chinensis parents and provide reliable estimates of additive genetic and environmental variance components and narrow-sense heritability in relation to susceptibility to Psa. For reasons explained above, we used GLMM methodology applicable to binary/binomial distributed data. The fundamentals of GLMM were developed some time ago, but its implementations in widely available statistical software happened much later. The R lme4 package  or tetraploid (4x = 116). Families included in this study came from a factorial mating design where four female parents  were crossed to 19 male parents (numbered 28:46). This is the standard notation used in kiwifruit breeding in New Zealand, with females in a cross named by two letters, males by numerals and the family by concatenating the two labels. Some crosses were untried or unsuccessful: female GO, with the exception of crosses to males 32 and 33; male 32, apart from the cross to female GO; and female GU to male 33. For fitting the GLMM, we left out parents GO and 32 to provide a nearly balanced factorial crossing structure  for the dataset with only one missing cross (GU × 33). Of the 55 full-sib families, 53 were retained in the subset that excluded one female and one male parent. All the parents used in crosses were tetraploid. The field experimentation was carried out at the Te Puke Research Centre in the Bay of Plenty (37.8°S 176.3°E), the major kiwifruit-growing region in New Zealand. On average about 36 seedlings from each full-sib family were field planted in spring (October) 2008 in a randomised block design with three replicates; each replicate comprised three consecutive rows except for the last replicate which had an additional row to accommodate extra seedlings of 8 families. The experimental layout, therefore, was not balanced. The family sizes varied from 24 to 50 depending on seed availability. The 10 rows in the experimental block ran in a south–north direction and contained posts spaced at 4 m within a row and with 3 m between rows. Progeny testing trials are usually planted at higher densities than commercial orchards. The experimental unit (plot) here was the bay between two posts within a row, where 12 unsexed seedlings of any one family were planted as a twin row, with seedlings alternating on either side of the trellis wire at six positions (a:f) placed at 60-cm intervals. There was a 40-cm distance between the seedlings in each twin row. A hedge of shelter trees was present on the west and south side about 4 m away from the experimental block. There were 19 bays (numbered 0:18) down each row of seedlings, with the first and last bays containing only two planting positions (a and b). Only half the seedlings planted were expected to be female, but both females and males can be assessed for Psa symptoms. Control plots of the A. chinensis cultivars ‘Hort16A’ and ‘Zesy003’ were included as end of row guards within the experiment.P and Q in terms of the additive and dominance genetic variances is Cov = rVA + uVD; and for diploids r = 2fPQu = fACfBD + fADfBC, where fPQ is the coancestry of P and Q etc., and A & B and C & D are parents of P and Q respectively  as follows: 0 = no symptoms, 1, 2, 3 = leaf spotting and damage at increasing intensity, 4 = oozing, with a qualifier to the score based on which plant part is involved: s – shoot, d - main leader and b – bud. 5, 6, 7, 8 = vine died and removed. The decision to remove plants with a Psa score higher than 4 was made to reduce inoculum levels and followed the recommendations of Kiwifruit Vine Health, the industry body charged with managing the disease. Monitoring the movement and progress of disease through the block was carried out fortnightly using the scale described, in 2010 and subsequent seasons. A preliminary analysis of the trends in disease score over time has been reported by Gea et al. . For theA heatmap is a graphical representation of data values in a matrix by a colour scheme. Figure The generalised linear mixed model for Bernoulli or binomial outcomes is of the formy is the n × 1 vector of outcomes (0 or 1), g(.) is the link function which relates outcome y to the linear predictor η, and g− 1(.) is the inverse of the link function. β and u are vectors of fixed and random effects respectively, with the corresponding design matrices X (N × p) and Z (N × q), and yi is distributed according to any one of the exponential family of distributions. The GLIMMIX procedure distinguishes between random effects in the linear predictor (G-side) and/or modelling correlations among the data directly (R-side) . In the case of GLMM, the distribution of data y is specified conditional on random effects u. The variance of y is a function of the mean for most distributions of the exponential family and is given by Var(y|u) = A0.5RA0.5, where A here is a diagonal matrix containing the variance functions  and ∑ y ~ Binomial  respectively, where Pr(y = 1) = π. We used the logit link function in both instances, i.e. loge(π/(1 − π)). The following is a detailed specification of the GLMM used for Psa data where the response variable is binomial. As described earlier, Psa susceptibility scores were categorised as binary (0/1) at the vine unit level. As such, our observations for each plot (i.e. row x bay) consisted of the number of seedlings that were diseased out of n total seedlings. The sample sizes were small. Of the 169 plots, 149 had a full complement of n = 12 seedlings; for the remaining plots n varied from 4 – 10. The resulting linear model for the disease count yij(klr) of the plot at the ith row and jth bay (or column), with the notation indicating it is the rth replicate sample plot of the family generated by crossing the kth female parent with the lth male, is:For the Psa data, appropriate distributions in the case of individual seedling and plot outcomes are r-th replicate full-sib sample of kl-th parents and A is the numerator relationship matrix between family samples; where the row effect Z to be generated outside the procedure. For BLUP breeding value estimation in the plot aggregate analysis, the genetic units of interest are the family replicate samples of the progeny, and the parents. We fitted what is generally called the ‘animal model’ to estimate the additive genetic variance. Replicate plots of a given family are different samples of the same full-sib family; hence the resemblance between sample means, given the two parents, is that of full-sibs. Writing the SAS code to construct variables for additive random effects of Z, we first used the GLMMOD procedure to generate dummy variables zx + 1, …, zx + N,where N = number of plots or vines depending on the model fitted, and x = number of parents. Variables were numbered in the same order as plots were arranged in the phenotype dataset, i.e. bays within rows. Then, we added x additional variables  with all elements equal to zero to represent the additive random effects of parents who had no phenotypic values, but who needed their breeding values to be predicted by the model. The coefficients of the covariance matrix of random variables z1 − zx + y + N, where y = number of ancestors of the parents, is given by the additive relationship matrix A. This was constructed using Proc INBREED as discussed in a previous section, followed by manipulations as described below to form a SAS dataset we named ‘Adata’. For this we first updated the A by deleting from it the corresponding rows and columns for the y number of ancestors of parents for which we were not interested in estimating breeding values. Secondly, we added in two numeric column variables named PARM and ROW before columns Col1-Col(x + N) of the updated A to form ‘Adata’. Since we were interested only in the additive variance, PARM had a value = 1 for all rows in ‘Adata’, which were numbered 1 to N in the ROW variable. Now, the GLIMMIX procedure statement for estimating the additive genetic variance and breeding values has the syntax: RANDOM Z1-Z(x + N)/TYPE = LIN(1) LDATA = Adata SOLUTION. Other random effects shown in the specified model were simply coded by a separate RANDOM statement. A similar application in genomic selection using SAS Proc MIXED and its LIN(1) structure is given by Piepho et al. . Unlike linear models in GLMM, the variance is a function of the mean, and as such the scale parameter is set to unity and there is no separate estimate of residual variance from the fitted model. The default link function for binomial data is logit, hence the residual error variance π2/3  = φ × Var(μ), and the scale parameter is estimated from the data. If the scale parameter is significantly different from unity, the residual variance of the above equation should be set to where variance components apart from As shown in Eq. u. The BLUP estimates output by analysis are on the liability scale (logit), which can be transformed to predicted probabilities after adding the estimate of the intercept, p. The breeding values as such given as predicted probabilities to disease susceptibility were used to compare parents.The key BLUP values of interest in this study were the breeding values of the parents. The GLIMMIX procedure produces estimates of prediction variability for random effects F = 0.25; its parents were full-sibs of unrelated ancestors 17 and 4). Male parent 37 descended from a cross between the male vine 20 and its brother’s (17) daughter (GT), which resulted in an F value of 0.125. Similarly, the male parents 41–46, which were full-sibs and whose parents (23 and 22) were half-sibs of the common female ancestor 4  ranged from 0 to 0.375 for the 21 parents. Female parent GU had no known relationship to any of the individuals in the pedigree  for each of two levels of the dataset, i.e. individual vine and family plot mean. After truncation, the ARM for the 1882 phenotyped vines and their parents was of dimension 1903. Similarly, the 169 plot aggregates gave a matrix which was much smaller (190). Because the mating design was a factorial and the parents forming the crosses were not all unrelated, the ARMs constructed were reasonably dense with a large proportion of non-zero elements.As seen in the pedigree Figure , there wGiven the relatedness matrices as calculated above, Proc GLIMMIX fitted the generalised mixed model as specified earlier to each of the individual and bay aggregate phenotypic datasets. The two models are hereafter called the Bernoulli and binomial, respectively. Both models estimated family × replicate interaction variance to be zero. Furthermore, the variance of the replicate effect was estimated as zero in the Bernoulli model, and very small and non-significant in the binomial model. Models were re-fitted by dropping out both these effects. Since rows were nested within blocks and both rows and bays were still in the model, we would not expect a loss of information on variation due to spatial effects in this reduced model. The GLIMMIX procedure code was implemented in SAS 9.4 (64 bit) installed on a laptop i5, 2.80 GHZ machine with 8 GB of RAM running the Windows 7 (64 bit) operating system. Both models achieved convergence; the binomial model took only 5.42 s user CPU time compared with the Bernoulli model with the much larger dataset, which took 13 h and 8 min. The GLIMMIX procedure provided information about the fitted model in terms of three statistics: − 2 Res Log Pseudo Likelihood, Generalized Chi Square and the ratio of the latter to its degrees of freedom. This ratio was close to 1, 0.80 and 0.92 respectively, for the Bernoulli and binomial, indicating no evidence for residual over-dispersion. Residuals are used to check if data meets the assumptions of the model and to detect any outliers and influential points. Plots of conditional studentized residuals of the binary model are shown in Figure Variance components estimated by the Bernoulli and binomial models are presented in Table A. chinensis populations, Cheng . This family also had the largest predicted random effect for Psa resistance, which accounted for any non-additive gene effects on family performance. A prediction based on the average eBV of the two parents and the family random effect provided an estimate of 0.31 for Pr(Psa) for this family.Estimated breeding values of the three female and 18 male parents, on the scale of probability of Psa incidence, are presented in Table The methods used in this study show that fitting GLMMs to binomial distributed data where disease status is categorised as 0/1 is a useful method to consider to obtain genetic parameter estimates for disease incidence. The GLIMMIX procedure in SAS is a useful tool for fitting GLMMs where the dataset is small to medium scale. As the scale grows, increasing memory is required and the CPU time can extend to many hours, as was evident when fitting the Bernoulli model in this study. Rescaling the data by aggregating counts at the plot level and using a binomial model instead made the convergence quicker, by as much as four orders of magnitude. We attempted to fit a GLMM that, in addition to the genetic covariance structure, included a spatial covariance. Spatial covariance structures can be specified in Proc GLIMMIX by the RANOM _RESIDUAL_ statement with options such as / TYPE = SP(EXP)(x y) where x and y here are the field coordinates for rows and bays. In principle one could also model spatial covariance on the G-side. The residual variance (R-side) would then correspond to a nugget effect. A detailed account of the application of spatial covariance in GLMMs with field experiments is given by Gotway and Stroup . Unfortun is small. A detailed analysis of the bias problem is given by Engel (n) are small. Gauss quadrature is more accurate to compute the likelihood but slower and can only be used with GLMM having a single random effect and hence could not help for the models considered in this study.Our analyses show, the use of GLMMs is not fool-proof, and there are several potential pitfalls. The failure to account for overdispersion is probably the most common problem. Convergence problems may be an issue and furthermore particularly with binary data GLMM fits may be severely biased, particularly when sample size by Engel  and a goby Engel . As the by Engel  algorithby Engel . The psux + c)/(1 − x + c)) / − x + c)x + c)/ − x + c)x + c)/ − x + c)n. Once calculated, the individual residual variances are averaged across all subjects and used in the heritability equation.In a GLMM framework, the definition of heritability of a binomial trait on the latent scale is problematic. The heritability equation  implies A. chinensis, our calculation of coancestry coefficients and estimation of the additive genetic variance component thereafter were based on the assumption of diploid inheritance. Autopolyploids such as tetraploid potato generally show polysomic inheritance compared with disomic inheritance in diploids and most allopolyploids. Even in allopolyploids, preferential pairing may not be perfect and some multivalent pairing is possible, which often leads to distortions in allele segregation. Recently, a detailed investigation of the pairing behaviour of tetraploid A. chinensis has been reported by Wu et al.  matrix as calculated in this study for genetic analysis of tetraploids should not give substantial bias in the additive variance estimate.Polyploid genetics is complex, with pairing patterns varying from polysomy to preferential to disomy and the possibility of double reduction in allele segregation, Estimators of even common population parameters become difficult to derive and often have to rely on large assumptions compared with those needed when dealing with diploids  described in this paper.standard error of heritability using delta method.Calculating the f = u/v, then the variance of the first order approximation of f(.) about their expected values μu and μv is given by Lynch and Walsh  = the vectors of coefficients of the numerator and denominator in the heritability equation, and Now let


Although genome-wide association studies (GWASs) have discovered numerous novel genetic variants associated with many complex traits and diseases, those genetic variants typically explain only a small fraction of phenotypic variance. Factors that account for phenotypic variance include environmental factors and gene-by-environment interactions (GEIs). Recently, several studies have conducted genome-wide gene-by-environment association analyses and demonstrated important roles of GEIs in complex traits. One of the main challenges in these association studies is to control effects of population structure that may cause spurious associations. Many studies have analyzed how population structure influences statistics of genetic variants and developed several statistical approaches to correct for population structure. However, the impact of population structure on GEI statistics in GWASs has not been extensively studied and nor have there been methods designed to correct for population structure on GEI statistics. In this paper, we show both analytically and empirically that population structure may cause spurious GEIs and use both simulation and two GWAS datasets to support our finding. We propose a statistical approach based on mixed models to account for population structure on GEI statistics. We find that our approach effectively controls population structure on statistics for GEIs as well as for genetic variants. Although genome-wide association studies (GWASs) have discovered numerous novel genetic variants associated with many complex traits and diseases, those genetic variants typically explain only a small fraction of phenotypic variance. Factors that account for phenotypic variance include environmental factors and gene-by-environment interactions (GEIs). Recently, several studies have conducted genome-wide gene-by-environment association analyses and demonstrated important roles of GEIs in complex traits. One of the main challenges in these association studies is to control effects of population structure that may cause spurious associations. In this paper, we show both analytically and empirically that population structure may cause spurious GEIs and use both simulation and two GWAS datasets to support our finding. We propose a statistical approach based on mixed models that can effectively correct for population structure when searching for GEIs. Over the past decade, genome-wide association studies (GWASs) have been a predominant approach to identify genetic variants involved in many complex traits and diseases.–3 While One major difficulty in association studies is that population structure can easily confound the studies. AssociatIn contrast to numerous studies that have analyzed effect of population structure on association statistics in real GWAS datasets, few studies have investigated its effect on GEI statistics empirically. There have been, however, a few studies that evaluated bias caused by population structure on GEI statistics through simulations. Wang et al. showed tIn this paper, we first show analytically that for the same reason that population structure causes spurious associations of genetic variants, it also causes spurious GEI associations based on the polygenic model. We show that disregarding sample structure can easily inflate test statistics for GEIs, leading to false positives. We then simulate a GxE GWAS using the 1000 Genomes Project dataset. This simIn addition to the simulation, we utilize two GxE GWAS datasets to show that population structure may cause serious effects on GEIs. One dataset is an expression quantitative trait loci (eQTL) study of the human aortic endothelia cell collected by Romanoski et al. and ErbiWe also propose a statistical approach based on a linear mixed model to correct for population structure on GEI statistics. We show that the traditional mixed model approach  that incK) from genotype data and the GxE kinship matrix (KD) from K and the environmental exposure. Phenotypes are generated such that the genetic kinship (K) explains 40% of phenotypic variance while the GxE kinship (KD) explains 20% (See λGC) should be close to one for both SNP and GEI statistics. We generate 100 replicates of simulation and measure inflation factors on SNP and GEI statistics of three different approaches. The first approach is one with no population structure correction on both SNP and GEI statistics (“OLS”), and another approach is a linear mixed model approach that incorporates the genetic kinship and accounts for population structure only on SNP statistics (“one RE”). The last approach is our proposed mixed model approach that uses both genetic and GxE kinship to correct for population structure on both SNP and GEI statistics (“two RE”).We generate a simulated GxE GWAS using two populations (GBR and TSI) of 1000 Genomes Project dataset. Each pop 20% See . There iλGC of the OLS approach is 1.032 and as high as λGC = 1.363. The results are similar when the ratio of exposed and unexposed individuals is different between the two populations. Population structure in the presence of GEIs may also cause inflation of SNP statistics, and λGC on SNP statistics tend to be higher than that on GEI statistics; the median inflation factor on SNP statistics is about 1.12. One of the reasons is that the genetic kinship (K) captures more phenotypic variance than the GxE kinship (KD) does in this simulation. The result demonstrates that both SNP and GEI effects are susceptible to false associations due to population structure.λGC becomes very close to one; the median λGC values on GEI statistics are 0.9969 and 0.9982 when the ratio of exposed and unexposed individuals between the two populations is the same and different, respectively. The maximum λGC values are also 1.026 and 1.0158, respectively. Interestingly, inflation factors on SNP statistics after applying our approach are even better than those after applying one RE; the median λGC with two RE is 0.9926 while that with one RE is about 1.02 when the ratio of exposed and unexposed individuals is the same. Hence, this shows that incorporating both kinship matrices also reduces inflation of test statistics on SNPs.The result of the simulation also indicates that we need to incorporate both genetic and GxE kinship matrices into the linear mixed model to correct for population structure on SNP and GEI statistics. While the one RE approach that uses only genetic kinship reduces inflation of test statistics on SNPs , it has λGC for each probe on SNP and GEI statistics of the three methods as in the previous simulation.To assess the influence of population structure on a real GxE GWAS, we first analyze the eQTL study of human aortic endothelial cell (HAEC)., 23 ErbiλGC values are greater than one than our approach. There are 2,687  and 2,509 (29%) probes with λGC > 1.02 according to OLS and one RE approaches, respectively, and the maximum λGC values are 1.492 and 1.498, respectively. After applying our approach, there are only 950 probes  with λGC > 1.02 and the maximum is 1.096. λGC values on SNP statistics are around or greater than 1.4 in the OLS approach have λGC < 1.4 after applying our method although the median λGC of our method is greater than one (1.0365).λGC and the variance of phenotype explained by the GxE kinship, denoted as r2 between λGC and λGC on SNP effects for phenotypes with higher r2 between λGC of the one RE approach (0.4620) is similar to that of the OLS approach  GWAS with population structure, we utilize HAPGEN2 software  to generK using the GCTA software from the simulated genotype data. To create a GxE kinship matrix KD, we first need to assign an environmental exposure to each individual. We assume a dichotomous variable where a half of individuals  are exposed and the rest  are unexposed. When assigning an environmental exposure to each sample, we consider two possible cases. One is that each population has the same number of exposed and unexposed individuals. In other words, each population has exactly 500 exposed and 500 unexposed individuals. The other case is that one population has more exposed than unexposed individuals. For example, the ratio between exposed and unexposed in one population is 0.6 while it is 0.4 in other population. This is possible in actual GxE GWASs when individuals in one population are more easily exposed to the environment than those in other population. We vary this ratio from 0.54 to 0.6 in one population. We consider both cases in our simulation to determine how they influence results. Once we decide on the number of exposed and unexposed for each population, we randomly assign the environmental exposure to each individual and create the GxE kinship matrix. The phenotype values are generated such that the genetic kinship matrix and GxE kinship matrix explain 40% and 20% of phenotypic variance, respectively. In other words, To generate phenotype values, we sample them from the following multivariate normal distribution.i.e. low GxE kinship) between 0 and 0.2, while holding the ratio between λGC) and the size of We also simulated our method at values of Because the sample sizes in the HMDP and HAEC datasets are rather low for GWAS, we wanted to demonstrate that our method can accurately detect the amount of GxE kinship.In order to do this, we performed simulations of populations with the same genetic kinship matrix as was computed from the HMDP dataset, and a phenotype distributed with an implanted GxE kinship component. The results, which demonstrate accurate estimation of this GxE component are in A alternate approach for correcting for population structure in gene-by-environmental interaction studies is to include principal components of the genetic relatedness matrix as covariates similar to the approach of EIGENSTRAT. We can extend such approaches to the scenario of gene-by-environment interactions by adding additional covariates of the form of the principal component times the environmental covariate (PCs × environment). We find that using PCs × environment does reduce inflation, but at a lesser extent than including a specifically GxE-based kinship matrix. This is consistent with comparisons of mixed models and principal components in traditional association studies . We demoMany of the phenotypes explored in the HMDP and HAEC datasets are close to normally-distributed but are not completely normal. We examine if this is a source of inflation by quantile normalizing the data. In We demonstrated that population structure may cause spurious associations of gene-by-environment interactions. Using the same argument that population structure can inflate test statistics for genetic variants in the polygenic model, we were able to derive analytically that the same phenomenon may occur for GEI statistics. We then used the 1000 Genomes simulation and two GxE GWAS datasets to observe the impact of population structure on GEI statistics. When the severe population structure exists as in the mouse GxE GWAS dataset, we observed very high inflation factors for GEI statistics. When the influence of population structure is relatively moderate as in the 1000 Genome simulation and the human eQTL GxE GWAS dataset, we found that test statistics for GEIs are nonetheless inflated, which may cause spurious associations. While Wang et al. showed tTo correct for population structure on GEI statistics, we proposed a linear mixed model approach that includes two random effects to take into account two types of similarities between individuals. One is the genetic similarity, and the other is the similarity caused by both genetic and environment. By incorporating two kinship matrices corresponding to the two similarities into linear mixed models, we were able to correct for population structure on GEI statistics successfully. We showed that accounting for only the genetic similarity controls the inflation of test statistics for SNPs, but not for GEIs. This is important because GWASs typically include only the genetic kinship matrix to correct for population structure. Sul and Recently, Zheng et al. studied The linear mixed model in our two RE approach is based on the GCTA GEI model , and we In this paper, we mainly focused on inflation of test statistics for measuring GEI effects of the three different approaches. We showed that only two RE approach achieves correct false positive rate while the two other methods do not. When comparing performance of different statistical tests, it is also important to compare power of tests in addition to measuring false positive rates. However, power comparison can only be made when all tests achieve the correct false positive rates. In our simulation and real data, it is not possible to compare power among the three different methods because OLS and one RE have incorrect false positive rate.We note that in our results, we are including the marker that we are testing in the kinship matrix. Recently, several studies have pointed out that including the tested marker in the kinship matrix effectively includes the marker twice in the statistical model and this is what causes the inflation factor to be one even when there are many genetic effects throughout the genome. Our resuM variants. Then, the standard genotype-phenotype model isyk is individual k’s phenotype value, μ is the mean of the phenotype, Xik is the genotype of individual k at variant i, βi is the effect of genetic variant i, and εk is the residual. The polygenic model assumes that there are many variants with small effects, which means that many βi’s are non-zero. The traditional association study considers each genetic variant individually and tests the effect of each genetic variant on the basis of the following model.β ≠ 0 since these are variants that putatively affect the phenotype. Note that we use different notations for the residual terms (εk in Xik). This violates the assumption of the traditional linear regression model in βr. Therefore, sample structure in GWAS datasets such as population structure or cryptic relatedness may cause inflation of test statistics (βr) for genetic variants.T and Xr = T where n is the number of individuals. u is the random effect in the mixed model that captures effect of population structure, and K is an n × n kinship matrix and I is an identify matrix of size n. Then, the total variance of phenotype is given as One approach to account for the sample structure is through the use of a linear mixed model., 30, 31 tructure., 26D. For simplicity, we assume it is a dichotomous variable. The exposure of an individual k to the environmental factor is denoted as Dk; Dk = 0 for the unexposed and Dk = 1 for the exposed. The model corresponding to δ represents the fixed effect of environmental factor D and each γj is the gene-by-environment interaction effect of variant j and environmental factor D. The goal of GxE association studies is to discover genetic variants or SNPs whose γj ≠ 0 because they have effects on the phenotype in the presence of the environmental factor. While it appears from Dk = 1, this is not always the case. For example, when βi = −γj and Dk = 1, a SNP does not influence the phenotype because SNP effects and interaction effects cancel each out. In this case, the SNP has effects on phenotype only for unexposed individuals (Dk = 0). Similar to the association study of genetic variants, we test the effect of each genetic variant and its GxE effect individually, which corresponds to fitting the following model.M: i≠rβiXik), and people who are genetically similar and are in the same environment, they would have similar value for the second sum (∑M: j≠rγjDkXjk). Hence, this equation shows that for the same reason population structure causes spurious genetic associations as shown in We extend the standard model to consider an environmental factor K), we define the matrix KD where each entry Di = Dj and KD describes how individuals are related both genetically and environmentally because a pair of individuals who are genetically related and share the same environment exposure have a non-zero kinship coefficient. We name KD “GxE kinship” and K “genetic kinship” to distinguish two kinship matrices. We propose the linear mixed model that incorporates both kinship matrices as followingD = T is a column vector of environmental exposures and D⋅Xr is the element-wise product. The random effect v accounts for the relatedness of individuals due to GEI effects and y is then given as two RE” because it uses two random effects to correct for population structure on both SNP and GEI statistics.We extend the linear mixed model approach to correct for population structure on GEI statistics by introducing an additional random effect that captures the similarity of individuals due to GEI effects. Given the kinship matrix  and a GEI effect (γr) and let X include all covariates including a SNP (Xr) and a GEI (D⋅Xr). Then, the estimated β isF-test for the null hypothesis βr = 0 and γr = 0 to obtain p-values for SNP and GEI effects, respectively. We provide a software package that implements the two RE approach at http://genetics.cs.ucla.edu/pylmm/. Our approach can efficiently be applied to standard GWAS datasets that contain thousands of individuals and hundreds of thousands of SNPs, similar to the linear mixed models for GWAS . Two of the most natural choices that satisfy the above recommendation are: R is the range of the environmental exposures, or μD and σD are the mean and standard deviation of the environmental exposures, and Φ is the standard normal cumulative distribution function. However, the best choice of f will depend on the scale of the environmental exposure.It is very attractive to extend this model to the case of continuous covariates; however, it is not straightforward to create Erbilgin et al. performeThe gene expression on 18,630 probes is collected using Affymetrix HT HG-U133A microarrays. The COMBAT software was utilized to correct for batch effects in the expression data [−4, and genotype completeness of 5%, and 575,042 SNPs in autosomes are tested for associations. Erbilgin et al. performed a principal component analysis to identify population structure among the 147 individuals with 11 HapMap3 populations and found that there are groups of individuals with different ethnicities.SNPs are genotyped using Affymetrix Genome-Wide Human SNP Array 6.0. We used the same QC filters as in the original paper : MAF of Hybrid Mouse Diversity Panel (HDMP) consistsS1 FigA, the same number of exposed an unexposed individuals were generated in each simuation; in B, fewer exposed individuals were simulated compared to unexposed individuals.Note that the scale is different from (TIF)Click here for additional data file.S2 FigNote that the scale is different from (TIF)Click here for additional data file.S3 FigNote that the scale is different from (TIF)Click here for additional data file.S4 FigThe GxE variance (TIF)Click here for additional data file.S5 FigA, we apply no quantile normalization; in B, the whole sample is first quantile normalized; in C, each environmental group is quantile-normalized separately. There is an apparent increase in performance of OLS and 1RE methods (their λGC becomes closer to 1) using quantile normalization, however, the performance is not as good as 2RE methods. There is little appreciable difference between the population-wide quantile normalization and within-group quantile normalization.In (TIFF)Click here for additional data file.S6 FigThe bias is low, and the variance of the estimate appears to be somewhat proportional to the size of (TIF)Click here for additional data file.S7 Fig(TIF)Click here for additional data file.S1 TableThe GxE heritability is highly significant for many phenotypes in (PDF)Click here for additional data file.


Most association studies focus on disease risk, with less attention paid to disease progression or severity. These phenotypes require longitudinal data. This paper presents a new method for analyzing longitudinal data to map genes in both population-based and family-based studies. Using simulated systolic blood pressure measurements obtained from Genetic Analysis Workshop 18, we cluster the phenotype data into trajectory subgroups. We then use the Bayesian posterior probability of being in the high subgroup as a quantitative trait in an association analysis with genotype data. This method maintains high power (>80%) in locating genes known to affect the simulated phenotype for most specified significance levels (α). We believe that this method can be useful to aid in the discovery of genes that affect severity or progression of disease. Current association studies focus primarily on disease susceptibility, searching for correlations between genetic variants and disease phenotypes. Studies looking for association with the severity or progression of a disease have been less frequent. This may be partially because these phenotypes require multiple data points across time, which require more time and money to collect properly. When longitudinal data are collected, however, studies show they can be used to accurately map genes. One example involves the progression of spine curvature in scoliosis .Historically, biological studies have been restricted in their use of longitudinal phenotypes. Breakthroughs in the field of growth mixture models, such as random effects modeling , have alIn this study, we present a novel method of mapping genes using longitudinal data that were obtained from Genetic Analysis Workshop 18 (GAW18). Our disease of interest is hypertension, and we use the simulated systolic blood pressure (SBP) values as our phenotype. We performed a population-based study using unrelated individuals and a family-based study using extended pedigrees. Our approach involved assigning individuals into trajectory subgroups and testing for association with SNPs using the BPP of being in the clinically relevant subgroup as a phenotype. Because hypertension is the disease of interest in this study, we define the subgroup with the highest SBP as the clinically relevant group. Power for given significance levels was then calculated on genes determined by GAW18 to affect the simulated SBP phenotype.Both phenotype and genotype data were provided by GAW18. We used the simulated SBP values as our phenotype, which contained SBP values at 3 time points for 850 individuals. GAW18 created 200 phenotype files (to serve as replicates), each containing the same 850 individuals with different SBP values at each time point. The analyses were performed on 2 studies, one was population-based and the other family-based. Genotype data included both sequenced and imputed data free of mendelian errors. The family-based study used related individuals from 20 extended pedigrees. The population-based study used 157 individuals extracted from the pedigrees determined to be genetically unrelated by GAW18. Three time-varying covariates  and 1 time-independent covariate (sex) were used. We performed 2 full analyses on each study, 1 with covariates and 1 without covariates. Thus we had 4 discrete analyses: population-based with covariates, population-based without covariates, family-based with covariates, and family-based without covariates. For each of the studies, we selected a set of 3 replicates by sampling without replacement from the pool of 200 simulated SBP phenotypes created by GAW18. The 3 replicates in a set represent 1 discovery data set and 2 confirmatory data sets. This was repeated 100 times, creating a total of 100 discovery sets and 200 confirmatory sets.k). Each model generated a different number of subgroup trajectories, where k = 1,...,6. Thus, the 1-subgroup model generated a single group while the 6-subgroup model created 6 subgroups from the data. SAS also allows for the specification of the polynomial order of each subgroup trajectory. The initial models used cubic polynomials for all subgroups. Outputs of the initial runs provided a p value for each polynomial coefficient. The order of the polynomials in each subgroup was adjusted to that of the highest order polynomials that were still significant at the 5% level in the first run. The analyses were repeated for a second and final run. For the analyses using covariates, the covariates were introduced into the models at this point. We then determined the optimum model  to be the model with the highest Bayesian information criterion (BIC).The replicates were then analyzed via SAS PROC TRAJ, which uses mixture modeling to assign longitudinal data into subgroups . Each reWith the optimum model selected, we obtained the BPP that an individual belongs to a particular subgroup. For this study, we determined that the clinically relevant group was the subgroup with the highest value at the last time point predicted by the corresponding polynomial. This correlates with individuals with the most severe hypertension. Thus, we define the highest subgroup as the subgroup with the highest SBP at the final time point. The BPP of each individual belonging to this subgroup was used in the association analyses. We note that by choosing the clinically relevant group based on the highest fitted value at the final time point we always get the group that contains individuals with the highest SBP values, regardless of the total number of subgroups determined by the BIC. We also note that clinical relevance is a function of the disease of interest and that the method can be modified to look at any subgroup.We performed association analyses using the BPPs as a quantitative trait. The population-based data were analyzed via PLINK's Wald test (assoc command) . We usedp value in the top x% (population-based) or below a given threshold, α (family-based). All other scenarios were considered NO. For the population-based data, x% was defined as the top 1%, 5%, and 10%. For the family data, α was defined as SNPs with p values at or below 0.1%, 1%, and 5%. We also estimated total power, which was defined as the average of the per-gene power and is intended as a cumulative assessment of the method. T tests were performed at each α level between the noncovariate and covariate analyses to determine whether the addition of covariates was significant. We also performed T tests between corresponding PLINK and TDT-HET to determine whether mean power  of one method was significantly higher than the other. In addition to the gene regions, we applied our method to 3 "null" regions of varying size. A null region was defined as a region not containing any of the simulated functional loci associated with SBP and was used to estimate our method's type I error rate. Figure Power calculations were then performed on the top 15 genes affecting the simulated SBP phenotype. Two types of power were calculated: per-gene power and total power. Per gene power was calculated for each of the 15 gene regions on a binary, or YES/NO, scale. To constitute a YES, all 3 replicates in a replicate set needed at least 1 marker within a given gene with a k) ranged from 3 to 6, mean: 4.78, median: 5, mode: 4. The fast subgroup had an average proportion of 8.89% individuals. When covariates were added to the study, the range shifted from 2 to 5, mean: 3.05, median and mode: 3. The average proportion of individuals increased significantly, to 40.52% (see Discussion). We saw a similar trend in the family studies. Without covariates, k ranged from 3 to 5 with mean: 4.22, median and mode: 4. The average proportion of individuals in the fast group was 8.98%. When covariates were included, the range expanded from 2 to 5 with mean: 4.18, median and mode: 4. The average proportion of individuals in the fast group increased to 36.61%. Thus, in both cases, covariates significantly increased the number of individuals within the fast group.For the population-based studies without covariates, the average number of subgroups . The population-based analyses showed that 12 and 11 genes had greater than 80% power at the 5% and 10% levels, respectively. Only a single gene had greater than 80% power at the 1% levels. We note that this gene was Results of our analyses suggest that our method of grouping longitudinal phenotypes into subgroups and using the BPP as a quantitative trait is a robust method for finding association with SNPs in gene regions. We were able to identify genes affecting the SBP phenotype with high power in both family-based and population-based studies. We also observed high per-gene power when using covariates. Our phenotype of interest was hypertension. For this reason, we used the highest group as our clinically relevant group. However, this choice is flexible and any subgroup could be used as clinically relevant. For example, researchers involved with scoliosis might also be interested in the fastest progression group. If a phenotype like renal failure was being investigated, though, researchers might be interested in the fastest decreasing renal function group.One interesting finding is the effect of covariates on the correct number of subgroups. Covariate inclusion increased the proportion of individuals in the fast group in both studies. Mathematically, this occurred because the covariate analyses tended to have more 2 and 3 subgroup models selected as compared with the noncovariate analyses that identified a larger number of subgroups. That could mean some of the subgroups in the noncovariate analyses are not distinct subgroups, but covariates helped to identify and collapse them.Another interesting finding was that no power was lost in the conversion of quantitative traits to dichotomous traits for TDT-HET. In fact, power was often gained. We believe this is a result of the bimodal nature of the BPP.MAP4, which accounted for 6% of the total SBP variance. We estimated 80% power using our method even at the 1% level.We note that our power at 1% level was less than 50%. We believe that this was caused by our low sample size, given that the effect size of some of the associated genes was known to be small for these data. This is evident in Finally, we wanted to compare the performance of our method to methods proven to identify SNP signals associated with longitudinal phenotypes. We chose to compare to the functional genome-wide association studies (fGWAS) software . We founOur method for analyzing longitudinal data produced high power for identification of association between BPP of group membership and SNPs in genes known to affect SBP. The method's power was high (greater than 80%) in multiple scenarios, including different genotype data and sample size. The flexible nature of this approach allows it to be used in a variety of tests, including exploratory analysis of the entire genome or confirmation analysis on a given region. Significant increases in power with covariate inclusion also show our method's ability to detect interactions between genetic and environmental factors. This provides the prospective researcher the potential to effectively analyze environmental covariates during an association study with longitudinal data. Based on our power results, we believe this method can be an effective and efficient approach to analyzing longitudinal data from a variety of different data sets and study designs.The authors declare that they have no competing interests.DG, DL, and TCM designed the overall study and supervised the analyses. AM performed all analyses except TDT-HET and drafted the manuscript. AQN performed the TDT-HET analyses. LZ performed quality control and data cleaning.


Experience from progeny-testing indicates that the mating of popular bull sires that have high estimated breeding values with excellent dams does not guarantee the production of offspring with superior breeding values. This is explained partly by differences in the standard deviation of gamete breeding values (SDGBV) between animals at the haplotype level. The SDGBV depends on the variance of the true effects of single nucleotide polymorphisms (SNPs) and the degree of heterozygosity. Haplotypes of 58 035 Holstein animals were used to predict and investigate expected SDGBV for fat yield, protein yield, somatic cell score and the direct genetic effect for stillbirth.Differences in SDGBV between animals were detected, which means that the groups of offspring of parents with low SDGBV will be more homogeneous than those of parents with high SDGBV, although the expected mean breeding values of the progeny will be the same. SDGBV was negatively correlated with genomic and pedigree inbreeding coefficients and a small loss of SDGBV over time was observed. Sires that had relatively low mean gamete breeding values but high SDGBV had a higher probability of producing extremely positive offspring than sires that had a high mean gamete breeding value and low SDGBV.An animal’s SDGBV can be estimated based on genomic information and used to design specific genomic mating plans. Estimated SDGBV are an additional tool for mating programs, which allows breeders to identify and match mating partners using specific haplotype information. Within the last years, dairy cattle breeding schemes have changed drastically with the availability of routine dense single nucleotide polymorphism (SNP) chips. Initially, research focused mainly on estimation of genomic breeding values-3 and moIn addition, genotyping large numbers of animals and dense SNP datasets makes it possible to characterize genetic variation at the chromosome and haplotype levels,11. ConsKnowledge on the mean (MGBV) and standard deviation of gamete breeding values (SDGBV) assuming normally distributed estimated breeding values allows the development of specific mating plans. For example, the probability that the breeding value of an offspring exceeds a certain threshold can be estimated. In addition, it is possible to predict the number of animals to be tested to produce an offspring with an estimated breeding value above a given threshold. Cole and VanRaden discusseThe objective of this study was to predict and investigate the expected SDGBV using genomic information and to demonstrate its usefulness to improve mating decisions.A total of 58 035 Holstein animals genotyped with the Illumina BovineSNP50 BeadChip  obtained from routine genomic evaluation for German Holsteins was usedFour traits  with different genetic architectures, heritabilities and genomic reliabilities were chosen. SNP effects were estimated with a BLUP model assuming trait-specific residual polygenic variance (for more details on the model see).The pedigree contained 58 035 genotyped animals  and their 136 477 ancestors. All sires and dams of the genotyped animals were known. The animals were born between 1960 and 2013 and were descendants from 2768 different sires and 32 416 different dams. Genomic inbreeding coefficients were calculated by setting up the diagonal elements of the genomic relationship matrix, as suggested by VanRaden (C being the number of chromosome segments). The mean recombination rate between the haplotype strands was set to 0.3, which is in line with the number of expected recombinations assuming one recombination per Morgan.where hn cattle, and n iThe MGBV of a parent was calculated as:ij is the ith parental or maternal haplotype breeding value.where N is the number of replicates of the simulation, H is the number of haplotypes, and hThe SDGBV of a parent was calculated as:G) and the pedigree (FP) inbreeding coefficients were computed for each trait. Furthermore, MGBV and SDGBV were tested for normality.Correlations between traits were analyzed for MGBV and SDGBV to investigate relationships between traits. To study whether selection, which should result in increased inbreeding and homozygosity per generation, had an antagonistic effect on MGBV and SDGBV, correlations of SDGBV and MGBV with the genomic  for fat yield, 0.54 σa, for protein yield, 0.22 σa for somatic cell score, and 0.09 σa for the direct genetic effect for stillbirth. A mean SDGBV of 0.47 σa was obtained for somatic cell score. The direct genetic effect for stillbirth had an average SDGBV of 0.25 σa. All plots show the presence of animals with equal MGBV but significantly different SDGBV. For example, for protein yield, bulls with an MGBV of 1.8 σa showed a maximum difference in SDGBV of 0.22 σa.Figure G) and the pedigree (FP) inbreeding coefficients. The correlation between MGBV was 0.66 for fat yield with protein yield and 0.15 for somatic cell score with the direct genetic effect for stillbirth. Correlation of SDGBV was lower with FG than with FP.Table P were negative. Correlations between SDGBV and FG were also negative for all traits and two to four times larger than correlations between SDGBV and FP.Correlations among SDGBV for the four traits are in Table a, a substantial deviation from the normal distribution was observed.The MGBV showed no difference between theoretical and sampled quintiles of the normal distribution function for any of the studied traits (results not shown). Figure a per year), followed by fat yield (-0.00087 σa per year).Changes in SDGBV over time are in Figure Table a and the number of animals to be tested to produce at least one animal with a breeding value exceeding a fixed threshold. Resulting distributions of the potential offspring were quite different between the two bulls. Mating of bull 1 with an average cow of the population is expected to produce animals with the highest mBV, i.e. 2.36 σa. The same mating of bull 2 will generate animals with a slightly lower expected mBV, i.e. 2.23 σa. However, a bull that has the highest mean does not guarantee the highest probability of producing offspring with a breeding value greater than 3 or 4 σa. In this case, bull 2 had the highest probability of producing such offspring, but its probability of having progeny with an extreme negative breeding value was also greater. Similarly, the number of animals to be tested to find at least one animal with a mBV higher than 2 σa was highest for bull 2. To produce extreme animals with a gamete breeding value higher than 3 or 4 σa, more progeny had to be tested for bull 1 than for bull 2. Choosing a poor or a superior dam instead of an average cow changed the mean breeding value of the potential offspring, but did not substantially change the likelihood of obtaining offspring with extremely low or high breeding values.Table The objective of this study was to predict the expected genetic standard deviation within groups of offspring using real data. The results indicate that gamete breeding values vary between animals and these results can be used to make specific mating decisions.MGBV and SDGBV for direct genetic effect for stillbirth were about half as high as for the three other traits Figure , which iIn comparison to the SNP-effect reference population, high MGBV for protein and fat yields can be explained by higher selection intensities and genetic gains than for somatic cell score and the direct genetic effect for stillbirth. Comparing the three different traits with similar reliabilities indicates that protein yield had the highest MGBV but the lowest SDGBV. This is explained by a higher selection intensity for protein yield, which is caused by a higher weight on this trait in the German Total Merit Index. HoweverG and SDGBV  and protein yield  for the 2003 birth cohort. This decrease is mainly caused by the offspring of three sires which pre-dominated in this birth year. On average, these groups had breeding values for fat and protein yields that were more than one σa lower than the pre-dominating groups of offspring in the birth cohorts in 2002 and 2004. In contrast to the gamete breeding values for fat and protein yields, no clear difference in gamete breeding values between maternal and paternal haplotypes was found for somatic cell score until the 2010 birth year. From birth year 2010 to 2013, the paternal haplotype was superior to the maternal haplotype. One explanation is that more and more genomically selected sires were used to produce animals born between 2010 and 2013. In contrast, due to genotyping costs, many dams were not genomically selected, which results in lower genetic gain on the female side. For gamete breeding values for the direct genetic effect for stillbirth, there was no genetic trend for either maternal or paternal haplotype breeding values because the direct genetic effect for stillbirth does not seem to be a trait under intense selection. However, Figure Simulated SDGBV can only be validated for sires that have large groups of offspring. A validation independent from genomic information is only possible by comparing the SDGBV of a bull with the standard deviation of the phenotype-based estimated breeding values of its sons. However, only some very popular sires have a large number of offspring with phenotype-based estimated breeding values. Using genomic information, many animals can be tested at a relatively low cost compared to the costs of progeny-testing of bulls, which makes it possible to investigate the standard deviation of genomic breeding values within groups of offspring. Another approach to investigate and validate the standard deviation within groups of offspring is to use daughter yield deviations corrected for the contribution of the dam. One benefit of this approach is that many sires have very large groups of female offspring because of artificial insemination. Figure Systematic genotyping of young Holstein Friesian candidates started in 2010. This implies that animals born before 2010 were selectively genotyped because of their importance for the breeding scheme and their contribution to the reference population. The within-family variance of older families could be affected by this selective genotyping. Genotyping more animals results in larger groups of offspring from randomly genotyped sires, which should result in improved future validations.Van Raden et al. and FritFigure a (707 kg) for fat yield, +32 σa (539 kg) for protein yield, +35 σa somatic cell score and +14.2 σa for the direct effect of still birth. Cole and VanRaden[a and a standard deviation of 0.76 σa. The probability to produce an offspring with a breeding value higher than 8 σa is 0.14%, which is only one third of the selection limit, which illustrates that animals from the current population are far from the selection limits.Haplotype information enables the estimation of selection limits. Summing up the best breeding value for each haplotype will give the theoretically best animal. The gamete breeding values of these hypothetical animals should reach +30 σVanRaden showed tVanRaden at the aFigure Finding the best combination of mating partners in mating programs that are based on genomic information requires time- and memory-intensive computing because of the large amount of data. A great benefit of the method described in this study is that MGBV and SDGBV need to be computed only once for each animal. After this step, it is computationally easy to find mating partners because mBV or vBV is the sum of maternal and paternal MGBV or SDGBV, respectively. Calculating the probability that an animal reaches a defined threshold is simple using normal distribution functions. Based on this methodology, a software tool for breeding associations was developed, which includes MGBV and SDGBV for a portfolio of bulls of interest and for genotyped cows. Given this information, the association can specify which breeding value threshold the offspring of a given cow should exceed and the tool provides a list of bulls that are expected to reach this criterion.Decreasing genotyping costs makes it possible to genotype whole commercial herds. ConsideDGAT1 mutation results in a higher SDGBV than expected. Furthermore, for all traits, SDGBV decreased slightly in recent years because of an increase in the level inbreeding. A genomic mating program was developed to find optimal mating partners with respect to expected MGBV and SDGBV. This approach also allows the probability of finding an offspring with a breeding value exceeding a chosen threshold to be calculated.The expected SDGBV of a potential parent can be estimated from genomic information. The SDGBV differs between animals and tend to be normally distributed in the absence of QTL with a large effect on the trait. For SDGBV for fat yield, a deviation from a normal distribution that is caused by the The authors declare that they have no competing interests.DS conducted the analyses and wrote the manuscript. FR helped to check the results and suggested improvements. ZL estimated the SNP effects. GT coordinated the project, added valuable comments and suggestions. All authors read and approved the manuscript.