Picea glauca (Moench) Voss x P. engelmannii Parry ex Engelm.) trees, representing 25 OP families, growing on three sites in Interior British Columbia, Canada. The use of the genomic realized relationship permitted genetic variance decomposition to additive, dominance, and epistatic genetic variances, and their interactions with the environment, producing more accurate narrow-sense heritability and breeding value estimates as compared to the pedigree-based counterpart. The impact of retaining (random folding) vs. removing (family folding) genetic similarity between the training and validation populations on the predictive accuracy of genomic selection was illustrated and highlighted the former caveats and latter advantages. Moreover, GBLUP models allowed breeding value prediction for individuals from families that were not included in the developed models, which was not possible with the ABLUP. Response to selection differences between the ABLUP and GBLUP models indicated the presence of systematic genetic gain overestimation of 35 and 63% for HT and WD, respectively, mainly caused by the inflated estimates of additive genetic variance and individuals’ breeding values given by the ABLUP models. Extending the OP genomic-based models from single to multisite made the analysis applicable to existing OP testing programs.The advantages of open-pollinated (OP) family testing over controlled crossing  are the potential to screen and rank a large number of parents and offspring with minimal cost and efforts; however, the method produces inflated genetic parameters as the actual sibling relatedness within OP families rarely meets the half-sib relatedness assumption. Here, we demonstrate the unsurpassed utility of OP testing after shifting the analytical mode from pedigree- (ABLUP) to genomic-based (GBLUP) relationship using phenotypic tree height (HT) and wood density (WD) and genotypic (30k SNPs) data for 1126 38-year-old Interior spruce ( A-matrix; Wright G-matrix) which can be used as a substitute to the A-matrix in quantitative genetics analyses   and the realized genomic relationship G-matrix (the observed relationships). More specifically, we evaluated the genomic selection predictive ability after removing genetic relatedness between the training and validation populations during the cross-validation process and determined the response to selection differences between the expected and realized relationships for the studied attributes. This OP progeny test permitted extending the single-site analysis to a more generalized multiple-site model that partitioned the genetic variance components into additive and nonadditive effects as well as accounting for and determining the extent of genotype × environment interaction, thus allowing covering a more complex genetic structure.Here, using 25 Interior spruce OP families grown in a replicated block design over three sites in British Columbia, Canada, we compared the genetic variance estimates generated from the average numerator relationship Picea glauca (Moench) Voss), Engelmann spruce (Picea engelmannii Parry), and their natural hybrids, and because of their similar growing habitat and silvicultural requirements, they are often collectively treated as one complex species ; WD (g·cm−3) was determined from bark-to-bark wood cores using X-ray scanning ; the cores were extracted from each tree at breast height in the north-south direction by 5-mm increment borers.Interior spruce is a complex of white spruce  was theZ is the rescaled genotype matrix following M − P; M is the genotype matrix containing genotypes coded as 0, 1, and 2 according to the number of alternative alleles; and P is the vector of twice the allelic frequency p ; a is the vector of random additive genetic effects following a ~ N interaction effects following axe ~ N is the vector of random block (replication) nested within the site effect following r(s) ~ N were obtained from the model described above, but the average numerator relationship matrix A is substituted by the marker-based relationship matrix Gadd. The extended model for the dominance term (GBLUP-AD) is performed as follows:d is the vector of the random dominance effect following d ~ N were obtained by solving the mixed models following:The narrow-sense heritability estimate was estimated as mentclass2pt{minimR2) was used , while the standard error (SE) of the predictions (SEPs) of the BVs was used to assess the precision of the BVs.Finally, to compare the relative quality of the goodness-of-fit for the said models, the variance explained by each model ) and the prediction accuracy  and predicted BVs from cross-validation (PBV-CV)) for the four models were estimated using 10-fold CV and five replicates. To assess the role of relatedness between the training and validation populations on the genomic selection predictive accuracy, two folding scenarios were used, namely, random  and family (removed relatedness as entire families were absent during the CV process) folding. In each replicate, the data was divided into 10-folds according to the used folding scenario, 9-folds was assigned as the training population, while the last fold was used as the validation population to estimate PBV-CV. The five replicates were used to estimate the SE of the correlation. Model pairwise prediction accuracy was also estimated between the four models in order to evaluate the ability of predicting each other. In this case, accuracy was estimated as the Pearson product-moment correlation between EBV-all of one model and PBV-CV of the other model (see above).Replications within-site variance components were consistent across the four models and accounted in each case for a relatively small variance component for both height HT 1.63–2.07%) and wood density (WD 6.91–8.62%)   can only be expressed through the terms “additive variance × environment,” and these were the first and third highest variance components, accounting for 34.73 and 16.98% of the total variance for HT and WD, respectively (Table R2) and the fitted line plots . Moving from ABLUP to the GBLUP-A was characterized by the lack of improvement for the two model comparison methods   model for HT , reflecting the presence of significant additive × additive genetic variances and indicating that GBLUP-ADE is the best model for WD. The R2 differences between HT and WD reflect the different genetic architecture of the two traits   for estimating genetic parameters and variance components; thus, the degree of genetic advance is dependent on the accuracy and precision of these parameters, specifically heritability and individuals’ breeding values. The utilized mating design determines mainly which genetic component can be generated (additive and dominance), and in some cases, additional efforts such as combining full-sib families with replicated clonal trials are required to estimate epistatic genetic variance (Foster and Shaw G-matrix) among individuals, irrespective of their genealogy, and substitute the A-matrix in estimating genetic variance components particularly in the population of forest trees (Denis and Bouvet R2 term (variance explained by the model) and the fitted line plots  Table . The threl Table ; Fig. 1 el Table .R2 value supporting better model fit (Table The GBLUP-AD model produced surprising results with a significant dominance variance for HT relative to the additive variance with a higher it Table . This wait Table . The sigit Table . Additioit Table  reportedP. glauca x P. engelmannii) in British Columbia , which was extended to include first-order interactions, gave exactly the same results as GBLUP-AD for HT indicating the absence of all kinds of epistatic interactions; this model subsequently showed no improvement in all goodness-of-fit measures and precision estimates Table  and 2. Tes Table , the fites Table  and 2. Aes Table  and in les Table . Thus, tes Table  and the The advantage of GBLUP models is their use of the realized genomic relationship among individuals regardless of their genealogy, while the ABLUP is mainly dependent on the pedigree-structure created by mating designs. Additionally, to capture the additive relatedness among individuals, the realized genomic relationship matrix is also capturing the linkage disequilibrium (LD) between the SNPs and quantitative trait loci (QTLs) and their co-segregation  and dominance as well epistasis variances (GBLUP-ADE in the case of WD) resulted in improving the breeding values’ estimates precision Fig. ; howeverThe observed predictive accuracy differences between the random (retaining genetic similarity) and family (excluding genetic similarity) folding are of great importance Table . Our resR = h2S; Falconer et al. The reported multisite genetic variance decomposition along with the selection of the best model for each studied trait is expected to improve the genetic variance partition (see above) as well as the individuals’ breeding values. We compared the ranking of the top 50 individuals for HT and WD between the pedigree-based ABLUP and the genomic-based GBLUP models  Fig. . Only 78The utilization of the genomic-based relationship in tree improvement testing and evaluation programs is implemented, and its usefulness, in particular for the simplest OP testing mode, is demonstrated. The proposed approach of genetic variance decomposition was illustrated, and additional estimates such as dominance and epistatic genetic variances as well as their interactions with the environment were obtained. The incorporation of genomic information in the analysis provided more accurate genetic variance estimates, particularly for those that could not be derived using the traditional OP pedigree-based analysis. The negative role of genetic similarity between training and validation populations on the predictive accuracy of genomic selection as well as examples of the unrealistic genetic gain estimates derived from pedigree-based analyses was illustrated. Finally, extending the GBLUP model to include multisite makes the analysis applicable to existing OP testing programs.


To evaluate statistical methods for genome-wide genetic analyses, one needs to be able to simulate realistic genotypes. We here describe a method, applicable to a broad range of association study designs, that can simulate autosome-wide single-nucleotide polymorphism data with realistic linkage disequilibrium and with spiked in, user-specified, single or multi-SNP causal effects.Our construction uses existing genome-wide association data from unrelated case-parent triads, augmented by including a hypothetical complement triad for each triad . We assign offspring qualitative or quantitative traits probabilistically through a specified risk model and show that our approach destroys the risk signals from the original data. Our method can simulate genetically homogeneous or stratified populations and can simulate case-parents studies, case-control studies, case-only studies, or studies of quantitative traits. We show that allele frequencies and linkage disequilibrium structure in the original genome-wide association sample are preserved in the simulated data. We have implemented our method in an R package (TriadSim) which is freely available at the comprehensive R archive network.We have proposed a method for simulating genome-wide SNP data with realistic linkage disequilibrium. Our method will be useful for developing statistical methods for studying genetic associations, including higher order effects like epistasis and gene by environment interactions.The online version of this article (10.1186/s12859-017-2004-2) contains supplementary material, which is available to authorized users. Most current methods for simulating extensive genome-wide data mimic evolutionary processes, either forward in time  profiles while allowing investigators to spike in (and then try to find) multi-SNP causal effects against a null background. The genetic simulation tools currently available take different approaches to simulation and offer different capabilities; the National Cancer Institute has provided a web resource that catalogues existing software packages and aids comparisons of their characteristics  or ba or bahtte e.g., , 5). Suc. Suchttpe e.g., , 7). Pro. ProhttpThe many available genetic simulators differ widely in their features and ease of use. We sought an approach that was conceptually straightforward and would deliver realistic LD structure. Those considerations led us toward a resampling-based approach. We sought an approach that would simulate genotype data for case-parents designs and for case-control designs. In addition, we wanted to be able to model traits flexibly -- either dichotomous or quantitative phenotypes -- and be able to include possible epistatic interactions and gene-environment interactions as contributing to phenotypes. No available simulator seemed to achieve all of those goals simultaneously.We propose a resampling-based simulation method that can generate genome-wide autosomal SNP genotypes under various risk scenarios. Our method requires existing autosomal genotype data from a genome-wide association study (GWAS) of case-parent triads as a starting point and largely preserves the allele frequencies and LD structure in that data. It creates simulated case-parents data by resampling genotype fragments sequentially from different families and concatenating them. Trait phenotypes, either dichotomous or quantitative, are then assigned to offspring at random based on a user-specified risk model. Though the method is applicable to multiple SNPs that act independently, we focus on risk models that involve one or more sets of interacting SNPs (to be referred to as “pathways”) with or without gene-environment interactions. If the available GWAS data contains identified subpopulations, the method can simulate either a homogeneous or a stratified population. Though the construction uses case-parents data, simulated samples from other study designs are achieved by retaining subsets of the simulated genotypes ; for example, population-based random samples for quantitative traits (with or without parents) and case-control samples are possible.We begin by briefly outlining some features of our R package followed by presenting our re-sampling algorithm for case-parents data and describing how we assign trait values to simulated offspring. We then document the performance of our approach with several simulations. We close with a brief discussion.https://cran.r-project.org/web/packages/TriadSim/index.html). The input files for the package are triad genotype data in the widely-used PLINK format. The output files are also in PLINK format. The user can nominate a single SNP or multiple SNPs in “pathways” (sets of SNP loci) through the input parameter “target.snp”. Alternatively, the user can specify a desired allele frequency for the SNPs in each pathway, the number of pathways and the number of SNPs in each pathway and allow the program to pick the SNPs in the pathways. The program allows for an array of user-specified parameters such as the number of simulated subjects, the number of break points to be used for each chromosome, exposure prevalence and the baseline disease prevalence among noncarriers. The input parameters also include a few Boolean variables to allow the user to perform simulations for different types of outcome: “qtl” for designating a quantitative trait rather than a dichotomous trait; “is.case” for simulating a case-triad rather than a control-triad. The user also needs to input risk parameters that quantify the effect of the genotype(s) on the trait. Statistical models for case-parents data estimate relative risks (RR), e.g. equation ′ is a vector of parameters and X = ′ is a vector of indicator variables. An indicator variable can be simple; for example, an indicator that the subject carries one or more copies of the variant at a particular SNP locus. Thus, X might encode indicators for p distinct SNPs that each contribute to the trait outcome. Our focus, however, is on epistatic scenarios where the risk is increased by inheritance of a particular combination of variant alleles in one or in multiple pathways. The indicator variables are then the product of a set of SNP-specific indicator variables. For example, a scenario may involve two pathways (p = 2), a 4-SNP and a disjoint 3-SNP pathway. Then, X1 would be the indicator that the subject carries at least one variant allele at each of the four loci in pathway 1, X2 would be the indicator that the subject carries at least one variant allele at each of the three loci in pathway 2, and β1 and β2 would assess the magnitude of each pathway’s influence on the trait. One can use the same software to generate simulations where risk depends on single SNPs by regarding them as 1-SNP pathways.For simplicity, all the trait models that we consider have as predictors some function of the offspring’s genotype. The function is a linear combination of X as:For a dichotomous disease phenotype, we model the penetrance among those with vector α is the log risk of disease among individuals who do not have a complete set of SNPs for any single pathway. As described above, each component of X is a product of locus-specific indicator variables and, for dichotomous traits, β is a vector of the log relative risks for the associated pathways. If two or more pathways are present in one individual, the model shown in (1) implies that their contributions combine multiplicatively on the relative risk scale. For case-parents triad data, only families with affected offspring are retained in the final data set. For case-only data, the user discards the parents. For control-parents data, only families with unaffected offspring are retained. For case-control data, the algorithm retains affected and unaffected offspring according to a user-specified ratio and the user discards parental genotypes.Here, For a quantitative trait, we model the trait value as:Y denotes a quantitative trait with a mean of α among noncarriers. Again, each component of X is a product of SNP-specific indicator variables, β is their corresponding vector of pathway-specific shifts of the mean, and ϵ is a normally distributed mean-zero random error term. With two or more pathways involved, we assume that their effects are additive on the original scale. The algorithm retains all offspring, regardless of trait value; though our software returns parental genotypes, they can be discarded subsequently.Here E, coded as 1 for present and 0 for absent. For dichotomous traits, we model penetrance as follows:For scenarios involving gene-environmental interactions, we consider only a dichotomous exposure, denoted α the log risk of the disease among the unexposed individuals who do not have a complete set of SNPs for any single pathway. β is a vector of the log relative risks for the associated pathways in unexposed individuals. θ is the log relative risk associated with exposure among individuals who do not have a complete set of SNPs for any single pathway (exposure main effect) and γ =  is a vector of the log interaction effects. The corresponding model for quantitative traits can be expressed similarly by including the terms for the exposure main effect and the interaction in formula .Provided the input GWAS data contain more than one identifiable genetically distinct sub-population , our implementation also allows for the simulation of a stratified population by sampling separately from GWAS data specific to each sub-population. Each sub-population has its own allele frequency distribution implicitly from the input data. In addition, the user specifies, separately for each subpopulation, its proportion in the underlying population targeted by the simulation, exposure prevalence (if relevant), and disease prevalence or mean trait value among (unexposed) non-carriers (we assume that other risk parameters are common across sub-populations). To simulate a setting where there would be bias due to population stratification, one should select alleles for the risk model that differ in frequency between the two identified sub-populations. Sub-population-specific disease prevalence or mean trait values are achieved by setting the To evaluate the performance of our software, we conducted simulations using the cleft consortium GWAS data downloaded from dbGaP as the input genotype source . These data included complete triad genotypes for 1899 families in two identified ethnic groups, 1028 Asian and 871 Caucasian. For these simulations, we set the number of break points at three for each chromosome.p < 5 × 10−8 . We used triad families of Asian and Caucasian origins in homogeneous and stratified scenarios. For homogeneous scenarios, all simulated triads are from just one ethic group; we provide results for Asian and Caucasian families separately. For stratified scenarios, we used both the Caucasian and Asian triads as the source population. The underlying proportion of the Caucasian population was set as 0.46 and the ratio of baseline disease prevalences was set as 1.3 (Caucasian to Asian). For each null scenario, we generated 2000 null data sets, each containing 1000 triads, a number close to the sample sizes of the two subpopulations in the original cleft study. Signals from the 14 loci reported at genome-wide significance level by the original GWAS study were all successfully obliterated in the simulated data as indicated by Type I error rates near the nominal per-comparison α-level of 0.05 when testing those loci for associations with risk ) between pairs of SNPs in the original data was well preserved in the simulated data. Among all SNP pairs, the correlation between pairwise LD measured in the original data and the average pairwise LD across 1000 simulated null data sets was 1.00. On average across 1000 simulated data sets, the absolute difference between correlations was less than 0.1 for 99.6% of SNP pairs. Among the exceptions, about 71% on average involved SNP pairs with low minor allele frequencies (MAF)  betwThe decay in average pairwise LD with increasing inter-SNP distance was similar in the original and simulated data Fig. . When reIn addition to preserving LD, our simulation method preserved the original allele frequencies, the correlation between the MAF in the original data and the average MAF for the same locus across the 1000 simulated null data sets approached 1 Fig. . Those vWe conclude that our simulation procedure provides simulated data that successfully mimics both the LD structure and the minor allele frequencies present in the original input data, though with some minor degradation among rare alleles.We also simulated data under different scenarios to verify that the trait-related pathways that we spiked in could be recovered analytically. For all these simulated scenarios, we used the same 10,279 loci and assumed two causative pathways, each with four interacting loci. We selected the eight pathway SNPs for these simulations from among SNPs with the targeted allele frequencies and selected SNPs that were widely spread across four chromosomes. First, we studied stratified null scenarios using 2000 simulated studies of 1000 triads each. We wanted to create stratification that would generate substantial bias under a naive analysis; consequently, we needed two subpopulations that differed in both allele prevalence and baseline disease rate. For these null scenarios, the two subpopulations were separately resampled from the Asian and Caucasian GWAS data, respectively. We selected SNPs to ensure that the allele frequency of each of the SNPs in the two pathways designated for testing was close to 0.15 and 0.5 in the two subpopulations, respectively. The underlying proportion of the second population was set at 0.46 (mimicking Caucasian proportion in the clefting data). The baseline disease risks in the two subpopulations were set to 0.17% and 0.5% for a dichotomous trait  that is positively associated with the offspring phenotype will have a slightly higher prevalence in the parental genotypes of the observed triads than in the source population from which they came; that enrichment will be propagated to the simulated triads. We selected three break points per chromosome in our simulations for convenience. Some researchers may prefer to take the chromosome size into consideration when picking the number of break points, and our R functions allow users to specify the number of breakpoints separately for each chromosome. The idea of using chromosomal fragments broken at recombination hotspots as part of a resampling scheme has been employed by others. It was used to simulate case-control data  and to iOur framework is more broadly applicable than our current software implementation supports. In addition to the study designs mentioned, one could simulate data based on outcome-dependent (extreme phenotype) sampling for a quantitative trait: after simulating offspring-parent data for the quantitative trait, the probability of inclusion into the simulation sample would depend on a user-specified function of the trait value. Additional risk models could be incorporated. For example, instead of the present multiplicative structure in eq. , one couOur approach does have some inherent limitations. Any re-sampling-based approach such as ours may be limited to an extent by the original data. Ideally the triads to be used as the raw material should include a large set of unrelated families. Also, to simulate stratified populations with our approach, the available data needs to contain distinct sub-populations. Also, unlike simulations based on mimicking evolutionary processes, resampling approaches cannot introduce new variants into a simulated population. For many purposes, this drawback is minor though it may be relevant when studying rare variants. Our approach may not be ideal for simulating rare variants. SeqSIMLA , a coaleAs always, the choice of a simulation method will depend on the goals of the project. If assessment of methods for studying genome-wide genetic associations, particularly those involving multi-SNP epistasis, is the goal, our method could serve this purpose well.We have provided a resampling-based method to simulate autosomal SNP genotypes for use in evaluating data-analysis methods. The required raw-materials input for these simulations is GWAS triad genotype data from individuals and their parents. Our approach can simulate both case triads but also control triads and offspring with quantitative traits (with or without their parents). Discarding parents from case triads provides case-only samples and discarding parents from both case and control triads provides case-control samples. We showed through simulations that our method produces simulated data sets that largely preserve the allele frequencies and the realistic SNP-pair LD structure that existed in the original data. Using our approach, one can simulate complex scenarios that involve multiple genetic pathways, each containing multiple interacting SNPs, pathways that possibly interact with dichotomous environmental factors.Project name: TriadSim.https://cran.r-project.org/web/packages/TriadSim/index.htmlProject home page: Operating system: Platform independent.Programming language: RLicense: GPL-3.


Complex traits with multiple phenotypic values changing over time are called longitudinal traits. In traditional genome-wide association studies (GWAS) for longitudinal traits, a combined/averaged estimated breeding value (EBV) or deregressed proof (DRP) instead of multiple phenotypic measurements per se for each individual was frequently treated as response variable in statistical model. This can result in power losses or even inflate false positive rates (FPRs) in the detection due to failure of exploring time-dependent relationship among measurements. Aiming at overcoming such limitation, we developed two random regression-based models for functional GWAS on longitudinal traits, which could directly use original time-dependent records as response variable and fit the time-varied Quantitative Trait Nucleotide (QTN) effect. Simulation studies showed that our methods could control the FPRs and increase statistical powers in detecting QTN in comparison with traditional methods where EBVs, DRPs or estimated residuals were considered as response variables. Besides, our proposed models also achieved reliable powers in gene detection when implementing into two real datasets, a Chinese Holstein Cattle data and the Genetic Analysis Workshop 18 data. Our study herein offers an optimal way to enhance the power of gene detection and further understand genetic control of developmental processes for complex longitudinal traits. Linear mixed models (LMM) have been widely applied in GWAS as they performed well in correcting environmental factors, controlling population stratification and accounting for relatedness between individualsetc. Analyzing such kind of data affords us an opportunity to investigate the heterogeneity of traits over time and early prediction of longitudinal traits or diseases8.So far, most of these commonly-used methods have been focusing on typical phenotypic data where single record per individual is collected. However, a different type of phenotypic data generated from longitudinal traits has seldom received attentions in GWAS. Longitudinal traits belong to a type of complex traits measured at various time points during a life cycle, such as blood pressures, daily gain, milk production, and residual feed intake, 10. The second one is based on phenotypic combination where multi-point measures of each individual are firstly fitted by some smoothing methods, and the estimated curve parameters, accumulated or average values for a period of time are then used as the alternative response variables13. The third one is based on varying coefficient model, which fits the coefficients of genetic and environmental effects as the linear regression on a set of splines or polynomials of time to model the time-varied effects17.In previous quantitative trait loci (QTL) linkage analysis on longitudinal traits, three statistical strategies are proposed as follows: The first one is based on repeatability model or multivariate model, which treats the multi-point measured trait as repeated measurements of the same trait or as different traits18. The strategy of phenotypic combination merely works when all effects are supposed to be constant over time19. Varying coefficient model is only suitable to well-structured dada where all individuals must be measured at the fixed time points. The drawbacks for these strategies limit their further application in the GWAS.These methods aforementioned have respective limitations. Specifically, repeatability/multivariate model is unable to explore time-dependent relationship between successive measures, and multivariate model is often difficult to apply in practice because it fits too many parameters in the model21. Nevertheless, it has been indicated that EBVs incorporating familial information would lead to higher false positive rates (FPRs)22. DRPs had adjusted for parental average effect23, but could still lead to higher FPRs when the EBVs were the results of repeated measurements22. In human disease studies, the interaction between Single Nucleotide Polymorphism (SNP) and time/age was incorporated in the analysis model26. However, Wu and Lin indicated that there were various dynamic patterns of genetic control 27, which could not be completely explained by the interaction model.In dairy cattle breeding, estimated breeding values (EBVs) or deregressed proofs of EBV (DRPs) are preferred as the response variable of GWAS28 provides a better way to model the time-varied measurements/traits, and has been widely used in genetic evaluation of dairy cattle30. Recent studies have proved that it increased the power to detect QTL compared with the combining phenotypes strategy, repeatability model and multivariate model in QTL mapping19. Random regression model was also suitable for QTL detecting in the presence of gene by environment interactions31. However, application of this model in GWAS has not been fully surveyed so far.Random regression modeli.e., functional GWAS model treating each SNP as the covariate (fGWAS-C) and functional GWAS model treating each SNP as the factor (fGWAS-F). A series of simulation studies were performed to investigate the properties of the proposed models, and to compare with previous developed models, i.e., genome-wide association studies where EBVs or DRPs were used as response variable with polygenic effects modelled (GWAS-EBV-P or GWAS-DRP-P), genome-wide association studies where EBVs or DRPs were used as response variable without polygenic effects modelled (GWAS-EBV-NP or GWAS-DRP-NP), and genome-wide association studies where estimated residuals were used as response variable . We further validated our model with a Chinese Holstein cattle data and the Genetic Analysis Workshop 18 (GAW18)32.In this study, we developed two models based on random regression model to model the time-varied SNP effect for the GWAS analysis, p value = 0.01 and 0.05) of the evaluated models were shown in Fig. The FPRs  achieved the highest power among all the models employed under all scenarios except when We also evaluated powers of all models using empirical thresholds based on different FPRs for null-effect SNP. The receiver operating characteristic (ROC) curves plotting the statistical powers against FPRs were shown in Figures p-values   Fig. . This inThe average of estimated cumulative additive effect (see equation 34 (AIC) as well as Bayesian Information Criterion35 (BIC) to determine the orders of basis functions. After model selection with AIC and BIC values, the model with a fifth-order basis functions for population mean, a third-order for additive genetic effects and a fifth-order for permanent environmental effects was best fit to the data for all the three traits (Table DGAT1 (diacylglycerol O-acyltransferase 1) gene, reported to be a major gene affecting milk production traits36, is located within this region.We used Akaike information criterionts Table . Manhatthttp://www.animalgenome.org/QTLdb). The detailed information of SNPs showing significant associations with the three traits, including their positions in the genome, p values, detected model, the nearest known genes and the PudMed IDs for nearest QTLs, were given in Tables For milk yield, 3 of 17 significant SNPs were located on chromosome 5, 12 and 15, respectively, while all the remaining SNPs were located between 1.48 and 4.36 MB of chromosome 14. For fat percentage, 126 significant SNPs were found, and most of which were located between 91.13 and 95.74 MB on chromosome 5 (15 SNPs) and between 1.41 and 8.45 MB on chromosome 14 (106 SNPs). For protein percentage, most of the total 113 significant SNPs were located in the region 1.65–4.47 MB on chromosome 14 (24 SNPs), 28.80–38.49 MB on chromosome 20 (62 SNPs), and 44.16–45.87 MB on chromosome 20 (7 SNPs). Meanwhile, the majority of genome-wise significant SNPs  were located within reported QTLs for three corresponding traits. Interestingly, two regions  were not overlapped with the known QTL regions and could be potential candidate QTL regions influencing the milk traits. QTL information of the three traits was obtained from Animal QTL Database , TMEM248 (within), RN7SL43P (782 bp away), VAV2 (within), UFM1 (53 kb away), and AP000959.2 (1.47 Mb away), respectively. Interestingly, both CDC42 and VAV2 genes participate in the biological process of blood coagulation, and CDC42 gene also participates in heart contraction.As higher order basis functions did not converge, the model with a second-order basis functions for all the time-varied effects was used to fit GAW18 data. Manhattan plots of 39. In current study, we proposed two models for the GWAS of longitudinal trait which could fit the time-varied QTN effects and directly use the raw longitudinal records. This can fully avoid the necessity of transforming phenotypes into pseudo-phenotypes, such as EBVs20, DRPs40, or estimated residuals. The simulation results indicated that our proposed models could capture genetic differences varied in the entire process of the time period, thereby increasing the statistical power of QTN detection. Although pseudo-phenotypes were substitutions for longitudinal records, the scales of them would be changed41. Therefore, the QTN effects predicted by these pseudo-phenotypes methods were biased. This might not influence the significance test, as the scales of corresponding estimated errors would also change. However, the pseudo-phenotypes methods could not directly predict the true proportions of the phenotypic variance explained by QTNs. As our fGWAS-C and fGWAS-F models directly used raw phenotypes and achieved the most accurate estimate of the QTN effects, they could be used to predict QTN heritability in practice. Overall, the proposed random regression-based methods clearly outperformed other traditional methods validated by extensive simulations.Recently, a growing number of studies indicated that the expression of genes was time-dependent23. However, the cryptic relationships between individuals still existed when the EBVs were estimated from repeated measurements22. In the simulation study, our results indicated that GWAS-EBV-P and GWAS-DRP-P models had similar performance in controlling the FPRs. The GWAS-DRP-P models lowered the power in some degree in our simulation. This may be because the cryptic relationships were corrected twice. One is removing the parental average information from EBVs, the other is including the polygenetic effects into the GWAS model. The EBVs or DRPs had removed the environmental effects and combined the repeated phenotypic values into a single one for each individual, which resulted in a much smaller dimension of the mixed model equation, and thus was more computationally efficient and feasible. Therefore, the GWAS-EBV-P and GWAS-DRP-P models are still an appealing alternative for GWAS study of longitudinal data when the computational efficiency is the primary consideration.Among the traditional GWAS models, while no polygenetic effects were fitted to account for cryptic relationships between individuals, the GWAS-EBV-NP and GWAS-DRP-NP models resulted in high FPRs. DRPs had adjusted for parental average effectet al. with method similar to GWAS-EBV-P20. Here, we expanded the population size from 2,093 to 6,619. Furthermore, the Animal QTLdb had collected 4,585 QTLs  for MY, FP and PP of dairy cattle since 1994, which made our study population be a suitable dataset for validating GWAS approaches for longitudinal traits. We mapped our significant SNPs to the Animal QTLdb and discovered that most of them (165/215) located within the reported QTL regions. Moreover, two novel regions  contained several significant SNPs (15 SNPs for the former and 7 SNPs for the later) in relatively narrow segments, and they were potential candidate QTLs regions for milk production traits. The estimated curves of additive effects, dominance effects and QTN heritabilities of SNP ARS-BFGL-NGS-4939 (within the DGAT1 region) for milk yield, fat percentage, and protein percentage were also predicted by our fGWAS-F model. The trend of these estimated curves implied that the genetic effects were not constant, and could depend on the data or environment.Meanwhile, we applied our two models to a Chinese Holstein cattle data and the fGWAS-F model to the GAW18 data. The GWAS for milk production traits of Chinese Holstein population had been implemented by Jiang et al. did not find any significant SNPs using admixture mapping analysis42, and Chung and Zou found four significant SNPs with extended EMMA1 model26. These studies indicated that human blood pressure might have a complicated genetic background, and there might be no major genomic region affecting it. In our studies, totally eight significant SNPs were discovered by our fGWAS-F model, and their nearest genes participated in the biological processes of nucleotide-excision repair, blood coagulation, heart contraction and so on, which closely related to heart disease. These eight significant SNPs could be candidate associated loci for blood pressure.For GAW18 dataset, Chen et al.43. One of the key differences between their model and our proposed ones is that we divided the time-varied mean values for SNP genotypes into two parts, time-dependent population mean and SNP effects, instead of fitting them directly. In this way, our models can be easily implemented by the popular ASReml software. The more important difference is that time-varied polygenetic effects are fitted in our models to control the FPRs. We applied the fGWAS software by Das et al. to our simulated data. The resulted FPRs at tabulated thresholds of p value = 0.01 and 0.05 were 0.046 and 0.125, respectively. As indicated by Xu44 in GWAS of non-longitudinal traits, the model ignoring polygenic covariance structure merely performed well for the simple experiment with one QTN. However, the signals became very noisy for complex experiment with multiple QTNs. In practical situation, quantitative traits of interest are controlled by more than one QTN45. Therefore, it should be beneficial to include polygenic effect in the model.Functional GWAS is not a novel conception, and has been proposed and carried out by Das 46. Luckily, our fGWAS-C model can also be applied to genotype dosages. This cannot be achieved by the fGWAS software by Das et al.The emerging next-generation sequencing technology impels us to find the “miss heritability” of complex traits. Along with the technological evolution, the availability of public data, such as the 1000 Genomes Project and 1000 Bull Genomes Project, provides opportunities to maximize the value of our existing data though genotype imputation. The number of variants can be increased and true QTNs may be located in this way. The genotype dosages (a continuous random variable between 0 and 2) can gain more powers than “best-guess” imputed genotype (genotypes with the highest probability) in GWAS47. However, a dense marker-based kinship matrix will increase the computational burden heavily. Zhang et al. suggested that a compression approach, which was called compressed mixed linear model, would decrease the effective sample size by clustering individuals into groups3. Meanwhile, for population with unknown degree of genetic relationship, Kang et al. developed a procedure for estimating the contribution of the polygenetic effects to the phenotypes and the polygenetic effects were not needed to fit in the GWAS model if they were tested non-significant2. Both approaches can be incorporated to improve our proposed models in our future endeavors.As expected, our proposed fGWAS-C and fGWAS-F models showed obvious computational inefficiency as the dimension of the mixed model equation was larger than other models. When the relationship matrix is established based on pedigree, the computational burden is less challenging as the numerator relationship matrix is relatively sparse. The marker-based kinship matrix can reflect the relationship between individuals more precisely. For example, the relationship among full siblings will be the same based on pedigree, but can be distinguishable with genetic markersIn conclusion, we proposed two models fGWAS-C and fGWAS-F using random regression for functional GWAS of longitudinal traits on a genome-wide scale. According to our simulation study results, the proposed models fitted longitudinal traits successfully and outperformed the models using EBVs, DRPs or estimated residuals as response variables. Using our proposed models, we have successfully found two novel regions which were significantly related with milking production traits for the Chinese Holstein data and some SNPs related with blood pressure for the GWA18 workshop dataset. Generally, functional GWAS models using random regression were useful and appealing in the GWAS for longitudinal traits.yi(t) is the phenotypic value of individual i at time t; μ(t) is the overall mean at time t; ai(t) and pi(t) are the time-varied additive genetic effect and permanent environmental effect respectively for individual i; ei(t) is the time-independent random residual for each measurement of individual i at time t. Here, μ(t), ai(t) and pi(t) can be denoted as the linear regression for a set of basis functions, i.e., splines or polynomials, below:nf, nr1, and nr2 are the orders of corresponding basis functions; φk(t) is the value of the kth basis function at time t; βk is the kth fixed regression coefficient; aik and pik are the kth random regression coefficients for additive genetic effect and permanent environmental effect of the ith individual. The orders of different basis functions can be determined by model selection criteria (such as AIC and BIC) suggested by Das et al.43. The matrix form of (1) can be represented as:The general expression of random regression model can be formulated as the time-dependent function:n individuals and the number of records for individual i is mi , then the total number of records for all individuals is y is a m × 1 vector of phenotypic values of all individuals; b is a [n(nf + 1)] × 1 vector of fixed regression coefficients; a is the vector of random regression coefficients for additive genetic effects with nr1 + 1 elements for each individual; p is the vector of random regression coefficients for permanent environmental effects with nr2 + 1 elements for each individual; X, Q and Z are the corresponding design matrices; e is the vector of random residuals.Here, we assume there are For equation , we haveA is the numerator relationship matrix based on pedigree information; I is the identity matrix; G is the variance–covariance matrix for random regression coefficients of additive polygenic effects with size of (nr1 + 1) × (nr1 + 1); P is the variance–covariance matrix of random regression coefficients for permanent environmental effects with size of (nr2 + 1) × (nr2 + 1); Here, i.e., functional GWAS model treating each SNP as the covariate (fGWAS-C), and functional GWAS model treating each SNP as the factor (fGWAS-F). To exploit the property of these two novel methods, several alternative models/strategies which used the EBVs, DRPs, or estimated residuals as the response variables for GWAS of longitudinal traits were also applied for extensive comparisons. Details of each model were specified below as well as listed in Table Under the framework of the random regression model, we proposed two detection methods for the association analysis of longitudinal traits, In our proposed fGWAS-C model, an additional fixed regression term is incorporated into equation  to explaxi is a genotype indicator which is assigned 0, 1 and 2 for genotype aa, Aa and AA, respectively; SNP(t) represents the time-varied additive effect for each marker and can be expressed as linear regression for a set of basis functions as mentioned before:kth basis function at time t; kth fixed regression coefficient for additive SNP effect; nf is the order of basis functions for the time-varied SNP effect. For convenience, we define the same order of time-varied population mean and SNP effect in this model.Here, Similarly, the fGWAS-F model is formulated as:SNPl(t) means time-varied effect for genotype l  of each marker and λlk is the kth fixed regression coefficient for genotype l. For fGWAS-F model, time-varied additive genetic effect, dominance genetic effect, and additive genetic variance of each SNP can be deduced as45:p and q are the allele frequencies for each locus.Here, i can be formulated as tmin and tmax are the first and last recording time of all individuals, then the accumulated EBV for each individual from tmin to tmax can be obtained as:Under such models, combined EBVs are firstly deduced through solving Equation . TherefoThe accumulated EBVs are then used as the latent response variable in the GWAS-EBV-P model.y is defined as the n × 1 vector of EBVs for all individuals; μ is the population mean; u is the vector of polygenetic effects with multivariate normal distribution MVN modelIn our study, the estimated residuals for multiple records corresponding to each individual were obtained by solving equation . Then, w48 was used to examine whether the SNP is associated with the trait. The Wald chi-squared statistic with a degree of freedom of dfw is given by:For each SNP, the incremental Wald statistic implemented by ASReml49 (RSS) or models with and without SNP effect. The symbol dfw is degree of freedom for the SNP effect. For fGWAS-C model, dfw = nf + 1, and for fGWAS-F model, dfw = 2(nf + 1), where nf is the order of basis functions for the time-varied SNP effect as defined above. For other models defined above, dfw = 1. The symbol Here, We performed extensive simulations to systematically compare the performance of two random regression-based GWAS models (fGWAS-C and fGWAS-F) proposed here and other multiple-step traditional linear mixed models  aforementioned. We evaluated statistical power, type-I error rate as well as the accuracy of SNP effect estimated for each GWAS method through 1,000 replication.50. The simulation started with a base population of 50 males and 50 females in generation −1,000, followed by 1,000 discrete historical generations  with the same population size and an equal sex ratio. After 1,000 historical generations, the recent population was generated from generation −1 to generation 0 with population size expanded from 100 to 1,000 . In the follow-up four recent generations (generation 1 to 4), 50 males were randomly selected from the last generation each mating with 500 females. Each female produced two offspring  at each recent generation. Finally, a total of 2,000 females from the last four recent generations were collected as the experimental population investigated. We determined 1,002 SNPs as the simulated genomic data, two of which were selected as independent target mutations. One contributed to the genetic variance (treated as the QTN) and the other had null effect on the longitudinal phenotype. These two SNPs were adopted to evaluate statistical powers and FPRs respectively across different models. The remaining 1000 SNPs were assigned the polygenic effects representing the genetic background of each individual, all genotypes of which were then removed in the final simulated data.Population and genomic data were simulated with QMSim softwareh2 was set to 0.3 and heritability of functional QTN The longitudinal phenotype observations were simulated using self-developed C program. The detailed description was given in Supplementary Methods. In the simulation, heritability of simulated trait In the simulation, the power and type-I error rate for each scenario were determined as the proportion of significant detections for functional QTN and null-effect SNP respectively among 1,000 replicates for each scenario.51 were used as the basis functions for the overall mean, additive genetic effect and permanent environmental effect. The orders of basis functions were evaluated based on the smallest AIC as well as BIC. As the effects contributed by most of SNPs were very small, we adopted the same variance components estimated by the reduced model of equation  data, were used to further validate performance of our proposed models. The detailed illustrations of the real data was provided in Supplementary Methods. For simplicity and conciseness, we merely employed the fGWAS-F model in the real data of longitudinal traits. Legendre polynomialsSupporting InfoSupporting InfoSupporting InfoSupporting Info


Mendelian randomization uses genetic variants to make causal inferences about the effect of a risk factor on an outcome. With fine‐mapped genetic data, there may be hundreds of genetic variants in a single gene region any of which could be used to assess this causal relationship. However, using too many genetic variants in the analysis can lead to spurious estimates and inflated Type 1 error rates. But if only a few genetic variants are used, then the majority of the data is ignored and estimates are highly sensitive to the particular choice of variants. We propose an approach based on summarized data only (genetic association and correlation estimates) that uses principal components analysis to form instruments. This approach has desirable theoretical properties: it takes the totality of data into account and does not suffer from numerical instabilities. It also has good properties in simulation studies: it is not particularly sensitive to varying the genetic variants included in the analysis or the genetic correlation matrix, and it does not have greatly inflated Type 1 error rates. Overall, the method gives estimates that are less precise than those from variable selection approaches , but are more robust to seemingly arbitrary choices in the variable selection step. Methods are illustrated by an example using genetic associations with testosterone for 320 genetic variants to assess the effect of sex hormone related pathways on coronary artery disease risk, in which variable selection approaches give inconsistent inferences. As an aside, fitted values from the first‐stage of the 2SLS method are equivalent (up to an additive constant) to values of an allele score . This implies that the optimal weights in an allele score with correlated variants are the conditional (multivariable) associations of the variants with the risk factor.1.2The 2SLS estimate can also be obtained using summarized data on genetic associations with the risk factor and with the outcome from univariable regression analyses of the risk factor or outcome on each genetic variant in turn. This is important as such summarized data from large consortia are often publicly available, enabling Mendelian randomization investigations to be performed on large sample sizes without the need for costly and time‐consuming data‐sharing arrangements  se(β^Xj), and with the outcome is β^Yj with SE se(β^Yj), and assuming that genetic variants are uncorrelated, then the causal estimate is  with the inverse‐variance weights [se(β^Yj)/β^Xj]−2. The variant‐specific estimates are combined using the standard formula for a fixed‐effect meta‐analysis −2 and with the intercept term set to zero. The IVW estimate is equivalent to the 2SLS estimate when the genetic variants are uncorrelated InveΩj1j2=se(β^Yj1)se(β^Yj2)ρj1j2, and ρj1j2 is the correlation between genetic variants j1 and j2. The causal estimate is:β^X,β^Y are vectors of the genetic associations, and T is a vector transpose. Again, this estimate is equivalent to the 2SLS estimate that is obtained using individual‐level data (see Appendix for proof). It therefore inherits the efficiency property of the 2SLS estimate as the optimally efficient causal estimate based on all the genetic variants.When genetic variants are correlated, the IVW method can be extended to account for the correlations between genetic variants  to ensure that all variants contribute to the analysis, but without introducing numerical instabilities. We discuss practical implications of these findings for applied Mendelian randomization investigations. Software code in the R programming language for implementing the analyses discussed in the paper is provided in the Appendix.2SHBG gene region. The associations of 325 individual SNPs with testosterone in 3,225 men of European ancestry are reported by Jin et al.  on adjustment for these six variants. In all analyses, correlations between variants are estimated using 1,000 Genomes Phase 3 data on 503 individuals of European descent as reference data. A further two variants were monomorphic in the reference data; analyses are conducted using the remaining 320 variants. As variants in the SHBG gene region are associated with circulating levels of both testosterone and SHBG, a positive Mendelian randomization finding would not distinguish which of these is a causal risk factor, but would suggest that sex hormone related mechanisms have a causal role in cardiovascular disease.We consider an example of Mendelian randomization analysis with serum testosterone as the risk factor and coronary artery disease (CAD) risk as the outcome using genetic variants in the n et al. ; associan et al. . Previoun et al.  demonstrP‐value for association with the risk factor in the conditional analysis. We proceed until no additional variants are associated with the risk factor at P<0.0001 or P<0.001. This approach is implemented using the GCTA software. Third, we perform a stepwise pruning approach  analysis. Once a variant is selected, all other variants whose correlation with the selected variant is greater in magnitude than a given correlation threshold  are removed from the analysis. We continue until each variant is either selected or removed. This ensures that a set of variants is chosen for each threshold correlation such that the variants are each marginally associated with the risk factor, and the pairwise correlations are all below the threshold correlation. Although a data‐driven approach to selecting variants to include in a Mendelian randomization investigation is often unwise . Second, we perform a stepwise conditional approach using the summarized associations reported by Jin et al., selecting at each step of the analysis the variant having the lowest The Mendelian randomization estimates are presented in Table 3To resolve the question of how to choose which variants to include in a Mendelian randomization analysis, we explore reasons why analyses that include too many or too few genetic variants may go wrong, and propose a solution that incorporates associations on large numbers of genetic variants into the analysis, but does not suffer from numerical instabilities.3.1A matrix is singular if it cannot be inverted—formally, if the determinant of the matrix is zero. This occurs when the rows or columns of a matrix are linearly dependent; that is, at least one column (or row) can be calculated as a linear sum of multiples of the other columns (known as multicollinearity). This will occur for the genetic correlation matrix when two genetic variants are in perfect linkage disequilibrium, or alternatively if a small number of haplotypes are present in the data (perfect multicollinearity can occur even if no pair of variants is highly correlated). In contrast, a near‐singular matrix can be inverted, but its determinant is close to zero. This occurs in a regression model when there is substantial, but not perfect, multicollinearity. As sample sizes for estimating genetic correlations increase, singular matrices will become less common, but near‐singular genetic correlation matrices are likely to become more common. This is because a discrepant allele count in a single individual (which could represent a genotyping error) can lead to a singular matrix becoming nonsingular. A near‐singular matrix is problematic as elements of its inverse can be very large. In the motivating example with correlation thresholds of 0.9 and 0.95, the maximal element of the inverse of the correlation matrix is over 10 million.If a matrix is exactly singular, then it cannot be inverted, and the analysis will report an error. If a matrix is near‐singular, then the analysis may report an estimate without giving any indication that the estimate may be misleading  criteria, then a different set of variants would have been obtained from the conditional analysis, resulting in a different Mendelian randomization estimate.3.3Ψj1j2=β^Xj1β^Xj2se(β^Yj1)−1se(β^Yj2)−1ρj1j2. The diagonal elements of this matrix are the inverse‐variance weights, and so each is equal to the precision of the causal estimate based on that variant alone.One potential solution for resolving the problem of multiple correlated variants is PCA. The use of PCA has been previously suggested for reducing the dimensionality of the instrumental variable space to resolve issues of weak instrument bias −2∝MAFj(1−MAFj) , where βXj is measured in standard deviation [SD] units.) Hence, if the variants were uncorrelated, then the first principal component would be the genetic variant that explained the largest proportion of variance in the risk factor, and so on. For correlated variants, the first principal component represents a linear combination of variants that explains the largest proportion of variance in the risk factor, and each subsequent principal component is the linear combination of variants that explains the next largest proportion of variance while being orthogonal to the previous principal components.Assuming that associations for all variants are estimated in the same sample size, these diagonal elements are proportional to the amount of variance in the risk factor explained by the genetic variant. This can be seen as the SEs of the associations with the outcome will be directly proportional to the SEs of the associations with the risk factor, which in turn relate to the minor allele frequencies se(β^Yj)−1) will be greater for this variant, and so it will (correctly) be preferentially selected. The number of principal components to be included in the analysis can be chosen based on a threshold of variance in the weighted genetic correlation matrix. Once the principal components have been selected, we multiply the vector of genetic associations with the risk factor by the matrix of principal components, we multiply the vector of genetic associations with the outcome by the matrix of principal components, and pre‐ and postmultiply the genetic correlation matrix by the matrix of principal components. The IVW method is then performed on the transformed vectors of genetic associations and the transformed correlation matrix. If the matrix Ψ=WΛWT, where W is the matrix of eigenvectors (or loadings), and Λ is the diagonal matrix with the eigenvalues λ1>…>λJ on the diagonal, then let Wk be the matrix constructed of the first k columns of W. Then we define:This choice of matrix should be advantageous for Mendelian randomization investigations over PCA approaches on the unweighted matrix of genetic correlations. If two variants are perfectly correlated, but the estimates for one are measured in a larger sample size, then the precision of the association with the outcome  and −0.045 (0.083), respectively. These estimates are similar in precision to that using the previous conditional analysis for variable selection, but less precise than those calculated using the GCTA method on the data under analysis or a liberal correlation threshold in the pruning method.Overall, the conclusion from this motivating example is that there is no strong evidence of a causal relationship between sex hormone related pathways and coronary heart disease risk on the basis of the genetic evidence presented here. The more extreme estimates suggesting a causal relationship come from the less reliable methodological approaches, and these estimates should not be trusted. A more detailed analysis could be performed using genetic variants previously associated with either SHBG or testosterone from other gene regions, although the specific relevance of other variants to sex hormone related pathways is not always clear. Additionally, it could be argued that these analyses should be performed in men and women separately. An authoritative analysis conclusively judging the causal relevance of sex hormone related pathways to coronary heart disease risk is beyond the scope of this methodologically focused paper.4We illustrate statistical issues arising from using too many and too few variants in a series of simulation studies based on the motivating example. Again, fixed‐effect analysis models are used throughout.4.1First, we repeated the analyses of the motivating example except using only 180 of the 360 genetic variants at a time. This represents a scenario in which only a subset of the genetic variants in the analysis were measured. Sets of 180 variants were chosen at random 10,000 times.4.2Second, we repeated the analyses of the motivating example except varying the correlation matrix. We took a bootstrap sample of the reference data , and calculated a correlation matrix based on this sample. This procedure was performed 10,000 times.ρ=0.8, causal estimates were unstable, and were particularly sensitive to changes in the correlation matrix. In contrast, estimates using the PCA approach were not so precise, but they were far less variable between iterations.For each of these simulation analyses, we performed the pruning method for selecting genetic variants at a threshold correlation of 0.2, 0.4, 0.6 and 0.8, and the PCA method using components that explained 99% and 99.9% of the variance in the summarized association matrix. Results are presented in Table 4.3ΩX, where ΩXj1j2=se(β^Xj1)se(β^Xj2)ρj1j2. The associations with the outcome are drawn from a multivariate normal distribution with mean zero and variance‐covariance matrix Ω, where Ωj1j2=se(β^Yj1)se(β^Yj2)ρj1j2 as defined above. This represents a null causal effect. We also set the mean of the distribution of the associations with the outcome as 0.1 times the associations with the risk factor, representing a causal effect of 0.1. We simulated 10,000 datasets for each value of the causal effect, and calculated the Mendelian randomization estimate using the same six approaches for variant selection as above. Additionally, we repeated the analyses but first rounding the genetic associations (and their SEs) to three and two decimal places.Finally, we simulated genetic associations with the risk factor and with the outcome directly. Genetic associations with the risk factor were drawn for 320 variants from a multivariable normal distribution with mean vector the measured genetic associations with testosterone from the motivating example and variance‐covariance matrix 4.4ρ=0.8 even when the association estimates were not rounded. Rounding exacerbated false‐positive findings, and inflated Type 1 error rates were present in all methods when associations were rounded to two decimal places. Coverage rates were least affected when pruning at a threshold correlation of ρ=0.2 or 0.4 and for the PCA approaches. With a positive causal effect, power increased as the threshold increased, although judging estimators by power estimates is misleading when Type 1 error rates are inflated. Power of the PCA approaches was similar to that using a pruning threshold of ρ=0.2, but lower than that at a threshold of ρ=0.4, and was greater using principal components that explained a greater proportion of the weighted correlation matrix.Results are presented in Table 5As the cost of high‐density genome sequencing continues to fall, additional signals are likely to be identified within known loci. There will be growing demand for methods to exploit correlated instruments in Mendelian randomization, as the addition of correlated variants can improve power to detect a causal effect. In this paper, we first connected previously known results together to show from theoretical arguments that genetic variants included in a Mendelian randomization analysis should be those that are associated with the risk factor in a conditional analysis. If the variants are combined in an allele score, then the conditional (multivariable) associations with the risk factor should be used as weights in the allele score to obtain the most efficient analysis. If only summarized data are available, then the same analysis can be replicated with the marginal (univariable) associations using an extension to the IVW method to account for correlations between variants.r2=0.64), and occasionally occurred at a threshold correlation of 0.6 (r2=0.36). We note as well that r2 is not always a good measure of correlation between genetic variants; near‐singular matrices can occur when the pairwise correlations measured by r2 are low, but there are haplotypes represented in the data, or when the minor allele frequencies of variants differ, but a common variant “tags” a rare variant .However, difficulties arise when there are many correlated genetic variants in a single gene region that are associated with the risk factor (fine‐mapping genetic data). Including too few genetic variants in an analysis means that estimates are less precise, but also highly variable, in that different approaches to choosing variants can lead to markedly different estimates. However, including too many variants can lead to numerical instabilities and overly precise estimates with inflated Type 1 error rates. These numerical instabilities are not computational issues, but arise due to inconsistencies in the data: for example, if association estimates are rounded to a fixed number of decimal places, or if association or correlation estimates are obtained in different samples. It is difficult in practice to judge at what threshold these numerical issues begin to occur, although in the simulation examples considered, problems regularly occurred when pruning variants at a threshold correlation of 0.8 ; however, they are less variable with respect to choices of how to implement the analysis (in particular the choice of variants).5.1ρ=0.8, adding 0.1 to the diagonal of the correlation matrix changed the causal estimate from −0.137 (SE 0.031) to −0.065 (0.057). Although the substantial change in the causal estimate is indicative of near‐singular behavior, it would seem preferable for estimation to simply use a stricter correlation threshold rather than misspecifying the correlation matrix .The IVW method presented here is a simple application of generalized weighted linear regression, and is not unique to Mendelian randomization. The same method has been used in a variety of contexts including discovery genetics  or a small number of gene regions. Another feature of Mendelian randomization is the prevalence of the summarized data and two‐sample settings, in which discrepancies in genetic associations are likely to arise.PCAs have been suggested before for fine‐mapping data, with Wallace demonstrating that 70% of the variance in the genetic correlation matrix could be explained by an average of seven components for 49 test gene regions Wallace, . A key iWe advocate the PCA method proposed in this paper as a worthwhile approach to analyze fine‐mapped genetic data for Mendelian randomization. It provides estimates that may be less precise compared with those from variable selection approaches such as GCTA, but are more robust to seemingly arbitrary choices in the variable selection step.


Picea glauca) consisting of 1694 trees from 214 open-pollinated families, representing 43 provenances in Québec, Canada. The results revealed that HBLUP bivariate analysis is effective in reducing the known bias in heritability estimates of open-pollinated populations, as it exposes hidden relatedness, potential pedigree errors, and inbreeding. The addition of genomic information in the analysis considerably improved the accuracy in breeding value estimates by accounting for both Mendelian sampling and historical coancestry that were not captured by the contemporary pedigree alone. Increasing family genotyping efforts were associated with continuous improvement in model fit, precision of genetic parameters, and breeding value accuracy. Yet, improvements were observed even at minimal genotyping effort, indicating that even modest genotyping effort is effective in improving genetic evaluation. The combined utilization of both pedigree and genomic information may be a cost-effective approach to increase the accuracy of breeding values in forest tree breeding programs where shallow pedigrees and large testing populations are the norm.Maximization of genetic gain in forest tree breeding programs is contingent on the accuracy of the predicted breeding values and precision of the estimated genetic parameters. We investigated the effect of the combined use of contemporary pedigree information and genomic relatedness estimates on the accuracy of predicted breeding values and precision of estimated genetic parameters, as well as rankings of selection candidates, using single-step genomic evaluation (HBLUP). In this study, two traits with diverse heritabilities [tree height (HT) and wood density (WD)] were assessed at various levels of family genotyping efforts  from a population of white spruce ( Maximizing genetic gain in tree breeding programs depends on the reliability of the estimated genetic parameters. Traits’ heritabilities and their genetic correlations are among the most critical genetic parameters determining potential genetic progress, therefore their precision is of great importance. Progeny testing is the vehicle by which these genetic parameters are estimated, while individual tree data, family composition, pedigree depth and connectedness, and field performance determine their reliability  (The advantages of using marker-based relationship estimates are: (1) bypassing the classical mating designs needed for generating structured pedigree (edigree) .Combining the The aim of this study is to compare the traditional ABLUP approach to the HBLUP approach by comparison of the precision of genetic parameters, accuracy of breeding values, and rankings of selection candidates using variable proportions  of genotyped individuals within the open-pollinated families.Picea glauca (Moench) Voss] provenance-progeny test site that was established in 1979 in Québec, Canada. The test site contains 214 open-pollinated families from 43 Québec provenances planted in randomly assigned five tree row plots at 2.4 m between row and 1.2 m within row spacing to attain a randomized block design with six blocks. A subset of 1694 trees were sampled from the site, the mean number of trees per family was ∼7.9, and the mean number of trees per family per block was ∼1.3. Complete details of the test site are available in 3) obtained via X-ray densitometry.Data for the present study was obtained from Single-nucleotide polymorphisms (SNPs) of the 1694 trees were genotyped using Illumina Infinium HD iSelect bead chip PgAS1  , and The genomic relationship matrix (GRM) , the matrix H , resultiined via :H−1=A−1+G were equivalent to those of i.e., ABLUP). At 100% genotyping effort, the A11 represents only the unrelated and nongenotyped maternal parents . To avoid possible sampling bias, a replicated resampling scheme was used to obtain parameter estimates at 25–75% genotyping effort. In detail, two, four, or six individuals per family, of a possible maximum of eight, were randomly removed from the Prior to calculation of The restricted maximum likelihood (REML) was used to estimate variance and covariance for the random effects in the bivariate mixed model (1), and were obtained with the ASReml-R v3.0 program (The single-trait narrow-sense heritability for of each trait from the bivariate ABLUP and HBLUP models were estimated as:ectively .The additive genetic correlation between the two traits was estimated as Pearson’s product moment:odel (1) .iF is the inbreeding coefficient of the ith individual. SEs of the breeding values were obtained from the bivariate model with variance components fixed to estimates from the same model with 100% genotyping effort for the respective traits, under the assumption that these were most accurate.The theoretical accuracy of breeding values  was computed to compare the fit of the bivariate ABLUP and HBLUP models. A smaller AIC value indicates better fit. Furthermore, Spearman rank correlations and the proportion of common candidates in the top 5% of mothers and progenies were also calculated to compare whether the predicted breeding values differed among the ABLUP and HBLUP models.The data used are available from the Dryad Digital Repository: doi:10.5061/dryad.6rd6f (A) and combined (H) relationship estimators presented the ranges of the H relationship groups to be larger than that of A . Samples were sorted according to family and provenance, then the pairwise relationship coefficients of the matrices were plotted. Historical relationships  not captured by the pedigree estimator A  were visually apparent in H with as little as 25% genotyping effort . Further, potential pedigree errors not accounted for in the contemporary pedigree were uncovered, these errors may be due to mislabeling or technical errors in the lab. Population structure and potential pedigree errors were further evidenced by the large deviations observed between the ranges maternal parent–parent, half-sib, and unrelated relationship coefficients between A and the H relationship estimators .Narrow-sense heritability estimates of the bivariate HBLUP models mirrored those of the additive genetic variance estimates . These estimates at 100% genotyping effort for HT and WD were intermediate to the polygenic and marker-based estimates described by H relationship estimator , using the stimator .H and A relationship estimators on mean breeding value accuracies for the maternal parents showed minor increases for HT  and no changes for WD (from 0.635 to 0.635) with increasing family genotyping efforts. The mean breeding value accuracy of nongenotyped progeny using the H estimator across all levels of genotyping effort was stable and equivalent to the mean breeding value accuracy of the progeny using the A estimator, for both HT and WD. As expected, the largest improvements over the A relationship estimator were noted for genotyped progeny using the H estimators. Increasing genotyping effort produced large increases in mean breeding value accuracy for both HT and WD over mean breeding value accuracies of the progeny using A , with the effect greater for HT .The mean breeding value accuracies for HT and WD are presented for three classes of individuals: maternal parents and genotyped and nongenotyped progeny . CompariA and H relationship estimator at 25, 50, 75, and 100% genotyping effort, for both HT and WD, are presented in The Spearman rank correlations for maternal and progeny breeding values, and the proportion of common candidates in the top 5% of maternal and progeny rankings, between the i.e., accuracy and precision of genetic parameters) are often a reflection of the extent of efforts invested into the design of the field experiment. Commonly, quantitative genetics analyses are based on the animal model implemented in mixed model theory (A matrix). The mixed model theory operates with the assumption that variance–covariance matrices of random terms used in the animal model are error free  is important to eliminate bias connected to the total reliance on using genotyped individuals only or error-prone multi-step methods. While the genomic information benefits from lower inbreeding build-up during the selection phases as compared to pedigree-based BLUP  . Our results confirm that hidden relatedness within the OP families and historical coancestry exists in this P. glauca population, and was better accounted for through the HBLUP method, with increasing genotyping effort helping to approach more realistic estimates of genetic parameters and better model fit as compared to the analysis performed on the complete pedigree alone  of genomic information per family produced differences in ranking the top 5% of both maternal and progeny candidates for both traits . The difference in contrasting heritabilities of the traits and these findings agree with previous discussions that genomic selection is expected to be more efficient in low heritability traits . Yet, thh traits .G matrix, but also the difference between them as it reflects the level of relatedness in the testing population . This result is likely due to the shallow pedigree structure and minor differences between the A22 and G matrices. Additionally, G, but only under strong selection  of genotyped individuals and when genotyping was across 10 generations. However, in the present study, the 25–75% of genotyped trees were randomly removed  and all these trees came from only one generation.The accuracy of estimated breeding values in HBLUP is not a function of only average diagonal and off-diagonal elements in the A were observed. Improvement in the results were continuous with increasing genotyping effort; however, improvements were also seen at the minimum level of 25% genotyping effort.As a preliminary investigation into the combined use of genomic and pedigree information in the applied genetic analysis of white spruce, the HBLUP method has proven to be a beneficial tool to forest tree breeders. The inclusion of nongenotyped and genotyped trees in a single analysis produced improvements in breeding value accuracy and model fit, particularly for the trait with low heritability (HT). Further, discordance in candidate rankings between the HBLUP and the traditional method that utilizes www.g3journal.org/lookup/suppl/doi:10.1534/g3.116.037895/-/DC1.Supplemental material is available online at Click here for additional data file.Click here for additional data file.


The study included observations on 2524 offspring from 155 full-sib families. Substantial contributions of dominance were observed (P < 0.05) for body depth (BD) and body weight at harvest (BWH) with estimates of d2 = 0.27 (s.e. 0.09) and 0.23 (s.e. 0.09), respectively in the current breeding population. In addition the study found maternal variance (P < 0.05) for BD, BWH, body thickness and fillet weight explaining ~10% of the observed phenotypic variance. For fillet yield (FY) and body length (BL), no evidence was found for either maternal or dominance variance. For traits exhibiting maternal variance, including this effect in evaluations caused substantial re-ranking of selection candidates, but the impact of including dominance effects was notably less. Breeding schemes may benefit from utilising maternal variance in increasing accuracy of evaluations, reducing bias, and developing new lines, but the utilisation of the dominance variance may require further refinement of parameter estimates.There are only few studies of dominance effects in non-inbred aquaculture species, since commonly used mating designs often have low power to separate dominance, maternal and common environmental effects. Here, a factorial design with reciprocal cross, common rearing of eggs and subsequent lifecycle stages and pedigree assignment using DNA microsatellites was used to separate these effects and estimate dominance (d Genetic variation can be partitioned into additive and non-additive components of variance, where the latter arises from the interactions among loci (epistasis) or between alleles within a locus (dominance). Although sustained genetic change in conventional breeding schemes depends only on the additive component at the time of selection, the non-additive components can be utilised in the short-term through mate selection to obtain favourable heterosis in the offspring cohort, and in the long-term to protect the genetic assets of the breeder through F2-breakdown, e.g., through selection within lines or through selection schemes like Reciprocal Recurrent Selection (RRS)  between 2014 and 2015. The test-groups studied were from the GST® strain which originated from the well-documented GIFT strain  by GenoMar AS  on Nile tilapia . The main reason for the low assignment rate is that the marker set do not have enough exclusion power for the family structure used in the cross-breeding scheme, which involved only more closely related breeders. Because of the factorial mating design, we had a lot of half-sib families, which made it harder to uniquely assign individuals. Therefore, some fish would fit equally well into 2 or more families. With no way of knowing which family was the correct one, these were set as unassigned. The complete distribution of offspring across parents and families is given in Table The established pedigree from generations 3 to 22 contained 4051 records  and other systematic effects such as batch (2 d.f.) and day of collection (7 d.f.) or filleting ; a is a vector of random additive genetic effects; d is vector of random dominance effects; m is vector of maternal effects; e is a vector of random residual errors and X, Z1, Z2 and Z3, are corresponding design matrices for fixed and random effects. For FW and FY, the fixed model also included filleter (2 d.f.)ASReml-4 : additive heritability (h2) = σ2A / σ2P, dominance ratio (d2) = σ2D / σ2P, maternal ratio (m2) = σ2M / σ2P. Goodness of fit was tested using likelihood ratio tests. The critical values for testing H0: σ2 = 0 against an alternative H1: σ2 > 0 with type 1 error of 0.05 was taken from the 90 percentile of χ12, i.e., 2.71.The estimated variance components were expressed relative to the total phenotypic variance (x and y (∆xy) was calculated following Lynch and Walsh  to incorporate corrections for inbreeding as per Harris , with VFsib constrained to be 0 in S and unconstrained in SF. Model SM and SMF had VSire and VDam unconstrained with VFsib constrained to be 0 in SM and unconstrained in SFM. The phenotypic variance was estimated as VP = VSire + VDam + VFsib. Heritabilities, maternal and dominance ratio were estimated as h2 = 4VSire / VP and d2 = 4VFsib / VP and m2 = (VDam − VSire) / VP.Secondly, the analyses were conducted with a simple diallel model used to decompose the variances, which were designated SFM models .Effects on the genetic evaluation was compared among the different models; by Pearson’s correlation between estimates of breeding values, ranking of the 100 best offspring  and then counting the numbers that would have been excluded from the selected group compared to the simple A model.Descriptive statistics for the six different traits are shown in Table Numerical differences between reciprocal cross means were not statistically significant, although B × A were observed to have greater sizes and weights and FY; ranging from 0.1% for FY to 0.4% for BWH.The outcomes of the likelihood ratio test for goodness of fit are presented in Table Estimates for the variance components and heritabilities for different traits obtained by the different models are shown graphically in Fig. 2 for all traits. The inclusion of dominance in the models decreased the additive variance in ADM and A*D*M* models but only marginally in SFM models. In contrast, including maternal effect decreased the additive genetic variance considerably for some traits. ADM and A*D*M* models gave similar results for all the traits.The simple models gave the greatest additive genetic variances, and greatest hFor BWH and BD, the two traits for which the best fit included dominance, the dominance ratio was found to be 0.06 ± 0.04 and 0.07 ± 0.04 using the SFM model, but was much greater, with corresponding greater standard errors, for ADM and A*D*M* models; 0.27 ± 0.09 and 0.23 ± 0.08, respectively for the A*D*M* models. The dominance deviation among and within the different full-sib families for BWH are presented in Fig. m2) was found (P < 0.05), the fraction was close to 0.1 for FW, BD and BWH; but was smaller for BT.For the four traits where evidence of maternal ratio  which showed only additive variation; small for BT and FW , where there was evidence of maternal effects but no dominance, and 0 for BD and BWH, which showed both dominance and maternal variation. In the latter case, estimates of h2 from SFM models were small  rather than 0.As shown in Fig. The difference in ranking of Estimated Breeding Values (EBVs) among the 100 best animals, as a result of different models and the use of different depth of pedigree, is presented in Table The correlation of the EBVs for all animals with observations among the different models are presented in Fig. However, including the maternal effects was found to be different for different traits, ranging from 0.76 for FW to 0.94 for BL. For FY, the correlation was 1, as the maternal effect was close to 0. As with the ranking, there was little impact from changing the base of generation 3 (ADM models) to generation 20 (A*D*M* models)To the best of our knowledge, these are the first published results on dominance ratios in tilapia, and are potentially important for commercial production, both for the accuracy of EBVs for use in selection and for the eventual utilisation of heterotic effects. These were obtained by separating out the additive and non-additive genetic effects from the maternal and common environmental effects. This was achieved using a factorial mating design, including reciprocal crosses, and exploiting the large full-sib family sizes possible in fish species, which is uncommon in livestock, and seldom used in commercial aquaculture. The scope of the trial encompassed both the commercially important morphological and weight-related traits, and the post-harvest measures of fillet weight and yield, which characterise the primary saleable product.® strain used for this study is derived from the GIFT strain that is the common ancestor to most tilapia populations used for commercial breeding, and the first 10 generations of GST® also correspond to the first 10 generations of the GIFT strain. The designated base generation of GIFT, which here is defined as generation 3 of GST®, was formed from four wild and four Asian strains crossed systematically over 3 generations to allow mixing of the strains before selection for growth was commenced , variances among dams (VDam), and the variances within full-sib families (VFsib). The supplementary information , if >0; and σ2D = 4VFsib; A*D*M* estimates with a base generation in generation 20, which most closely correspond to random mating in the current population; and ADM estimates which project back to generation 3, the GIFT base. Since each emerge as scaling of the same set of core components, the standard errors and uncertainties reflect the magnitude of the scaling factors applied. The near-equal scaling factors from using generation 3 (ADM) or 20 (A*D*M*) as the base, demonstrate that the scaling observed for estimates of σ2D in ADM models is a consequence of the design rather than the additional pedigree. There are additional approximations in the use of the fraternity matrix to assess dominance, as it is an approximation of the full dominance model , and itVFsib can be interpreted as dominance variance, an assumption common to many other studies. Although our design has separated out the maternal effect and minimised common environmental effects through the management described in the Materials and Methods, this interpretation cannot be certain. The results show that maternal variance is still detectable for four of the six traits (not for BL and FY) despite this management. These effects might be related to the size and quality of the eggs or mitochondrial effects. Large eggs have more yolk reserves and have been shown to be positively correlated to the growth and development of fry  for coho salmon (Gallardo et al. There have been no previous estimates of dominance ratios in tilapia, but very few in other fish species, including the more intensively studied trout and salmon (Winkelman and Peterson The maternal component, shown to be present in all but two of the traits, has practical consequences for the genetic evaluation. This source of variance is not always fitted, however, as shown in Table The phenotypic data and the pedigree are available as Supplementary Supplementary 1 Design of the studySupplementary 2 Correspondence between SFM, ADM and A*D*M* modelsSupplementary 3 Tables of variance component estimates for all the modelsSupplementary 4 Literature reviews for heritabilities and common maternal and environmental effects in tilapiaSupplementary 5 - Phenotypic dataSupplementary 6 - Pedigree data