Estimating relationships among subjects in a sample, within family structures or caused by population substructure, is complicated in admixed populations. Inaccurate allele frequencies can bias both kinship estimates and tests for association between subjects and a phenotype. We analyzed the simulated and real family data from Genetic Analysis Workshop 19, and were aware of the simulation model.We found that kinship estimation is more accurate when marker data include common variants whose frequencies are less variable across populations. Estimates of heritability and association vary with age for longitudinally measured traits. Accounting for local ancestry identified different true associations than those identified by a traditional approach. Principal components aid kinship estimation and tests for association, but their utility is influenced by the frequency of the markers used to generate them.Admixed families can provide a powerful resource for detecting disease loci, as well as analytical challenges. Allele frequencies, although difficult to adequately estimate in admixed populations, have a strong impact on the estimation of kinship, ancestry, and association with phenotypes. Approaches that acknowledge population structure in admixed families outperform those which ignore it. Estimates of the kinship coefficient, defined as the probability that 2 alleles randomly sampled from 2 subjects are identical by descent, have many uses . These iWe explored whether genetic associations with systolic blood pressure (SBP) changed over time using longitudinal data. We estimated heritability and performed a kinship-adjusted half-genome-wide association test at the first and third visits for real SBP. We discuss the similarities, differences, and potential foundations for those changes.Through both selection and drift, different populations can have different variants influencing the same trait, or very different frequencies of shared risk alleles. Admixture mapping takes advantage of these differences to identify risk loci . Using tWhen testing for association between genotype and phenotype, principal components (PCs) are often included as covariates to minimize the effects of population structure. We investigated how well PCs estimated on different subsets of the data were able to capture ancestry proportions. Performance was evaluated with the coefficient of determination.http://www.1000genomes.org/category/exome).We used GAW19 genome-wide association study (GWAS) data for odd-numbered autosomes for association testing and admixture mapping. Data were available on 959 subjects from 20 pedigrees. Sex-averaged map positions (cM) were converted by the Haldane map function from the Rutgers framework map with all of the Single Nucleotide Polymorphism database 134 (dbSNP134) variants . Sex-ave2 ≤ 0.2), applied filters described below, and pruned for LD with SNPRelate [2 ≤ 0.1). The Agnostic design includes every 100th variant: 21,484 WGS variants. Agnostic variants are rare: 58 % have founder MAF of 1 % or less. The Selective design includes variants with MAF 5 % or greater: 64,389 WGS and 7215 exome variants. With an allele frequency spectrum comparable to the Selective design, our Homogenizing design includes the 30,710 WGS variants with alternate allele frequencies that minimally vary across the African (AFR), Native American (AMR), Asian (ASN), and European (EUR) populations [We pruned WGS variants for LD in PLINK  , maximuWe examined the role age may play on genetic effects on SBP by fitting a linear mixed-effects model to the longitudinal data for 916 individuals and 2189 SBP observations. Using the pedigree-based kinship matrix, we estimated the additive variance as a function of cubic splines of age, with log of the environmental variance similarly fit. SBP was adjusted using fixed effects for medication use, smoking status and gender, as well as smoking status and gender-specific curves for age.−7 .We performed a half-genome scan for association with adjusted SBP at the first and third visits using Efficient Mixed-Model Association eXpedited (EMMAX) . EMMAX ig + Iσe, where I is the identity matrix. Ancestry for CEU, YRI, and AMR were each fit separately. For admixture mapping, we use twice the whole-genome-wide nominal p value of 7 × 10−6, which has previously shown a type I error of 0.05 [−7.Admixture and association analysis of the simulated SBP at the first visit, adjusted as described above, included 955 subjects with GWAS data. Local ancestry was estimated by RFMix . We phas of 0.05 . For ass2, the coefficient of determination, from a linear regression model with 10 PCs from a PC analysis (PCA) as predictors and CEU, AMR, and YRI ancestry proportions from a supervised ADMIXTURE [i, j)th entryS is the number of variants, Gis and Gis are the number of minor alleles  that individuals i and jhave at marker s, and ps is the MAF at marker s. Unlike the EIGENSTRAT method [We began with the WGS data for 102 unrelated subjects from the GAW19 pedigrees. We created four subsets of variants based upon their MAF: rare (MAF <0.01 or <0.05) and common (MAF >0.01 or >0.05). There were 7,407,452 SNPs . We estimated RDMIXTURE  analysisDMIXTURE . We perfThe MoM approach and Agnostic design provide the least-reliable kinship estimates see Fig. , whereasN = 831; 2060 SBP measurements), we found the heritability estimates were robust to kinship estimators . The adjusted first- and third-visit SBP values are quite different, with a correlation of 0.48, and the heritability estimate from EMMAX for the first-visit SBP is 0.26 and 0.13 for the third-visit SBP.Environmental variance generally increased as a function of age, shown in the narrow-sense heritability estimates presented in Fig. p values in Fig. LRRC31 gene) had a p value of 5 × 10−10.The EMMAX approach found no significant associations with SBP at the first visit Fig. , but ide−5. If we reduce the significance threshold to 5 × 10−4, we have 17 % power across 200 simulated replicates to detect the variant in the P2RX5 gene on chr17 and 81 % power to detect the variant in the COL5A3 gene on chr19. For association mapping, we observed 100 % power across 200 replicates to detect the causal variant in the MAP4 gene on chr3 at the significance level of 10−7. Figure p values for each approach for the first simulated replicate.Admixture mapping had no power at the significance level of 1.4 × 10As shown in Table We have shown that the frequency of alleles included in kinship, association, and population structure estimation have strong influences on their results. Kinship estimation is most accurate when markers are restricted to common variants that are not ancestry informative, and the moment estimator showing the least bias incorporated ancestry-informative PCs. Admixture mapping and association testing each identified different causal genes for the simulated adjusted SBP, likely a consequence of differences in frequency of risk alleles in the AMR and CEU reference populations. This warrants a future analysis evaluating a single test of ancestry at each locus using a 2-degrees-of-freedom test, as opposed to treating each ancestry separately. When estimating population structure, the marker data must be selected using the frequency of alleles in multiple relevant reference populations in order to adequately capture the complexity of ancestry in admixed populations.Association testing found little evidence for real adjusted SBP loci, and the inflation of EMMAX results with third-visit SBP was likely caused by the increase in environmental variance in the trait as age increased, consistent with the change in heritability estimates over time. There could be shared environmental factors acting on third-visit SBP that are not being modeled. This would cause the systematic inflation of test statistics across the genome.Variants with imprecise allele frequencies bias estimates of kinship, PCA, and association testing. Admixture mapping and association testing proved complementary. The influence of environment on estimates of heritability and association appear to have been revealed by analysis of longitudinal data.


Cortisol concentrations in hair are used to create hormone profiles spanning months. This method allows assessment of chronic cortisol exposure, but might be biased by hair pigmentation: dark hair was previously related to higher concentrations. It is unclear whether this association arises from local effects, such as increased hormone extractability, or whether the association represents systemic differences arising from population stratification. We tested the hypothesis that hair pigmentation gene variants are associated with varying cortisol levels independent of genetic ancestry. Hormone concentrations and genotype were measured in 1674 children from the Generation R cohort at age 6. We computed a polygenic score of hair color based on 9 single nucleotide polymorphisms. This score was used to predict hair cortisol concentrations, adjusted for genetic ancestry, sex, age and corticosteroid use. A 1-standard deviation (SD) higher polygenic score (darker hair) was associated with 0.08 SD higher cortisol levels . This suggests that variation in hair cortisol concentrations is partly explained by local hair effects. In multi-ancestry studies this hair pigmentation bias can reduce power and confound results. Researchers should therefore consider adjusting analyses by reported hair color, by polygenic scores, or by both. Each cm of proximal scalp hair represents ca. 1 month cortisol exposure, which makes the measurement of relatively long-term profiles of cortisol and cortisone, a metabolite and precursor of cortisol, feasible5. Hair cortisol assessment is therefore an attractive addition to repeated plasma or saliva measurements.In the last decade, studies demonstrated that hair is a useful medium to measure chronic cortisol secretion over a period of 3–6 months2. Hair pigmentation might directly affect the potential to extract cortisol from hair and thus measured differences may mirror local effects only. Second, the hair cortisol differences might reflect genetic differences in subpopulations (population stratification). For example, since hair color is strongly linked to genetic ancestry, it might be a marker of genetic variations related to cortisol metabolism or sensitivity3. Third, color might be a marker for minority status and the related stress, which would explain higher systemic cortisol levels4. Distinguishing between these scenarios is important for observational hair cortisol research, since an association between hair color and hair cortisol might introduce a confounding bias.While hair samples are a compelling method, there is concern that hair color might bias measurements. We reported previously that hair color was associated with cortisol and cortisone levels in the Generation R Study, specifically that higher cortisol levels were found in darker hair7. We created a polygenic score of hair color, which predicts hair lightness/darkness on the basis of these pigmentation SNPs. We computed a genetic score as opposed to solely using reported hair color, because the score allows a continuous assessment of hair pigmentation, is objective, and is not affected by the environment. Such a genetic score potentially represents hair pigmentation more accurately. This way we tested the main hypothesis that a genetic score of hair color is associated with hair cortisol and cortisone levels independent of genetic ancestry in children.To explore the nature of the hair color and hair cortisol association, we investigated whether single nucleotide polymorphisms (SNPs) associated with hair pigmentation are associated with hair cortisol levels in childhood independent of genetic ancestry. For this purpose we selected 16 SNPs from 10 genes included in the HIrisPlex system previously developed to predict hair and eye color from DNA9. All parents gave informed consent for their children’s participation. The Generation R Study is conducted in accordance with the World Medical Association Declaration of Helsinki and study protocols have been approved by the Medical Ethics Committee of the Erasmus Medical Center, Rotterdam.This study was conducted in Generation R. Generation R is a population-based birth cohort aiming to identify early environmental and genetic determinants of development and healthHair color and genetic information was available in 3262 children. To avoid overfitting of the polygenic score, we split the sample into children with cortisol or cortisone information n = 1697), the validation sample, and a training sample with neither cortisol nor cortisone (n = 1565) information. Selection and weights of the SNPs for the polygenic pigmentation score were determined in the training set. Hair cortisol or cortisone measurements were available for 1697 children (1674 had cortisol and 1656 had cortisone available). See Fig. 697, the −7). Excess heterozygosity, gender accuracy, and relatedness were tested. We used MACH 1.010 to impute to the 1000 Genomes Iv3 reference In Generation R DNA was extracted from whole blood at birth and analyzed using Illumina 610 K/660 W. We filtered for sample (≥97.5%) and SNP call rates (≥95%), minor allele frequency ≥1% and deviations from Hardy-Weinberg equilibrium (p < 102 < 0.3) and 2 due to a minor allele frequency below 1%  were calculated for the whole Generation R sample (n = 5731) and subsequently used in the subsample of children with available hair color and hormones data, the training and validation samples. Participants exceeding 4 SDs difference with the mean European reference level (HapMap CEU) on any of the first four principal components were classified as non-northwestern European. For analyses restricted to children with northwestern European ancestry, the PCA were recalculated in that Generation R subsample (n = 2830). Again the whole Generation R sample was used for the estimation of PCA. Figure Multidimensional scaling was used for the investigation of genetic ancestry based on the genome-wide SNP data2. Hair color was categorized into 7 categories: “sandy red” (1), “red or chestnut” (2), “blond” (3), “dark blond” (4), “brown” (5), “dark brown” (6), “brownish black or black” (7), analyzed as continuous variable (ranging from 1–7) indicating pigmentation intensity. See Table Hair color of the children was obtained by parent report and when not available, scored with photographs and videos taken during the research center visit. Inter-coder reliability was calculated with 50 overlapping observations using Krippendorff’s alpha. Alpha was 0.79 between the investigators and 0.69 between the investigators and parentsndy red” , “red or5. Briefly, steroids were extracted using LC-grade methanol at 25 °C for 18 h in the presence of deuterium labeled steroids as internal standard. Samples were centrifuged and cleaned using solid phase extraction, after which steroids were quantified by liquid chromatography-tandem mass spectrometry (LC-MS/MS) , using positive electrospray ionization. See Table Cortisol and cortisone concentrations were measured in the proximal three cm scalp hair, as described previously2 = 0.92). The model was adjusted for 20 PCA ensuring that only SNPs are selected which have explanatory power beyond being markers for genetic ancestry. Using the regression coefficients of the 9 nominally significant (α = 0.05) SNPs . Rs16891982 was not included in the training model due to high multicollinearity (Variance Inflation Factor (VIF) = 16.5) caused by strong linkage disequlibrium with rs28777 , next to sex, age and corticosteroid use (parent-reported yes/no). Two to ten PCA are commonly recommended to correct for population stratification, depending on the trait and ancestry admixture2 of the fully adjusted pigmentation effect14. Finally, we also stratified the main analysis by four ethnic subgroups . Given that these classifications are based on national origin rather than genetic data and that the sample sizes are low, we interpret these analyses exploratory.The ancestry corrected analysis using a polygenic score of hair pigmentation was used to test the main hypothesis. However, we performed additional analyses for exploratory and comparative purposes. We also tested individual hair pigmentation SNPs in separate and in a mutually adjusted model. Next, we related observed hair color (treated continuously) to cortisol and cortisone. As the polygenic score was calculated correcting for ancestry, all models were rerun without the additional genetic ancestry adjustments in the regression analyses. Furthermore, we performed sensitivity analyses in the subsample of children of European ancestry as defined by genetic data. We calculated the power of this subsample to detect effect sizes found in the multi-ancestry sample. For this purpose we used the local Cohen’s FThe hair cortisol and cortisone regression analyses yielded skewed residual distributions. We therefore applied box-cox transformations. The best fitting lambda was −0.26 for cortisol and −0.06 for cortisone, based on the main model. Transformed values were multiplied with −1 to keep directionality. The polygenic score, cortisol and cortisone were standardized to facilitate interpretation.et al.15. Estimates and standard errors (SE) were extracted for 9 SNPs from ancestry-adjusted models. We meta-analyzed using inverse-variance weighting to calculate Q and I2 statistics after orienting the SNP effects. A significant Q or high I2 indicate heterogeneity in the effect estimates and can be an indication that the SNP associations are not solely explained by hair color. Heterogeneity is problematic if the pleiotropic effects are in the same direction for a majority of SNPs, which can be detected visually as asymmetry in a funnel plot or by a significant asymmetry test16.To investigate potential pleiotropic effects of the hair pigmentation SNPs, i.e. whether the SNPs are associated with hair cortisol via pathways unrelated to hair color, we investigated the heterogeneity of the single SNP estimates as described by Burgess 17. The package MASS 7.3–4518 was used for box-cox transformations, metafor 1.9–919 for heterogeneity analyses and funnel plots, pwr20 for power analysis, psych 1.6.921 for descriptives and foreign 0.8–6722 for reading external files.Statistical analyses were performed in R 3.3.22). An increase in 1-standard deviation (SD) of the polygenic score was associated with a 0.83-level darker hair . For comparison, a polygenic score based on 13 SNPs from a model without genetic ancestry adjustment explained 49% of the variance in this sample.In the training set 9 of the 16 SNPs showed nominally significant (α = 0.05) associations with hair color independent of genetic ancestry as expected  and 0.06 SD higher cortisone levels  (Tables The polygenic score explained 4.2% of the hair cortisol variance and 0.7% of hair cortisone in a simple regression (genetic ancestry adjusted in training step), whereas hair color explained 4.5% and 0.6% respectively (no genetic ancestry adjustment). For comparison, a polygenic score based on 13 SNPs from a training model without genetic ancestry adjustment explained 5.8% and 0.8% variance. Genetic ancestry explained 8.0% of the cortisol and 3.8% of the cortisone variance. In contrast, national origin (dummy coded) explained 6.1% and 3.5% respectively.2 = 0.006, power = 86%). Repeating the analysis within ethnic minorities revealed a significant association of the polygenic score with hair cortisol in children of African national origin   and cortisone  . The funnel plot showed no asymmetry , hair color certainly adds additional variance to hormonal measurements, which can increase standard errors, and thus adjustment for hair color or genetic markers of hair color could be beneficial. In genetically homogeneous European samples, bias introduced by hair color is small, as shown here, and may be ignored.A hair color bias could occur in observational cortisol studies with predictors or outcomes, which are associated with hair color. Such population stratification by hair color is conceivable in studies of metabolic traits, psychological stress, and cortisol genetics among others.In studies of psychological stress in Western multi-ethnic populations for example, the scenario is possible that dark hair is associated with minority status and consequently increased stress exposure. The observed effects of stress on hair cortisol, however, would then be inflated as the association represents the effects of stress on systemic levels as well as those of dark hair pigmentation on hair cortisol levels. In contrast, if light hair is related to higher stress exposure, associations would be deflated. Such studies are typically adjusted for ethnicity, however, given that ethnicity assessments are imperfect and will not be able to account for all hair color biases, further adjustments for hair color are likely useful. Specifically, polygenic scores are beneficial given that their association with hair hormones is partly independent of genetic ancestry. One might even consider adjusting for a polygenic score only instead of ethnicity to reduce chances of overadjustment for true stress effects, though some degree of overadjustment may not be avoidable.Other research situations, in which hair color might cause misleading results, are future genome-wide association or heritability studies of hair cortisol. These study designs might find genetic effects for hair cortisol, which could be completely driven by hair pigmentation genes and their association with local hair hormone levels. Strategies to counter this phenomenon include the exclusion of hair pigmentation genes before analysis, adjustments for hair cortisol genes in the analysis, or the examination of the linkage disequilibrium between genetic association loci and pigmentation variants after analysis.We used strict adjustments for genetic ancestry in this study. However, residual confounding by ancestry cannot be completely ruled out and hair pigmentation may remain a marker of ancestry even after controlling for principal components. If this scenario were to explain the observed associations, this study would suggest that adjustments for both principal components and genetic markers of hair color are necessary to correct for population stratification in hair cortisol studies.The possible implications of this study can be summarized as follows: in genetically heterogeneous study populations hair pigmentation bias can reduce power and lead to confounded associations. Researchers should therefore consider adjusting analyses by (reported) hair color, by polygenic scores or both.Supplementary Tables and Figures


Genomic prediction using high-density (HD) marker genotypes is expected to lead to higher prediction accuracy, particularly for more heterogeneous multi-breed and crossbred populations such as those in sheep and beef cattle, due to providing stronger linkage disequilibrium between single nucleotide polymorphisms and quantitative trait loci controlling a trait. The objective of this study was to evaluate a possible improvement in genomic prediction accuracy of production traits in Australian sheep breeds based on HD genotypes  compared to prediction based on 50k marker genotypes. In particular, we compared improvement in prediction accuracy of animals that are more distantly related to the reference population and across sheep breeds.Genomic best linear unbiased prediction (GBLUP) and a Bayesian approach (BayesR) were used as prediction methods using whole or subsets of a large multi-breed/crossbred sheep reference set. Empirical prediction accuracy was evaluated for purebred Merino, Border Leicester, Poll Dorset and White Suffolk sire breeds according to the Pearson correlation coefficient between genomic estimated breeding values and breeding values estimated based on a progeny test in a separate dataset.Results showed a small absolute improvement  in prediction accuracy of purebred animals from HD genotypes when prediction was based on the whole dataset. Greater improvement in prediction accuracy (1.0 to 12.0% and on average 5.2%) was observed for animals that were genetically lowly related to the reference set while it ranged from 0.0 to 5.0% for across-breed prediction. On average, no significant advantage was observed with BayesR compared to GBLUP. The development of high-throughput genotyping based on single nucleotide polymorphisms (SNPs) in livestock species has made the implementation of genomic evaluation more practical. In genomic prediction, the breeding values of selection candidates are evaluated according to their genotypes and a prediction equation derived from a reference population with both phenotypes and genotypes . The accResults of simulation studies in livestock show various degrees of improvement in genomic prediction when using HD genotypes compared to genotypes from moderate-density SNP panels such as 50k. For example, based on simulation studies, Meuwissen and Goddard  reportedAnalyses of real data are available from dairy cattle and show zero to relatively small increases in prediction accuracy from HD genotypes. Solberg et al.  reportedInitially, the first factor that was suggested to affect the accuracy of genomic prediction was the LD between genome-wide SNPs , 2. HoweThe objective of this study was to compare the accuracy of genomic prediction for weight, ultra-sound scanned fat and muscle traits, and wool quality and quantity traits in Australian sheep breeds based on both observed and imputed HD genotypes (600k Illumina Ovine SNP) to accuracies based on moderate-density SNP genotypes (Illumina ovine SNP50k). Using a reference set comprised of purebred, crossbred or mixed crossbred and purebred animals, prediction accuracies were compared for purebred industry sires for which very accurate estimated breeding values based on a progeny test were available. Furthermore, we contrasted accuracy of genomic prediction within a breed between animals with low and high genetic relatedness to the reference set as well as prediction within and across breeds.The genomic prediction reference set consisted of about 20,000 animals that were recorded for a large number of production traits measured in the “Sheep Cooperative Research Centre Information Nucleus Flock” (INF) and “Sheep Genomics Flock” (SGF). The INF consisted of eight flocks that are located across different regions of Australia and are linked to each other because artificial insemination with common sires was used between 2007 and 2011 . The SGFA validation population was used to find the empirical accuracy of genomic prediction. The validation population was a group of industry purebred sires with accurate estimated breeding values (EBV) (accuracy ranging from 0.70 to 0.99 and on average 0.92), which were calculated based on progeny records. The phenotypes of INF and SGF animals (genomic prediction reference set) were not used in the calculation of EBV of the validation sires.−15). Furthermore, an individual sample was removed if the correlation of its genotypes  with those of another sample was equal or greater than 0.98.The reference and validation populations were genotyped using a 50k SNP panel . This 50k SNP panel provided 48,559 SNP genotypes after applying quality control based on the following criteria: individual SNP genotypes were removed if their call rates were lower than 90%, or if the GenCal (GC) scores were <0.6, if the heterozygosity rate for a given SNP deviated more than 3 SD from the population mean, if the SNP minor allele frequency was lower than 0.01, and for SNPs located on chromosomes X and Y or SNPs that deviated from Hardy–Weinberg equilibrium  ovine SNP panel. This SNP panel provided 510,174 SNPs after applying the same quality controls as above. Using all HD genotyped animals as imputation reference set, the un-typed genotypes of the rest of the population were imputed to HD genotypes using the software program FImpute . The accy is a vector of phenotypes, b is a vector of fixed effects, a is a vector of random additive polygenic effects, w is a vector of random maternal effects, q is a vector of random breed effects, s is a vector with random sire by flock interaction effects and e is a vector of random residual effects. X, W and Q is a matrix with breed proportions for each animal derived from pedigree data. Up to 28 breed effects, including those of the three Merino strains, were estimated via the Q matrix, however the major breeds were Merino, BL, PD and WS. All random effects are identically and independently distributed except for a which is distributed as: A is a numerator relationship and A was replaced by G, where G is a genomic relationship matrix calculated based on 50k or HD SNP genotypes using VanRaden’s [M is a matrix of the size n × m  with coefficients equal to A1A1), (A1A2) and (A2A2) of the jth SNP genotype respectively, A1 for the jth SNP genotype. m refers to the random effects of SNPs, For the analysis based on pedigree relationships, the following mixed model was fitted using ASReml 3.0 :\documennRaden’s  equation (BayesR ) in whic , as the Pearson correlation coefficient between GBV and an accurate EBV estimated from progeny test. Correlations were estimated for each breed separately, while an effect due to the Merino strain was fitted to avoid GBV accuracy to be biased upward for merinos by evaluating accuracy across strains. The size of the validation set for different traits was 341 to 389 sires for Merino, 79 to 88 for BL, 161 to 188 for PD and 189 to 204 for WS. We also contrasted the accuracy of GBV for animals with high or low genomic relationships with the reference set. Animals with high genomic relatedness were those for which the average value of their 30 highest genomic relationships to the reference population was at least 0.20. Animals with low genetic relatedness were those for which the genomic relationship with any of the individuals in the reference set was not higher than 0.10.Table Variance components estimated by using HD genotypes resulted in larger additive genetic variance, smaller residual variance and hence higher heritability across all studied traits, when compared to 50k genotypes. However, the increase in additive variance and heritability was small . Variance components and heritability estimates were similar between models that used HD genotypes and pedigree. Less than 1% differences were found between heritability estimates based on HD genotypes and pedigree when averaged across all weight, carcass scan and wool traits.Tables When using HD genotypes, the accuracy of genomic prediction was very similar between GBLUP and BayesR across all traits, with an average absolute value of the difference in genomic prediction accuracy between GBLUP-HD and BayesR of −0.008, −0.006 and 0.03 for Merino, Maternal and Terminal breeds, respectively.Table Table The data in Table Figures Table A and G matrices are not necessarily on the same scale (e.g. the G matrix is derived as a genomic relationship) so these estimates cannot be directly compared. Haile-Mariam et al. [This study investigates the possible improvement in accuracy of genomic prediction of breeding values for weight, scanned carcass and wool quantity and quality traits in Australian sheep when using high-density SNP genotypes. First, we compared the variance components that were estimated based on relationships derived from 50k and HD genotypes to those based on pedigree relationships. Estimated additive genetic variances based on HD genotypes were larger than those based on the 50k SNP panel, which suggests that the HD panel captures more genetic variation; this is likely due to higher LD between SNPs and QTL. Estimated genetic variances based on the HD panel were similar while the estimates based on the 50k panel were lower than those based on pedigree data. However, the m et al.  also repm et al.  argued tHD SNP panels provided higher prediction accuracies but the increase had only practical significance for individuals that were not closely related to the reference population. The average improvement in prediction accuracy was small, ~2.2% which is likely due to the effect of closer relationships providing information that is not much improved by higher marker density. SNPs can capture co-segregation of alleles (family relationships) as well as the LD between SNPs and QTL , 21, 22.Previous reports based on real data in dairy cattle also showed a very limited improvement in prediction accuracy when using HD genotypes , 8, whicGenomic prediction in a multi-breed reference set could potentially benefit from across-breed prediction when using HD genotypes, as has been suggested in various studies , 26, 27.Some studies have shown that using moderate-density SNP panels ~50k) provide a more marked improvement in genomic prediction accuracy over low-density SNP panels in different livestock species. Moghaddar et al.  compared0k providThe regression coefficient of EBV on GBV was on average higher (less biased) based on HD SNPs than on 50k SNPs. This could be related to the larger additive genetic variances that were estimated when using HD genotypes and are more similar to the estimates of additive genetic variance based on pedigree data. Bias could also occur if selected SNPs were used for genomic prediction. To some extent, the BayesR method uses selected SNPs, in the sense that it uses some priors to emphasize a larger effect for some SNPs by giving them more weight. However, regression coefficients did not differ between GBV based on GBLUP using HD genotypes and GBV based on the BayesR method, which suggests that this explanation is less likely.G-matrix not being expressed at the same scale as the numeric relationship matrix (A) used in the genetic evaluation that produces the EBV, or because of differences in the method for accounting for genetic groups in the reference and validation populations. The A-matrix is based on pedigree relationships whereas GBV are calculated with a G-matrix that uses relationships across various subpopulations within the population. Since this study was mainly aimed at evaluating genomic prediction accuracy, we did not attempt to rescale the G-matrix, since accuracy is calculated as a correlation which is independent of scale. Furthermore, the averages of diagonal and off-diagonal elements of A and G were similar  as was suggested by Legarra [Regression coefficients of EBV on GBV were generally lower than 1.00 (0.74 to 0.94). This may be due to the  Legarra  as a reqOur results show that the use of high-density (600k) SNP genotypes for the genomic prediction of weight and wool production traits in a multi-breed sheep population resulted in a small improvement in accuracy compared to a moderate SNP density (50k). Improvement in accuracy was greater for individuals that were distantly related to the reference set. Prediction accuracy based on a reference set from other breeds was low and showed limited improvement with HD genotypes. Results of GBLUP and BayesR were not significantly different.


This study compares five models of GWAS, to show the added value of non-additive modeling of allelic effects to identify genomic regions controlling flowering time of sunflower hybrids.Genome-wide association studies are a powerful and widely used tool to decipher the genetic control of complex traits. One of the main challenges for hybrid crops, such as maize or sunflower, is to model the hybrid vigor in the linear mixed models, considering the relatedness between individuals. Here, we compared two additive and three non-additive association models for their ability to identify genomic regions associated with flowering time in sunflower hybrids. A panel of 452 sunflower hybrids, corresponding to incomplete crossing between 36 male lines and 36 female lines, was phenotyped in five environments and genotyped for 2,204,423 SNPs. Intra-locus effects were estimated in multi-locus models to detect genomic regions associated with flowering time using the different models. Thirteen quantitative trait loci were identified in total, two with both model categories and one with only non-additive models. A quantitative trait loci on LG09, detected by both the additive and non-additive models, is located near a GAI homolog and is presented in detail. Overall, this study shows the added value of non-additive modeling of allelic effects for identifying genomic regions that control traits of interest and that could participate in the heterosis observed in hybrids.The online version of this article (doi:10.1007/s00122-017-3003-4) contains supplementary material, which is available to authorized users. Currently, several tools are available to geneticists and breeders to identify the genetic control of traits of interest and to improve the performance of animals and plants. A powerful tool for mapping the genes controlling complex traits, association genetics essentially evaluates statistical correlations between the alleles at a given locus and the observed phenotype  is a key trait in plant biology. Its evolution has been crucial for the domestication of many crop species and their dissemination into new climatic regions  from various French experiments conducted in 2013 by private partners  and by the French National Institute for Agricultural Research (INRA) as part of the SUNRISE project. Five experimental sites in different environments of regions in Southwestern France were planted with different hybrids from a set of 452 hybrids (between 303 and 444 hybrids per environment). Hybrids for this study were obtained by crossing 36 males and 36 females in an incomplete factorial design. They were chosen so that every parent was represented equivalently in the hybrid population (between 12 and 15 hybrids per parent).We collected data on the flowering time of sunflower , a replicate fixed factor if necessary, an independent random genetic factor, and the residual error.p-value as a fixed regressor in a model that contains a random polygenic effect, as in classic GWAS model of Yu et al.  relatedness.L is the total number of markers, L, coded as XRQ allelic dose), and The formula for biallelic markers , which is generally used, is not strict enough for model selection in large model space .To calculate the effects of SNPs selected by eBIC, the model Linkage disequilibrium was studied to compare and pool the discovered SNPs among models and environments. It was calculated between all pairs of SNPs selected by eBIC, using the classic The use of QTLs instead of SNPs allows us to identify regions of interest rather than specific loci. A QTL is defined as a group of SNPs located on the same chromosome with linkage disequilibrium greater than the predefined significance threshold  ranged from 0.78 to 0.94 Table . The proFlowering time is not known as a highly heterotic trait. However, some hybrids exhibited a visible heterotic phenotype, when looking at the hybrid performances according to their parents , particularly in 13EX01 and 13EX02 environments. As examples, the hybrid resulting from the cross between the female SF301 and the male SF336 on 13EX01 and that from the cross between the female SF217 and the male SF324 on 13EX02, showed later flowering time than expected if the trait was governed only by additive behavior. Dominance was recorded in Table The correlations among environments are high , ranging from 0.68 to 0.85. These correlations were calculated only on common hybrids between the tested environments, i.e., between 297 and 425 hybrids. Environment 13EX02 correlates with the others the least, with correlation coefficients between 0.679 and 0.697. This result can be explained by the fact that the sowing date for this environment was 7–20 days after the other sowing dates. In addition, measurement in this environment was performed less regularly. Despite the good correlation among environments, we analyzed each one independently to capture environment-specific associations.AD, FM, and FMI), the eBIC selection only retains a single SNP, despite the fact that the non-additive parts of variance are significant for some environments (Table S1). In total, among all models and all environments, 31 unique SNPs are associated with the flowering time.Table The MLMM approach selects a single SNP, i.e., the most associated one, to explain the effect of the causal polymorphism in this genomic region. However, several SNPs could be in LD with the causal polymorphism and different sources of errors (phenotypic and genotypic), and missing data could lead to the selection of different SNPs to explain the same causal polymorphism in our different experiments. Therefore, we grouped the SNPs to define QTLs and refer to regions rather than specific positions. This grouping was achieved using linkage disequilibrium between SNPs and positions on the sunflower genomic reference sequence  is a gene involved in flowering time   , since this QTL was only detected with non-additive models despite having a strong impact on flowering time, as illustrated by its effects and p-values in the FMI model. In addition, models AD and FMI, which include intra-locus interaction by modeling dominance or parental allelic interaction, both found the most strongly associated regions indicated by FT09.199 and FT11.47. The first exhibits a clear deviation from an additive behavior, in contrast to the latter, for which an additive behavior cannot be rejected. FT11.47 was not found by additive models, because there is linkage disequilibrium between it and the strong FT16.167 on LG16 detected by additive models. In our forward detection procedure of the additive models, this phenomenon led to the addition of FT16.167 first, which likely decreased the signal of FT11.47 and prevented its detection. Performing GWAS with different models allowed us to increase both the number of associated QTLs and the confidence in the detected regions. Non-additive models can highlight regions with non-additive behavior even for a trait such as flowering time, which is notably genetically additive . The restricted maximum likelihood (REML) used for this calculation [on the basis of the calculation of Segura et al.  and the other models was observed. The two breeding pools of sunflower  have undergone neither the same trait improvement nor the same selection pressure . We also detected height QTLs, found associated in only one environment. These two types of QTLs are usually observed in multi-environment GWAS, as in sunflower Supplementary material 2 (eps 922 KB)Supplementary material 3 (png 181 KB)Supplementary material 4 (png 201 KB)Supplementary material 5 (png 180 KB)Supplementary material 6 (png 186 KB)Supplementary material 7 (png 175 KB)Supplementary material 8 (eps 684 KB)Supplementary material 9 (pdf 150 KB)Below is the link to the electronic supplementary material.


A multivariate genome-wide association test is proposed for analyzing data on multivariate quantitative phenotypes collected from related subjects. The proposed method is a two-step approach. The first step models the association between the genotype and marginal phenotype using a linear mixed model. The second step uses the correlation between residuals of the linear mixed model to estimate the null distribution of the Fisher combination test statistic.The simulation results show that the proposed method controls the type I error rate and is more powerful than the marginal tests across different population structures (admixed or non-admixed) and relatedness (related or independent). The statistical analysis on the database of the Study of Addiction: Genetics and Environment (SAGE) demonstrates that applying the multivariate association test may facilitate identification of the pleiotropic genes contributing to the risk for alcohol dependence commonly expressed by four correlated phenotypes.This study proposes a multivariate method for identifying pleiotropic genes while adjusting for cryptic relatedness and population structure between subjects. The two-step approach is not only powerful but also computationally efficient even when the number of subjects and the number of phenotypes are both very large. If exindependent subjects. Taken together, advanced methods that can handle multivariate phenotypes and related subjects simultaneously are highly desirable.Many complex diseases such as mental health disorders have multiple phenotypic traits that are correlated . These mThe crucial problem in GWAS is to deal with confounders such as population structure, family structure, and cryptic relatedness. Astle and Balding  reviewedFor related subjects with multivariate phenotypes, there are two sources of correlations between multivariate phenotypes: one is the correlation arising from genetically related subjects whose phenotypes are more highly correlated because of shared genotypes; and the other is the correlation between multiple phenotypes which exists even when independent subjects are employed. This study proposes a new statistical method that can model both sources of correlations. We also compare the performance of the proposed method with that of the mvLMMs method. The rest of this paper is organized as follows. In the “R different phenotypes and run an assay with S SNPs. The resulting measurements can be organized with two data matrices. The genotype data are stored in a S×N matrix where N is the total number of subjects and each element of the matrix is coded as 0, 1, or 2 copies of the reference allele. The phenotype data are stored in an N×R matrix where each row records the individual’s multivariate phenotypes. Studying the association between genotypes and phenotypes, thus, involves measuring and testing the association between each row of the genotype matrix and the entire phenotype matrix. Since one SNP is consider at a time, the association test is repeated S times for all SNPs. Specifically, Let β1,…,βR be the effect sizes of a candidate SNP on R different phenotypes. The null hypothesis of testing the pleiotropic gene is H0 is rejected, we claim that the corresponding SNP is associated with the pre-determined multivariate phenotypes.Suppose that for each subject, we measure When the phenotype is univariate, the association test for GWAS data can be carried out using commonly adopted software such as R  or PLINKi be the index of individuals. Define rth phenotype of individual i  and sth genotype of individual i . Therefore, the vector rth marginal phenotype collected from N individuals and the vector sth SNP from N individuals.To illustrate the method proposed by Yang et al. (2016) , we defiR=1 , a regression model is commonly adopted to model yr as a function of gs with covariates in the model to adjust for confounding factors or to increase the precision of estimates. When R>1 , Yang et al. (2016) [r, a marginal p-value, prs, is derived from a likelihood ratio test under a linear regression of yr on gs. The next step is to test association between R multivariate phenotypes and the sth SNP based on these marginal p-values of ps1,…,pRs. The Fisher combination function is used to combine them and the test statistic is defined as When . (2016)  proposeds is claimed to be associated with the R multivariate phenotypes if ξs is statistically significant. Although −2 log(prs) follows a chi-square distribution with 2 degrees of freedom, ξs, which is a sum of dependent chi-square random variables, does not follow a chi-square distribution with 2R degrees of freedom. The permutation method may be adopted to calculate the p-value of ξs but it is computationally too expensive in the context of GWAS  [The SNP . (2016)  for detaξs is the sum of dependent chi-square statistics. Thus, the null distribution of ξs follows a gamma distribution [k and the scale parameter θ: Under the the null hypothesis, the statistic ribution , 24 withApplying the method of moments, we can derive the following equations based on the first two sample moments: Yang et al. (2016)  showed tc1=3.9081, c2=0.0313, c3=0.1022, c4=−0.1378 and c5=0.0941. This approximation is very accurate as the maximum difference is less than 0.0001. Thus, we can efficiently estimate k and θ using Eqs. (cov(·) in Eq.  is a vector of error terms. When the subjects are genetically related, we modify the model to be a linear mixed model: The multivariate method described in the previous section only applies to independent subjects. When multivariate phenotypes data are collected from genetically related subjects, there are two types of correlations: 1) the correlation between multivariate phenotypes (even when the subjects are independent); and 2) the correlation due to genetically related subjects (even when the phenotype is univariate). The approach described in the previous section only addresses the first type of correlation. To address both correlations in the regression model, the marginal regression model in the first step needs to be modified to account for genetically related subjects. Recall that the original regression model has the form of zr is a random effect and it follows K is called the genetic relationship matrix (GRM) [where the added term ix (GRM) .σ2 under the null model while the GRM K was estimated from all the SNPs. To test H0:βr=0, the estimates of the random effects  under the null model were plugged in for the estimation of the variance of the H0, this statistic follows an asymptotical chi-squared distribution with 1 degree of freedom; and the corresponding marginal p-value indicates the strength of association between the SNP and a marginal phenotype.Direct calculation of the best linear unbiased estimates of the fixed effects and the best linear unbiased predictors of the random effects for a large sample size is extremely slow and may be beyond the memory capacity of most computers. Many flexible and efficient methods have been developed to carry out GWAS using linear mixed models. For example, the efficient algorithm implemented in GCTA  uses thep-values, ps1,…,pRs, can then be combined together using the Fisher combination function in Eq.  [Although the GRM associated with the random effect l in Eq.  by addin. (2009)  showing . (2009) , this adWe simulated genotype data based on two different population structures (parents from the same population or parents from two different populations) and two different relatedness structures (independent subjects or related subjects) so there were four different types of data sets reflecting all possible combinations. We generated a set of allele frequencies  from uniform random numbers between 0.1 and 0.9 to represent Population I; and another set of allele frequencies to represent Population II. Given a set of population allele frequencies, we can generate the genotypes of parents from the particular population. Through random mapping, we can generate three types of parents (1/3 each): (1) both parents from Population I; (2) both parents from Population II; and (3) one parent from Population I and the other parent from Population II.Once we had simulated parents’ genotypes, the genes were dropped down the pedigree according to Mendel’s law to simulate children’s genotypes. Our procedure ensured that children from different families represented independent subjects and children within a family represented strongly related subjects. To generate a sample of independent subjects, we simulated 1000 families with one child from each family. To generate a sample of related subjects, we simulated 250 families with four children in each family. Depending on whether parents’ genotypes were generated from one population (either Population I or Population II) or from two different populations, we had four scenarios of children genotype samples: 1) independent samples from non-admixed/isolated population (Non-admixed Independent); 2) related samples from non-admixed/isolated population (Non-admixed Related); 3) independent samples from admixed population (Admixed Independent); and 4) related samples from admixed population (Admixed Related).e be the genetic effect size. The mean value of the marginal phenotype μr was −e if the genotype was AA; 0 if the genotype was AB; and e if the genotype was BB. The specific model to simulate phenotypes is: To evaluate the methods for estimating the correlation between phenotypes Σ is a 2×2 symmetric matrix with the diagonal elements being 1 and the off-diagonal element ρ. The value of ρ determines the correlation between the phenotypes. For each data set, the values of ρ ranged from 0 (independent) to 0.9 (highly dependent), and the values of e ranged from 0 (no effect) to 1 (large effect).where Each configuration was simulated 1000 times. In each simulated data set, we calculated the estimate for The third one was a näive method that did not adjust for the correlation due to related subjects and thus was expected to overestimate e was 0 (no effect) or when the phenotype correlation was highly correlated (near 0.9), all three methods performed well. On the other hand, when the effect size was large (e=1) and the phenotype correlation was small (near 0), all three methods over estimated the true correlation. However, in this situation, the methods using residuals from linear mixed models performed better than the näive method. To our surprise, adding principal components in the linear mixed model did not substantially improve the accuracy of estimates. Because of the poor performance of the näive method, it was not used for the simulations evaluating the type I error rate and power.The simulation results based on the three methods of estimating mentclass2pt{minime, were 0 (no effect), 0.1 (medium effect), and 0.2 (large effect) and the values of the correlation ρ were 0 (independent), 0.4 (moderate correlated), and 0.8 (highly correlated). Each configuration was repeated 1000 times.Because the proposed method was designed to identify pleiotropic genes, evaluating the performance of the multivariate association test in terms of the type I error and power is essential. We simulated four correlated phenotypes using multivariate normal (MVN) random variables. The values of the genetic effect size, p) for different values of correlation ρ and genetic effect size e. Large values of − log10(p)-value are equivalent to small p-values. Thus, when the effect size was large, we expected − log10(p) to be large. Based on our configuration to generate phenotypes, there was no difference between the four marginal p-values. Hence, we only presented the distribution of marginal p-values corresponding to the first marginal phenotype. The multivariate p-values were derived using the proposed method with the correlation estimated by the residuals from the linear mixed model with the first ten principal components as covariates. The findings from this simulation study were summarized as follows: e=0), both the marginal and multivariate methods produced uniform p-values distributions which reflected the null distribution of p-values. When the genetic effect size increased, the value of − log10(p) increased. Therefore, the simulation showed that both marginal and multivariate tests were unbiased.When there was no genetic effect  for various multivariate testing statistics .When the genetic effect was not zero, the proposed multivariate method was more powerful than the marginal test in all situations. The advantages of using the multivariate method was most evident when the correlation between phenotypes was small to moderate. But even when the correlation between phenotypes was as large as 0.8, the multivariate method was still more powerful than the marginal tests. Therefore, combining multivariate phenotypes could increase the power of test.When the sample size was held constant , the difference in power between admixed and non-admixed samples or between independent or related samples were very small.Figures Σ was a 4×4 symmetric matrix with the diagonal elements being 1 and the off-diagonal element ρ. We manipulated the value of ρ to be 0.1 (weak correlated) or 0.5 (moderate correlated). Let e= be the genetic effect sizes corresponding to the phenotypes Y1,…,Y4. We considered the following combinations: e=;Small effect sizes: e=;Increasing effect sizes: e=.Medium effect sizes: We further evaluated the performance of the proposed method in comparison to a competing method, the multivariate linear mixed model (mvLMMs) method, that has been implemented in the GEMMA  softwaree=) because both methods have been shown to control the type I error.We did not consider the situation of no effect  for different values of the correlation ρ and the genetic effect sizes e. A powerful method should result in small p-values ). The findings form this simulation study were summarized as follows: The power of both methods depended on the effect sizes. When the effect sizes were increased from small to medium, the power of both methods increased. More importantly, the scale of such increase was larger for the proposed method.When the effect sizes were fixed, both methods had higher power when the correlation between phenotypes was weak.When the effect sizes were not equal among marginal phenotypes, the proposed method still maintained its high performance.p) compared to the mvLMMs method in 5 our of the 6 configurations. The mvLMMs only achieved the same level of performance when the phenotypes had a medium correlation and the effect sizes were increasing.Overall, the proposed method was more powerful than the mvLMMs method. The proposed method had a larger median value of − log10 due to computational and statistical barriers of the EM algorithm .We demonstrate the application of the proposed method by conducting analysis on the data from the Study of Addiction: Genetics and Environment (SAGE).p-value >10−5), and frequency of missingness per SNP (<0.05) [The SAGE is a study that collected data from three large scale studies in the substance abuse field: the Collaborative Study on the Genetics of Alcoholism (COGA), the Family Study of Cocaine Dependence (FSCD), and the Collaborative Genetic Study of Nicotine Dependence (COGEND). The total number of subjects in all three studies was 4121. Each subject was genotyped using the Illumina Human 1M-Duo beadchip which contains over 1 million SNP markers. From the original 4121 individuals, some subjects were genotyped twice so we eliminated duplicate samples and the sample size was reduced to be 4112. Although dbGap provided a PED file to show pedigree and relationship among participants, we used the KING program  to verif (<0.05) . The finage_first_drink:the age when the participant had a drink containing alcohol the first time.ons_reg_drink: the onset age of regular drinking (drinking once a month for 6 months or more).age_first_got_drunk:the age when the participant got drunk the first time.alc_sx_tot: the number of alcohol dependence symptoms endorsed.Because our research aimed to identify the SNPs associated with the risk for alcohol dependence, four correlated phenotypes were used for the analysis: To deal with missing values in any of these four phenotypes, we imputed them using the mi package  from R sx contains the participant’s first ten principal components, gender, age at interview, and self-identified race, and the corresponding regression coefficients ηr are treated as fixed effects. The QQ-plots of the p-values for the marginal association tests are shown in Fig. p-values for the proposed multivariate tests is displayed in Fig. We conducted marginal genome-wide association tests on each of these four phenotypes using the GCTA program to account for relatedness among subjects. We also added the first ten principal components to increase the precision of estimates. In addition to these principal components, the participant’s gender, age at interview, and self-identified race were included as covariates in the model. The regression model for the marginal phenotype is p-values were less than the significance level of 10−6. Based on the marginal tests, two SNPs  were associated with age_first_drink; one SNP  was associated with ons_reg_drink; one SNP  was associated with age_first_got_drunk; and one SNP  was associated with alc_sx_tot. On the other hand, using the proposed multivariate method, two SNPs  were significantly associated with these four correlated phenotypes for the risk of alcohol dependence.To identify the SNPs associated with the risk for alcohol dependence, we declared significant SNPs if their rs11157640) was also found to be associated with ons_reg_drink in the marginal test. Hence, combining multiple phenotypes not only can increase the power of identifying SNPs that may not be identified by marginal tests but also can provide insight into the pleiotropic genes contributing to the common risk expressed by multivariate phenotypes.Based on the findings in the marginal tests, we identified five SNPs each of which was associated with an individual phenotype; none of these five SNPs was associated with two or more phenotypes. Thus, the marginal tests were limited in terms of finding pleiotropic genes associated with multivariate phenotypes. In contrast, using the proposed multivariate method, we identified two SNPs associated with the four phenotypes together and one of these two SNPs  and relatedness (related or independent). The proposed method is also computationally efficient. The first step takes advantage of the efficient program GCTA to carry out marginal tests under linear mixed models. In practice, a few hours are sufficient to derive all marginal


Longitudinal genetic data provide more information regarding genetic effects over time compared with cross-sectional data. Coupled with next-generation sequencing technologies, it becomes reality to identify important genes containing both rare and common variants in a longitudinal design. In this work, we adopted a weighted sum statistic (WSS) to collapse multiple variants in a gene region to form a gene score. When multiple genes in a pathway were considered together, a penalized longitudinal model under the quadratic inference function (QIF) framework was applied for efficient gene selection. We evaluated the estimation accuracy and model selection performance under different model settings, then applied the method to a real dataset from the Genetic Analysis Workshop 18 (GAW18). Compared with the unpenalized QIF method, the penalized QIF (pQIF) method achieved better estimation accuracy and higher selection efficiency. The pQIF remained optimal even when the working correlation structure was mis-specified. The real data analysis identified one important gene, angiotensin II receptor type 1 (AGTR1), in the Ca2+/AT-IIR/α-AR signaling pathway. The estimated effect implied that AGTR1 may have a protective effect for hypertension. Our pQIF method provides a general tool for longitudinal sequencing studies involving large numbers of genetic variants. Next-generation sequencing technologies provide the opportunity to study the role of rare variants in complex diseases, as evidenced by reports in the literature6.Longitudinal data are often observed in biomedical studies with repeated measures of the same subject over time. Coupled with the recent development of genome-wide SNP genotyping and next-generation sequencing technologies, longitudinal designs are becoming popular in genetic association studies because of their increased power over cross-sectional designs to detect genetic associations. Most longitudinal genetic association analyses have focused on the evaluation of associations at common variants8. Methods for detecting rare variants have been developed and can be broadly classified into three categories: (1) burden tests, for example, the weighted sum statistic (WSS) methods9; (2) variance component-based tests represented by the sequence kernel association test (SKAT)10; and (3) dimension-reduction based tests such as functional principal components analysis (FPCA)11 and the adaptive ridge regression method12. For a comprehensive review of the design and analysis of rare variants in association studies, please refer to Lee et al.8.Because of the extremely low minor allele frequency (MAF) of rare variants (defined as variants with a MAF < 5%), the commonly-used single-variant association analysis is underpowered16. Yan et al.15 extended the kernel machine method to handle longitudinal continuous trait from family samples in the linear mixed model (LM) framework. Chien et al.16 extended burden test and kernel association tests to genetic longitudinal studies in the generalized estimating equations (GEE) framework. Wu et al.14 and Chiu et al.13 summarized the rare variants longitudinal studies, where most of the statistical models were based on GEE and LM models. These methods face computational challenges with limited sample size and missing data. Furthermore, large numbers of gene variables increase the complexity of modeling, especially because most genes have no association with the studied disease trait. The challenge particularly increases when the number of genes or SNP markers becomes larger than the number of samples. As such, the classical methods faces estimation instability issues when the number of variants is large. This motivates us to adopt a penalized regression method for better parameter estimation, and achieving gene selection in the mean time.All the rare variants association tests methods described above are available for phenotypes measured at a single time point. Very few methods have been developed or extended to detect rare variants associated with longitudinal disease traits19. Various penalty functions have been developed for different purposes, such as the group LASSO for selecting a group of variables20 and the network-constrained penalty for selecting variables with a graph structure21. In fact, penalized regression methods have been applied to rare variants association analysis when a univariate disease trait is considered24.When a large number of gene variables are modelled simultaneously in a regression model, high-dimensional variable selection strategies become essential for a genetic association study. Variable selection methods with a univariate response in linear and generalized linear regression analyses have been studied extensivelyet al.25 can handle cases where the number of covariates have the same order as the number of individuals. Cho and Qu26 proposed the penalized quadratic inference function (pQIF) method for a diverging number of regression parameters, and showed that pQIF outperformed pGEE in various settings. Given the improved performance of a longitudinal design in identifying genetic variants, it is essential to develop a variable selection strategy to improve estimation accuracy and gene selection efficiency in a longitudinal study. In this work, we extended the pQIF method to a longitudinal genetic association study with rare variants and evaluated its performance with extensive simulation studies. Practical suggestions were obtained for real applications. We further applied the pQIF method to a hypertension dataset from the Genetic Analysis Workshop 18 (GAW18)27 and focused on the Ca2+/AT-IIR/α-AR signaling pathway to identify genes associated with the development of hypertension.For multivariate responses in a longitudinal study, variable selection methods have also been developed. Under the population-averaged  model framework, several variable selection methods have been developed for a diverging number of parameters. For example, the penalized generalized estimating equations (pGEE) method developed by Wang In statistical methods section, we briefly describe the quadratic inference function (QIF) method for longitudinal data, then describe the pQIF method. We then review the methods used for rare variants detection, focusing on the collapsing method that was applied in this work. The simulation studies are reported in simulation study section. In real data analysis section, we describe the application of the pQIF method to a real dataset focusing on the Ca2+/AT-IIR/α-AR signaling pathway, followed by a discussion section.Yit is the phenotype measured for subject i at time t; g(·) is a known link function ; Xit contains both time-varying covariates and time-invariant genetic variants; and β represents unknown coefficients. In this study, we focused our analysis on a longitudinal binary disease trait.For repeated outcome or exposure measurements under a longitudinal design, the model can be expressed as:Yit. GEE is a classic population-averaged  model that requires only a working correlation for Yit to be specified in order to obtain consistent estimates for the mean parameters, even when the correlation structure is mis-specified28. However, the estimator of the regression parameter is inefficient under mis-specification of the correlation structure29. The quasi-score equation in GEE is defined as:Ai being a diagonal matrix of marginal variances for Yi, and R(ρ) being a common working correlation with nuisance parameter, the intra-class correlation coefficient ρ, Ri, such as independent, exchangeable, or AR(1).It is generally difficult to specify a full likelihood function for the correlated responses ρ is consistently estimated. When such a consistent estimator does not exist, Qu et al.29 suggested that the inverse of R(ρ) can be represented by a linear combination of a class of basis matrices such as R−1(ρ) ≈ a0I + a1M1+, …+aSMS, where I is an identify matrix and M1, …, MS are known symmetric matrices. Under the QIF framework of Qu et al.29, we defined the score functions as:ψi), i.e.,The GEE method has the advantage that the estimators of the regression coefficients are consistent even when the correlation structure is mis-specified, given that Under certain conditions, the estimation consistency of the estimator a > 2 and θ > 2. The SCAD penalty function can select variables and estimate coefficients simultaneously, and possesses an “oracle property”18. For pQIF, the penalized score function is defined as:QIF was extended to a high-dimensional regression setup where the number of predictors can be large. The pQIF is based on the non-convex SCAD penalty given by:kth iteration of the non-zero components. Qn(βk), andBecause the SCAD penalty function is non-convex, the penalized score function in equation  can be ak + 1)th iteration of the non-zero component The Newton-Raphson algorithm can be applied to get Un(β) in equation  which is analogous to the Bayesian information criterion and is based on QIF as the objective function. The BIQIF is defined as:equation  for a gi30. For example, the cohort allelic sum test (CAST)31 collapses multiple rare variants into one binary variable, which indicates whether there are any rare variants. Morris and Zeggini32 extended CAST by counting the total number of minor alleles. The combined multivariate and collapsing method33 first collapses the variants into several subgroups based on some predefined criteria , and then performs a multivariate test. The WSS method weights all variants differently when determining the genetic load of an individual. So, by weighting the signals from each variant, the WSS accentuates variants that are rare in an individual9. The variable threshold method34 selects the optimal rare frequency threshold on a grid of points, and estimates the p-value by a permutation procedure. All these burden tests assume all the variants share the same effect direction and magnitude (after incorporating weights). Thus, any violation of this assumption can result in a loss of power35. To overcome the limitations of the burden tests, the data-adaptive sum test (aSum) was proposed36. Specifically, the aSum method first estimates the direction of effect for each variant using a marginal regression model, then it changes the coding of variants accordingly, and finally uses the same method as the burden test to test for any association. However, aSum is computationally intensive because it obtains the p-value via permutations. Moreover, the estimation of regression coefficients for single rare variants is typically difficult and unstable8.Rare variants association studies typically focus on multiple variants in a specific genomic region  rather than on individual variants separately. The gene- or region-based methods can be broadly categorized into three classes: (1) burden tests, (2) variance component-based tests, and (3) dimension-reduction based tests. Burden tests simply collapse multiple variants into a single genetic score10) assume the effect sizes of rare variants follow a normal distribution, and then test for the variance components. It has been demonstrated that burden tests were more powerful than SKAT when most of the rare variants were causal and had the same directions, whereas SKAT outperformed burden tests when the effects of rare variants were heterogeneous37. This motivated the development of some hybrid methods such as SKAT-O38 and MiST39, which combined the benefits of the burden tests and SKAT. These hybrid methods were more robust across a range of scenarios, but were less powerful than either one of these tests on their self-underlying assumptions30.Variance-component based methods  and concluded that, among them, FPCA was the most powerful. However, the performance of the dimension reduction techniques and variance components-based tests is not clearly known. Borrowing the idea of the WSS, we proposed to adopt the collapsing idea to collapse both rare and common variants over a gene or region into a single genetic score for further estimation and gene selection.Other dimension-reduction techniques are available for rare variants analysis, such as FPCAJ is the total number of variants in a gene. Let Gij be the number of disease alleles for variant j in individual i, and Gij = 0, 1, 2 under an additive genetic model. Then, each individual is scored by a single weighted average of the number of alleles in a given gene as:wj is the weight given as the inverse of the standard deviation for the minor allele, i.e., pj is the MAF of variant j. This weighting function assumes that rare variants have larger effect sizes than common variants9. A weighted gene score can be obtained for each gene. The gene-based scores are then fitted into the pQIF model to select the genes associated with a longitudinal disease trait.The WSS method jointly analyses a group of SNP variants in a gene or region. Without loss of generality, here we focused on a gene to describe the method. Suppose Ekt is the kth time-varying or time-invariant environmental variable and Cj is the weighted sum score for the jth gene, which is time-invariant. This mean model is then fitted with the pQIF method for further estimation and gene selection.After collapsing multiple (common and rare) variants in each gene with the weighted sum, the longitudinal model can be defined as:Hi can be applied for each subject to fit the pQIF model26. For each fully observed individual without missing data, Hi is expressed as an m × m identity matrix for the ith subject, where m is the total repeated time point. For the ith subject with missing measurements, Hi is generated by deleting the columns that correspond to the missing measurements. Consider a study with a total of three time points. For an individual i with the 3rd time point missing, the transformation matrix Hi can be expressed as Hi = I3×3 if no measurements are missing. Then ψi(β) in equation , and exchangeable).The simulation was based on the GAW18 real sequencing data. The GAW18 dataset was based on a longitudinal study design consisting of whole-genome sequencing of 1043 individuals in the San Antonio Family studies with pedigrees. Among the 1043 individuals, 142 are unrelated and had both real phenotype data and imputed sequence data. The sequencing data for GAW18 were provided only for markers on odd-numbered autosomes. Two phenotype datasets were provided: one was the real phenotype data including systolic blood pressure and diastolic blood pressure along with other covariates such as current use of antihypertensive medications, sex, age, and smoking status up to four time points; the other was the simulated longitudinal phenotype data that were based on the real genotype data. Along with both datasets, a list of “functional loci” associated with the simulated phenotypes were also provided, thus the true functional mechanism is known for the simulated data.Yit based on the following model:Cij is the weighted score for gene j, chosen from the above five genes. We also simulated noisy gene variants with no genetic effect. Each noisy gene consists of 10 SNP variants with the proportion of rare and common variants set as 6:4. An additive coding {0, 1, 2} for each SNP variant was used. Both the rare and common variants were collapsed over genes as a weighted score using the WSS method. Ages were taking from the original dataset, and missing age values at exams were filled in by adding or subtracting 3.9 years between exam 1 and exam 2 and 6.9 years between exam 1 and exam 3. Tobacco smoking was generated as follows: 22.9% of individuals were selected randomly to be smokers at exam 1, and 1.45% were deemed to have quit smoking during each exam. This follows the same quitting rate as in the real dataset. All the variables were standardized to have mean zero and standard deviation one before further analysis. The R package mvtBinaryEP was used to generate the longitudinal binary responses. Under each scenario, 200 simulation runs were conducted.Here we focused on the 142 unrelated individuals in both the simulation and real data analyses. In the simulations, we chose the top five influential genes provided in the GAW18 dataset, MAP4, TNN, NRF1, LEPR, and FLT3, as the true effect genes in our simulation studies. Because the sample size (142) was not large enough to demonstrate the performance of the pQIF, we bootstrapped additional samples assuming that the 142 individuals represented the population. For each bootstrapped sample, we fixed the original sequencing data, but simulated new binary longitudinal responses p is the dimension of β and β in the jth simulation run. We also calculated the mean squared error (MSE) for noisy gene variants (NMSE) in the same way as we calculated TMSE. True positive (TP) and false positive (FP) rates were calculated to evaluate the model selection performance.To evaluate the estimation accuracy, we calculated the total mean squared error (TMSE) asn = 142 unrelated samples from the GAW18 dataset, and n = 250 and n = 500 based on the bootstrapped samples. The total number of covariates (p) including environmental variables and genes (both effective and noisy ones) were assumed to be 20 and 40, and the number of effective variables was assumed to be q = 4 and 6. Data were simulated assuming an AR(1) correlation structure and were subsequently analyzed by applying an AR(1) correlation structure (assuming the true correlation structure was known). In the first simulation scenario, the true coefficients were assumed to be β = T, where the nonzero coefficients corresponded to covariates age and three genes . In the second scenario, β = T, where nonzero coefficients corresponded to covariate age and five genes . Two intra-class correlation coefficients were considered with ρ = 0.4 in scenario 1 and ρ = 0.7 in scenario 2. The optimal tuning parameter λ was chosen by a grid search based on a sequence of 100 values of equal interval in , where λmax is the value for which all coefficients were shrunk to zero. λmax was set differently under different sample sizes. Here we set λmax = 0.35, 0.25, 0.2 for sample sizes n = 142, 250, and 500, respectively. We set the tolerance level tol = 10−12 in the QIF method, and tol = 10−10 in the pQIF method (the tol of pQIF has to be larger than QIF) to control the FP rates in the simulation studies.We compared the model performance under three different sample sizes: p = 20, the pQIF chose most of the TP variables, even when n = 142, and the FP selection rate was very low under different model conditions. For the increased sample sizes, the TP selection rate also increased. The TP selection rate for ρ = 0.4 was higher than for ρ = 0.7 for n = 142 but the difference in the TP selection rate between the two ρ values diminished as the sample size increased. The detailed estimation accuracy of the pQIF under different model setups is listed in Table n = 142 when p = 40 because the pQIF did not converge well for the larger p value in many simulations runs. Thus, in real applications, when the gene dimension is large, the pQIF may not be useful because of computational limitations, especially when the sample size is small.Figure p = 20 and p = 40).Compared to the unpenalized method, the TMSE of the penalized approach was much smaller Table . As the In a short summary, when data were analyzed assuming the true covariance structure was known, the pQIF performed well with a low FP selection rate. However, if the sample size is small relative to the variable dimension, the pQIF may not converge well due to computational issues.n = 300, p = 20, and ρ = 0.5. We considered one covariate age and three gene variables, MAP4, TNN, and NRF1. The coefficients for the four variables were set as, β = T and the rest were set as zero. To choose the best tuning parameter λn, we set the sequence as 100 values of equal interval in . We simulated data under each correlation structure and analyzed data separately assuming independent, AR(1), and exchangeable correlations. Our aim was to assess the selection performance under a mis-specified working correlation.We next examined the performance of the pQIF when the covariance structure was mis-specified, under three different correlation structures: independent, AR(1), and exchangeable. Because we had already evaluated its performance with different sample sizes and different data dimensions, here we evaluated it only with Figure In summary, the pQIF remained optimal even when the working correlation structure was mis-specified. No significant differences in the MSEs were found when the data were analyzed assuming different correlation structures, which implies the pQIF method was robust even when the correlation structure was mis-specified.st exam, 89 attended the 2nd exam, 92 attended the 3rd exam, and 37 attended the 4th exam). Because the 4th time point had a large number of missing values, imputation would be unreliable. Thus, we focused our analysis on the first three time points. The hypertension diagnosis was based on the criteria of systolic blood pressure >140 and diastolic blood pressure >90, or on antihypertensive medications being taken at the time of diagnosis. A description of the related phenotype data is summarized in Table We applied the pQIF method to the GAW18 real dataset to identify important genes associated with hypertension. We focused on the binary hypertension trait (yes = 1/no = 0) of 142 unrelated individuals, measured over four time point . Part of the adrenergic signaling in cardiomyocytes pathway, which we defined as the Ca2+/AT-IIR/α-AR signaling pathway  working correlation structure in the analysis. The gene score for each gene was collapsed with multiple rare and common variants using the WSS method. Table AGTR1 has 80 common variants and 215 rare variants. Because the WSS approach emphasized rare variants, the pQIF result implies that rare variants in AGTR1 may play important roles for hypertension. The negative coefficient obtained for AGTR1 indicates that the synergistic effects of multiple variants on AGTR1 are protective for hypertension by preventing elevated blood pressure due to angiotensin II. Adjusting for age, gender, and smoking effect in the pQIF model, every one unit increase in gene score of the risk variants in this gene will result in 41% decrease in the risk of hypertension. Because we analyzed only genes in odd numbered autosomes in the Ca2+/AT-IIR/α-AR signaling pathway and because we chose only one gene encoding each protein, we could have missed other important genes in this pathway. Nevertheless, our gene-based longitudinal association analysis indicates the important protective role of AGTR1 on hypertension.p = o(n1/4). This might explain the poor convergence rates of the pQIF in the simulation studies when p = 40 and n = 142. Compared to the poor convergence when p = 40, n = 142 and ρ = 0.4 in the simulation studies, when the sample size n increases from 142 to 200, nearly all the 200 runs can converge. The convergence of ρ = 0.4 performs better than ρ = 0.7, indicating that the convergence is better when the intracluster correlation is low. Computation using the pQIF method is fast. For example, for p = 20, n = 142, and ρ = 0.4, the average running time for the pQIF in each simulation run was about 3 mins. In addition, the total MSE of the pQIF method was much smaller compared with the unpenalized QIF methods, indicating the relative gain of fitting a penalized model.Next-generation sequencing data are generated routinely in many laboratories in order to identify common and rare variants associated with complex diseases. With longitudinally collected disease traits, it is possible to understand disease progression as well as the underlying dynamic genetic mechanisms. However, very few studies have reported the association of rare variants with longitudinal traits, especially in a high-dimensional regression setup. In this work, we explored gene-based association studies for next-generation sequencing data with longitudinal measures of binary phenotypic traits using the pQIF method. We evaluated the performance of the pQIF method based on extensive simulation studies. The results indicated that the pQIF worked well when the sample size was relatively large but the method had convergence issues with a small sample size. The pQIF model is proposed for diverging numbers of covariates and holds for 16. The kernel machine method based on the LM framework15 was extended to rare variants in longitudinal data for a family-based study, but only applicable to continuous traits. The SKAT method proposed by Chien et al.16 was built under the GEE framework and was applicable to longitudinal binary data. However, the SKAT approach fits genes one at a time, and can be computationally expensive15. Analyzing multiple genes simultaneously in a regression framework can greatly enhance the association study performance involving rare variants44. With a limited sample size and a large number of genes in sequencing data, we approached the problem based on the pQIF method to conduct gene selection and estimation simultaneously for a diverging number of regression parameters. Our simulation studies provide a practical guidance to implement the method for longitudinal sequencing association studies with sequencing data.In this paper, we focused on next-generation sequencing longitudinal binary data analysis, with the special feature of dealing with rare variants, intracluster correlations, and high dimensions. Although longitudinal genetic data analyses have been reported previously, only a few of these reports focused on rare variants46. Hypertension is a complex and multifactorial polygenic disease. The risk loci that have been discovered so far are very limited and explain only a small part of hypertension heritability48. Polymorphisms in AGTR1 associated with hypertension have been studied, but the results were inconsistent and conflicting51. Mottl et al.47 concluded that analyses that focused on single variants were fruitless and multiple variants analysis was needed. Our analysis improves the current approaches by integrating multiple common and rare variants in a gene (or region). However, further biological experiments are needed to verify the real biological function of the identified gene.In the real data analysis, the pQIF identified one important gene in the Ca2+/AT-IIR/α-AR signaling pathway, which further confirmed that the angiotensin II receptor protein AGTR1 had important physiological functions such as vasoconstriction, cellular proliferation, and growth8. This collapsing method can be improved further by adopting more powerful strategies to detect heterogeneous effects such as the aSum statistic method. Second, given the large numbers of genetic variants in a pathway or in genome-wide data, the pQIF may be limited by the amount of data it can handle. Other methods such as the penalized participant-specific  model could be alternatives to the pQIF for longitudinal binary data. For example, Groll and Tutz52 proposed generalized linear mixed models by L1-penalized using a gradient ascent algorithm to maximize the penalized log-likelihood. Schelldorfer et al.53 proposed GLMMLasso, which can handle problems where the number of variables is in the thousand using an efficient coordinate gradient descent algorithm. Third, to further evaluate the robustness of the pQIF approach, independent longitudinal sequencing data sets should be included and tested. In addition, we did not allow the coefficients of time-invariant covariates in the pQIF to vary over time, thus changing patterns of genetic effects over time could not be captured. This can be improved by adopting a penalized varying-coefficient model under the QIF framework54, and will be investigated in our future work.Although the pQIF method provides a powerful tool for analyzing longitudinal sequencing data, there are some limitations in this work. First, we describe our strategies using the WSS to collapse multiple variants in a gene region for simplicity. The WSS collapsing method, which gives a single weighted score incorporated by collapsing rare and common variants in a gene, could suffer from power loss if the assumption of same effect direction and magnitude for all variants is violatedIn conclusion, our research sheds light on the analysis of next-generation sequencing longitudinal binary data. We found that the penalized models were more efficient than the unpenalized models with interpretable regression coefficients by achieving variable selection and estimation simultaneously. The pQIF together with the collapsing methods provides a powerful tool to evaluate the synergistic effects of both rare and common variants in a gene or a genetic region with next-generation sequencing data in a longitudinal design.Supplementary Figure S1. Ca2+/AT-IIR/α-AR signaling pathway