Genetic association analysis on complex phenotypes under a longitudinal design involving pedigrees encounters the problem of correlation within pedigrees, which could affect statistical assessment of the genetic effects. Approaches have been proposed to integrate kinship correlation into the mixed-effect models to explicitly model the genetic relationship. These have proved to be an efficient way of dealing with sample clustering in pedigree data. Although current algorithms implemented in popular statistical packages are useful for adjusting relatedness in the mixed modeling of genetic effects on the mean level of a phenotype, they are not sufficiently straightforward to handle the kinship correlation on the time-dependent trajectories of a phenotype. We introduce a 2-level hierarchical linear model to separately assess the genetic associations with the mean level and the rate of change of a phenotype, integrating kinship correlation in the analysis. We apply our method to the Genetic Analysis Workshop 18 genome-wide association studies data on chromosome 3 to estimate the genetic effects on systolic blood pressure measured over time in large pedigrees. Our method identifies genetic variants associated with blood pressure with estimated inflation factors of 0.99, suggesting that our modeling of random effects efficiently handles the genetic relatedness in pedigrees. Application to simulated data captures important variants specified in the simulation. Our results show that the method is useful for genetic association studies in related samples using longitudinal design. Genetic studies show that genes influence both the mean level and the rate of change of anthropometric traits. For example, substantial genetic contributions to the rate of change have been detected for both body mass index (BMI) ,2 and bohttps://dsgweb.wustl.edu/llfs/), supported by the National Institute on Aging, collects high-resolution genome-wide genotype data in families with longevity probands and their offspring. With support from the NIH, Genetic Analysis Workshop 18 (GAW18) provides whole genome sequencing data and longitudinal blood pressure measurements on large pedigrees. In fact, genetic analysis on longitudinal measurements from pedigrees is an important topic calling for novel statistical modeling [Just as traditional genetic epidemiology studies are frequently conducted in large pedigrees because disease-causing mutations segregate within families, so the relative ease in SNP genotyping has led to genetic association analysis in large pedigrees. For example, the Long Life Family Study  measured longitudinally over 4 times in a period of 30 years. In total, 246 individuals have 1, 183 have 2, 309 have 3, and 194 have 4 measurements. Besides blood pressure, information concerning hypertension diagnosis , antihypertension medicine intake, and tobacco smoking is also collected at each examination. Participants entered the study at different ages, ranging from 16 to 94 years, with a mean age of 39.6 years.The data contains SNP genotypes on odd-numbered autosomes for individuals in the 20 pedigrees obtained using different versions of the Illumina Infinium Beadchips. Raw genotype data were processed using standard quality control procedures and cleaned for mendelian errors. This paper focuses on chromosome 3 data, which contains 65,519 SNPs, to illustrate the application of our proposed method.GAW18 also provides simulated phenotype data for 200 replicates, each with 849 individuals from the real pedigrees. As with the real data, longitudinal blood pressure measurements were simulated for 3 time points with 3.9 years between exams 1 and 2 and 6.9 years between exams 1 and 3. The simulation included age and sex as fixed covariates and hypertension diagnosis, medication use, and tobacco smoking as time-varying covariates.The HLM is a complex form of regression analysis ,14, refeij Yis SBP measured for individual i at age j; ij Xis age at measurement for individual i; 0,i βand 1,i βare the intercept and slope parameters for individual i; ij eis a random error associated with individual i at age j that is normally distributed with E(ije) = 0, ij) var associating genotypes with 0,i βand 1,iβ; 0i Uand 1i Uare the random effects for 0,i βand 1,i βafter adjusting for genotype i Gthat are normally distributed with 0i) = E(U1i) = E and (3), i Gin equations (2) and (3) as a fixed-effect variable and 0,i Uand 1,i Uas random effect variables that can be decomposed into 2 parts: a heritable component  and a nonheritable component  such thatThe HLMs described so far are most appropriate for unrelated individuals from a general population. When longitudinal data are collected in pedigrees, correlation arises among the random effects for individual trajectories described above. In human pedigrees, the heritable random variation is distributed following a kinship matrix defined according to the pairwise genotypic similarity of individuals, which can be readily derived from pedigree structures or whole genome data. Explicitly modeling the pedigree structure in a mixed-effect model is shown to improve performance in GWAS by bringing the genomic control factor very close to 1 ,15. Here0 uis a vector representing the heritable component in the random variation in individual intercepts, 0,iβ; 1 uis a vector representing the heritable component in the random variation in individual slopes, 1,iβ; K is a matrix of kinship coefficients calculated from pedigree structures. In the actual estimation of the level 2 models, extra fixed-effect variables, such as sex, can be included so that effects of these fixed variables can be estimated together with that for each SNP with the possibility of estimating interactions between them. The level 2 models are fitted using the lmekin function in the free R package, kinship.In equation (6), We started with regressing SBP values on 2 time-varying variables, smoking and intake of antihypertension medicine, to remove their effects, and then fitted the level 1 models to the age trajectories of SBP measured over time. Figure p values for effects of each SNP on SBP(42) (the left panel) and on rate of change (the right panel). Although no SNP reached genome significance , some SNPs on chromosome 3 may tend to affect the mean level of SBP  to (6) and included sex as an additional fixed-effect variable. The Q-Q plot is shown in Figure MAP4 gene on chromosome 3. The simulation included 14 functional variants  accounting for 7.7% of the total variance in SBP. A total of 87 SNPs in the GAW18 GWAS data were mapped to the MAP4 gene. We performed level 2 analysis on all 200 replicates, focusing on 62 SNPs with MAF >0.01. Based on the results on simulated replicates, statistical power was calculated as defining type I error rate α = 0.05 and α = 0.05/number of SNPs tested (~0.0008). With a sample size of 849 individuals, the highest power in detecting the effect on mean SBP or SBP(42) was achieved by SNP rs11711953 (MAF = 0.028) with power estimates of 1 and 0.70 for α values of 0.05 and 0.0008. This SNP is the variant in MAP4 assigned the highest effect on SBP, accounting for 2.78% of the variance in SBP. Interestingly, we also obtained the highest power for detecting association with rate of change from the same SNP (0.75 for α = 0.05), suggesting that rs11711953 influences both the mean level (negative correlation) and the rate of change over ages (positive correlation) of the simulated SBP. Table MAP4 gene specified in the simulation.The application to original data has shown that the influence of pedigree structure on statistical testing can be nicely controlled by our method. We next use the simulated phenotype data to validate performance of our model and to see whether important genetic variants can be captured for given sample size in the simulation. As with the real data, we focused only on SBP and used regression models to adjust for effects of time-varying variables (smoking and medication) and fitted level 1 model to the residuals. Considering the enormous computational load in performing GWAS for 200 replicates, we limited our analysis on simulated data to all SNPs in the In longitudinal investigations, researchers are interested in studying interindividual differences in intraindividual changes. Such studies are inherently hierarchical in nature. The nested structure in data includes multiple or repeated observations within an individual, and a collection of multiple individuals in a sample, which can be modeled by HLM. The nested modeling is actually equivalent to the mixed-effects model from a theoretical prospective. By incorporating equations (2) and (3) into equation (1), we obtainThe combined model has fixed effects for age, genotype and their interaction, and the composite error including random effects for the age trajectory and for the mean SBP level. Unlike the mixed model that corrects phenotype correlation within pedigree, our modeling strategy allows a direct adjustment of kinship correlation in the mean level and the rate of change of SBP. In a combined model, however, this is somewhat more difficult to implement. As shown by the application example, our hierarchical modeling strategy enables integration of kinship matrix to explicitly account for the genetic correlation both in the level and in the rate of change of SBP in general pedigrees using readily available software packages. For comparison, we performed similar association analysis on chromosome 3 but using the linear mixed model with individual rate of change as dependent variable and assuming random effect for each pedigree. The estimated inflation factor becomes 1.07, indicating a moderate degree of sample substructure. This is in contrast with a lambda of 1 from the lmekin model, suggesting that explicitly modeling the kinship correlation structure can help to improve model performances for GWAS in pedigree studies.Our hierarchical modeling procedure also offers an easy way to detect outliers in longitudinal studies , which iFinally, although only chromosome 3 data are used as an example, application of the method to genome-wide data is straightforward. The fast fitting of the mixed model with a predefined kinship matrix enables computational feasibility for large GWAS data. With the increasing popularity of GWAS, we hope that the proposed method can be helpful in analyzing data of related individuals using both cross-sectional and longitudinal designs.Hierarchical linear modeling of longitudinal pedigree data can handle relatedness in detecting genetic variations that affect the mean level or the rate of change for a phenotype of interest in genetic association analysis.The authors declare that they have no competing interests.TAK, KC, and JBH designed the overall study; QT, AKJ, and JZ conducted statistical analyses. QT, MT, and LC drafted the manuscript. All authors read and approved the final manuscript.


Genomic selection estimates genetic merit based on dense SNP (single nucleotide polymorphism) genotypes and phenotypes. This requires that SNPs explain a large fraction of the genetic variance. The objectives of this work were: (1) to estimate the fraction of genetic variance explained by dense genome-wide markers using 54 K SNP chip genotyping, and (2) to evaluate the effect of alternative marker-based relationship matrices and corrections for the base population on the fraction of the genetic variance explained by markers.Two alternative marker-based relationship matrices were estimated using 35 706 SNPs on 1086 dairy bulls. Both pedigree- and marker-based relationship matrices were fitted simultaneously or separately in an animal model to estimate the fraction of variance not explained by the markers, i.e. the fraction explained by the pedigree. The phenotypes considered in the analysis were the deregressed estimated breeding values (dEBV) for milk, fat and protein yield and for somatic cell score (SCS).When dEBV were not sufficiently accurate (50 or 70%), the estimated fraction of the genetic variance explained by the markers was around 65% for yield traits and 45% for SCS. Scaling marker genotypes with locus-specific frequencies of heterozygotes slightly increased the variance explained by markers, compared with scaling with the average frequency of heterozygotes across loci. The estimated fraction of the genetic variance explained by the markers using separately both relationships matrices followed the same trends but the results were underestimated. With less accurate dEBV estimates, the fraction of the genetic variance explained by markers was underestimated, which is probably an artifact due to the dEBV being estimated by a pedigree-based animal model.When using only highly accurate dEBV, the proportion of the genetic variance explained by the Illumina 54 K SNP chip was approximately 80% for Brown Swiss cattle. These results depend on the SNP chip used and the family structure of the population, i.e. more dense SNPs and closer family relationships are expected to result in a higher fraction of the variance explained by the SNPs. Genome-wide dense marker arrays that are available for livestock populations cover all chromosomes with dense single nucleotide polymorphism (SNP) markers . Many daIdentity by descent (IBD) alleles refer to alleles that descend from a common ancestor in the base population . The coeHowever, the relationship matrix can also be estimated from genome-wide genetic markers such as panels of SNPs -12. MethCmiss) represents the variance that cannot be used by GS and affects the maximum accuracy that can be achieved by GS [Cmiss: (1) increasing the sample size in order to also detect genes with smaller effects, (2) expanding the studies to non-European samples in human genetics, (3) enlarging the collection of phenotypes to explore gene-gene interactions, (4) changing the structure of the training population, mainly in terms of the relatedness of the included individuals, and (5) moving to the genomic selection approach instead of estimating the marker effect for each SNP individually [The proportion of the genetic variance not captured by markers  and PLINK v1.07 [A total of 1092 Italian Brown Swiss bulls were genotyped with the Illumina Bovine54K chip . These bulls were born between 1963 and 2002. Figure NK v1.07 . At the The phenotypic data available were the EBV for fat yield (FAT), milk yield (MILK), protein yield (PROT) and somatic cell score in milk (SCS) for each bull, which were calculated by the Italian National Association of Brown Swiss (ANARB). The EBV were deregressed as proposed by Garrick , in ordeThree subsets were formed according to the reliability of EBV as follows: animals with a reliability of at least 50% for each trait; animals with a reliability greater than 70% for each trait; animals with a reliability of at least 90% for each trait.A) with an adapted version of the procedure proposed by Meuwissen and Luo [A pedigree file was extracted from the Italian Brown Swiss herd book. Pedigree was traced back five generations and the pedigree file included 6826 entries. The completeness in the pedigree was 100% up to the grandparents, and decreased to ~90% thereafter. The equivalent number of known generations as calculated by the software Pedig  was on a and Luo .G) were computed for all genotyped animals. The first GV was based on the method proposed by VanRaden [M be the marker-genotype matrix with number of individuals (n) and number of loci (m) as dimensions. The elements in the matrix M were coded as -1  0 (heterozygous) and 1 for . The nxm matrix P contains columns with all elements 2(pi-0.5), where pi is the frequency of the second allele at locus i. The matrix P was subtracted from M to give Z = M - P. Finally, matrix GV was calculated as:Two genomic relationship matrices (VanRaden . Let M bGY) was computed as:The second genomic relationship matrix ; u is the vector of random additive genetic effect using the genomic relationship matrix, with u ~ N; and finally, e is the vector of random residual effects. Because the number of daughters per bull was high for all bulls, the reliabilities of the dEBV were high and varied little between bulls, and a homogeneous error variance structure was assumed.where A is an unbiased estimate of G, and write G = A + D[D is a matrix of deviations from pedigree relationships due to the segregation of a finite number of chromosome segments in the genome, the genetic variance of the records becomes V(g) = Gσu2 + Aσa2 = A(σu2 + σa2) + Dσu2. Hence, as in a model that fits only pedigree relationships (y = 1μ + Z1a + e), the total genetic variance is explained by the A matrix and the segregation of chromosome segments that are traced by the markers is explained by σu2. The fraction of genetic variance not captured by the markers on the SNP chip (Cmiss) was thus estimated as:If we assume that G = A + D, where Dσ2g is the total genetic variance, σ2u is the variance due to marker-based relationships and σ2a is the variance due to pedigree-based relationships.where Cmiss2) as follows: Cmiss2  has the advantage that σ2a0 is known to yield an unbiased estimate of the genetic variance, but it has the disadvantage that σ2u0 is likely to include more genetic variance than that explained by QTL that are in LD with the markers [σ2a, because the pedigree-based relationship matrix more closely resembles the family relationships at the unmarked chromosomes than at the marked chromosomes, which may show relationships that (randomly) deviate from the pedigree. Thus, Cmiss2 is expected to underestimate the fraction of missing genetic variance.The two additive genetic variances were also estimated by fitting each separately: the additive genetic animal variance using the pedigree-based relationship matrix  and traits . Results are in Table Cmiss was 0.373 ± 0.068 based on GV and 0.363 ± 0.069 based on GY. The estimates of Cmiss were smaller for the dFAT70 subset than for the dFAT50 subset. For dFAT90, the estimate was 0.305 ± 0.074 GV, while the GY matrix did not result in converged variance component estimates. Algorithms other than the AI-REML algorithm might have converged , but the convergence difficulties are probably due to the small size of the dataset, thus resulting variance component estimates would have been unreliable.The fraction of genetic variance not explained by molecular markers based on Cmiss2 through the additive genetic variances was estimated separately for all datasets and traits  for dMILK90 and 0.206 (±0.098) for dPROT90 when using GY. These estimates were not significantly different from those obtained with the larger datasets for the same traits (dEBV50 or dEBV70), although they were systematically lower for all traits.Results for dMILK, dPROT and dSCS were similar to those described above for dFAT for both genomic relationship matrices. Estimates of Cmiss were obtained for dSCS50, with 0.532 (±0.091) for GV. When using GY, the corresponding Cmiss estimate was lower (0.486 ± 0.095). The smallest Cmiss estimate was obtained for dSCS90: 0.061 (±0.197) using GY. The variance component analysis with GV on the same dataset did not converge. This was the smallest dataset and, although the average reliability was the highest, estimates of Cmiss were not significantly different from 0.The highest estimates for Cmiss2 decreased as the reliability of the dEBV increased. Estimates of Cmiss2 differed from estimates of Cmiss, probably because Cmiss2 is expected to underestimate the fraction of the missing genetic variance.In general, estimates of Cmiss) based on the Illumina 54 K SNP chip for complex traits in dairy cattle. The results showed that the estimates of Cmiss depended on the reliability of the phenotypic traits considered, i.e. the dEBV used as response values. When the accuracy of the dEBV increases, i.e. when the correlation between dEBV and the true breeding value increases, the proportion of the genetic variance explained by SNPs tended to increase. When the reliability of the dEBV is low, the family/pedigree information greatly contributes to the estimation of the EBV, which results in a larger fraction of the variance being explained by A and, in turn, in upward biases of Cmiss. Because the estimates of the Cmiss values, are expected to be overestimated due to the use of (family information in) dEBV, the best estimates of Cmiss are obtained for data sets with high reliabilities, which resulted in estimates around 0.2. This implies that the maximum accuracy of GEBV is √(1-Cmiss) ≈ 0.9, which agrees with the result of Daetwyler [We estimated the fraction of the genetic variance not accounted by SNPs in the marker panel , the results were virtually the same.For all production traits, the fraction of the genetic variance not explained by the SNPs was significantly different from 0, even when the phenotypes were very accurate (reliability > 90%), and were, therefore, very close to the true breeding values. Correction for the base population did not affect the fraction of the genetic variance explained by markers for any of the marker-based relationships here used. The differences in Cmiss, instead of dEBV, the upward biases mentioned above are not expected to occur. The error variances would be higher than when using dEBV, but the value of σ2a would not be inflated, because family information does not contribute to own phenotype (in contrast to dEBV phenotypes).If original performance records of production and SCS phenotypes are used to estimates Cmiss for each phenotype [The sources of phenotypic information used in genomic analyses are very heterogeneous and vary from individuals with highly reliable information, i.e. progeny-tested bulls, and animals with phenotypes with low levels of accuracy, i.e. young cows. To take into account these differences in reliability in a weighted analysis, it is necessary to know the value of henotype . In addihenotype .A and G into a single matrix (H), according to Legarra et al. [The base population correction of the genomic relationship matrix generally affected neither the proportion of genetic variance captured by markers, nor the genetic variance captured by the pedigree-based relationship matrices, which agrees with ,30 but na et al. , Christea et al. , and Meua et al. .Cmiss2 using the pedigree-based estimate of genetic variance. The denominators of Cmiss and Cmiss2 were significantly different from each other but both estimates revealed that the genomic relationship matrix could explain more than 95% of genetic variance if sufficiently reliable phenotypes are used (with reliabilities greater than 95%).We also estimated Cmiss and Cmiss2 depend on the SNP chip used, i.e. more dense SNP chips are expected to yield lower estimates of Cmiss and Cmiss2 (a larger fraction of the variance is explained by the SNPs), and also on the family structure of the population [Cmiss. The population structure of the Italian Brown Swiss population reflects that of a typical dairy breeding population, and, thus, our results probably apply also to other dairy breeding populations.It should be noted that the estimates of pulation . PopulatCmiss) was equal to 0.2, which was estimated based on the most accurate phenotypes. This value agrees with other values reported in the literature. Correction of the genomic relationship matrix for the variance of the allele frequency of each locus (GY) instead of the average frequency of heterozygotes (GV), hardly explained any additional genetic variance. Our estimate of Cmiss of 0.2 implies that about 80% of the genetic variance is explained by the Illumina 54 K SNP chip. Values for Cmiss are expected to depend on the density of the chip (a larger SNP chip is expected to explain a larger fraction of the genetic variance) and on family relationships in the population, i.e. closer family relationships are expected to reduce Cmiss.The fraction of genetic variance explained by genetic markers from high-density SNP panels was significantly different from 0 for the complex traits analyzed when the phenotypes are not highly accurate. The minimum fraction of the genetic variance not explained by the markers (The authors declare that they have no competing interests.SIRP performed the study and drafted the manuscript. ABS contributed to writing the draft. SIRP, MAD and AB prepared the genotypic and phenotypic data. THEM planned and coordinated the whole study, and contributed to writing the manuscript. All the authors read and approved the final manuscript.


Statistics in Medicine published by John Wiley & Sons Ltd.Mendelian randomization is the use of genetic instrumental variables to obtain causal inferences from observational data. Two recent developments for combining information on multiple uncorrelated instrumental variables (IVs) into a single causal estimate are as follows: (i) allele scores, in which individual‐level data on the IVs are aggregated into a univariate score, which is used as a single IV, and (ii) a summary statistic method, in which causal estimates calculated from each IV using summarized data are combined in an inverse‐variance weighted meta‐analysis. To avoid bias from weak instruments, unweighted and externally weighted allele scores have been recommended. Here, we propose equivalent approaches using summarized data and also provide extensions of the methods for use with correlated IVs. We investigate the impact of different choices of weights on the bias and precision of estimates in simulation studies. We show that allele score estimates can be reproduced using summarized data on genetic associations with the risk factor and the outcome. Estimates from the summary statistic method using external weights are biased towards the null when the weights are imprecisely estimated; in contrast, allele score estimates are unbiased. With equal or external weights, both methods provide appropriate tests of the null hypothesis of no causal effect even with large numbers of potentially weak instruments. We illustrate these methods using summarized data on the causal effect of low‐density lipoprotein cholesterol on coronary heart disease risk. It is shown that a more precise causal estimate can be obtained using multiple genetic variants from a single gene region, even if the variants are correlated. © 2015 The Authors.  These data are increasingly becoming available from large consortia, such as the Global Lipids Genetics Consortium (GLGC) for lipid fractions In this paper, we review and extend the literature on IV estimation methods with summarized data, currently described in disparate sources. In Section For reference, a summary of methods for IV estimation discussed in this paper is given in Table 2In this paper, the situation of a continuous risk factor and a continuous outcome will be assumed, although the binary outcome case can be handled in a similar way. We assume that the causal effect of the risk factor on the outcome is linear and homogeneous in the population without effect modification. We also assume that the associations of the IVs with the risk factor are linear and homogeneous in the population without effect modification. As shown previously, these assumptions lead to the identification of the causal effect If the IV–risk factor and IV–outcome associations are estimated in different datasets  that can be represented as random variables taking the values 0, 1 or 2, denoting the number of risk factor‐increasing alleles in the genotype of an individual. An unweighted allele score is constructed as the total number of risk factor‐increasing alleles for an individual across multiple genetic variants. If an individual If the weights in an allele score are derived from the data under analysis, then they will be the same asymptotically as the coefficients from a multivariable regression of the risk factor on the IVs (under the assumption that the IVs are uncorrelated). Values of the weighted score for each individual would therefore equal the fitted values of the risk factor from that regression (up to an additive constant), meaning that the allele score and (multivariable) 2SLS estimates would coincide F statistic from regression of the risk factor on the IVs – remains fixed as the sample size increases), confidence intervals (CIs) from the 2SLS method using standard asymptotic approximations are overly narrow and coverage rates are below nominal levels Under weak instrument asymptotics , outcome (Y) and IVs  in all participants. We assume that the estimate of association for IV k = 1,…,K with the risk factor is β^Xk with standard error σXk, and the estimate of association with the outcome is β^Yk with standard error σYk. These estimates are typically obtained from linear regression (or logistic regression for associations with a binary outcome). Although the standard errors are estimated, we assume that they are known without error. This may lead to slightly overprecise estimates, but coverage levels have been shown to be close to nominal levels in realistic simulations Z=∑kwkGk) used as a single IV, and writing cov for the sample covariance and var for the sample variance, the IV estimate is β^Yk=cov/var(Gk) for each k = 1,…,K (similarly for each β^Xk). The weights wk are assumed to be pre‐specified and are typically taken as the association estimates of each IV with the risk factor in an independent dataset. If the IVs explain a small proportion of variance in the outcome, then var(Gk) is approximately proportional to σYk−2, and so the allele score estimate based on summarized data (β^SSw) is We assume the context of a one‐sample IV analysis in a single dataset with data on the risk factor  and then taking the ratio of the pooled estimates. Even in this unweighted case, the directions of the IV associations with the risk factor are required to be specified, even if the magnitudes of the associations are unknown. There have been reports of genetic variants having different directions of association with a risk factor in different datasets We note that at no point in this calculation have we made use of the fact that the genetic variants are uncorrelated. With equal weights, this is equivalent to performing separate inverse‐variance weighted meta‐analyses of the genetic associations with the outcome and of the genetic associations with the risk factor . The standard errors of the association estimates σXk and σYk are used to specify the variance–covariance matrix for the normal distribution and as before are assumed to be known. Model parameters  can be estimated either by numerical maximization of the log‐likelihood function or in a Bayesian framework A likelihood‐based method has also been proposed, in which the IV associations with the risk factor and with the outcome for each IV are modelled directly by a bivariate normal distribution, with correlation θL is due to the IV associations with the risk factor and with the outcome being estimated in the same data. There is likely to be little information on this parameter in the data θL will be zero.The correlation 3.5We investigate the properties of estimates from these methods in a simulation study with two specific goals. The first is to demonstrate that estimates from the allele score method are similar whether they are calculated using individual‐level or summarized data. The second is to compare the behaviour of the summarized data methods  in a two‐sample setting with external weights. We have previously shown that the summary statistic and likelihood‐based methods give similar estimates and standard errors in a one‐sample setting i, xi is the risk factor of interest, ui a confounder, yi the outcome, and gik=0,1,2 is the kth IV , representing the number of risk factor‐increasing alleles of a genetic variant: βX=0.2 throughout. The risk factor‐increasing allele frequency πk and strength of association of the kth IV with the risk factor αk are allowed to vary between the IVs. We set α = 0.05,0.1,0.2, and consider scenarios for K = 15 IVs with positive (βU=+1) and negative (βU=−1) confounding. As genetic variants are defined arbitrarily with respect to either the risk factor‐increasing or risk factor‐decreasing allele, the restriction to consider only positive values of αk does not result in any loss of generality. The mean proportion of variance in the risk factor explained by the IVs varies from 1.0% to 10.2%, corresponding to mean F statistics from 3.3 to 37.9. 10000 simulations were undertaken for each set of parameter values.Data on 5000 individuals were generated from the following model in which, for subject wk=β^Xk) and equal weights (wk=1), we also consider external weights, corresponding to a two‐sample IV analysis. The external weight for the kth IV is generated by sampling from a normal distribution with mean αk and variance 1Nπk(1−πk). This is equivalent to estimating genetic associations with the risk factor in a separate dataset of size N generated under the same model In addition to crude weights  Comparison of allele score methodsN = 5000) score using individual‐level data and using summarized data on the genetic associations with the risk factor and outcome as in equation α = 0.05 and is reduced for larger values of α. We additionally provide mean estimates and mean standard errors for this simulation in Table We calculated the allele score IV estimates for a crudely weighted, equally weighted and externally weighted (var(Gk)∝σYk−2) are reasonable. Estimates with crude weights showed the same pattern of weak instrument bias as those from a 2SLS method; estimates with equal and external weights were unbiased. Standard errors calculated from summarized data, obtained from equation βU=+1) were sometimes narrower but on average wider than those calculated from individual‐level data. With negative confounding (βU=−1), median standard errors based on summarized data were considerably smaller. The same phenomenon with the average size of standard errors depending on the direction of confounding has been observed previously K = 5 and K = 25 IVs to investigate the behaviour of these estimates across different numbers of IVs; similar results were observed.Allele score estimates calculated using individual‐level data and summarized data were equal to at least the first three decimal places for almost all simulated datasets. This suggests that the approximations used in calculating the allele score estimate using summarized data  Comparison of summarized data methodsN= 5000 (imprecise weights), 50 000 (precise weights) and using the true ‘oracle’ weights. The oracle weights are the αk parameters, corresponding to a notional sample size of infinity. The Monte Carlo error for the coverage rate is 0.3% in all scenarios and for the empirical power is 0.5% or lower. Estimates are not provided for the likelihood‐based method with oracle weights as the uncertainty in the weights (σXk) cannot be expressed; otherwise, we took σXk=1Nπk(1−πk). The likelihood‐based method failed to report a standard error using imprecise weights with α = 0.05 for nine of the 10000 simulated datasets when βU=1 and for five datasets when βU=−1; these results are omitted from Table Table Allele score estimates using external weights were unbiased, with median estimates close to the true value of 0.2. With positive confounding, coverage levels were conservative compared with the nominal 95% level, whereas with negative confounding, coverage levels were often slightly below nominal levels. With imprecise weights, the summary statistic estimates were biased towards the null; this bias disappeared as the weights become more precise. However, coverage rates were close to nominal levels. Further investigations showed that the summary statistic estimates with external weights were unbiased under the null Table . The sliThe summary statistic and allele score methods had equal power estimates and rejected or accepted the null together. The likelihood‐based method dominated the other methods in terms of power with imprecise weights; with precise weights, power was very similar between the methods. The power of estimates increased as weights became more precise, although the increase from using weights estimated in a sample size of 50000 to the oracle weights was not substantial.3.7It is likely that the choice of weights will have a greater impact on the findings of Mendelian randomization investigations than differences between analysis methods. For choosing weights in practice, we echo the advice of Burgess and Thompson 4The use of multiple genetic variants in a Mendelian randomization analysis is often necessary to give clinically relevant results because of low power. In some circumstances, including multiple potentially correlated variants from a single gene region is likely to lead to a more reliable analysis than one using variants from multiple gene regions. Although including additional genetic variants that are perfectly correlated will not increase the precision of a Mendelian randomization analysis, the inclusion of multiple variants in partial linkage disequilibrium can explain a greater proportion of variance in the risk factor. If the variants explain additional variation in the risk factor, this will lead to more powerful Mendelian randomization analyses. We do not make any assumption on the underlying genetic architecture leading to multiple correlated genetic variants that each explain independent variation in the risk factor but note that a genetic variant is not required to be a ‘causal variant’ for use in Mendelian randomization β^Xk and β^Yk estimates, with the variance–covariance matrix incorporating the correlation between the IVs (see Appendix A.3 for details). We proceed to consider extensions to allow for correlated variants in the summarized data allele score and summary statistic methods.Extensions to allow for correlated IVs have been discussed in the context of the likelihood‐based method 4.1An allele score composed of genetic variants that are valid IVs will be a valid IV regardless of correlation between variants. However, if the weights for the score are taken from univariate regression analyses, IV estimates may be inefficient. For instance, if two sets of highly correlated variants have the same strength of association with the risk factor for each genetic variant, taking weights from univariate regression analyses will assign weight overall in proportion to the number of variants measured in each set. A better approach would be to take weights from a multivariable regression of the risk factor on all the IVs in the analysis model. However, unless the weights are obtained in an independent dataset, this is likely to exacerbate weak instrument bias as overfitting is a greater problem when the predictors in a regression model are correlated. Moreover, in practice, it is unlikely that genetic association estimates from a multivariable regression model in an independent dataset would be generally available, and so, we do not pursue complex strategies for weighting allele scores further in this manuscript. We restrict our attention to equally weighted and externally weighted scores, where the external weights are modelled as univariate weights from an independent dataset.ρk1k2 is the correlation between IVs k1 and k2. Only the first term from the delta method expansion corresponding to equation As noted previously, correlation between IVs should not affect allele score estimates based on pre‐specified weights equation , althoughttp://www.broadinstitute.org/mpg/snap/ldsearch.php) Estimates of correlations between genetic variants can be obtained from the published literature if they are not otherwise available, for example, using the SNP Annotation and Proxy Search , is exactly the same as the test statistic for the allele score method using summarized data evaluated using equation β^SSwse(β^SSw), and the two methods will therefore reject or not reject the null hypothesis together. However, the expression in equation Similarly, if the summary statistic method in equation β^Yk parameters on the β^Xk parameters with no intercept term, we can perform a weighted generalized linear regression of the β^Yk parameters on the β^Xk parameters using the σYk−2 parameters as inverse‐variance weights and taking into account the correlation between the IVs.Alternatively, in the same way as the summary statistic method with uncorrelated IVs in equation Ωk1k2=σYk1σYk2ρk1k2, then the estimate from a weighted generalized linear regression is β^SSc does not equal the estimate in equation If 4.3We perform a further simulation study to investigate the properties of estimates from these methods with correlated IVs. Other than as specified, all parameters and features of the simulation study are the same as those in the simulation with uncorrelated IVs. As previously, the main goals are to consider the similarity of allele score estimates calculated using individual‐level and summarized data and to compare the behaviour of the summarized data methods  with external weights. We do not consider the summary statistic method  were mostly (78%) positive, with an average first quartile of 0.06 and third quartile of 0.30 across all pairwise correlations. A further simulation analyses was also performed (not reported) in which the off‐diagonal elements of Λ0 were all 0.2; findings were substantially the same as those reported in this paper.The data‐generating model is as follows: βX=0.2 and βX=0. Rather than generating external weights using a random draw from a normal distribution, we generated independent data from the same data‐generating model for 5000 participants and used univariate regression for each IV in these individuals to derive external weights. This procedure should closely mirror an applied two‐sample analysis, particularly one using published summarized data on genetic associations with the risk factor.We considered two values of the causal effect 4.4(a) Comparison of allele score methodsTable βX=0 was around 5%, meaning that coverage rates (type I error rates) were at correct nominal levels.The pattern of results for the allele score methods is very similar with correlated IVs as with uncorrelated IVs. Estimates in each simulated dataset calculated using individual‐level data and summarized data were equal to at least three decimal places and were unbiased both with a positive causal effect and with a null causal effect. Median standard errors with a positive causal effect using summarized data were slightly larger with positive confounding, and slightly smaller with negative confounding, compared with those with individual‐level data. With a null causal effect, median standard errors were almost the same using individual‐level and summarized data and did not vary with the direction of confounding. The power to detect a causal effect with (b) Comparison of summarized data methodsN = 5000). The likelihood‐based method failed to report a standard error with α = 0.05 for between 1 and 9 of the 10000 simulated datasets for each set of parameter values and weights; these results were omitted from Table Table βX=0.2.Results for the summarized data methods were similar with correlated IVs as with uncorrelated IVs. Estimates with external weights were unbiased under the null with nominal coverage rates preserved. With a positive causal effect, estimates from the weighted generalized linear regression method were slightly biased towards the null, reflecting the uncertainty in the IV associations with the risk factor. For the likelihood‐based method, there was bias towards the null with the strongest IVs and slight undercoverage under the null with the weakest IVs. Examination of the values of the optimized log‐likelihood function revealed lack of convergence for a small number of datasets, particularly with The simulations were also repeated with crude weights Table . With cr4.5Allele score and weighted generalized linear regression estimates using summarized data showed good statistical properties with correlated genetic variants, particularly under the null hypothesis. With a non‐null causal effect and imprecisely estimated weights, the weighted generalized linear regression estimates were slightly biased towards the null. This suggests that Mendelian randomization analyses can be performed using summarized data on correlated genetic variants, provided that data on the correlations between the variants are available. Although the likelihood‐based method has good theoretical properties, caution should be taken with the method in practice to ensure that the optimization routine has converged appropriately. A sensitivity analysis can be undertaken by repeating the analysis in a Bayesian analysis framework; software code for a two‐sample analysis is provided elsewhere 5PCSK9 gene region. We consider rs11206510, previously shown to be associated with LDL‐c concentration and CHD risk To illustrate the methods described earlier, we estimate the causal effect of LDL‐c on CHD risk using genetic variants from the http://www.sph.umich.edu/csg/abecasis/public/lipids2013. Genetic associations with CHD risk were taken from the CARDIoGRAM consortium http://www.cardiogramplusc4d.org/downloads. Estimates from both GLGC and CARDIoGRAM were obtained using data on individuals of European descent, mostly of working age, and so the two datasets should be similar; in fact, several studies are included in both consortia. After pruning for linkage disequilibrium at r2>0.8, all variants in a 10‐kilobase pair region around rs11206510 available in both the GLGC and CARDIoGRAM consortia were included in the analysis; 10 variants in total were included. Associations with LDL‐c and CHD risk are given in Table θL in the likelihood‐based method  in a one‐sample setting is approximately equal to a commonly used summary statistic (inverse‐variance weighted) estimate. This summary statistic estimate is equivalent to an estimate from a weighted linear regression analysis. Both the allele score  and summary statistic estimates with crude weights are approximately equal to an estimate from a 2SLS method. This means that allele score and summary statistic estimates using crude weights suffer from weak instrument bias and are biased in the direction of the observational association. In contrast, allele score and summary statistic estimates using equal or externally derived weights give valid tests of the null hypothesis of no causal effect. Estimates from the summary statistic method using external weights are conservatively biased towards the null when the external weights are imprecise estimates of the true weights.We have also provided formulae for calculating an allele score estimate using summarized data with correlated IVs, which in Mendelian randomization correspond to genetic variants in linkage disequilibrium. Alternatively, a causal estimate can be calculated from summarized data using weighted generalized linear regression. These methods enable researchers to perform IV analyses with correlated IVs using summarized data and in particular Mendelian randomization analyses with more than one genetic variant in a given gene region. If the multiple variants explain more of the variance in the risk factor than any single variant, then power to detect a causal effect will be improved. Inclusion of multiple genetic variants from a single gene region may provide a better way of improving power in Mendelian randomization investigations than inclusion of genetic variants from multiple gene regions, as variants from a single candidate region may be more likely to satisfy the IV assumptions.β^Xk is given in standard deviation units, then the proportion of variance in the risk factor explained by the SNP (the R2 statistic) is approximately equal to 2β^Xk2×MAF×(1−MAF), where MAF is the minor allele frequency. The F statistic can then be calculated from the R2 statistic as F=N−K−1KR21−R2, where N is the sample size and K is the number of genetic variants. An overidentification test can be performed by considering the ratio estimates from each IV individually and performing a heterogeneity test, such as Cochran's Q test β^Yk on the β^Xk that is similar to the one discussed in this paper but with an intercept term. The intercept term represents the average association of an IV with the outcome in the absence of association with the risk factor. Under the IV assumptions that the association of each IV with the outcome is mediated via the risk factor, this intercept term should be zero. If the estimated intercept term is different from zero, there is evidence of direct effects of IVs on the outcome not via the risk factor that do not average out; this is known as directional pleiotropy.There are several practical considerations to take into consideration when using summarized data. If the IV assumptions are violated even for one genetic variant in a Mendelian randomization analysis, then causal estimates will be biased and type I error rates will be inflated, as previously demonstrated in the context of allele scores As a recommendation for using summarized data, we suggest either the allele score or weighted  linear regression methods. If the likelihood‐based method is used, care should be taken to check whether the optimization algorithm has converged appropriately. The allele score method is desirable as estimates are unbiased with either equal or externally derived weights, and the summary statistic method has good intuitive justification from weighted linear regression, although there is some bias towards the null with imprecise externally derived weights. The formulae provided in this paper mean that evaluating a causal estimate based on multiple IVs that does not suffer from weak instrument bias is relatively simple; ensuring that the choice of IVs and the interpretation of the causal analysis are appropriate is the difficult step.


The objective of this study was to identify genomic regions that are associated with meat quality traits in the Nellore breed. Nellore steers were finished in feedlots and slaughtered at a commercial slaughterhouse. This analysis included 1,822 phenotypic records of tenderness and 1,873 marbling records. After quality control, 1,630 animals genotyped for tenderness, 1,633 animals genotyped for marbling, and 369,722 SNPs remained. The results are reported as the proportion of variance explained by windows of 150 adjacent SNPs. Only windows with largest effects were considered. The genomic regions were located on chromosomes 5, 15, 16 and 25 for marbling and on chromosomes 5, 7, 10, 14 and 21 for tenderness. These windows explained 3,89% and 3,80% of the additive genetic variance for marbling and tenderness, respectively. The genes associated with the traits are related to growth, muscle development and lipid metabolism. The study of these genes in Nellore cattle is the first step in the identification of causal mutations that will contribute to the genetic evaluation of the breed. Bos taurus indicus) account for 80% of this herd and the Nellore breed corresponds to 90% of these animals [Bos taurus indicus meat is inferior when compared to the meat of Bos taurus taurus [Brazil is one of the world’s largest exporter of beef and possesses the second largest herd . Zebu an animals . Studiess taurus ,4. Meat s taurus . In thiss taurus . Anothers taurus . This atThe application of traditional selection to these traits is difficult, since they are expensive and difficult to measure because it requires the slaughter of the animals. Genome-wide association studies (GWAS) allow the identification of single-nucleotide polymorphisms (SNPs) associated with large-effect genes that influence these traits, providing a better biological understanding of the trait and a list of candidate genes for fine mapping.At first, classical methods were used for GWAS, which were based on testing one marker at a time. However, using those methods increases the chance of false-positives due to multicolinearity . FurtherBos taurus taurus, are also polymorphic in Nellore cattle [Some association studies of SNPs with meat quality traits in taurine cattle have identified different regions associated with marbling, which are located on chromosomes 2, 3, 7, 8, 10, 17, 22 and 27 and with meat tenderness, which are located on chromosomes 7, 8, 9, 10 and 29 –15. Genee cattle . In GWASe cattle  identifiIn previous studies, it is unclear whether these genes contribute to the additive genetic variance of these traits in Zebu cattle. Therefore, the objective of this study was to perform GWAS in order to identify genomic regions that are associated with meat quality traits in the Nellore breed.Tenderness and marbling were the meat quality traits used in this study. Data of male Nellore animals born between 2008 and 2010 were collected among herds located in different regions of Brazil, which belong to three breeding programs . The animals were finished in feedlots for approximately 90 days and slaughtered at an average age of 731 ± 81 days.The approval of the ethics committee of the Faculty of Agrarian Sciences and Veterinary of Sao Paulo State (FCAV-UNESP) was not necessary, because the slaughter of animals was done in commercial slaughterhouses , located in several regions of Brazil. Since such slaughterhouses have animal welfare department, staffed by professionals trained by WAG , ensuring that the animals are killed humanely, by use of captive bolt pistol in the stunning process.post-mortem and longissimus dorsi muscle (tenderloin) samples were detached with bone between the 12th and 13th rib of the left half-carcasses. All samples were frozen and none of them was aged. Animals from the same farm and year of birth were slaughtered in the same slaughterhouse and slaughter date.The carcasses were maintained in a cold chamber for 24 to 48 h Longissimus dorsi samples measuring 2.54 cm in thickness were obtained for analysis of tenderness. The procedure standardized and proposed by [posed by  was adopposed by , where 1There were a total of 7,436 animals in the relationship matrix. Data from 1,875 animals with tenderness and marbling records were available. Measures that were 3.5 standard deviations above or below the mean of the contemporary group  were excluded from the analyses. Additionally, contemporary groups containing fewer than three animals were eliminated. Thus, there were records of 1,822 animals with tenderness measures and of 1,873 animals with marbling measures . The her−5, GenCall score ≤ 0.70, call rate ≤ 0.98, and r² (correlation between SNPs) > 0.995 with adjacent SNPs within a window of 100 SNPs. Samples with a call rate ≤ 0.92 and without a valid phenotype were also excluded. After quality control, 1,630 animals genotyped for tenderness, 1,633 genotyped for marbling, and 369,722 SNPs remained.The animals were genotyped using a high-density panel containing 777,962 SNPs (Illumina High-Density Bovine BeadChip). The criteria for exclusion of SNPs were: non-autosomal SNPs, SNPs at the same position, with a minor allele frequency ≤ 0.02, a p value for Hardy-Weinberg equilibrium ≤ 10y is the vector of phenotypic observations; X is an incidence matrix relating the phenotypes to the fixed effects; b is the vector of fixed effects, including the contemporary group  and slaughter age and days between slaughter and physicochemical analysis of meat as covariates (linear effect); Z is an incidence matrix relating the animal to the phenotype; a is the vector of effects of the animals, and e is the vector of residual effects.The ssGWAS method proposed by  was useda and e can be written as:σ2a is the direct additive genetic variance; σ2e is the residual variance; H is a matrix that combines pedigree and genomic information as proposed by [I is an identity matrix. The inverse of matrix H is:A is the pedigree matrix for all animals; A22 is the relationship matrix for genotyped animals, and G is the genomic relationship matrix, which was calculated as described by [The variances of posed by , and I iribed by .g is the animal effect of genotyped animals; Z is a matrix that relates the genotypes of each locus; D is a diagonal matrix of the weights of SNP variances. For this study the weights of SNP was not used, so D = I (identity matrix). G is the genomic relationship matrix.The effects of the SNPs (ȗ) were obtained using the equation described by :u⌢=λDZ'rding to ; ȃg is thttp://www.ncbi.nlm.nih.gov/projects/mapview/map_search.cgi?taxid=9913&build=103.1), was used to identify the genes. The DAVID software [The analyses were carried out using the BLUPF90 family of programs . The ressoftware  was usedFigs The windows are located on chromosomes 5, 15, 16 and 25 for marbling and on chromosomes 5, 7, 10, 14 and 21 for tenderness and explained 3.89% and 3.80% of the additive genetic variance for marbling and tenderness, respectively.. This cluster of genes participates in the change of GDP (guanosine diphosphate) to GTP (guanosine triphosphate), which are regulators of G proteins. GDP and GTP can be used as energy source and are responsible for transferring energy within the cell [Describing the results obtained from the genes for marbling and tenderness Tables  and 3, ithe cell . Anotherthe cell . Furtherthe cell . Some stthe cell , and conTCEB2 gene was detected by proteomic analysis of hepatic fat droplets, which showed that high production of this protein protects against diabetes in mice [TIGD7 gene has been associated with body mass index and plasma glucose levels in humans [APPL2 gene has been shown to be associated with obesity in humans [NEU3 gene promotes the accumulation of triglycerides in mice, which may facilitate the deposition of body fat [MFNG gene is responsible for causing bile duct abnormalities in mice, which can compromise bile flow and the consequent emulsification of fat for absorption [CHST11, LALBA and PGD genes form a cluster and are involved in the chemical reactions resulting in the formation of carbohydrates.For marbling, the genes described in  in mice . The TIGn humans . The APPn humans  and to pn humans . The NEUbody fat . The MFNsorption . The lisBos taurus taurus [The genes associated with marbling in the present study are not the same as those reported in other GWAS for the same trait in s taurus –41. In ss taurus , the autPLAG1, SLC27A2, RB1CC1, HDC, LYPLA1, XKR4 and TMEM68). The PLAG1 gene has been associated with meat tenderness in cattle [For meat tenderness, genes related to growth and muscle development were identified  is an important gene associated with tenderness, which is known to inhibit calpain (CAPN1). These genes (CAST and CAPN1) are considered responsible for the process of meat tenderization during the postmortem period [CAPN1 determines an increase in final meat tenderness, while CAST acts in the opposite manner, inhibiting this tenderizing process. In the present study, the window where CAST was located explained 0.46% of the additive genetic variance in tenderness. The CAST gene has been associated with meat tenderness in GWAS of Australian cattle [CAST gene and tenderness. The authors identified 56 genes for different tenderness measures , but these genes only explained a small percentage of the additive genetic variance (less than 0.2%).The calpastatin Click here for additional data file.S1 TableaNCBI Symbol .(DOCX)Click here for additional data file.S2 TableaNCBI Symbol .(DOCX)Click here for additional data file.


Longitudinal data enables detecting the effect of aging/time, and as a repeated measures design is statistically more efficient compared to cross-sectional data if the correlations between repeated measurements are not large. In particular, when genotyping cost is more expensive than phenotyping cost, the collection of longitudinal data can be an efficient strategy for genetic association analysis. However, in spite of these advantages, genome-wide association studies (GWAS) with longitudinal data have rarely been analyzed taking this into account. In this report, we calculate the required sample size to achieve 80% power at the genome-wide significance level for both longitudinal and cross-sectional data, and compare their statistical efficiency. Furthermore, we analyzed the GWAS of eight phenotypes with three observations on each individual in the Korean Association Resource (KARE). A linear mixed model allowing for the correlations between observations for each individual was applied to analyze the longitudinal data, and linear regression was used to analyze the first observation on each individual as cross-sectional data. We found 12 novel genome-wide significant disease susceptibility loci that were then confirmed in the Health Examination cohort, as well as some significant interactions between age/sex and SNPs. Disease prognosis and personalized medicine require the identification of genetic and non-genetic risk factors and, with the rapid improvement of genotyping technology, more than ten thousand genome-wide association studies (GWAS) have been conducted to discover disease susceptibility loci. Since the first such successful study in 2005 , more thEven though there are few GWAS using longitudinal data ,9,10, cop-values for Hardy-Weinberg equilibrium (HWE) less than 10−6, or with minor allele frequencies (MAF) less than 0.01, were eliminated. Furthermore, individuals with tumors, gender inconsistencies, or whose heterozygosity rates were more than 30%, or identity in state (IBS) more than 0.8, were excluded from the analysis [The KARE cohort consists of a total of 10,038 individuals . Participants ranged from 40 to 69 years old, and their phenotypes were consecutively measured with two-year intervals from 2001 to 2005. Among the 10,038 participants, 10,004 individuals were genotyped for 500,568 SNPs with the Affymetrix Genome-Wide Human SNP array 5.0. Individuals and SNPs with call rates less than 95% were excluded from the analysis. SNPs with analysis . In totap-values for HWE less than 10−6, genotype call rates less than 95%, or MAF less than 0.01, were excluded, and the remaining SNPs for 3703 individuals were used for the analysis. In particular, the HEXA cohort is a cross-sectional study, and if there are some SNPs which have a progressing effect on phenotypes, results from HEXA and KARE cohort could be heterogeneous.Independent individuals in the HEXA cohort were from a second population based cohort sample provided by the Health study. This study combines subjects from the Wonju, Pyeong Chang, Gangneung, Geumsan, and Naju regional cohorts in Korea. There are 120,000 participants in the HEXA cohort, 4302 of whom, between 40 and 69 years old, were randomly selected for genotyping with the Affymetrix Genome-Wide Human SNP array 6.0. Individuals in the HEXA cohort were used to replicate the significant findings found in the KARE cohort, and individuals with tumors, large heterozygosity rates, gender inconsistencies, evidence of non-Asian ancestry, or whose IBS was more than 0.8 or call rates were less than 95%, were excluded from analysis [−8 significance level were calculated in both the presence and absence of population substructure. We denote the number of individuals and measurements for each individual by n and t, respectively. We assume that t is same for all individuals. The additively coded value of the genotype for individual i is denoted by iX. We assume Hardy-Weinberg equilibrium and each individual’s genotype is assumed to follow a trinomial distribution. The effect of the disease allele is assumed to be β. We assume a matrix of environmental effects that does not contain time-varying covariates for individual i, denoted by iZ, and its coefficient is assumed to be the column vector α. The columns of iZ denote each covariate. The phenotype for individual i at time-point j is denoted by ijY, and the corresponding t-dimensional vector by Yi. Letting 1t be the t-dimensional column vector with elements 1, Yi was assumed to be:ci indicates the random effect which explains the similarity of repeated measurements for each individual attributable to a non-polygenic effect. We denote pσ2 = gσ2 + cσ2 + εσ2 and ρ = cσ2/pσ2. Thus ρ measures the proportion of the variance of ic relative to the phenotypic variance. In particular, we included in the model to provide a polygenic effect for individual i, and assumed as an approximation 0 and variance-covariance matrix Φ corresponds to the genetic relationship matrix. If R of Yi becomes compound symmetric. The null hypothesis H0 is β = 0, and β is assumed to be βa under the alternative hypothesis.Sample sizes required to achieve 0.8 power at the 10Zi is 1t. Then if we let lπ = P = P(iX = δl) for the coded genotype δl where δ1 = 0, δ2 = 1 and δ3 = 2, and K = π1(π2 + 2π3)2 + π2(π3 − π1)2 + π3(1 + π1 − π3)2, the required sample size for 1 − ϕ power at the significance level α can be derived to be:iX is binary and our results are based on their derivations. We extend their result to iX with arbitrary many levels. For αn, we assume that σc2 + σ ε2 = 1 and:For sample size calculation, we assumed that there is no covariate effect other than the genotypes, and then we could assume that the environmental variable g2 is assumed to be larger than 0. The required sample size cannot be directly calculated, and we calculated an by simulation studies based on a Monte Carlo method. In our simulations, we assumed that σg2 + σc2 + σ ε2 = 1 and the effect of disease the allele, β, was calculated with the following assumptions:Φ was assumed to be a compound symmetric matrix with off-diagonal elements 0.1. The off-diagonal elements are asymptotically equivalent to twice the kinship coefficient between two individuals, and 0.1 means that the individuals are genetically remote relatives.In the presence of population substructure, σIn the Korean Association Resource (KARE) data, eight phenotypes  were observed every two years from 2001 to 2005, so that three observations were available for each phenotype. Genotypes and phenotypes for 8842 individuals were initially available. However, there were some missing phenotypes for follow-up observations, and only 7568 and 6675 individuals were observed for the second and third time-points, respectively. Reasons for dropout were not known, but may include death, immigration, and non-response.i is denoted by Yi which is a three dimensional vector. The matrix Zi indicates a covariate vector for environmental effects, including the intercept as the first column, and sex, age and age2 at the first time-point were included as covariates for all eight phenotypes. In particular, weight is known to be related to glucose levels, and thus it was included as an additional covariate for the GWAS of GLU0 and GLU120 [Zi is denoted by a vector α. The effect of the time interval can be understood as the effect of aging, and it was denoted by the vector w. Here, w is a three-dimensional row vector and its coefficient is η. The population substructure between individuals was adjusted for with the EIGENSTRAT approach [i and its coefficient vector are PCi and γ, respectively. The additively coded value of the genotype for individual i is denoted by iX. The effect of the disease allele is assumed to be β. The variance-covariance matrix for εi is denoted Σ and assumed to be an unstructured symmetric matrix. For longitudinal analysis, our final model is:i = 1,2, …, 8842.GWAS using longitudinal data were performed by generalized least squares using the nlme package in the R software. The phenotype for individual d GLU120 ,15. The approach  and the approach . In partapproach  and we ui = 1,2, …, 8842. The results from longitudinal and cross-sectional data were compared.Furthermore, we conducted GWAS using cross-sectional data for comparison with the GWAS using longitudinal data. For this we took the phenotypes at the first-time point in the KARE cohort, and the GWAS were conducted with linear regression. For the cross-sectional analysis, we used the same covariates except for time interval, with the linear model:We tested whether there exist any interaction effects between our significant findings and environmental variables by adding interaction terms as covariates. We considered age, sex and time interval as environmental variables, and their statistical interaction with the SNPs were tested. Significant results were further tested for replication in the HEXA cohort . Finally, we tested all the significant findings from the KARE cohort, as a discovery dataset, in the HEXA cohort, as a replication dataset.−8 in both the absence and presence of population substructure. αn, in the absence and presence of population substructure, as a function of the number of time points t and the common correlation ρ between phenotype measures on the same person. We found the required sample size αn is proportionally related to t and inversely related to ρ. Sample size is minimized for small ρ and large t, and the effect of t on sample size is maximal when ρ = 0. These results illustrate the practical efficiency of GWAS with longitudinal data. For instance, if ρ = 0.4 and t = 3, then 5158 individuals are sufficient to achieve 0.8 power at the genome-wide significance level and, compared to cross-sectional data, genotyping costs for 3438 individuals can be saved vs. the cost of obtaining 6878 phenotypes.We calculated the sample size required to achieve 0.8 power at the genome-wide significance level α = 10The results of the GWAS with longitudinal data in the KARE cohort were compared with the results from GWAS using cross-sectional data. For the cross-sectional data we used the phenotypes at the first time-point, applying linear regression. Population substructure was adjusted for with the EIGENSTRAT method, and five principal component (PC) scores were included as covariates in both the longitudinal and cross-sectional data analyses. We found that five PC scores explain roughly 75% of the kinship matrix, and p-values from either the cross-sectional or longitudinal data analysis less than 10−6 were selected for the replication studies. For the discovery analyses, the genome-wide significance level by Bonferroni correction is 1.4E − 07. For replication, we calculated the one-sided p-value for the direction from the longitudinal analysis using the KARE data, and used 0.05 as the significance level. Whenever the results from the two cohorts were in different directions, the p-values from the HEXA cohort were larger than 0.5. In We calculated the correlations between the different time-points for each phenotype and they are presented in rs7197218 in the XYLT1 gene, which is related to corneal astigmatism , was genp-values for gene×environment interaction less than 10−6. p-values, and the combined p-values by Fisher’s and Liptak’s methods [p-values become smaller. We also performed association analysis to detect gene×environment interaction, and SNPs that interact with aging, sex and time interval were identified by using the longitudinal data in the KARE cohort.  methods ,41. It h methods . Table 9In summary, we can conclude that GWAS with longitudinal data provide an efficient strategy, and our overall results show that the improvement in power is substantial, its effect being inversely proportional to ρ.It is well known that longitudinal analysis is useful to detect aging effects, and statistically efficient for detecting significant associations. In this report, we numerically calculated the sample sizes required to achieve statistical power at the genome-wide significance level, and our results showed that the power is proportionally related to the number of observations on each individual and inversely related to the correlation between the pairs of observations on an individual. In a large-scale genetic analysis, genotyping cost may be larger than the phenotyping cost, and then we can conclude that analyzing longitudinal data is an efficient strategy to improve the rate of false negative findings. However if the proportion of missing data is large, statistical power loss can be substantial; and if the missingness is not at random, even a small proportion of missing phenotypes can generate a serious bias . In spitp-values from replication studies strengthened the possibility that they are causal. In particular, GWAS with longitudinal data showed that rs3025047 is significantly associated with DBP even though it is not significantly associated in GWAS with cross-sectional data. The MAF of rs3025047 is 0.01, so it is a variant with relatively low frequency. In the HEXA cohort, rs3025047 was not available, nor were any SNPs in linkage disequilibrium with it. Even though further studies are necessary to confirm whether rs3025047 is a true causal variant, our analysis results illustrate that GWAS using longitudinal data can be an efficient strategy for rare variant association analysis.Furthermore, we performed GWAS with both longitudinal and cross-sectional data, and significant results from a longitudinal data analysis in the KARE cohort were further tested in the HEXA cohort. 12 SNPs that have not been reported elsewhere were identified, and the significant During the last decade, more than ten thousand GWAS successfully identified disease susceptibility loci, and these findings increase our understanding of diseases. However, the so-called missing heritability  reveals Analyzed as a repeated measure design, the power of longitudinal data is proportionally related to the number of observations on each individual and inversely related to the correlation between the multiple observations on an individual. This facilitates finding causal SNPs and their interactions with environmental variables, as well as with age and sex. In two Korean cohorts it enabled us to find 12 novel genome-wide significant SNPs associated with eight phenotypes, and significant gene × environment interaction. Therefore, we can conclude that longitudinal data seem to provide efficient strategies for GWAS.


The kernel score statistic is a global covariance component test over a set of genetic markers. It provides a flexible modeling framework and does not collapse marker information. We generalize the kernel score statistic to allow for familial dependencies and to adjust for random confounder effects. With this extension, we adjust our analysis of real and simulated baseline systolic blood pressure for polygenic familial background. We find that the kernel score test gains appreciably in power through the use of sequencing compared to tag-single-nucleotide polymorphisms for very rare single nucleotide polymorphisms with <1% minor allele frequency. This is appealing if the loci carry rare variants or are thought to belong to a specific gene set, such as a pathway or genes with similar functions. The kernel score statistic is a specific type of such a global covariate-adjusted multilocus association test. Depending on context, it has received different names, such as SKAT (sequence kernel association test) . Kernel Until very recently, kernel methods were only applied to independent data (see Ref.  for a reMAP4). From group discussions, we knew that MAP4 associates with simulated SBP and that simulated trait Q1 mimics the segregation of genes within families, but otherwise represents the genetic null hypothesis.GAW18 provided blood pressure trait data from Mexican Americans participating in the San Antonio Family Heart Study or the San Antonio Family Diabetes/Gallbladder Study, and in the Type 2 Diabetes Genetic Exploration by Next-generation sequencing in Ethnic Samples (T2D-GENES) Project 2 study . The sami = F−1((ri-3/8)/(nobs+1/4)). F−1 denotes the standard normal distribution quantile; nobs the number of observed SBP values; and ri their respective rank. Individual traits Yi for i = 1, ..., n subjects were modeled to depend on fixed effects b of covariates Xi , on random effects c for shared familial polygenic background, and on a semiparametric model h(G) of a genetic marker set G.We analysed baseline SBP, which has a right-skewed distribution toward large extremes. This was amended by a priori rank-normalization (Blom transformation ) of SBP Xi and Zi represent the subject rows of respective design matrices X, Z for fixed covariate effects and random family effects. The semiparametric model h(G) can be written as h=KaT with n × n dimensional kernel matrix K and multivariate normally distributed random effects estimates a~N [0: τ = 0. Most conveniently, it can be computed based on restricted maximum likelihood parameter estimates of the genetic null modelVectors ~N . The kerob, random pedigree effects oc, polygenic component variance s2fam and residual variance s2, e~N. Extending the kernel score statistic to adjust for familial dependencies, we obtain test statisticwhich estimates only fixed covariate effects Vo = s2I+s2famZZT. With fixed effects estimate o=b(XTVo1−X)1−XTVo1−Y, we rewrite test statistic (3) as quadratic form Q=RTMR of standard normally distributed residuals R= Po1/2Y with matrix M=(Po1/2K Po1/2)/2 and null projection matrix o=Vo−1 Po−1X(XTVo−1X)−1XTVo−1- V. The p values for test statistic (3) can be calculated by the Davies exact method [M. We adjusted for polygenic familial background based on the kinship coefficient matrix Φkin=ZZT using R-packages kinship2 and coxme with R-function lmekin for genetic null model (2). We employed a linear kernel on allele dosages. This allows for additive genetic models on tested markers. The kernel matrix entry Kij=giTgj for 2 subjects i, j is the scalar product of their subject-specific vectors gi, gj of allele dosages for NSNP SNP marker. SNPs were included for testing if their Hardy-Weinberg equilibrium test p values ≥10−5 (rounding imputed dosages for this purpose only). We contrasted test performance for the same subjects on 2 SNP panels, sequence  versus GWAS .with matrix t method  with theMAP4 SNPs. Simulated trait Q1 was used to test the quality of polygenic adjustment. We analyzed Q1 analogously to SBP on gene MAP4 and obtained p values ≤0.05 with rate 0.02  >5%), rate 0.05 (for MAF 1% to 5%), and 0.06 (for MAF <1%). This is within 95% tolerance limits  for true value 0.05 on 200 replicates.Permutation testing demonstrated overall test validity: 50,000 permutations of subject-genotype assignment (permuting across families) yielded type I error 0.049 ± 0.0019, 0.00046 ± 0.00019 at significance level 0.05, 0.0005 for simulated SBP on all sequenced rare MAP4 was detected with 100%, 62.5%, and 21.5% power at significance levels 0.05, 10−4, and 10−8 on 200 study replicates when subjecting the MAP4 sequence to the global test. Power diminished by only 0% to 2.5% when using the GWAS panel instead, or only MAF >5% SNPs. Figure MAP4 association based on MAF ≤5% SNPs. We varied the significance level to compare test performance on the sequence (filled circles) with the considerably sparser GWAS panel (open circles). Sequenced data are powerful and outperform GWAS data particularly for very rare SNPs (MAF <1%) where GWAS had no power . GWAS outperformed sequence data on SNPs with MAF 1% to 5%  with just 11 tag-SNPs compared to 105 sequenced SNPs.For simulated SBP, associated gene P values do not withstand Bonferroni correction. Note, however, the difference between sequence (p = 0.150) and GWAS (p = 0.031) for MECOM SNPs with MAF 1% to 5%. For AGTR1, sequence data tend to reduce p values compared to GWAS, as well as for SLC4A7 on rare variants. We partitioned the large MECOM, ULK4 sequences based on number of SNPs, as we expect that power improves by changing from a situation where we test over many more SNPs compared to number of subjects (full gene) to a situation with fewer SNPs than subjects (gene parts).Table P values for genes are dominated by their common SNPs , without type I error inflation. Ignoring familial dependency would inflate type I error to 16%, 6% at nominal 0.05, 0.01 levels (applying the independent subjects test statistic [MAP4 on polygenic trait Q1). We found that sequencing can improve test power appreciably, particularly for MAF <1% variants . For more frequent SNPs, the comparison was not decisive and tag-SNPs proved to be strong competitors. With sequencing, even a single, large gene may provide more SNP variants than available study subjects. The kernel score test remains valid because of the Bayesian nature of the underlying nonparametric genetic model [p ≤0.05 on 10% of 200 study replicates for all MAP4 SNPs, inflation increases with decreasing MAF). This, in turn, would yield similar but overly optimistic estimates for Figure AGTR1 p values in Table The kernel score statistic is a global covariance component test. tatistic  to familThe authors declare that they have no competing interests.DM designed the study concept, developed the statistical method, conducted data extraction, all statistical analyses and data interpretation, and drafted the manuscript. SF and AR provided the SNP mapping to positions (NCBI build 37) and genes. HB contributed throughout with discussions. All authors read and approved the final manuscript.