CASR gene region is 0.044 . In contrast, using our method to account for the correlation between variants, the corresponding estimate using 17 genetic variants is 0.022 , a more clearly positive causal effect.Finding individual-level data for adequately-powered Mendelian randomization analyses may be problematic. As publicly-available summarized data on genetic associations with disease outcomes from large consortia are becoming more abundant, use of published data is an attractive analysis strategy for obtaining precise estimates of the causal effects of risk factors on outcomes. We detail the necessary steps for conducting Mendelian randomization investigations using published data, and present novel statistical methods for combining data on the associations of multiple (correlated or uncorrelated) genetic variants with the risk factor and outcome into a single causal effect estimate. A two-sample analysis strategy may be employed, in which evidence on the gene-risk factor and gene-outcome associations are taken from different data sources. These approaches allow the efficient identification of risk factors that are suitable targets for clinical intervention from published data, although the ability to assess the assumptions necessary for causal inference is diminished. Methods and guidance are illustrated using the example of the causal effect of serum calcium levels on fasting glucose concentrations. The estimated causal effect of a 1 standard deviation (0.13 mmol/L) increase in calcium levels on fasting glucose (mM) using a single lead variant from the The online version of this article (doi:10.1007/s10654-015-0011-z) contains supplementary material, which is available to authorized users. Mendelian randomization is a technique which uses genetic variants to assess whether a risk factor, such as a biomarker, has a causal effect on a disease outcome in a non-experimental  setting , 2. We aMendelian randomization is a valuable approach for identifying risk factors as potential targets for clinical or behavioural intervention . EvidencThe steps involved in a Mendelian randomization investigation are: 1) specification of the dataset(s) for analysis, (2) search for candidate instrumental variables, (3) validation of the instrumental variable assumptions, (4) estimation of the causal effect (if appropriate), (5) supplementary and sensitivity analyses. A schematic diagram of the relevant components in a Mendelian randomization analysis is given in Fig.  specificTraditionally, Mendelian randomization analyses have been performed on a single study or studies containing data on genetic variants, and both the risk factor and outcome of interest. The main advantages of using published data rather than individual-level data are their size and scope. The associations of these variants with the risk factor and outcome in large consortia are likely to be more precisely estimated than in a single study. However, it is unlikely that published data on the genetic associations with the risk factor, with the outcome, and with potential confounders are available on the same set of studies.Two-sample Mendelian randomization is a strategy in which evidence on the associations of genetic variants with the risk factor and with the outcome comes from non-overlapping data sources . The limIn any case, it is likely that the sets of individuals used in the gene-risk factor and gene-outcome arms of the analysis will not be identical. An important assumption to ensure the validity of the analysis is that the two sets represent samples taken from the same underlying population. If this is not the case, then inferences may be misleading, as the association of the genetic variants with the risk factor may not be replicated in the set of individuals in which the association with the outcome is estimated, or a variant may not be a valid instrumental variable in both sets.Genetic variants are sought which are associated with the risk factor of interest. These can be obtained from available individual-level data or from the catalogues of genetic variants identified by genome-wide association studies (GWAS) that have been compiled . It is iIn two-sample Mendelian randomization, any bias from weak instruments  is in the direction of the null , so the The instrumental variable assumptions for a genetic variant, or set of variants, are vitally important to the validity of any Mendelian randomization investigation. However, the assumptions are not all empirically testable. This means that, while the assumptions should be interrogated as far as possible, they cannot be entirely verified and must be justified as much by biological understanding as they are by statistical testing.the variant must be associated with the risk factor of interest;the variant must be independent of confounders of the risk factor–outcome association;the variant can only affect the outcome through the risk factor—if the value of the genetic variant changes, but not that of the risk factor, then the outcome is unchanged .With regard to biological understanding, if the function of the gene in which the variant is located is known, this may give a clue as to whether the variant is a plausible instrumental variable. For example, variants in the CRP gene are likely to be valid instrumental variables for CRP. However, few genetic variants discovered in GWAS investigations are located within coding regions or have functional follow-up ascribing their association to a particular gene, and so the functional relationship between a variant and the risk factor may not be clear.The assumptions necessary for a genetic variant to be a valid instrumental variable are:With regard to statistical testing, the simplest and perhaps most effective way of assessing the instrumental variable assumptions is to test the association of the candidate genetic variants with a range of covariates which are potential confounders using individual-level data. While there is no way of testing the association of the variants with unknown or unmeasured confounders, for several diseases many of the covariates having the strongest association with the outcome  are known and often measured in epidemiological studies. Associations with several covariates can also be assessed from the literature, for example by searching for associations of the variants in a GWAS catalogue . HoweverOne difficulty with this assessment of the instrumental variable assumptions is the problem of multiple testing. If there are many covariates and multiple genetic variants, then a hypothesis testing approach that accounts for the multiple comparisons may lead to a lack of power to detect any specific association. Additionally, as several covariates (or the genetic variants) may be correlated, a simple Bonferroni correction may be an over-correction. A second difficulty is that genetic variants can be associated with a covariate without violating the instrumental variable assumptions. If, for example, a genetic variant which is a candidate instrumental variable for body mass index (BMI) is also associated with blood pressure levels, this may be due to the causal effect of BMI on blood pressure and not due to a pleiotropic effect of the variant (pleiotropy means that a variant has multiple effects). If the genetic association with a covariate is entirely mediated through the risk factor of interest, then the instrumental variable assumptions are not violated. In this case, taking the example above, the coefficient in the regression of blood pressure on the genetic variant should be substantially attenuated on adjustment for BMI. However, attenuation may not be complete, due to possible measurement error in the intermediate variable , and as the genetic variant is not independent of blood pressure conditional on BMI due to the presence of confounding factors between BMI and blood pressure .p value for the association of each variant with a covariate is greater than a pre-specified level (say p > 0.01), and a ‘liberal’ set, for which the minimum p value for each variant is greater than the Bonferroni corrected p value (V is the number of covariates tested). If this approach is followed, to minimize the possibility of bias due to pleiotropy, the Mendelian randomization estimate using the ‘conservative’ set of variants should be regarded as the primary analysis and the estimate using the ‘liberal’ set as the secondary analysis.A practical way to proceed is to specify two sets of genetic variants to be used as instrumental variables: a ‘conservative’ set, for which the minimum Other violations of the instrumental variable assumptions, such as population stratification, are more difficult to test using only summarized data. This particular issue is discussed in the Web Appendix in the context of the applied example.p values) are available for the genetic associations with the risk factor and with the outcome. Initially we assume that the scenario is two-sample Mendelian randomization and all the genetic variants considered are uncorrelated (in linkage equilibrium). These assumptions are later relaxed.We assume that estimates and standard errors  . The estThe estimate from the likelihood-based method .If the genetic variants are correlated, then estimates from the inverse-variance weighted method will overstate precision. If estimates are available of the correlations between variants, then the likelihood-based method can be modified by assuming a multivariate normal distribution for the genetic associations with the risk factor Again, if the data sources for the association estimates are overlapping then a joint normal model for the genetic associations In addition to the primary analysis to estimate the causal effect of the risk factor on the outcome, a number of additional analyses can be performed, which fall into the categories of supplementary or sensitivity analyses.If there are multiple mechanisms by which the risk factor may affect the outcome, and if genetic variants can be categorized as relating to one or other of these mechanisms, then separate Mendelian randomization estimates can be obtained using each category of variants. For example, variants may be associated with BMI by various mechanisms, such as suppressing appetite or increasing metabolic rate. A Mendelian randomization estimate constructed using variants associated with BMI through appetite suppression more closely represents the causal effect of intervening on BMI via appetite suppression. Differences in the causal estimates using genetic variants associated with different mechanisms may be informative in understanding the aetiology of the disease, and may highlight specific mechanisms to prioritize for pharmacological intervention.If there are variants whose status as instrumental variables is uncertain, then sensitivity analyses can be performed using a more conservative and a more liberal set of genetic variants, as described in step 3. Additionally, if there is no pleiotropy and the effects of the risk factor on the outcome associated with changes in the genetic variants are homogeneous for all variants, the genetic association estimates with the risk factor and with the outcome should follow a linear relationship passing through the origin. By plotting the genetic association estimates with the risk factor and with the outcome, any points which are not compatible with a straight-line through the origin  can be investigated for potential pleiotropy of the variants or for heterogeneity of the causal effect (perhaps due to different mechanisms of association with the risk factor).A formal test for heterogeneity is known as an overidentification test . ExampleCalcium is the most abundant mineral in the body, with a wide range of vital functions in human biology, including bone development and maintenance, muscle contraction, neurotransmitter release, and exocytosis. Indeed, insulin secretion is a calcium dependent process , and totwww.magicinvestigators.org . The allele score based on variants from the SR locus . HoweverCASR gene region. The restriction to a single genetic region means that the causal estimate is likely to apply only to a single mechanism by which calcium levels affect fasting glucose, and therefore may not be generalizable to other mechanisms. However, as the genetic region has a plausible mechanistic association with calcium levels, it is more likely to be a valid causal estimate than one based on variants from many genetic regions with unknown functional relevance to calcium levels and clear evidence of pleiotropy.We proceed to consider causal estimation only using the genetic variants in and around the The genetic associations with calcium levels and with fasting glucose are displayed in Fig. ent bias ; hence tdf), df, Parameters in the likelihood based model  were estWe conclude from this example that there is evidence that increases in calcium levels lead to increases in fasting glucose. The lack of availability of data on important covariates (in particular vitamin D levels), the potential for bias by population stratification, and the reliance on genetic variants from a single region mean that the evidence that intervening to lower serum calcium levels would decrease fasting glucose concentrations is suggestive, but not conclusive.In this discussion, we highlight some extensions of the approach discussed in this paper, as well as issues in its implementation and interpretation.In some cases, genetic variants are associated with several related risk factors, such as multiple lipid fractions ) in such a way that it is not possible to find variants specifically associated with each risk factor which are not associated with the related risk factors . By consIf the data on the genetic associations in a Mendelian randomization investigation are taken from multiple studies, then the association estimates may represent pooled estimates from a meta-analysis, as with the data on gene-outcome associations in the example of this paper. If the individual-level or summarized data are available at a study level, then these can be incorporated into the analysis using hierarchical models, as has been previously proposed for the analysis of individual-level data . This caIn a hierarchy of evidence, Mendelian randomization investigations have been advocated as providing “critical evidence” on risk factor–outcome relationships . HoweverIn conclusion, we have here explained why Mendelian randomization is a useful approach for the assessment of risk factors as potential targets for clinical intervention. We have demonstrated how published data enable efficient analysis strategies for Mendelian randomization experiments. This is a timely development in view of the increasing public availability of genetic association estimates in large datasets. The efficiency of these analyses can be improved by using multiple variants in each gene region, but correlation between the variants must be accounted for.Supplementary material 1 (pdf 149 KB)Below is the link to the electronic supplementary material.


The study also aimed to identify QTLs for the four trait groups as composite traits using multivariate least square interval mapping (MLSIM) to further understand the genetic control of these traits.Interaction and genetic control for traits influencing the adaptation of the rice crop to varying environments was studied in a mapping population derived from parents (Moroberekan and Swarna) contrasting for drought tolerance, yield potential, lodging resistance, and adaptation to dry direct seeding. A BCqDTY3.2 locus. These included QTLs for traits such as bleeding rate, shoot biomass, stem strength, and spikelet fertility. Multivariate QTLs were identified at loci supported by univariate QTLs such as on chromosomes 3 and 4 as well as at distinctly different loci on chromosome 8 which were undetected through CIM.Significant correlations between drought- and yield-related traits at seedling and reproductive stages respectively with traits for adaptation to dry direct-seeded conditions were observed. CIM and MLSIM methods were applied to identify QTLs for univariate and composite traits. QTL clusters showing alignment of QTLs for several traits within and across trait groups were detected at chromosomes 3, 4, and 7 through CIM. The largest number of QTLs related to traits belonging to all four trait groups were identified on chromosome 3 close to the Rice requires better adaptation across a wide range of environments and cultivation practices to adjust to climate change. Understanding the genetics and trade-offs related to each of these environments and cultivation practices thus becomes highly important to develop varieties with stability of yield across them. This study provides a wider picture of the genetics and physiology of adaptation of rice to wide range of environments. With a complete understanding of the processes and relationships between traits and trait groups, marker-assisted breeding can be used more efficiently to develop plant types that can combine all or most of the beneficial traits and show high stability across environments, ecosystems, and cultivation practices.The online version of this article (doi:10.1186/s12863-015-0249-1) contains supplementary material, which is available to authorized users. Rice growing environments are highly diverse and are affected by fluctuations in environmental conditions during crop growth. Water shortages due to climate change, increased competition for fresh water from industries and domestic usage, and increasing labor and fertilizer costs threaten the sustainability of the transplanted system of rice cultivation –3. With To adapt to such variable conditions, ideally the plant should possess a combination of morpho-physiological traits such as high yield potential, early and uniform emergence, better weed competitiveness, and lodging resistance . Apart fThe cultivars Moroberekan and Swarna showed high contrast for plant type and other key traits that determine performance in terms of the four trait groups considered in this study Fig. . AdditioThe phenotypic variation among the parents was transferred to the progeny and significant genetic variation for a large proportion of traits studied under each trait group was observed. Significant differences among the progeny for 64–100 % of the traits were observed for the different trait groups Fig. . AdditioCorrelations among the traits belonging to the four different trait groups are presented in Additional file qDTY3.2 region. For the drought tolerance trait group, QTLs were seen for traits related to drought and grain yield. However, higher numbers of QTLs were identified for drought-related traits as compared to yield-related traits. QTL clusters were observed at chromosome 3 at the qDTY3.2 region, including a QTL for grain yield under drought. However, the yield-enhancing allele in this case came from the susceptible parent. While this QTL is known to affect the flowering time along with its effect on grain yield under drought, the staggered seeding of the progeny from different maturity groups may explain Swarna’s contribution of the yield-enhancing allele at this locus. The advantage of having the Moroberekan allele at this locus can be seen through its effect on several other drought-related traits affecting plant function , and (2) those influencing few specific traits (such as MVQTL2.1 for drought and MVQTL2.2 for lodging resistance). While class 1 MVQTLs were observed for the trait groups on drought tolerance and lodging resistance, class 2 MVQTLs were observed for the trait groups on yield potential and adaptation to direct seeding.Unlike CIM, MLSIM allowed the identification of QTLs for composite traits representing a group of individual traits such as drought tolerance, yield potential, lodging resistance, and adaptation to direct seeding. A total of 3, 15, 8, and 12 multivariate QTLs (MVQTLs) were identified for the four trait groups, respectively , Los Baños, Laguna, Philippines  in the dry season (DS) and wet season (WS) of 2013 , depending on the maturity group , yield potential (9 traits), lodging resistance (8 traits), and adaptation to direct seeding (22 traits) were recorded .X) of the following equation:Shoot growth and groundcover dynamics were monitored according to NDVI measured around mid-day using a Greenseeker Hand-held Sensor . Canopy temperature was measured at three locations per plot using a hand-held data-logging infrared (IR) sensor  after stress initiation. The increase in canopy temperature throughout the stress period was calculated as the slope  divided by the leaf dry weight. Days to flowering (DTF) was recorded when about 50 % of the plants in the plot had flowered. Plant height (PH) of three plants from each plot was measured at maturity from ground level to the tip of the tallest tiller and averaged to get the mean PH for analysis. At physiological maturity, three hills were sampled in each plot for the measurements of the yield components including number of tillers and panicle, panicle length (cm), spikelet fertility (%), 1000-grain weight (gm) and rachis-stem-leaf dry weight (gm). Grain yield was measured from a sampled area of 1.5 m2 and dried to 14 % moisture content. The weight of grains was then used to calculate the kg ha−1 yield for each plot for further analysis.Root samples were taken 2–3 days after re-watering following severe stress symptoms at the grain-filling stage using a 4 cm-diameter core sampler  to a depth of 60 cm. Soil cores were divided into 15-cm segments, and roots were washed by repeatedly mixing the soil with water in a container, and pouring the root-water suspension over a 1-mm plastic sieve. Only roots identified as living rice roots were retained for measurement. All samples were dried and weighed. Root mass density was calculated as the root mass per 15-cm soil core segment divided by the volume of the soil core segment. Percentage of deep roots was calculated as the mass of roots in the core below 30 cm divided by the total mass of roots inside the core. Bleeding rate measurements were carried out as described by Morita and Abe  and meas−1) and yield-related traits such as number of panicles and tillers at harvest, spikelet fertility percentage (by weight), panicle length (cm), biomass (kg ha−1), DTF and plant height (cm) were classified in Trait group 2: Yield potential. These traits were recorded under lowland well-watered conditions (Experiment 2). The harvested area for grain yield was 1.5 m2.Grain yield . These included stem strength, stem thickness, and fresh and dry weight per plant. Stem strength was measured using the prostrate tester . All data were recorded from three plants from each plot. At physiological maturity, plants were cut off at 40 cm height, with the prostrate tester set perpendicularly at the middle (20 cm), and the pushing resistance of the lower part of the plant was measured by pushing the plants to the point at which the stem broke and the scale displacement (mm) due to pushing resistance was recorded. The stem diameter was measured from the same three plants at a height of 40 cm using a screw gauge. The plants were then harvested from the base to measure fresh weight per plant and then oven dried for three days at 70 °C and weighed to estimate the dry weight per plant. Average values for all parameters were calculated and used for further analysis.Visual observations of the time (DAS) to first emergence (when the first seedlings of a plot emerged) and full emergence (most of the seedlings in each plot had emerged) were recorded in the seed bed nursery of Experiment 1 and of the direct-seeded experiments (3 and 4). Shoots from five seedlings per plot were sampled at a two-week interval (18 DAS and 32 DAS) to determine the relative growth rate (RGR) in Experiment 2. Shoots were dried and weighed to determine the biomass for each sampling date and the RGR was calculated as:B2 is the shoot biomass on date 2, B1 is the shoot biomass on date 1, D2 is date 2 and D1 is date 1.Where, 2, respectively.In addition, grain yield and traits related to yield potential and phenology under direct-seeded conditions such as DTF, panicle number at harvest, tiller number at harvest, spikelet fertility, 1000-grain weight, and panicle length were recorded in Experiment 3. The harvested area for grain yield in Experiment 3 and 4 was 1 and 0.25 mStatistical analysis for the computation of means and standard error of difference (SED) were conducted using CROPSTAT version 7.2.3. A mixed model analysis of data from individual years was carried out using the model:ijk is the measurement recorded in a plot, μ is the overall mean, gi is the effect of the ith genotype, rj is the effect of the jth replicate, bk(rj) is the effect of the kth block within the jth replicate, and eijk is the error. Genotypic effects were considered fixed and the replicates and block effects were random.where yCorrelations between the traits were estimated using the ‘cor’ function in R 3.1.0 . For bethttps://www.integratedbreeding.net/482/communities/genomics-crop-info/crop-information/gcp-kaspar-snpmarkers). A total of 2015 SNP markers were screened for polymorphism between the two parents. Out of these 2015 SNPs, 591 polymorphic SNP loci were identified. The genotypic data from a set of 193 polymorphic SNP markers was used to generate the genotypic profile of the population.Fresh leaves for all lines were collected and freeze-dried. DNA was extracted from the freeze-dried leaf samples by a modified CTAB method in deep-well plates. The DNA was then quantified and purified and lines were genotyped using KASPar SNP assays. These SNPs were selected as subsets from the set of 1536 and 44 K SNP chips , 35, conComposite interval mapping (CIM) was conducted using QTL Network 2.1  based onThe data sets supporting the results of this article are included within the article and its additional files.


Estimates of dominance variance in dairy cattle based on pedigree data vary considerably across traits and amount to up to 50% of the total genetic variance for conformation traits and up to 43% for milk production traits. Using bovine SNP (single nucleotide polymorphism) genotypes, dominance variance can be estimated both at the marker level and at the animal level using genomic dominance effect relationship matrices. Yield deviations of high-density genotyped Fleckvieh cows were used to assess cross-validation accuracy of genomic predictions with additive and dominance models. The potential use of dominance variance in planned matings was also investigated.Variance components of nine milk production and conformation traits were estimated with additive and dominance models using yield deviations of 1996 Fleckvieh cows and ranged from 3.3% to 50.5% of the total genetic variance. REML and Gibbs sampling estimates showed good concordance. Although standard errors of estimates of dominance variance were rather large, estimates of dominance variance for milk, fat and protein yields, somatic cell score and milkability were significantly different from 0. Cross-validation accuracy of predicted breeding values was higher with genomic models than with the pedigree model. Inclusion of dominance effects did not increase the accuracy of the predicted breeding and total genetic values. Additive and dominance SNP effects for milk yield and protein yield were estimated with a BLUP (best linear unbiased prediction) model and used to calculate expectations of breeding values and total genetic values for putative offspring. Selection on total genetic value instead of breeding value would result in a larger expected total genetic superiority in progeny, i.e. 14.8% for milk yield and 27.8% for protein yield and reduce the expected additive genetic gain only by 4.5% for milk yield and 2.6% for protein yield.Estimated dominance variance was substantial for most of the analyzed traits. Due to small dominance effect relationships between cows, predictions of individual dominance deviations were very inaccurate and including dominance in the model did not improve prediction accuracy in the cross-validation study. Exploitation of dominance variance in assortative matings was promising and did not appear to severely compromise additive genetic gain. The dominance deviation for a given genotype at the locus is the difference between genotypic value and breeding value, and is equal to -2q2d, 2pqd and -2p2d for genotypes A1A1, A1A2 and A2A2, respectively [Dominance arises when the effects of alleles at a locus are not only additive, but interact so that the value of the heterozygous genotypes deviates from the mean of the values of the homozygous genotypes. With Until recently, studies on dominance deviations were sparse because without genomic information, the availability of large datasets with sufficient proportions of individuals with non-null dominance effect relationships, such as full-sibs, is essential for accurate estimation of dominance variance. Estimates of dominance variance in dairy cattle that are based on pedigree data range from 7.3% to 49.8% of the total genetic variance for conformation traits ,3 and frAt the individual animal level, dominance is hardly used in animal breeding , althougWith the availability of SNP genotypes, dominance at a marker locus can be readily determined, dominance effects of markers can be estimated ,10 and cIn this work, we explored the possibilities of including dominance effects in genomic evaluation and furthermore in planned matings in dairy cattle. We estimated variance components, including dominance variance, in a dataset of genotyped Bavarian Fleckvieh cows, analyzed the predictions of breeding and total genetic values using cross-validation, and predicted total genetic values of specific matings.P < 10-5) from the Hardy-Weinberg equilibrium, and SNPs that were not annotated (UMD3) on the autosomes or on the pseudo-autosomal region of the X-chromosome were excluded from the analysis. A total of 629 028 SNPs remained in the dataset after editing. High-density SNP genotypes and yield deviations (YD) for nine traits  from 1996 Bavarian Fleckvieh cows were available to (a) estimate variance components, including dominance variance and (b) perform cross-validation in order to evaluate the predictive ability of a model with dominance effects in comparison to a purely additive model. Both studies were done within a GBLUP framework. YD were calculated based on test-day observations adjusted for non-genetic effects, but not for permanent environmental effects, for each lactation and interpolated by the method of best prediction [First-lactating cows from 145 Bavarian dairy herds , born in 2008 and 2009, were genotyped with the Illumina BovineHD Genotyping BeadChip that includes 777 962 SNPs. SNPs with a call rate lower than 0.9, a minor allele frequency higher than 0.005 and a highly significant deviation  by the number of loci (m), with elements that are equal to 2-2pk and -2pk for opposite homozygous and 1-2pk for heterozygous genotypes, pk is the minor allele frequency of locus k, and qk =1-pk. Matrix G* was scaled so that the means of diagonals and off-diagonals are the same as in A[A to G = 0.95 G* + 0.05 A in order to improve numerical stability. The variance matrix of residual effects was F is a diagonal matrix with reciprocals of the EOP as weights. Extending model MG with dominance effects leads to model MGD:where matrix e as in A,19 and tv is a vector of dominance deviations of cows. V(u) and V(e) are defined as in model MG. The covariance matrix of dominance effects is D is the genomic dominance relationship matrix and D* was calculated as:where Wd has dimensions of the number of individuals (n) by the number of loci (m), with elements that are equal to 1A1, 2pkqk for genotype A1A2, and 2A2. Matrix D* was then combined with the identity matrix I as D = 0.95 D* + 0.05 I to improve numerical stability.where 2-distributions with 0 and 1 degree of freedom [th sample included in the posterior analysis. Convergence to the final distribution was checked with the Geweke diagnostics [Estimation of variance components was performed with REMLF90 . Goodnes freedom . In addi freedom  in ordergnostics  of the Rgnostics ,24.Additive and dominance variance components at the marker level  and total genetic values (g), that include dominance deviations, were calculated for the offspring of all possible matings between 1996 cows and 50 bulls for milk yield and protein yield. The bulls were genotyped and selected for the respective trait on their conventional breeding value after progeny test (including the records of 1996 genotyped cows) from the German-Austrian genetic evaluation. SNP effects a and d were estimated in a BLUP model  using GS3. Variance components Genotype probabilities and expectations of purely additive breeding values  and D were equal to 0, which implies that the population was in Hardy-Weinberg equilibrium. The standard deviation of off-diagonals of G was equal to 0.036, which is five times larger than the standard deviation of off-diagonals of D, i.e. 0.007. The proportion of off-diagonals that were smaller than -0.05 or larger than 0.05 was 6.27% for G but only 0.02% for D. Therefore, matrix D was less informative than G.Figure Estimated variance components for model MGD are in Table 2-distributed [2-distribution function can take only non-negative values because it is defined as a sum of squared values. For two traits (stature and udder score), the likelihood ratio test statistics were negative (but very close to 0), which was due to numerical rounding or not finding the mode of the likelihood exactly.For all traits, model MG, which exploited genomic information, fitted the data better than model MA, which included pedigree information only. The superiority of model MGD, which included a dominance effect, compared to model MG was significant for milk yield, fat yield, protein yield, somatic cell score and milkability, based on the likelihood ratio test. Likelihood measures and statistics of the likelihood ratio test between models MG and MGD are in Table tributed . The χ2-Mean accuracies of predicted breeding values [For models MA and MG, For milk yield, 16 bulls were chosen as mating partners when matings were selected on Expected total genetic superiorities and additive genetic gains obtained with the selected matings are in Table k defined as (2pkqkd)2[This study analyzed the importance of dominance variation for several milk production and conformation traits in the Fleckvieh breed using the GBLUP methodology. Additive and dominance genomic relationship matrices were calculated similar to Su et al. , except (2pkqkd)2,12. Thisu and v is the classical treatment [u and v. However, this independence is contradictory with the phenomena of inbreeding depression and hybrid vigor; presence of inbreeding depression indicates that dominance is directional, e.g. [a priori dependencies between a and d (which would result in dependencies between u and v) and Bayesian regression models that could accommodate those dependencies. The treatment of dependencies between breeding values and dominance deviations is rather complex and the computational requirements are large, thus, we did not consider this method although it should be a field of further research.Independence between reatment  and it ial, e.g. . Wellmanal, e.g. ,29 revieEstimates of dominance variance varied from 3.3 to 50.5% of total genetic variance for the analyzed traits. Estimated dominance variance  was greater for milk production traits than for conformation traits. These results agree with those of Misztal et al. , who fouAlthough moderate changes in estimates of additive variance were observed between pedigree and genomic models, estimates of additive variance were consistent for genomic additive and dominance models, except for milkability. Su et al.  reportedG and D. However, in other studies the ratio between relative standard errors of dominance and additive variances was even larger, i.e. 4.1 in Misztal [Both Gibbs sampling with model MGD and at the marker level with the MGD-SNP model resulted in estimated variance components that were comparable with REML estimates for most traits. The relative standard error  of dominance variance was on average 2.7 times larger than the relative standard error of the estimated additive variance, which is expected based on the properties of  Misztal  and 4.5  Misztal . In ordeDespite the large estimates of dominance variance for most analyzed traits (significantly larger than 0 for five traits), prediction accuracy of breeding values and total genetic values did not change when dominance effects were included in the model. Estimates of additive variance did not differ much between models MG and MGD, which means that additive variance is already captured quite accurately in the additive model. Thus, additive effects are relatively well predicted, whether the dominance effect is modeled or not. The accuracy of predictions of total genetic values in cross-validation was not higher with the dominance than with the additive model because the proportion of full-sibs and dominance effect relationship coefficients between the training and validation datasets were small. Thus, little information was transferred from the reference to the validation group in cross-validation for prediction of dominance effects. Su et al. , who anaThe regression coefficient of YD on predicted breeding values was generally close to 1, with a few exceptions. With the dominance model, this regression coefficient was slightly closer to 1 for most traits but differences were small, which is similar to the data reported by Su et al. , i.e. 0.The results show that selection of matings on d Varona  found thd Varona  pre-seleEstimates of genomic variance due to dominance in Fleckvieh cattle ranged from 3 to 50% of the genetic variance and were within the range of published pedigree-based estimates for dairy cattle. The computational complexity and modeling were straightforward. Predictive ability of breeding and total genetic values by cross-validation was not improved when dominance effects were included in the prediction model, probably because of the limited size of the dataset and the small proportion of full-sibs. There is potential to exploit dominance variance in planned matings in order to increase total genetic value of the offspring (i.e. future performance) without compromising additive genetic gain. Use of planned matings could also be a way to motivate farmers that are otherwise not interested in using genomic breeding values for breeding schemes.The authors declare that they have no competing interests.JE performed the analysis and drafted the manuscript. JE, AL and KUG designed the study. AL, ZGV and LV developed methods. CE and RE prepared phenotypic and genotypic data. AL, ZGV, LV, CE, RE and KUG revised the manuscript. All authors read and approved the final manuscript.


Mycobacterium avium, subspecies paratuberculosis (MAP). Control of Johne's disease is based on programs of testing and culling animals positive for infection with MAP while concurrently modifying management to reduce the likelihood of infection. The current study is motivated by the hypothesis that genetic variation in host susceptibility to MAP infection can be dissected and quantifiable associations with genetic markers identified. For this purpose, a case-control, genome-wide association study was conducted using US Holstein cattle phenotyped for MAP infection using a serum ELISA and/or fecal culture test. Cases included cows positive for either serum ELISA, fecal culture or both. Controls consisted of animals negative for the serum ELISA test or both serum ELISA and fecal culture when both were available. Controls were matched by herd and proximal birth date with cases. A total of 856 cows (451 cases and 405 controls) were used in initial discovery analyses, and an additional 263 cows (159 cases and 104 controls) from the same herds were used as a validation data set. Data were analyzed in a single marker analysis controlling for relatedness of individuals (GRAMMAR-GC) and also in a Bayesian analysis in which multiple marker effects were estimated simultaneously (GenSel). For the latter, effects of non-overlapping 1 Mb marker windows across the genome were estimated. Results from the two discovery analyses were generally concordant; however, discovery results were generally not well supported in analysis of the validation data set. A combined analysis of discovery and validation data sets provided strongest support for SNPs and 1 Mb windows on chromosomes 1, 2, 6, 7, 17 and 29.Paratuberculosis, or Johne's disease, is a chronic, granulomatous, gastrointestinal tract disease of cattle and other ruminants caused by the bacterium  Mycobacterium avium subspecies paratuberculosis (MAP). The clinical signs of disease in cattle are pipestream diarrhea, weight loss, edema due to hypoproteinemia caused by protein-losing enteropathy MAP test-positive cows produce less milk and are culled earlier in their productive life Paratuberculosis, or Johne's disease, is a chronic, granulomatous, gastrointestinal tract disease of cattle and other ruminants caused by the bacterium MAP infection varies between European countries and is greater than 60% in some regions according to a recent review MAP in Crohn's disease in humans, which at the current time remains uncertain The disease occurs worldwide in dairy cattle and other ruminants. Control programs for paratuberculosis have been established in some nations including Australia MAP infection can contribute to disease control programs by facilitating genetic selection for a less susceptible population to reduce incidence of infection in the future. The opportunity for genetic improvement in susceptibility to infection is evidenced by estimates of heritability of MAP infection in dairy cattle ranging from 0.03 to 0.28 MAP infection of their daughters There currently is no cure for Johne's disease, and vaccination is problematic. Routine testing combined with culling currently provide the best opportunity for controlling the disease. Knowledge concerning genetics of susceptibility to MAP infection in Holstein cattle and develop tools for genomic selection for lesser susceptibility to MAP infection.Genetic factors affect susceptibility to common diseases, and genome wide association studies (GWAS) provide a tool for identifying these genetic factors The University of Wisconsin-Madison College of Agricultural and Life Sciences Animal Care and Use Committee approved this research.MAP. Population 1 consisted primarily of twelve paternal half-sib families of daughters of sires heavily used within the US Holstein population. Collection of these samples has been described previously . Samples were obtained mainly during 2001 to 2003 from 300 herds from across the US with Wisconsin and California herds together accounting for approximately 48% of the total samples. Population 2 consisted of cows from six commercial Holstein herds in Wisconsin that were cooperators in a Johne's disease control project. Blood samples for disease testing and DNA extraction were obtained from all cows in these herds over a period of 15 months in 2006-7 Two resource populations of approximately 5,000 Holstein cows each were used to identify genomic regions associated with susceptibility to infection by MAP infection in Population 1 was based on both fecal culture of MAP and evidence of antibody titer to MAP as based on a serum ELISA test. Fecal culture for MAP was measured over 12 weeks using a radiometric BACTEC method Phenotype for −6) or with minor allele frequencies below 0.02 were also not included in statistical analyses. After exclusion for these various reasons, a total of 33,484 SNPs remained.Samples from both populations were genotyped with the Illumina Bovine SNP50 BeadChip for discovery analysis and Illumina BovineLD BeadChip for validation analysis. A combined total of 856 of 890 samples (451 cases and 405 controls) from Populations 1 and 2 were successfully genotyped by either Illumina, Inc. , GeneSeek Inc.  or the University of Wisconsin Biotechnology Center (UWBC). These samples are referred to herein as the discovery data. The initial genotyping by Illumina and GeneSeek MAP infection based on ELISA test results and fecal culture test results if available) were genotyped by DNA Landmarks, Inc. , using the BovineLD BeadChip. These samples plus 160 samples from Population 1 and 2  previously genotyped with the SNP50 BeadChip and masked for all but the 6,844 SNPs in common between the 50 K and LD BeadChips constituted the validation data set.For validation analyses, 103 samples from Populations 1 and 2  as implemented within the GenABEL package MAP infection was discrete, and consequently the data were analyzed using a categorical threshold analysis in GenSel. Individual SNP effects were estimated from a mixture model with the value of π set at 0.999, meaning that one out of a thousand SNPs were included in the model in any particular iteration of the Bayesian analysis and given a non-zero effect estimate. This high π value has been shown to give faster convergence in the model averaging procedures A second GWAS analysis was implemented with a Bayes C model averaging approach using the GenSel program Effects of an underlying (ungenotyped) causative locus may be divided between multiple SNP loci in the Bayesian analysis. To more effectively identify genomic regions harboring quantitative trait loci (QTL), association of groups of consecutive SNPs in 1 Mb windows were analyzed using the windowsBV option in GenSel. Windows were defined as spanning consecutive, non-overlapping 1 Mb regions across the genome. Genetic variance for a window as a proportion of total genetic variance across the genome was used to identify the most informative regions. Posterior probability of inclusion (PPI), which is the proportion of samples of the Markov chain in which a given SNP window was included in the model with a non-zero effect was calculated and those exceeding an arbitrary threshold of PPI>0.20 are reported. For both GenABLE and GenSel analyses, the analysis was applied to the 50 K discovery data, the validation data imputed to 50 K and combined discovery and validation data sets.The accuracy of genomic predictions was evaluated by pooling estimates using a 5-fold cross-validation strategy. Genotyped animals were divided randomly (within cases and controls) into five exclusive groups. In each training analysis, four of the five groups were combined to comprise a training data set to estimate marker effects using Bayes C analysis. Results from this analysis were then used to predict genomic values of individuals from the omitted group (testing set) using the GenSel program. Additionally, a sixth analysis was performed in which the discovery data comprised the training data set and the validation data comprised the testing data set.MAP infection with h2 = 0.10. A threshold was applied to achieve frequencies of cases corresponding to 10%. Probability of correct classification based on the known, simulated genetic effects was then calculated in the same manner as above for the real data with the simulation repeated 1,000 times.Prediction efficacy was evaluated by Receiver Operating Characteristic (ROC) analysis Regarding diagnostic evaluations, examination of a multi-dimensional scaling plot  for the MAP infection phenotype  are presented in −5 on chromosomes 15 and 7 , low accuracy of prediction for a single event is to be expected. Even with the simulation study in which genetic values were known with complete accuracy, probability of correct classification was 0.67. This is directly attributable to the low heritability for the trait, and the value of 0.67 realized in the simulation provides an upper bound for correct classification in this scenario. A more meaningful scenario would be a comparison of the genomic predictions of sires with their average progeny susceptibility based on large numbers (hundreds) of offspring.Results from the cross-validation analysis suggest low accuracy for the genomic prediction, but this conclusion must be tempered by the recognition that the comparison here is genomic prediction versus a realization of a single event, i.e. did an individual animal manifest evidence of http://www.genome.gov/gwastudies/). Comparing the corresponding bovine genomic locations of these Crohn's risk loci with results of the current study suggest two potential overlaps, both on BTA7. SNP ARS-BFGL-NGS-7756 at 71 Mb on BTA7 is approximately 2 Mb from interleukin 12B (IL12B), a subunit of cytokines (interleukins 12 and 23) that activate transcription activator STAT4 and stimulate production of interferon gamma. IL12B has been associated with Crohn's disease in three independent GWAS 2<0.6) were separated by a median distance of ∼1 Mb Johne's disease in cattle shares similar manifestations to Crohn's disease in humans, thus it is interesting to compare results of the current study with results from GWAS of Crohn's disease in humans. Individual studies and subsequent meta-analyses have identified 131 unique risk loci for Crohn's disease based on the tabulation found at “A Catalog of Published Genome-Wide Association Studies”  from a gene (IL12B) known to be associated with Crohn's disease in humans. However, additional potential candidate genes are in closer proximity to this region, specifically T-cell immunoglobulin and mucin domain containing 4 (TIMD4) and IL2-inducible T-cell kinase (ITK). TIMD4 is implicated in regulation of T-cell proliferation Figure S1Multi-dimensional scaling plot for the combined discovery and validation data sets. Case and control samples are indicated as two different colors. PC1 and PC2 are the first two principal components obtained from genomic kinship matrix. Distance between points represents the genetic distance between animals.(TIF)Click here for additional data file.Figure S2Quantile-quantile plot for results from the GRAMMAR-GC analysis of the combined discovery and validation data. The Y-axis represents observed P-values and the X-axis represents expected P-values under a null hypothesis  of no association.(TIF)Click here for additional data file.Dataset S1Genotype data for discovery and validation samples. There are 38,434 genotypes per individual; missing genotypes have been imputed.(ZIP)Click here for additional data file.


Me). The theoretical formula, the new proxy, and existing proxies were compared for simulated data, and the results point to the validity of our approach. The calculations were also illustrated on a new perennial ryegrass set  genotyped for 24,957 single nucleotide polymorphisms (SNPs). In this case, most of the proxies studied yielded similar results because of the lack of markers for coverage of the entire genome (2.7 Gb).Genomic selection is focused on prediction of breeding values of selection candidates by means of high density of markers. It relies on the assumption that all quantitative trait loci (QTLs) tend to be in strong linkage disequilibrium (LD) with at least one marker. In this context, we present theoretical results regarding the accuracy of genomic selection, i.e., the correlation between predicted and true breeding values. Typically, for individuals , breeding values are predicted by means of markers, using marker effects estimated by fitting a ridge regression model to a set of training individuals. We present a theoretical expression for the accuracy; this expression is suitable for any configurations of LD between QTLs and markers. We also introduce a new accuracy proxy that is free of the QTL parameters and easily computable; it outperforms the proxies suggested in the literature, in particular, those based on an estimated effective number of independent loci ( During the last decades, investigators have mainly concentrated on linkage analysis to detect the regions of DNA, so-called quantitative trait loci (QTLs), responsible for quantitative variation. The linkage analysis (LA) is specific because it relies on family data and on pedigrees: segregation of a QTL is studied within a family by means of information related to the family.In this context, the most popular statistical method for QTL mapping is interval mapping . It invoMore recently, researchers moved on to genome-wide association studies (GWAS) that are based on unrelated individuals, in contrast to LA. A GWAS allows researchers to analyze individuals without knowing their pedigree. One of the most popular methods relies on the model proposed by . HundredNonetheless, both approaches have a drawback: QTLs with very small effects are difficult to detect. Note that most traits of interest can be characterized as complex traits: they are presumably governed by a large number of small-effect QTLs –16. In aAt present, genomic selection (GS) is focused on prediction of breeding values of selection candidates by means of high density of markers. In contrast to LA and GWAS, the main goal of GS is not to detect QTLs anymore but to predict the future phenotype of young candidates as soon as their DNA has been collected. GS relies on the expectation that some QTLs will be in strong linkage disequilibrium (LD) with at least one marker . From a Me into the original formula of ).X′V−1Q that has an impact . This weighted LD can be viewed as an extension of the work of [With the help of our general  work of  where thnTRN and the effective number of segments Me were found to have a strong influence on the accuracy. Besides, a “big formula effect” was observed, and those authors were unable to demonstrate superiority of one method to the others. One criticism voiced by those authors was that the different formulas in the literature did not take into account the relation between TRN and TST populations. Our theoretical formula now involves explicitly the link between the two populations.Our present study can be viewed as an answer to the analysis of , where tThe Theoretical accuracy depends on unknown parameters such as the QTL effects and their locations. It may be useful to first perform an association study on the TRN population in order to identify the QTLs. After that, the detected QTLs could be plugged into our theoretical Me into the original formula of [Me, using either the effective population size , or, orMe inze  are correlated with forage productivity measured in a dense canopy . In than 1kb .X′V−1Q, which can be viewed as LD corrected for the relatedness. On the other hand, GBS can also be considered as limiting factor for the accuracy. Although 20,000 markers were obtained by GBS, the number of independent markers is actually much smaller . Recall that a large number of markers is required to capture the genetic variability from many QTLs in the genome of perennial ryegrass. Finally, the accuracy was found to be strongly affected by the configuration of TRN and TST sets: the same phenomenon was observed in other studies . In this specific population, LD was expected to decrease slowly due to relatedness. In that case, we thought that a relatively small number of markers would be sufficient to cover the genome if the LD was large enough. According to our study, the mean accuracy is approximately 0.35. This value is similar to the results obtained on rice  and whea on oats ).For all these reasons, in order to improve the accuracy of GS in perennial ryegrass, we propose to perform denser genotyping, and to select individuals to be phenotyped by minimizing the quantity S1 Text(PDF)Click here for additional data file.S2 Text(PDF)Click here for additional data file.S1 Table(PDF)Click here for additional data file.S1 Fig(PDF)Click here for additional data file.


Genetic pleiotropy refers to the situation in which a single gene influences multiple traits and so it is considered as a major factor that underlies genetic correlation among traits. To identify pleiotropy, an important focus in genome-wide association studies (GWAS) is on finding genetic variants that are simultaneously associated with multiple traits. On the other hand, longitudinal designs are often employed in many complex disease studies, such that, traits are measured repeatedly over time within the same subject. Performing genetic association analysis simultaneously on multiple longitudinal traits for detecting pleiotropic effects is interesting but challenging. In this paper, we propose a 2-step method for simultaneously testing the genetic association with multiple longitudinal traits. In the first step, a mixed effects model is used to analyze each longitudinal trait. We focus on estimation of the random effect that accounts for the subject-specific genetic contribution to the trait; fixed effects of other confounding covariates are also estimated. This first step enables separation of the genetic effect from other confounding effects for each subject and for each longitudinal trait. Then in the second step, we perform a simultaneous association test on multiple estimated random effects arising from multiple longitudinal traits. The proposed method can efficiently detect pleiotropic effects on multiple longitudinal traits and can flexibly handle traits of different data types such as quantitative, binary, or count data. We apply this method to analyze the 16th Genetic Analysis Workshop (GAW16) Framingham Heart Study (FHS) data. A simulation study is also conducted to validate this 2-step method and evaluate its performance. In genetics, the phenomenon that a single gene or locus influences more than one trait is known as pleiotropy. Genetic pleiotropy plays a crucial role in many complex diseases. One of the most well-known examples is the phenylketonuria (PKU) disease. The defect of a single gene supposed to code for enzyme phenylalanine hydroxyls results in multiple malfunctioned phenotypes such as mental retardation, eczema, and skin pigment defects. These phenotypes characterize the PKU disease Lobo,  and infop-values from the individual association test with each trait. However, these methods generally find SNPs that are consistently significant among multiple independent tests for each trait on the same dataset or via meta-analyses on different datasets. Hartley et al.  based on nonparametric functions. In functional mapping, all measurements are utilized to capture the trajectories of the development or progression of a trait and thus a more powerful approach to unravel the genetic association with these trajectories.To effectively investigate the development of disease, longitudinal cohort studies are designed to obtain repeated measures of a variety of disease-related traits within an individual over time. For example, in CHD studies, repeated measurements of cardiovascular risk factors such as systolic blood pressure are taken over time as well as information on other covariates such alcohol consumption or smoking status. Despite the availability of multiple time point measurements for each subject, many genetic analyses only use one single time point measurement or an average over all time points for each subject. For example, Levy et al.  regresseOn the other hand, mixed effects models have been a popular choice for modeling longitudinal data. Gauderman et al.  summarizIn this paper, we propose a 2-step strategy for analyzing the association of a genetic variant with multiple longitudinal traits. In the first step, a mixed effects model is used to analyze the repeated measurements for each trait individually. The subject-specific random effect is used to extract the component of variation that includes genetic factors contributing to the trait for each individual subject. Throughout this paper, we refer to this random effect as the subject-specific effect. The fixed effects account for observed confounding factors such as environmental factors and some time-dependent variables. In the second step, we treat the estimated subject-specific effect as a phenotype. We propose to regress the genotype of a genetic variant on all estimated subject-specific effects for the traits and test the association between the genetic variant and these subject-specific effects simultaneously through the score test or the likelihood ratio test.The remainder of our paper is organized as follows. In Section 2, we describe the proposed method and the details of the simulation study. We also apply our method to analyze data from the 16th Genetic Analysis Workshop (GAW16) Framingham Heart Study. Results from simulation study and data analysis application are presented in Section 3. Discussion and possible future study follow in Section 4.This section consists of three subsections. The first subsection describes our proposed 2-step method. A simulation study is present in the second subsection. In the third subsection, we apply our method to analyze the data from GAW16 Framingham Heart Study.In this subsection, we begin by defining a generalized linear mixed effect model for each of the multiple longitudinal traits of interest. Each model can include time-dependent or time-independent covariates together with a random effect for subject-specific effect. This constitutes Step 1 of the proposed method. Then, we introduce a binomial regression model that treats the genotype as the response variable and includes multiple subject-specific genetic effects obtained in Step 1 for each longitudinal trait as the explanatory covariates. This constitutes Step 2 of the proposed method.n independent subjects in our study. For each subject i, i = 1, 2, …, n, we collect repeated measurements on J different traits. Let Xij = ′ be a vector that represents the ijT measurements of the jth trait for subject i and so ijtX is the tth measurement of the jth trait for subject i. A general form of a GLMM can be expressed asIn longitudinal study designs, repeated measurements of phenotypic traits and covariates are taken for each subject over time. GLMMs are useful for modeling phenotypes of different data types, such as quantitative, binary, and count data. Suppose we have a sample of jg(.) is the link function for the jth trait, Zijt is a vector of covariates associated with the jth trait for the ith subject at time t, αj is the vector of fixed effects for covariates Zijt, γij is the random effect representing the ith subject-specific effect on the jth trait, and μijt is the conditional mean of ijtX given ZTijt and γij. Here, the associated covariates can be time-dependent or time-independent. Examples of time-dependent covariates include treatment status and age at each measurement time. Time-independent covariates such as sex are treated as constants over time. We allow different sets of covariates to be considered for different traits and the number of measurements T can be different for each subject as well. The subject-specific effect γij can be interpreted as the influence of subject i on his/her repeated measurements on the jth trait and it typically includes genetic effects on the trait. So, the γij's can capture the effects of unobserved major genes and polygenes; the latter refers to the combined effects of a large number of genetic variants that each make a small contribution to trait variation. For each trait, say the jth trait, we assume the γij's follow a normal distribution with a mean of 0 and a trait specific variance σ2jγ.where For a quantitative trait, a linear mixed effects model can be used, for exampleijt is assumed to follow a N  distribution. Then, jg(.) is an identity link with j(μijt) = μijtg. For a binary trait, a logistic link can be used with ij's will be treated as phenotypic traits for the association analysis in Step 2. The fixed effects associated with confounding factors can be estimated using the “lme4” package as well.where random error ϵjth trait being considered is hypertension status, γij can be interpreted as the underlying genetic risk factors of subject i that affect the log-odds for the risk of hypertension. When the jth longitudinal trait of interest is the daily seizure count of an epilepsy patient, γij can be interpreted as the underlying genetic risk of subject i that affects the log of the daily seizure rate.For different longitudinal data types, we interpret the associated subject-specific effect accordingly. When a longitudinal trait is binary, for example if the Y =  ′ be a vector of observed proportions of allele 1 of a given SNP for n unrelated subjects. So, iY takes values of 0, p = ′ be a vector of the expected frequency of allele 1 in this SNP for n subjects and 0 < ip < 1 for all i. Then, under the Hardy-Weinberg equilibrium, 2iY follows a binomial distribution and the log-likelihood function over n unrelated subjects has the formSingle nucleotide polymorphisms (SNPs) are the most common genetic variants in human and animal genomes. Because association studies are nearly all conducted using SNP data, our method will focus on applications to SNP association studies. Most SNPs are biallelic so, without loss of generality, for each SNP, we label the two alleles as “0” or “1”; the possible genotypes for this SNP are 0, 1, or 2 for the count of copies of the less frequent allele 1. Let γ be an n × (J + 1) design matrix of the formLet j+1)th column represents the subject-specific effects corresponding to the jth longitudinal trait for all subjects and the ith row, γi, contains a 1 for the intercept and the J subject-specific effects for subject i. With a logistic link,where the  is a vector of the score functions computed under the null hypothesis and β0 = 0. The subscript of U0−β indicates the removal of the first term  from U .  is the observed information matrix of β computed under the null hypothesis and β0 = 0. The subscript of −10−β indicates the removal of the first row and the first columns corresponding to β0 from −1. Based on Equation (2), we derive an explicit form of score statistics as follows,where γ−1 indicates the removal of the first column of design matrix γ, (γTγ)−1−1 represents the removal of the first row and the first column of (γTγ)−1, and 1 is a vector of 1's. Under H0, sW follows an asymptotic χ2J distribution with J being the number of traits to be tested.where l and (4), we plug in the estimated subject-specific effects ij to replace the γij's.where G1, G2, and G3, at different levels. In the first study, each SNP affects all three traits. In the second study, each SNP can affect a different number of traits, as specified in Table To assess the performance of the proposed method, we conducted simulation studies evaluating the type I error rate and the power of the association tests. Our simulation studies accommodate two different designs. In both studies, we consider two quantitative traits and one binary trait. These three traits can be affected by three SNPs, denoted by n independent subjects. For each subject i, we generate genotypes of three independent trait-causal SNPs, G1, G2, and G3, and genotypes of three SNPs, M1, M2, and M3, that are in LD with G1, G2, and G3 respectively. To generate SNP genotypes, we generate a haplotype for each pair of associated SNPs. Let Hr =  be the haplotype for SNPs rG and rM for r = 1, 2, 3. The haplotype Hr is generated from a bivariate Bernoulli distribution with mean vector πr = ′ and covariance matrixIn many situations, trait-causal SNPs may not be genotyped but instead, SNPs that are close to or in linkage disequilibrium (LD)/associated with these causal SNPs are available in the study. So, in our simulation study, we consider testing on both the causal SNPs and SNPs that are associated with these causal SNPs. Suppose we generate a sample of 2rG = πrG(1 − πrG), σ2rM = πrM(1 − πrM), and σ2r,MrG = ρrσrGσrM with ρr being the correlation between the SNPs rG and rM. πr is a vector of frequencies of allele 1 for SNPs rG and rM. We set π1 = ′, π2 = ′, and π3 = ′. We then specify the correlations with ρ1 = 0.95, ρ2 = 0.9, and ρ3 = 0.85. A pair of Hr are generated to make up the genotypes of rG and rM. We also simulate an independent SNP M for the purpose of Type I error rate assessment. The genotype of SNP M is simulated from binomial.where σZit1 and Zit2 for subject i at the tth measurement. The covariates can be time-varying or time-invarying. When the covariate is time-invarying, it will be a constant with respect to t. Here, we generate time-varying covariates for both Zit1 and Zit2. We let the total number of measurements be T = 5 for each subject. We let Zit1 be a binary covariate generated from Bernoulli(0.3) and let Zit2 be a quantitative covariate generated from N . We let μ = 40 and σ = 7 to mimic the age distribution of patients, which is a typical time-varying covariate in longitudinal data. Thus, the Zit2's are sorted in ascending order such that Zi12 < ··· < Zit2 < ··· < ZiT2.Then we generate two general covariates Given the generated covariates and the genotypes of causal SNPs, we generate measurements of each trait for each subject by first computing the linear predictors given byi = 1, …, n, j = 1, 2, 3, and t = 1, …, 5. The two quantitative traits, Xi1t and Xi2t, are generated from N  and N  with identity links ηi1t = μi1t and ηi2t = μi2t, respectively. The binary trait, Xi3t, is generated from Bernoulli(μi3t), where for α1 = T = T and b1 = T = T for the first quantitative trait Xi1t. For the second quantitative trait Xi2t, we set α2 = T and b2 = T. For the binary trait i3tX, we set α3 = T and b3 = T such that the simulated sample consists of about 40% cases and 60% controls. In simulation study 2, the fixed effects αj's that are associated with the covariates Z's in study 2 remain the same as in study 1. However, we set b1 = T, b2 = T, and b3 = T, such that, SNP G1 affects all three traits, SNP G2 affects two traits, and SNP G3 affects one trait only.In simulation study 1, we set n = 100, 200, and 300 and, for each specified sample size, we simulate 1000 data sets. For each data set, we first fit the GLMM to obtain an estimate of γij for each trait and each subject. In the GLMMs, both covariates, Zit1 and Zit2, are included. For each SNP, we then perform a simultaneous test on all three estimated subject-specific effects; 1, 2 and 3, where each j = T is treated as a phenotype. Because we simultaneously test on three phenotypes, both sW and Λ test statistics follow a χ23 distribution asymptotically under the null hypothesis. We reject the null hypothesis if the test statistic is greater than the (1 − αF)th quantile of the χ23 distribution. We let αF = 0.05, 0.01, and 0.001. We also perform individual association tests between each SNP and each subject-specific effect for each trait. We reject the null hypothesis if the test statistic computed for only one estimated subjected-specific effect has a value greater than the (1 − α)th quantile of χ21 distribution. Here, α is given by αF = 1 − (1 − α)3 and αF is the family-wise error rate (FWER) controlling at 0.05, 0.01, and 0.001 levels.For each simulation study, we generate samples of size We also consider different sets of covariates and fixed effects in our simulation studies. The results demonstrate similar patterns in terms of power and empirical type I error rates of association tests when different scenarios for fixed effects are considered. Please see the Supplementary Material for other simulation models and their corresponding results.Our proposed method is used to analyze the 16th Genetic Analysis Workshop (GAW16) Framingham Heart Study (FHS) data. The GAW16 FHS data are drawn from the FHS under the direction of the National Heart, Lung, and Blood Institute. The FHS aims to identify risk factors that contribute to cardiovascular disease (CVD). Data from families from the town of Framingham, Massachusetts (USA) were collected between 1948 and 2005 to a maximum of three generations. The FHS consists of three cohorts. The first cohort consists of the original participants in the first generation. The second cohort is the offspring recruited from children of the original participants and the spouses of these children. The third cohort consists of the third generation, which are the offsprings of the second generation. Most participants have repeated measurements on phenotypic traits from four examinations. Among the three cohorts, the offspring cohort possesses the most complete genotype data and phenotype information from the four exams.Our analysis in this paper focuses on the offspring cohort. From this cohort, we select a subset of 1817 unrelated children using an algorithm as described in the R function“pedigree.unrelated” in the package “kinship2”  have genetic effects on all three longitudinal traits. When testing the association between each causal SNP and all three subject-specific effects simultaneously, the power is consistently higher than the power obtained from the union of three individual tests. When testing SNPs M1, M2, and M3 that are in LD with the causal SNPs, the simultaneous tests are also consistently more powerful than the union of individual tests. Certainly, the power is diluted in comparison with the tests on the trait-causal SNPs. In study 2, SNP G1 affects all three longitudinal traits, SNP G2 affects the first and the third longitudinal traits (Xi1t and i3tX), and SNP G3 affects one longitudinal trait only (i2tX). We observe that when the SNPs are associated with more than one trait, the simultaneous test is consistently more powerful than the union of individual tests for different sample sizes. The power gain is more obvious when the SNP is associated with more traits. When the SNP is associated with one trait, the power of the simultaneous trait test is similar to the individual trait test. Again, when testing on SNPs M1, M2, and M3 that are in LD with causal SNPs, the power is generally diluted. However, similar patterns to those obtained from tests of causal SNPs are observed. Note that in Tables G2 in study 2 is not associated with the second longitudinal trait, so, its empirical rejection rate corresponds to the type I error rate.The empirical power for each causal SNP and associated SNP are reported in Tables p-values, are presented in Table p-values . The time-invariant covariate diabetes  is also strongly significant for log(SBP), log(HDL), and log(TG). Their p-values are all close to 0. Time-variant covariates bmi and smoking status are significantly associated with four longitudinal traits. Covariates age and alcohol consumed (in ounces/week) are found to have a very significant effect on log(SBP), log(HDL) and log(TG). The number of cigarettes per day has very significant effect on LDL and log(TG). Treatment for lipid (cholrx) significantly reduces the log(SBP), LDL, and log(TD), and treatment for hypertension (htnrx) significantly reduces log(HDL). Note that the R function “bfFixefLMER_F.fnc” is used to select covariates to be included in the GLMM. So, the entry with “−” in Table In fitting the GLMMs to the four longitudinal traits: log(SBP), log(HDL), LDL, and log(TG), the estimated fixed effects for each confounding covariate, and their associated standard errors (SE) and p-value < 1.0 × 10−5 from the simultaneous tests (either score test or LRT test) are summarized in Table p-value associated with each individual trait are adjusted via Bonferroni procedure for multiple testing. For easy comparison, results are also presented in Figure p-values) or equivalently a lower p-value. SNPs that are significantly associated with only one trait have a comparative −log or equivalently a similar level of significance in p-value. On chromosome 8, nine SNPs are found by the simultaneous test to have a strong and significant association with at least one of the four traits. These SNPs are in the LPL gene or very close to this gene. The LPL gene encodes lipoprotein lipase, a triglyceride hydrolase that acts as a ligand factor for receptor-mediated lipoprotein uptake. According to the individual tests, these nine SNPs are significantly associated with HDL and TG but their p-values based on the union of the individual tests are consistently larger than the p-values based on the simultaneous test. Note that a larger p-value means a less significant level. These significant findings are consistent with other FHS analyses reported by Piccolo et al.  and the union of the individual tests . The CEPT gene mediates the transfer of cholesterol ester from HDL to other lipoproteins. So, not surprising, this SNP is strongly associated with HDL in the individual test . This result is also confirmed by Sull et al.  gene which is known to associate with maturity-onset diabetes of the young.On chromosome 11, 3 SNPs  are found to be significantly associated with HDL and/or TG traits. These SNPs are either in or close to the In this paper, we proposed a two-step procedure for a genetic association analysis with multiple longitudinal traits. In the first step, a GLMM is used to analyze each longitudinal trait individually. This allows us to flexibly incorporate different covariate sets that are relevant to different longitudinal traits and also to flexibly handle traits of different data types. In the GLMMs, unmeasured subject-specific genetic effects are packed into the random effects term while accounting for the fixed effects of other confounding factors. With a longitudinal study design, repeated measurements on each subject enable the estimation of subject-specific effects. This has been validated by our simulation study that included genetic effects. With the 2-step approach, the method has the advantage of being able to efficiently and simultaneously test a large-scale genome-wide SNP associations with multiple traits in the second step, with the fixed effects of the potential confounding factors for each trait taken into account in the first step. Then, subsequent individual tests would be performed on a much smaller subset of significant SNPs found by this two-step procedure to further investigate which particular traits are associated with the SNPs.Our proposed method opens several avenues for future research. For example, a specific gene-environmental interaction can be modeled by the introduction of a random slope term in the GLMM. However, there are a small number of repeated measurements and many possible gene-interacting environmental factors. Therefore, it is worthwhile to investigate an efficient procedure to incorporate gene-environmental interaction terms in the GLMMs, and perform genome-wide association tests for these interactions. The proposed method is for a single marker test only. When there are gene-gene interactions, but only one of the markers is tested marginally, the power to detect genetic association may be comprised. Moreover, our proposed method is a logistic regression method. It is reported that the probability of a rare event can be underestimated by logistic regression  are nested within subjects (level 2) and subjects are nested within families (level 3). When testing the association between a SNP and subject-specific effects, the response in the binomial regression model is the allele frequency of the SNP, so observed responses are no longer independent due to the relationship among related subjects. The current score test and likelihood ratio test based on the independent subjects assumption would no longer be applicable. A modified method such as the quasi-likelihood based method could be considered.ij s are not observable. In the analysis, we replace the true random effect γij by the estimated random effect, ij, obtained in the first step. Since the random effect is treated as a covariate in the second step, issues of measurement errors may be of concern. Based on a Taylor expansion, Rosner et al.  mechanism, our method is applicable for subjects that have different numbers of measurements. However, when the MCAR assumption is invalid, methods of handling missing data under a different mechanism, such as missing not at random (MNAR), should be considered in order to obtain unbiased estimates for fixed effects of covariates as well as the subject-specific random effects.Weiqiang Wang and Zeny Feng developed and implemented the method, and performed simulation studies and FHS data analysis. Zeny Feng supervised the project. Shelley B. Bull and Zuoheng Wang provided constructive comments and suggestions. All authors drafted, read and approved the final manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


We consider analysis of Genetic Analysis Workshop 18 data, which involves multiple longitudinal traits and dense genome-wide single-nucleotide polymorphism (SNP) markers. We use a multivariate linear mixed model to account for the covariance of random effects and multivariate residuals. We divide the SNPs into groups according to the genes they belong to and score them using weighted sum statistics. We propose a penalized approach for genetic variant selection at the gene level. The overall modeling and penalized selection method is referred to as the penalized multivariate linear mixed model. Cross-validation is used for tuning parameter selection. A resampling approach is adopted to evaluate the relative stability of the identified genes. Application to the Genetic Analysis Workshop 18 data shows that the proposed approach can effectively select markers associated with phenotypes at gene level. The Genetic Analysis Workshop 18 (GAW18) data consists of multiple longitudinal traits and dense genome-wide single-nucleotide polymorphism (SNP) markers. A commonly used approach for identifying markers associated with traits is to conduct single-variant analysis and then adjust for multiple comparisons on each trait. However, for complex polygenic traits, single-variant analysis methods may not be appropriate as they fail to take into account the accumulated and/or joint effects of multiple genetic variants on the traits. In addition, analyzing each trait separately does not take into account the correlation among traits, and thus can be ineffective. To overcome these limitations, we developed a joint analysis approach referred to as the penalized multivariate linear mixed model (PMLMM). This approach takes into account covariance of both random effects and residuals and uses a group minimax concave penalty (MCP) approach [N subjects, p genetic variants, and q environmental exposure covariates. Here a genetic variant can be a single SNP marker or a score representing a group of SNPs. For subject i, suppose that there are m traits. Let i. Let Y be the N subjects, where Y is i. Let Consider a genetic association study with longitudinal measurements on er et al .Consider the multivariate linear mixed modelB is a p genetic variants on m traits, and ith subject. We treat D is the row covariance matrix, and each where Then iu et al .From Dawid , the neg∑. We estimate D by using the estimates from m univariate linear mixed models and subsequently get the estimate V as B, which is Hastie et al.  suggest approach . The ovejth row of B and λ and regularization parameter γ[where rameter γ.γ are better at retaining the unbiasedness of the MCP penalty for large coefficients, but they have the risk of creating objective functions that have problems with nonconvexity [λ. Simulation studies by Breheny and Huang [λ using 5-fold cross-validation.For computation, we use a group coordinate descent algorithm . The groonvexity , are difnd Huang  suggest http://www.scandb.org. SNPs in each gene are scored using weighted sum statistics to generate gene-level measurements [The GAW18 data set consists of dense genome-wide markers with longitudinal measurements on systolic and diastolic blood pressure (SBP and DBP) and other covariates. Other measurements include gender, age, year of examination, use of antihypertensive medications, and tobacco smoking at up to 4 time points. In this study, we analyze the 157 unrelated individuals using SBP and DBP as traits and other medical and demographic covariates as random effects. Gene annotations for SNP data are obtained from urements . After qWe apply the proposed PMLMM to identify genetic variants that are associated with both SBP and DBP at the gene level. As a benchmark, we also analyze each trait separately using a penalized linear mixed model (PLMM) approach. Table B, ∑, and V. In variant selection, our method is designed to search for genes associated with all the traits considered. When different sets of genetic variants are suspected to be associated with different phenotypes, the sparse group penalization approach [In this study, we analyze the GAW18 data and develop a PMLMM approach. A multivariate linear mixed model is used to model variance components among traits and longitudinal measurements. A penalization approach is adopted for variant selection. In the estimation procedure, it can be considered heuristic to use approach  can be aWe have presented a penalized multivariate linear mixed model (PMLMM) for detecting pleiotropic genetic associations among multiple traits in the presence of pedigree structures. The proposed approach combines the advantages of mixed models that allow for elegant correction for pedigree-based family data and integrative analysis of multiple traits. Compared with PLMM which considers one trait at a time, the proposed PMLMM can achieve better performance when the pleiotropic effect is appropriately modeled.The authors declare that they have no competing interests.All authors were involved in study design. JL conducted the numerical work. All authors were involved in manuscript preparation, and read and approved the final manuscript.


We surveyed 26 quantitative traits and disease outcomes to understand the proportion of phenotypic variance explained by local ancestry in admixed African Americans. After inferring local ancestry as the number of African-ancestry chromosomes at hundreds of thousands of genotyped loci across all autosomes, we used a linear mixed effects model to estimate the variance explained by local ancestry in two large independent samples of unrelated African Americans. We found that local ancestry at major and polygenic effect genes can explain up to 20 and 8% of phenotypic variance, respectively. These findings provide evidence that most but not all additive genetic variance is explained by genetic markers undifferentiated by ancestry. These results also inform the proportion of health disparities due to genetic risk factors and the magnitude of error in association studies not controlling for local ancestry. Two statistical techniques to map disease risk variants are widely used with genome-wide genetic data, association testing and admixture mapping. Genetic association refers to a correlation of phenotype and genotype. In association studies, population structure can be a confounder, leading to both false positive and false negative associations. Population structure is typically described by two models, population stratification and admixture. Population stratification refers to systematic differences in allele frequencies between subgroups of the sample, also called strata. Each individual belongs to one stratum. Methods to identify and control for population stratification include genomic control . We estimated the effective number of tests in admixture mapping using autocorrelation of local ancestry to be 373.1  is a population-based observational study of African Americans from Washington, D.C. Ethical approval was obtained from the Howard University Institutional Review Board. All subjects gave written informed consent in accordance with the Declaration of Helsinki. Data are available upon collaboration with Dr. Charles N. Rotimi. HUFS comprised 1976 individuals, 1055 of whom were unrelated  is a prospective study of atherosclerosis and cardiovascular disease. We obtained approval for data access from dbGaP (Accession phs000280.v2.p1). We retrieved data from the GENEVA sub-study (phs000090.v2.p1), including phenotype data (pht000114) and genotype data (phg000035). ARIC included 2,600 unrelated African Americans from Forsyth County, North Carolina or Jackson, Mississippi. Genotyping was performed using the Affymetrix Genome-Wide Human SNP Array 6.0, with quality control as described previously ; this scaling induces an inverse relationship between pi and effect size β2, with p equaling the mean local ancestry at rs2814778 in the ARIC data and β equaling the effect size under the additive model, and random noise normally distributed with mean 0 and variance 1 − σ2. We tested for bias using the one-sample Wilcoxon signed rank test.To investigate bias in the random effects models implemented in GCTA and GEMMA, we simulated phenotype data based on the white blood cell count data from ARIC. Phenotype data were simulated as the sum of signal normally distributed with mean 0 and variance σhttp://www.complextraitgenomics.com/software/gcta/. GEMMA is available at http://www.xzlab.org/software.html.GCTA is available at ACKR1 (formerly known as DARC) that causes the Fy-/- phenotype. Note that white blood cell count is not unique for having a major effect gene in admixture mapping of African Americans, as rs73885319 in the gene APOL1 defines a major effect for end-stage kidney disease  of the phenotypic variance of white blood cell count  on chromosome 18 that explained 0.6% of the phenotypic variance  of the additive genetic variance is variance due to local ancestry  perform conventional admixture mapping in order to identify loci with large effects and (2) model these loci using fixed effects rather than random effects in a mixed effects model  Study was carried out as a collaborative study supported by National Heart, Lung, and Blood Institute contracts HHSN268201100005C, HHSN268201100006C, HHSN268201100007C, HHSN268201100008C, HHSN268201100009C, HHSN268201100010C, HHSN268201100011C, and HHSN268201100012C. Funding for the ARIC Gene Environment Association Studies  was provided by National Human Genome Research Institute grant U01HG004402 (Eric Boerwinkle). Support for the Howard University Family Study was provided by National Institutes of Health grants S06GM008016-320107, S06GM008016-380111, and 2M01RR010284. Genotyping support was provided by the Coriell Institute for Medical Research. This research was supported by the Intramural Research Program of the Center for Research on Genomics and Global Health (CRGGH). The CRGGH is supported by the National Human Genome Research Institute, the National Institute of Diabetes and Digestive and Kidney Diseases, the Center for Information Technology, and the Office of the Director at the National Institutes of Health (Z01HG200362).The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.