P<5 × 10−8) that maps to TOMM40-APOE, a region previously implicated in lipid metabolism and Alzheimer's disease. We have also confirmed previous reports that single-nucleotide polymorphisms at the TCF7L2 locus demonstrate the greatest extent of heterogeneity in allelic effects between ethnic groups, with the lowest risk observed in populations of East Asian ancestry.Genome-wide association studies (GWAS) have traditionally been undertaken in homogeneous populations from the same ancestry group. However, with the increasing availability of GWAS in large-scale multi-ethnic cohorts, we have evaluated a framework for detecting association of genetic variants with complex traits, allowing for population structure, and developed a powerful test of heterogeneity in allelic effects between ancestry groups. We have applied the methodology to identify and characterise loci associated with susceptibility to type 2 diabetes (T2D) using GWAS data from the Resource for Genetic Epidemiology on Adult Health and Aging, a large multi-ethnic population-based cohort, created for investigating the genetic and environmental basis of age-related diseases. We identified a novel locus for T2D susceptibility at genome-wide significance ( A common approach is to apply principal components analysis (PCA) to a genetic relatedness matrix (GRM) between individuals. The first eigenvectors of the GRM represent linear combinations of single-nucleotide polymorphisms (SNPs), or axes of genetic variation (AGV), that best distinguish genetically dissimilar individuals, such as those of diverse ancestry. For example, application of PCA to reference samples from the International HapMap Project11 There is increasing evidence from these studies that common variant association signals for complex traits are, in fact, shared across ancestry groups, and there is relatively little heterogeneity in allelic effects between populations.12 Furthermore, methodology to account for any such heterogeneity is well developed,2 and the differential structure of linkage disequilibrium (LD) between ancestries may, in fact, be beneficial for localising causal variants in trans-ethnic fine-mapping studies.13With the increasing availability of GWAS data from diverse populations, trans-ethnic meta-analysis of association summary statistics across ancestry groups may offer increased power to detect novel loci for complex traits through increased sample size.In this article, we have developed and evaluated an approach for the analysis of GWAS of complex human traits in a multi-ethnic cohort to: (i) test for association whilst accounting for confounding with population structure by adjusting for AGV; and (ii) test for heterogeneity in allelic effects between ethnicities by considering an interaction with the first two AGV that distinguish broad ancestry groups. We have also applied this approach to the detection of loci associated with type 2 diabetes (T2D), and characterisation of the effects of lead SNPs across ancestry groups, through application to GWAS data from the multi-ethnic Resource for Genetic Epidemiology on Adult Health and Aging (GERA) cohort (database of Genotypes and Phenotypes (dbGaP) phs000674.p1).y and G, respectively. To account for population structure, within and between ethnicities, we considered AGV, denoted x, that were constructed through PCA of the GRM obtained from G. To test for association of the jth variant with phenotype, we considered a generalised linear modelling framework, given byConsider a sample of unrelated individuals from a multi-ethnic cohort, with phenotypes and genome-wide genotypes denoted by g(.) is the link function. In this expression, β is the effect of the jth variant on phenotype, with genotypes coded under an additive model, and γ is a vector of regression coefficients for the AGV. A likelihood ratio test with one degree of freedom was then formed by comparing the maximised log-likelihood of the unconstrained model (1), with that obtained under the null hypothesis of no association, β=0.where jth SNP, we extended model (1) to include an interaction between genotype and the first two AGV, x1 and x2, which we expect to distinguish between individuals from diverse ethnicities, given byTo test for heterogeneity of allelic effects between ethnicities at the λ1 and λ2 are regression coefficients for the interaction between the first two AGV and genotypes at the jth variant. A likelihood ratio test with two degrees of freedom was then formed by comparing the maximised log-likelihood of the unconstrained model (2) with that obtained under the null hypothesis of no heterogeneity, λ1=λ2=0.In this expression, 5 incorporating haplotypes of African, East Asian, South Asian, Hispanic and European ancestry, to simulate a multi-ethnic cohort of 20 000 individuals. Five of the reference populations  are admixed, in which individuals have varying continental ancestry proportions, whilst the remainder  are relatively homogeneous.5 We began by generating genome-wide genotype data for 2000 individuals from each of the reference populations using HAPGENv2.14 We constructed a GRM from pair-wise identity by descent metrics estimated from LD pruned  autosomal SNPs with MAF⩾1%. We applied PCA to the GRM to obtain: (i) 10 ‘multi-ethnic' AGV to account for structure between and within ancestry groups; and (ii) four ‘population-specific' AGV to account for structure within each population.We performed a detailed simulation study to investigate the type I error rate and power of the generalised linear modelling approach, with and without adjustment for AGV, to detect: (i) association between a variant and a case-control phenotype; and (ii) heterogeneity of allelic effects on the phenotype at the variant. We considered the 10 reference populations from Phase 3 of the International HapMap Project,We considered a range of models of association of a causal SNP with the dichotomous phenotype across ethnic groups, parameterised in terms of the allelic effect (log-odds ratio) of the alternative allele in each population . These m5 and obtained the alternative allele frequency in each population. Genotypes were then simulated in the required number of cases and controls in each population, according to the population-specific allele frequency and odds ratio. For each replicate of data, we tested for association between the causal SNP and phenotype: (i) with no correction for population structure; and (ii) by inclusion of the 10 ‘multi-ethnic' AGV as covariates in the logistic regression model. We also tested for heterogeneity in allelic effects of the causal SNP across ethnicities by including an interaction with the first two AGV as additional covariates. For comparison, we tested for association between the causal SNP and phenotype within each of the 10 populations, separately, by inclusion of the four ‘population-specific' AGV as covariates in the logistic regression model. We combined association summary statistics across populations through fixed-effects meta-analysis with inverse-variance weighting of effect sizes. We also assessed evidence for heterogeneity in allelic effects between populations in the meta-analysis by means of Cochran's Q statistic. Each test of association and/or heterogeneity was evaluated at nominal significance thresholds of P<0.05 and P<0.01, and at the traditional genome-wide standard of P<5 × 10−8.For each scenario, we generated 1000 replicates of genotype data for the causal SNP for the multi-ethnic cohort of 20 000 individuals. For each replicate, we selected the causal SNP at random from those present in the Phase 3 HapMap reference panel,16 We undertook quality control of these genotype data, removing individuals from known pedigrees and/or with call rate (<97%), and excluding SNPs on the basis of call rate (<95%) and extreme deviation from Hardy–Weinberg equilibrium . We constructed a GRM from pair-wise identity by descent metrics estimated from LD pruned  autosomal SNPs shared across the four genotyping arrays, and with MAF⩾1%, after exclusion of those in high LD and complex regions, and those mapping to established T2D loci. We defined related individuals on the basis of pi-hat>0.2, and removed those from each family set with the lowest call rate.We applied the methodology developed and evaluated in this article to identify and characterise loci associated with T2D susceptibility using GWAS data from GERA, a large multi-ethnic population-based cohort, created for investigating the genetic and environmental basis of age-related diseases (dbGaP phs000674.p1). T2D status is based on ICD-9 codes in linked electronic medical health records. Participants in the GERA cohort have previously been genotyped using one of four custom arrays, which have been designed to maximise coverage of common and low-frequency variants in non-Hispanic white, East Asian, African American and Latino ethnicities.P<0.05) were retained as covariates in downstream SNP association analyses.We applied PCA to the GRM to obtain AGV, the first 20 of which were tested for association with T2D in a logistic regression model to determine confounding with disease status. AGV correlated with T2D at nominal significance  or if they were palindromic (AT/GC) to avoid strand errors. For each of the four genotyping arrays separately, we constructed a scaffold for imputation after excluding SNPs with MAF<1%, which was then pre-phased using SHAPEITv2.5.17 The resulting haplotypes were imputed, separately for each genotyping array, up to the 1000 Genomes Project reference panel  using IMPUTEv2.3.18 Data were then merged across the four genotyping arrays, and SNPs excluded from downstream association analyses if rare and/or poorly imputed .SNPs passing initial quality control were lifted to NCBI build GRCh37 (UCSC hg19 assembly) of the human genome, and were excluded if they then had unknown position. SNPs were then removed if alleles did not match those reported in the 1000 Genomes Project multi-ethnic reference panel19 to test for association of T2D with each SNP in a logistic regression framework under a dosage model, and after adjustment for sex and AGV (described above) as covariates to take account of trans-ethnic and ancestry-specific population structure. The genomic control inflation factor20λ1000 was calculated to assess the evidence for residual population structure that is not accounted for by covariate adjustment. Genome-wide significance was defined by the traditional threshold of P<5 × 10−8. We tested for heterogeneity in allelic effects on T2D between ancestry groups by including an interaction between genotypes and the first two AGV from PCA in the logistic regression model.We used SNPTESTv2.5Under the null hypothesis of no association with the phenotype, type I error rates were substantially inflated in the presence of population structure (moderate or extreme) without adjustment for AGV, as expected . HoweverP<5 × 10−8) with the causal variant as a function of the allelic effect size: (i) for the generalised linear model after adjustment for 10 ‘multi-ethnic' AGV as covariates for all populations combined; and (ii) for fixed-effects meta-analysis across populations with inverse-variance weighting of effect sizes, obtained after adjusting for four ‘population-specific' AGV. The four panels correspond to alternative models of heterogeneity of allelic effects between ancestry groups , as a function of allelic effect size, assessed by: (i) including interactions of the variant with the first two AGV in the generalised linear regression model; and (ii) Cochran's Q statistic in the fixed-effects meta-analysis. The four panels correspond to alternative models of heterogeneity of allelic effects between ancestry groups , and were retained as covariates in downstream SNP association analyses  and IGF2BP2 .After quality control, a total of 71 604 unrelated participants, including 9747 T2D cases, were retained for analysis. The majority of GERA participants (79.3%) were self-reported non-Hispanic white, although there was considerable variability in ethnicity according to genotyping array . As expeanalyses . The genjustment . Variantjustment , with thTOMM40-APOE . This association signal maps ~800 kb upstream of the previously reported locus for T2D susceptibility at GIPR21 , as an additional covariate in the logistic regression model, the novel association signal at TOMM40-APOE was not attenuated (pCOND=3.2 × 10−9), and thus demonstrated to be distinct.Amongst the loci attaining genome-wide significance was a novel association signal for T2D susceptibility, mapping to t GIPR21 . HoweverpINT<0.05) only at the lead SNP, rs34872471, at the TCF7L2 locus (pINT=0.012). Specifically, the odds ratio at the SNP increased with larger values of the first AGV, which corresponds to a smaller effect on T2D susceptibility in individuals of East Asian ancestry  and attains nominal T2D association in GERA (P=1.6 × 10−6). The association signal for this SNP was replicated in meta-analyses of T2D GWAS of up to 12 171 cases and 56 862 controls of European ancestry from the DIAGRAM Consortium21 (P=0.0025). There was no evidence of heterogeneity in allelic effects between GERA and DIAGRAM (Cochran's Q P=0.95), with consistent odds ratios observed in both studies  is only in moderate LD with that which we report here for T2D susceptibility (EUR r2=0.334 with rs157582).Encouraged by the results of our simulation study, we applied the methodology developed and evaluated in this article to identify and characterise loci associated with T2D susceptibility using GWAS data from a large multi-ethnic population-based cohort. Amongst the loci attaining genome-wide significance in our multi-ethnic analysis was a novel association signal for T2D susceptibility, mapping to  studies . Common TOMM40-APOE locus has also previously been associated with late onset Alzheimer's disease (LOAD). Compared with the common APOE ɛ3 allele, ɛ4 increases the risk of LOAD and lowers the age at onset of AD in a dose-dependent fashion, whereas the ɛ2 allele confers a protective effect.24APOE alleles are tagged by two SNPs in the region, rs429358 (NC_000019.9:g.45411941T>C) and rs7412 (NC_000019.9:g.45412079C>T), which are not present on the majority of commercial genome-wide genotyping arrays, but can be well imputed using reference panels from the 1000 Genomes Project.25 In particular, rs429358, which defines the ɛ4 LOAD risk allele, is in strong LD with our lead SNP (EUR r2=0.524 with rs157582), and can consequently partially explain the T2D association signal in conditional analyses , we were unclear whether our observed diabetes association at this locus might thus reflect under-representation of LOAD patients amongst GERA cases and confounding with age. To investigate the influence of this potential ascertainment bias, we considered the impact of age on the allelic effect of rs157582 on T2D susceptibility. We observed no evidence of an interaction (P=0.83) between age and the SNP, after adjusting for the main effects of both variables in the logistic regression model. We also observed consistent allelic effects of rs157582 on T2D susceptibility after stratifying participants according to their year of birth (TOMM40-APOE locus cannot be fully explained by LOAD ascertainment bias. However, to fully disentangle the association signals for T2D and LOAD at this locus would require case-control status for both diseases in the same sample of individuals, which is beyond the scope of this study.Variation mapping to the analyses . Given tof birth . Taken t26 Amongst lead SNPs at loci attaining genome-wide significance in our multi-ethnic analysis, we detected nominal evidence of interaction with the first two AGV only at TCF7L2, where a smaller allelic effect on T2D susceptibility in individuals of East Asian ancestry was observed than in other ethnic groups. We recognise that the odds ratios at established T2D susceptibility loci are towards the lower end of those considered in our simulation study, and thus that power to detect heterogeneity in allelic effects between populations is limited. However, our results are consistent with previous findings from trans-ethnic meta-analysis of T2D susceptibility GWAS that highlighted heterogeneous allelic effects between ancestry groups at TCF7L2, where the smallest odds ratios for the disease were observed in populations of East Asian descent.26Large-scale trans-ethnic meta-analyses have also demonstrated that many T2D GWAS loci are shared across diverse populations and that the allelic effects of lead SNPs at common variant association signals are predominantly homogeneous across ancestry groups.The choice of analysis strategy in multi-ethnic GWAS will often be restricted by study design. When aggregating individuals from diverse populations from distinct GWAS, it may be impractical to exchange individual level genotype data, and trans-ethnic meta-analysis will be the only practical approach. On the other hand, in large, multi-ethnic GWAS, which include individuals of admixed descent, such as GERA, it will likely be difficult to define homogenous population groups for ethnic-specific analyses, and self-reported ancestry may not always be reliable. Our results would indicate that power to detect association is equivalent through meta-analysis of GWAS undertaken in homogeneous ancestry groups or multi-ethnic studies with adjustment for population structure. However, we have demonstrated increased power to detect heterogeneity in allelic effects between diverse populations in a combined multi-ethnic GWAS that incorporates interaction with AGV that distinguish major ancestry groups.2 This Bayesian ‘hybrid' of fixed- and random-effects meta-analysis allows for heterogeneity in allelic effects between populations according to a model of relatedness between them. In principal, the multi-ethnic GWAS analysis proposed here could be extended to a ‘joint test' of the main effect of the SNP and the interaction with the first two AGV, thereby allowing for heterogeneity in allelic odds ratios between diverse populations in evaluating the evidence of association with the trait. As with MANTRA, this approach would be particularly applicable in the context of trans-ethnic fine-mapping, where heterogeneity in allelic effects is driven by differential patterns of LD of SNPs with the causal variant in diverse populations, but is leveraged to localise the association signal.Specialist meta-analysis approaches for aggregating association summary statistics from multi-ethnic GWAS have also been developed, including MANTRA.TOMM40-APOE, a region previously implicated in lipid metabolism and Alzheimer's disease. We have also confirmed previous reports of heterogeneity in allelic effects at the TCF7L2 locus, where the lead SNP demonstrates the lowest risk for T2D in populations of East Asian ancestry. Further research will be required to determine the cause of this heterogeneity, but allelic effect size differences could reflect interaction of the causal variant with environmental risk factors that differ in exposure between ethnicities. With the increasing availability of large-scale multi-ethnic cohorts, our results will also help to inform future GWAS design and application of methodologies for complex trait locus discovery and characterisation.In summary, through multi-ethnic association analysis in the GERA cohort, we have identified a novel locus for T2D susceptibility mapping to


Genome‐wide association studies (GWAS) have detected large numbers of variants associated with complex human traits and diseases. However, the proportion of variance explained by GWAS‐significant single nucleotide polymorphisms has been usually small. This brought interest in the use of whole‐genome regression (WGR) methods. However, there has been limited research on the factors that affect prediction accuracy (PA) of WGRs when applied to human data of distantly related individuals. Here, we examine, using real human genotypes and simulated phenotypes, how trait complexity, marker‐quantitative trait loci (QTL) linkage disequilibrium (LD), and the model used affect the performance of WGRs. Our results indicated that the estimated rate of missing heritability is dependent on the extent of marker‐QTL LD. However, this parameter was not greatly affected by trait complexity. Regarding PA our results indicated that: (a) under perfect marker‐QTL LD WGR can achieve moderately high prediction accuracy, and with simple genetic architectures variable selection methods outperform shrinkage procedures and (b) under imperfect marker‐QTL LD, variable selection methods can achieved reasonably good PA with simple or moderately complex genetic architectures; however, the PA of these methods deteriorated as trait complexity increases and with highly complex traits variable selection and shrinkage methods both performed poorly. This was confirmed with an analysis of human height. The availability of genomic data has revolutionised the statistical analysis of human diseases and traits. The development of methods that can accurately predict the genetic risk associated with these diseases and complex human traits can have a great impact on public health . These studies have identified unprecedented numbers of variants associated with important complex traits and diseases. In some cases, the variants identified so far explain a sizable proportion of the variance of the trait or disease. Examples of these include Crohn's disease, age‐related macular degeneration and Type I diabetes  have been conducted  was used; this ridge‐regression‐type method induces homogeneous shrinkage of marker effects. Second, a The performance of these methods was assessed with simulated and real data. Our simulation comprised different scenarios pertaining to the complexity of the trait (in terms of number of large‐effect loci) and the pattern of LD between markers and causal or QTL. The results obtained from simulation studies were validated by analysis of human height measured on distantly related individuals.yi is described as a sum of three components: the population mean (μ), a random component reflecting the genetic factors, the so‐called genetic value ui, and a random model residual (ɛi) usually assumed to be identically and independently normal distributed with zero mean and variance σɛ2.In the classical quantitative genetic model, a continuous trait ui are approximated using regressions on marker genotypes. For instance, in an additive model one can set ui=∑j=1pXijβj, where Xij∈0,1,2 represents the allele dosage at the jth locus of the ith individual and βj represents the corresponding marker effect. Thus, the model for p markers can be expressed as:In genomic models, the genomic values p≫n). Thus, the estimation of effects in the model described above requires the use of some type of regularised regression procedure such as penalised or Bayesian regression. In Bayesian regressions, the type and extent of shrinkage of estimates of effects is controlled by the choice of prior for marker effects.In WGR methods, the number of effects to be estimated can vastly exceed the number of data points  and an intermediate one (Bayes A) that induce differential shrinkage of estimates of effects.βj∼N,j=1,...,p. This approach yields estimates equivalent to those from ridge regression, where all effects are shrunk toward zero to a similar extent. Using the expectation of ith phenotype yi (given the genotypes and marker effects), and the genomic value ui=∑j=1pXijβj, we rewrite equation yi=ui+ɛi,i=1,...,n. Thus, the genomic value is also normal: u∼N with a genomic relationship matrix, which is obtained as a cross product of genotype readings G={Gik}=1∑j2pj(1−pj)XX′ [pj is the minor allele frequency (MAF) at the jth locus] and a genomic variance component σu2=∑j=1p2pj(1−pj)σβ2. Therefore, the GBLUP could be implemented in Bayesian settings as a random effect model with a variance–covariance structure represented by σu2G+σɛ2I, assuming for example a scaled inverse χ2 density as a prior distribution for variance components σu2 and σɛ2.The GBLUP model is obtained by assigning independent identically distributed (IID) normal before the marker effects, that is: G∝XX′. Some authors ; we refer to this method as to GBLUP‐ldak.Above we described the GBLUP model that one obtains by regressing phenotypes on markers using IID normal priors for marker effects. This model can be fitted by either regressing phenotypes on markers explicitly, or using an equivalent model based on a genomic relationship matrix t densities . In practice it is convenient to represent this density as an infinite mixture of scaled‐normal densities: t=∫Nχ−2∂σβj2, where N is a normal density with null mean and variance σβj2 and χ−2 is a scaled‐inverse χ2 density with degree of freedom df and scale parameter S =πN+(1−π)N, where σβ12 reflects the variability in large effects and σβ22 is the variance component of small effects. An example for π=0.15 is represented in Figure S1.In the Spike‐Slab model, the prior assigned to marker effects is a mixture of two distributions: one (the spike) with small variance concentrated around zero that corresponds to small or no effects and the other (the slab) is a flat distribution with large variance that is linked to large marker effects. The spike can be represented by a continuous distribution centred at zero and with very small variance or by a point mass at zero. We concentrate on the prior introduced by George and McCulloch , a mixtuαj∼N, assigned to all markers and sparse effects γj∼πN+(1−π)δ0 , which are assigned to a proportion of markers π, so that the total effect of the jth SNP βj=αj+γj is a mixture of normal distributions πN+(1−π)N. Zhou et al. .All simulations as well as subsequent statistical analyses of simulated and real data were implemented in R , which is a consortium of 16 genome wide association studies. We used a subset of GENEVA consisting of data from the Nurses’ Health Study and the Health Professionals’ Follow‐up Study studies. Samples were genotyped using the Affymetrix Genome‐Wide Human SNP Array 6.0 with about 780 K SNPs. The GENEVA data set contains phenotypic and genotypic records of n = 5,961 individuals  with average age of 57.2 years (SD = 7.7 years) and average height 170.2 cm (SD = 9.6 cm). For the real data analysis we used adult height  as the phenotype.The genotypes used for simulation and in the real data analysis came from by NIH‐funded gene–environment association studies  and simulated phenotypes.r2 between genotypes at the QTL and those at the two flanking markers.SNPs were randomly divided into two subsets: 350K SNPs were designated as markers and the rest (∼323K) were used as a pool for sampling subsets of QTL . The 5K QTL were sampled from the pool of 323K loci either completely at random (RAND) or by oversampling among the loci with low minor allele frequency (LOW‐MAF). In this case sampling probabilities were set to target 75% of the QTL with MAF < 0.05, 25% of the QTL with MAF between 0.05 and 0.15, no QTL had a MAF > 0.15. In the LOW‐MAF scenario, the distributions of allele frequencies at markers and at QTL were expected to be different, and this was expected to influence the extent of LD between markers and QTL. Therefore, for each replicate, we used PLINK . In the second and third scenarios we randomly chose p = 50 or 250 SNPs, respectively, and sampled their effects from a normal density with a large (see next) variance, the rest of the QTL effects were sampled from a normal density with a smaller variance. We set the variance parameters of the two normal densities used to sample effects in scenarios 2 and 3 to target a heritability (h2) of 0.5 and a partition of the genetic variance  where large effect QTL explain either 25% or 75% of genetic variance in scenarios 2 and 3.We assumed that only a subset of QTL had large effects, whereas the rest of them had small effects. We considered three different scenarios: in the first one all QTL effects were sampled from IID normal densities yi=∑j=15000Zijβj+ɛi for i=1,...,n, where model error ɛi and marker effects βj follow normal distributions with zero mean and Zij are the genotype readings at causal loci. The variance of the residual term V(ɛi)=0.5 was kept fixed across all scenarios, whereas the variance of marker effects V(βj) varied from scenario to scenario, depending on the number of large effect QTL, amount of genetic variance explained by these large effects QTL, and the distribution of MAFs in QTL.The phenotypes were constructed according to an additive model We analysed the simulated data using markers, QTL or markers and QTL. The first scenario involved imperfect LD between markers and QTL, the last two contained the causal variants in the panel and therefore were perfect LD scenarios.hG2=σg2σg2+σɛ2 was defined as the ratio between the variance explained by genomic factors, σg2, and the phenotypic variance, σp2=σg2+σɛ2; in the G‐BLUP, hG2 was estimated based on posterior samples collected using the BGLR‐package.For the GBLUP, the estimated genomic heritability 2 distribution is assigned to the variance of the effects βj. Therefore, we have E(σβ2)=S0df−2; using this we can define the genomic variance as follows: σg2=∑j=1p2pj(1−pj)S0df−2, where pj stands for allele frequency at locus j. With this, the genomic heritability can be defined as hG2=∑j=1p2pj(1−pj)S0df−2∑j=1p2pj(1−pj)S0df−2+σɛ2. We also estimated this parameter using posterior samples collected using the BGLR‐package.For Bayes A, the BGLR‐package did not provide the estimates of genomic heritability directly. In this model, a scaled‐inverse χPVE=V(Xβ+u)V(Xβ+u)+τ−1  and random effect (u). Essentially. this quantity meets definition of genomic heritability, we used posterior mean of PVE to obtain the estimate of genomic heritability. In addition to estimates of genomic heritability, we report the R2 between phenotypes and predictions in the training data set as a measure of goodness of fit. This was only done for the GBLUP and Bayes A because GEMMA does not provide predictions for the training data set.GEMMA provided posterior samples of  cor  in the validation set.To assess prediction accuracy, in both the simulated and real data, we replicated 30 times a training–testing (TRN–TST) validation design  replicates in the RAND‐scenario was 0.624 with a standard deviation (SD) of 0.286. However, the average of pairwise. r2 in the LOW‐MAF‐scenario was three times smaller.LD is allele‐frequency dependent; therefore, based on results of Table The average (over MC replicates) estimated genomic heritabilities obtained by simulation scenario , statistical method , information used  and genetic architecture are shown in Figure pve = 0.75, p = 50 in Fig. pve = 0.75, p = 50 in Fig. When only QTL genotypes were used to fit models to data simulated with the RAND scenario Fig. C, the GBIt is important to note that, due to imperfect marker‐QTL LD when only markers are used in the analysis, the true proportion of variance that can be explained by regression on markers . In analyses including markers, (either markers only or markers+QTL), adding QTL to the set of loci used to compute the G matrix increased prediction accuracy when Bayes A or Spike‐Slab were used, whereas the GBLUP methods did not benefit from having the QTL loci within the set of markers used to compute the G matrix. As expected, the prediction accuracy obtained in the RAND scenario was higher than the one obtained in the LOW‐MAF scenario; this pattern was observed across statistical methods.For each MC replicate we computed differences in prediction accuracy, measured by differences in correlations Figure G . The correlations between phenotypes and predictions were low (0.16–0.17) for all methods, and only slightly higher for the GBLUP methods. These correlations are in agreement with what we obtained in the simulation study in the LOW‐MAF scenario when QTL were not used in the model . However, for complex traits and diseases, the variants identified so far usually explain a small fraction of inter‐individual differences in a trait or in disease risk, a problem referred to as the missing heritability of complex traits  and GBLUP‐ldak showed a clearly downwardly biased estimate in the RAND scenario.When the analysis was carried out using QTL genotypes only, the Spike‐Slab and GBLUP methods yielded estimates very close to the simulated heritability, while Bayes A and GBLUP‐ldak yielded substantial biases. In the case of Bayes A, the estimate was downwardly biased in scenarios where a few QTL made a substantial contribution to genetic variance ; we believe that this is not a limitation of the model per se but a consequence of the degree‐of‐freedom parameter being fixed. Estimating this parameter from the data, as done, for instance in Yi and Xu . The GBLUP and Spike‐Slab methods performed well in the RAND scenario, but had clear problems in the LOW‐MAF scenarios . However, GBLUP‐ldak exhibited some clear problems in the RAND scenarios  or upwardly biased estimates in the LOW‐MAF analysis based on markers and QTL). Finally, Bayes A showed somewhat erratic behaviour, especially with simple genetic architectures . This indicates that if one is able to narrow down the influential genetic regions of a trait to a limited number (5000 loci in our simulation), regularised regressions like the one used here can yield relatively high prediction accuracy. In these scenarios, the prediction accuracy of the GBLUP and GBLUP‐ldak methods was not affected by the genetic architecture and tended to be poorer than that of Bayes A and the Spike‐Slab methods. Bayes A and Spike‐Slab performed similarly and clearly better than any of the GBLUP methods in scenarios where a limited number of QTL  explained a sizable proportion of the genetic variance. However, with increase in trait complexity there was a decrease in prediction performance of these two methods, to the point that the three methods performed very similarly when the most complex genetic architecture was considered . Overall, our results are in agreement with previous studies in animal and plant breeding  of the genetic variance, the benefits of using methods that perform variable selection (Spike‐Slab) or differential shrinkage of estimated effects (Bayes A) relative to the GBLUP methods were pronounced. In the scenario with the simplest genetic architecture (50 QTL explaining 75% of the genetic variance) these methods, especially the Spike‐Slab were able to achieve levels of prediction accuracy comparable to those obtained when only QTL genotypes were used, illustrating the “oracle” property  and on the statistical model used. We believe that this model‐genetic architecture dependency has been overlooked so far. Importantly, the model that yields the highest estimated genomic heritability is not necessarily the one that yields the best prediction accuracy.Second, the assessment of prediction accuracy suggests that for traits in which a limited number of regions explain a sizable proportion of genetic variance, the use of WGR methods that perform variable selection or differential shrinkage of estimates of effects is strongly recommended over ridge‐regression type methods such as the GBLUP. However, for very complex traits such as human height, all the methods evaluated yield low prediction accuracy. It remains to be determined whether significant increases in sample size (which likely should be by orders of magnitude) will also yield substantial gains in prediction accuracy.Disclaimer: Supplementary materials have been peer‐reviewed but not copyedited.Figure S1 Prior distributions commonly used in Bayesian regression models.Click here for additional data file.Figure S2R2 statistic in training data sets.Click here for additional data file.Figure S3 Difference in prediction accuracy by scenario and data used.Click here for additional data file.Figure S4 Differences between GBLUP methods in the analysis of human height.Click here for additional data file.Table S1 Genomic heritability estimates obtained with the GBLUP method in the RAND scenario by: genetic architecture simulated, data used and Monte Carlo replicate.Click here for additional data file.Table S2 Genomic heritability estimates obtained with the GBLUP method in the LOW‐MAF scenario by: genetic architecture simulated. data used and Monte Carlo replicate.Click here for additional data file.Table S3 Average  correlation between simulated phenotype and predictions in Training data sets by method, simulation scenario and data used for the analysis.Click here for additional data file.Table S4 Average  R2 in validation data sets by simulation scenario, data used for analysis and estimation method.Click here for additional data file.Table S5 Correlation and R2 between human height and genomic predictions in testing data sets by method and testing set.Click here for additional data file.


Developments in genome‐wide association studies and the increasing availability of summary genetic association data have made application of Mendelian randomization relatively straightforward. However, obtaining reliable results from a Mendelian randomization investigation remains problematic, as the conventional inverse‐variance weighted method only gives consistent estimates if all of the genetic variants in the analysis are valid instrumental variables. We present a novel weighted median estimator for combining data on multiple genetic variants into a single causal estimate. This estimator is consistent even when up to 50% of the information comes from invalid instrumental variables. In a simulation analysis, it is shown to have better finite‐sample Type 1 error rates than the inverse‐variance weighted method, and is complementary to the recently proposed MR‐Egger (Mendelian randomization‐Egger) regression method. In analyses of the causal effects of low‐density lipoprotein cholesterol and high‐density lipoprotein cholesterol on coronary artery disease risk, the inverse‐variance weighted method suggests a causal effect of both lipid fractions, whereas the weighted median and MR‐Egger regression methods suggest a null effect of high‐density lipoprotein cholesterol that corresponds with the experimental evidence. Both median‐based and MR‐Egger regression methods should be considered as sensitivity analyses for Mendelian randomization investigations with multiple genetic variants. Over the past decade, Mendelian randomization has become an established tool for probing questions of causality when characterizing the etiology of disease  . However, a revolution in the field is under way led by the identification of increasing numbers of genetic variants robustly associated with particular traits, and the public release by many large consortia of summary association estimates for hundreds of thousands of genetic variants with exposures and disease outcomes  then the ratio estimates from each genetic variant can be combined into an overall estimate using a formula from the meta‐analysis literature  estimator . If the total number of genetic variants is odd (J=2k+1), the simple median estimator is the middle ratio estimate β^k+1. If it is even (J=2k), the median is interpolated between the two middle estimates 12(β^k+β^k+1). In terms of notation, we assume in this manuscript that genetic variants are ordered according to their ratio estimates.The IVW estimate is an efficient analysis method when all genetic variants are valid IVs. Unfortunately, it will be biased even if only one genetic variant is invalid. For this reason, the IVW estimate can be said to have a 0% breakdown level. However, an estimator exists that enjoys a 50% breakdown level; that is, it provides a consistent estimate of the causal effect when up to (but not including) 50% of genetic variants are invalid. This simple estimator is the median ratio estimate Han, . Specifiγ^j) and gene‐outcome (Γ^j) association estimates for an Mendelian randomization study with a finite sample size th percentile. For all other percentile values, we extrapolate linearly between the neighboring ratio estimates. The contribution of the jth genetic variant to the empirical distribution is proportional to its weight wj. The simple median estimator can be thought of as a weighted median estimator with equal weights. Although the simple median provides a consistent estimate of causal effect if at least 50% of IVs are valid, the weighted median will provide a consistent estimate if at least 50% of the weight comes from valid IVs. We assume that no single IV contributes more than 50% of the weight, otherwise the 50% validity assumption is equivalent to assuming that this IV is valid . Some technical remarks on weighted medians are given in Supporting Information Appendix 1.The simple median estimator is inefficient, especially when the precision of the individual estimates varies considerably. In order to account for this, a weighted median can be defined as follows. Let β^3 and β^4, but will be closer to β^4 as the percentile p4 is closest to 50%. The exact weighted median estimate in this case will beβ^1 for variant 1 appears 100×w1 times, ratio estimate β^2 for variant 2 appears 100×w2 times, and so on. R code to calculate weighted median estimates, confidence intervals, standard errors and P‐values is provided in Supporting Information Appendix 2.As an illustration, two sets of weights are given in Table wj=wj′∑jwj′. The unstandardized weights are identical to those used in the IVW estimator. Only the first‐order term from the delta expansion is used here; further terms could be considered, although we found that they did not affect estimates or standard errors from the weighted median method substantially.Analogously to the IVW method, we suggest using the inverse of the variance of the ratio estimates as weights:In Figure β^ to be the IVW estimate  approximately have chi‐squared distributions with 1 degree of freedom. So as not to distort the weightings of the majority of variants, we propose penalization using the one‐sided upper P‐value (denoted qj) on a chi‐squared 1 distribution corresponding to Qj, by multiplying the weight by the P‐value multiplied by 20 . The (unstandardized) penalized weights (wj*) are thereforeOne way of minimizing this problem is downweighting the contribution to the analysis of genetic variants with heterogeneous ratio estimates. Heterogeneity between estimates can be quantified by Cochran's Q statistic:Γ^j on the gene‐exposure coefficients γ^j:γ^j associations are orientated to be positive , and the weights in the regression are the inverse variances of the gene‐outcome associations (σYj−2). Reorientation of the variants is performed as the orientation of genetic variants is arbitrary , and different orientations of genetic variants change the estimate of the intercept, as well as the sign and magnitude of the pleiotropic effect of the genetic variant. If there is no intercept term in the regression model, then the MR‐Egger slope estimate β^E will equal the IVW estimate  assumption. If the association of the jth genetic variant with the outcome Γj=βγj+αj, where αj is the pleiotropic (direct) effect of the variant, then the InSIDE assumption states that the pleiotropic effects αj must be distributed independently of the instrument strength parameters γj  The InSIDE assumption is likely to be satisfied if pleiotropic effects on the outcome are direct . There is some empirical evidence supporting the proposition that genetic effects on separate exposures are independent ; the simulations are repeated in Supporting Information Appendix 3 in a one‐sample setting. Only the summary data on the genetic associations with the exposure and with the outcome (and their standard errors) are used as data inputs in the analysis methods . Details of the data‐generating model and the parameters used in the simulation study are given in Supporting Information Appendix 3.We generated 10,000 simulated datasets for each scenario in a two‐sample setting with two values of the causal effect , estimates from the IVW method are biased with inflated Type 1 error rates. Estimates from the weighted median methods are less biased, although there is a consistent bias in the direction of the pleiotropic variants. Nominal Type 1 error rates are maintained when only 10% of genetic variants are invalid IVs, although Type 1 error rates are above the expected 5% rate when 20% or more genetic variants are invalid IVs. However, even though they are inflated, Type 1 error rates are far lower for the median‐based methods than those from the IVW method. The penalized weighted median estimates are less biased when a few genetic variants are invalid, but when 30% of genetic variants are invalid, it seems that the invalid IVs are often retained in the analysis (as their estimates are homogeneous with each other) and valid IVs are excluded. Estimates from MR‐Egger regression are close to unbiased, and Type 1 error rates are at nominal levels, suggesting its utility as a sensitivity analysis method when the InSIDE assumption is satisfied. However, the wide confidence intervals and low power with a true causal effect (close to 5%) mean that MR‐Egger regression is not likely to be a discerning sensitivity analysis, but rather giving conservative findings as to whether a causal effect is present or not. Although power to detect a causal effect is greater in the IVW method, this is achieved at the cost of the Type 1 error rate; comparisons of power that do not make reference to the Type 1 error rate are meaningless.In Scenario 3 , all methods suffer from bias and inflated Type 1 error rates. Most concerning are results from the MR‐Egger method, which are more biased than those from the IVW method and have similar Type 1 error rates. The weighted median methods, and in particular the penalized weighted median method, give lower Type 1 error rates, suggesting their potential use as a sensitivity analysis method when InSIDE is not satisfied, as well as when it is satisfied.In Supporting Information Tables A1 and A2, results are presented in a one‐sample setting. Each of the methods suffers from weak instrument bias in this setting, and Type 1 error rates are inflated. Estimates from MR‐Egger regression are substantially more biased than those from other methods. However, the median‐based methods remain a reasonable sensitivity analysis, as Type 1 error rates are similar to those of the IVW method in Scenario 1, and substantially lower in Scenarios 2 and 3. In Supporting Information Table A3, results are presented in a two‐sample setting for the simple median estimator, and a weighted median estimator using inverse‐standard error weights rather than the inverse‐variance weights used above. Simple median estimates are slightly less precise than those from the weighted median methods, and have similar bias and Type 1 error properties in Scenarios 1 and 2, but much improved Type 1 error rates in Scenario 3. This is because the invalid variants in this scenario are stronger than the valid variants, and so receive more weight in the weighted analyses. It is unclear that this would occur in applied practice, although it suggests that the simple median method is a worthwhile additional sensitivity analysis. The inverse‐standard error weighted median estimator has improved Type 1 error properties compared with the IVW median estimator particularly in Scenario 3 , indicating the potential to use different choices of weights in median‐based methods to provide further sensitivity analyses.LDL‐c is observationally positively associated with CAD risk . Second, we restrict to genetic variants for which the P‐value for association with the target exposure  is less than the P‐values for association with the nontarget exposures . We perform each of the methods presented in the manuscript . The data are presented as scatter plots in Figure Taking each of LDL‐c, HDL‐c, and additionally triglycerides as the exposure variable in turn, we consider two analysis strategies. First, we take all genetic variants associated with the exposure at a genome‐wide level of significance , hence pleiotropic effects on CAD risk may operate via LDL‐c and triglycerides. This may be the reason why the MR‐Egger estimate changed sign in the analysis using variants having primary association with HDL‐c.In this paper, we have introduced a weighted median method for the estimation of a causal effect using multiple IVs. Unlike other methods commonly used in Mendelian randomization, this method can give consistent estimates when some of the genetic variants in the analysis are not valid IVs. We have shown how the method performs in a simulation study and in an applied example.A summary of the median‐based and other methods considered in this paper is given in Table Although the only mechanism for generating invalid IVs considered in this paper was pleiotropy, median‐based methods are agnostic to the mechanism by which the invalid IVs violate the IV assumptions. Consistent estimates would be guaranteed if some genetic variants were invalid IVs due to other mechanisms, such as linkage disequilibrium, population stratification, or differential survival  provides a sensitivity analysis that either adds to or questions the robustness of a finding from a Mendelian randomization investigation. If, as in the case of the effect of LDL‐c on CAD risk, a causal effect is reported across all methods, then a causal finding is far more plausible than if the methods give contradictory findings. Our advice in Mendelian randomization investigations using multiple genetic variants where the IV assumptions are in doubt for some or all genetic variants, would therefore be to perform and report results from a range of sensitivity analyses using robust methods, including the simple median, weighted median, and MR‐Egger regression methods, in addition to the main analysis result.Disclaimer: Supplementary materials have been peer‐reviewed but not copyedited.Supporting Information S1Click here for additional data file.Web Table A1: Results from simulation study in one‐sample setting with null causal effectWeb Table A2: Results from simulation study in one‐sample setting with positive causal effectWeb Table A3: Results from simulation study in two‐sample setting for addi‐ tional methodsWeb Figure A1: Scatter plots of genetic associations for each pair of lipid fractions  in turn. Left side: all genetic variants, right side: genetic variants having primary association with the target exposure.Web Figure A2: Funnel plots of instrument strength (defined as genetic association with exposure divided by standard error of genetic association with outcome: γ^jσYj) against ratio estimates (defined as genetic association with outcome divided by genetic association with exposure: Γ^jγ^j) for each genetic variant, and each exposure  in turn. Left side: all genetic variants, right side: genetic variants having primary association with the target exposure.Click here for additional data file.Figure 1: Illustrative diagram representing the hypothesized relationships between ge‐ netic variant Gj , exposure X, disease Y and confounders U when Gj is a valid instrumental variable (IV). Crosses indicate violations of assumptions IV2 and IV3 that potentially lead to invalid inferences from conventional methods.Figure 2: Fictional example of a Mendelian randomization analysis with 10 genetic variants – 6 valid instrumental variables (hollow circles) and 4 invalid instrumental variables (solid circles) for finite sample size (left) and infinite sample size (right) showing inverse variance weighted  and simple median (dashed line) estimates compared with the true causal effect (dotted line).Figure 3: Empirical distribution functions of ordered ratio instrumental variable es‐ timates (β^j) used for calculation of the simple median estimate (black) and two weighted median estimates (shown in red and blue) using the weights given in Table I.Figure 4: Scatter plots of genetic associations with the outcome  against genetic associations with the exposure . Left side: all ge‐ netic variants, right side: genetic variants having primary association with the target expo‐ sure. Solid line represents IVW estimate, dashed line represents weighted median estimate, and dotted line represents MR‐Egger estimate.Click here for additional data file.Supporting Information S1Click here for additional data file.


To date, little consideration has been given to the performance of gene-based tests  when the number of SNVs in the gene or in the pathway is very large and the genotypes come from sequence data organized in large pedigrees. We consider recently proposed gene-based tests for rare variants from complex pedigrees that test for association between a large set of SNVs and a qualitative phenotype of interest  as well as 2-stage approaches. We find that many of these methods show inflated type I errors when the number of SNVs in the gene or the pathway is large (>200 SNVs) and when using standard approaches to estimate the genotype covariance matrix. Alternative methods are needed when testing very large sets of SNVs in 1-stage approaches.Pathway analysis approaches for sequence data typically either operate in a single stage  or in 2-stages (gene-based  Until recently, the majority of methodological approaches for the analysis of common, single-nucleotide variants  involved analysis of microarrays using either single-marker or multiple-marker approaches, with single-marker approaches, by far the more common of the 2 approaches in practice. Typically, large genome-wide association studies will conduct hundreds of thousands (or millions) of single-marker analyses. Although less common in practice, considerable methodological development has taken place in the area of multiple-marker analysis in which signals from multiple common SNVs are aggregated into a single test of association for the set of SNVs of interest  technology, there has been a tidal wave of methodological developments to analyze such data. At the heart of most analytic approaches for NGS data is the desire to appropriately handle rare variants. Because statistical tests of individual rare variants lack power and multiple testing penalties quickly become unwieldy because of the preponderance of rare variants, most recently proposed NGS data analysis approaches attempt to aggregate multiple SNV association signals so as to increase power. These rare SNV-set methods can be viewed as a special type of multiple-marker test that is particularly useful when SNVs are rare. For a broad overview and classification of these rare variant approaches for case-control studies see Liu et al .There are few family-based tests of rare variant association that are appropriate for complex pedigrees analyzing dichotomous phenotypes for association with rare variants  or not hypertensive based on whether the individual was classified as hypertensive at any of up to 4 measurements (waves). For our analysis of the simulated data we focused on simulated phenotype 1. We had knowledge of the simulation answers for our analysis. SNVs were mapped to genes using a custom version of ANNOVAR, where a SNV is assigned to a gene if its physical location is within the start-stop position of the gene ) or uses eigenvectors from the genotype matrix to reduce matrix dimensionality . All 1-stage pathway analysis tests are of the null hypothesis that no SNVs in the set are associated with the phenotype, with an alternative hypothesis that at least 1 SNV in the set is associated with the phenotype.p values for each test-set combination using Fisher's combined probability test . Fisher's combined probability test is defined as i pis the p value from the ith gene in the set and p value for each of 2T, CMC5%, CMC0.5% , FPCA, and p values being combined are independent . All 2-stage analyses have a null hypothesis that none of the genes in the set of genes contain any SNVs associated with the phenotype, with an alternative hypothesis that at least 1 gene in the set of genes contains at least one SNV associated with the phenotype.Second, we applied each of the 5 methods to all 5 individual genes contained within each of the 800 sets. We then combined the 5 gene When the Zhu and Xiong approaches are applied to the 800 sets of SNVs, a dramatic inflation of the type I error rate is observed. In particular, among the 5 single-stage approaches, type I error rates ranged from 43.5% to 99.5%, and among the 5 two-stage approaches, type I error rates ranged from 7.5% to 90.5%. Although we do not provide detailed results here, we note that the magnitude of the type I error rate increased as the number of SNVs in the set increased; in particular, the inflated type I error rates only occurred on sets containing more than 200 SNVs. We determined that a potential cause of the inflated type I error rate was in the approach taken when estimating the SNV genotype covariance matrix [10]. When analyzing large sets (>200) of SNVs and given the relatively small sample size (n = 849), prior research shows that the use of the maximum likelihood estimates (MLEs) in covariance matrix estimation may yield unstable results, and that a shrinkage covariance estimator may perform better [10].We modified the covariance estimation procedure for all methods, replacing the MLEs with the shrinkage estimator. Although we lost the guarantee of the analytic null distribution derived by Zhu and Xiong, we explored the use of the shrinkage estimator using the null distribution in Zhu and Xiong. We found that this approach provided increasingly overconservative results as the set size increased (detailed results not shown). Nonetheless, we still expect that large values of the test statistic should lead to rejection of the null hypothesis.p values for the "null" sets  separately as a function of the number of SNVs in the set for each of the test statistics. In particular, i,m, I = 1...kp, are the p values for test statistic m  when the set size is n. To obtain robust estimates of F(t), we binned sets of similar set size; there were 5 bins in total, representing quintile breaks in the null set size distribution. To provide appropriate control of the type I error rate and allow evaluation of the different methods, we computed a modified p value, p values necessarily control the type I error rate for null sets, allowing us to get a sense of the performance of the different methods if we had an appropriate null distribution for the new test statistics. Although in practice we do not have knowledge of the null set of pathways, this serves as an exploratory analysis to motivate derivation of a closed form null distribution for the modified Zhu and Xiong statistic.To account for the effect of set size on the behavior of the statistic, we estimated the empirical cumulative distribution function of the We applied each of the 10 tests described earlier to all 800 sets of genes. Table 0.5%, while other methods showed little difference in the percent of significant null sets and nonnull sets.By design, sets containing no causal genes have empirical type I error rates of approximately 5%. The reason for some deviation from 5% in Table Few analyses have considered methods to analyze pathways  for association with rare variants in family studies. Previous research showed that, depending on the underlying genetic architecture, either single-stage or 2-stage approaches to pathway testing may provide a powerful testing approach. However, this result has not been rigorously established across the large class of pathway testing approaches, especially rare variant pedigree data. In this article, we considered both 1-stage and 2-stage approaches to pathway analysis by applying methods proposed by Zhu and Xiong to very large multi-SNV sets (1-stage), or a 2-stage approach using Fisher's combined probability test in conjunction with the methods of Zhu and Xiong.As with most gene-based testing approaches the authors of the primary methods considered here only evaluated their method on "small" sets of SNVs 189 SNVs) [ SNVs [9]p values, which allowed us to examine performance of the 10 methods.In our consideration of 1-stage approaches, we found a substantially inflated type I error rate, which increased as the number of SNVs increased. Further analysis pinpointed the issue to the use of MLEs when estimating the genotype covariance matrix (as in the Zhu and Xiong code). Schafer and Strimmer [10] showed that a shrinkage estimator provides robust covariance estimates as the number of SNVs increases relative to the sample size. When we implemented the shrinkage estimator, the results were overly conservative  when using the null distribution for MLEs from Zhu and Xiong. This pattern of findings also held true for 2-stage approaches when any genes in the set were large (containing more than 200 SNVs). To address these limitations, we applied an empirical correction factor to the Further research is necessary to develop alternative covariance estimation procedures and corresponding null distributions for large (more than 200 SNVs) sets. We note that, in both a companion paper  and here∞ norm) will perform better than 2T-based (L2 or L1 norm) type tests (see Liu et al [Assuming that appropriate finite sample null distributions can be derived when applying SNV-set methods to very large sets of SNVs, the true underlying genetic disease architecture will play a significant role in determining which statistical methods will perform best in practice. A recent article evaluating the differences in rare variant tests of association illustrated that for sets of SNVs where the proportion of noncausal SNVs is very large, as will likely be the case in 1-stage approaches for rare variants, tests like the minimum iu et al  for detaFew methods exist for the analysis of potential relationships between binary phenotypes and rare genetic variation; fewer still have considered how such methods will perform on very large sets of SNVs that may span multiple genes. Our analysis identified substantial inflation of the type I error rate using a standard approach, and so we implemented an empirical approach to evaluate the relative performance of different methods. Further research is necessary to explore robust test statistics and analytic strategies for large SNV sets in complex pedigrees across a wide variety of genetic architectures.The authors declare that they have no competing interests.All authors participated in design of the overall study; AL and AB carried out various aspects of data preprocessing, including gene mapping and building preliminary data analysis files; BG estimated the kinship matrices; CA and AH implemented the 4 FBAT tests; AL, BG, and NT ran statistical tests, analyzed data, and conducted literature review. AL investigated a novel statistic; NT drafted the final manuscript. All authors read and approved the final manuscript.


The consequences of preterm birth are a major public health concern with high rates of ensuing multisystem morbidity, and uncertain biological mechanisms. Common genetic variation may mediate vulnerability to the insult of prematurity and provide opportunities to predict and modify risk.To gain novel biological and therapeutic insights from the integrated analysis of magnetic resonance imaging and genetic data, informed by prior knowledge.We apply our previously validated pathway‐based statistical method and a novel network‐based method to discover sources of common genetic variation associated with imaging features indicative of structural brain damage.PPAR (peroxisome proliferator‐activated receptor) signaling was the highest ranked pathway once degree of prematurity was accounted for. Within the PPAR pathway, five genes were found by Graph Guided Group Lasso to be highly associated with the phenotype: aquaporin 7 (AQP7), malic enzyme 1, NADP(+)‐dependent, cytosolic (ME1), perilipin 1 (PLIN1), solute carrier family 27 (fatty acid transporter), member 1 (SLC27A1), and acetyl‐CoA acyltransferase 1 (ACAA1). Expression of four of these  is controlled by a common transcription factor, early growth response 4 (EGR‐4).Lipid pathways were highly ranked by Pathways Sparse Reduced Rank Regression in a model examining the effect of prematurity, and PPAR signaling.This suggests an important role for lipid pathways in influencing development of white matter in preterm infants, and in particular a significant role for interindividual genetic variation in  Preterm birth accounts for 11% of all births  provides measures of white matter structure that are correlated with neurodevelopmental outcome , posing a statistical and analytical problem. Two current approaches are to either increase n significantly or find a principled way to reduce p while preserving the underlying signal. We have addressed this problem by developing a pathways‐driven sparse regression method (PsRRR) ; b value: 750 sec/mm2; SENSE factor: 2. T1‐weighted 3D MPRAGE were acquired with parameters: TR = 17 msec, TE = 4.6 msec, inversion delay = 1500 msec, flip angle = 13°, acquisition plane = sagittal, voxel size = 0.82 × 1.03 × 1.6 mm, FOV = 210 ×167 mm and acquired matrix = 256 × 163. T2‐weighted fast spin‐echo: TR × 8700 ms, TE = 160 msec, flip angle = 90°, acquisition plane = axial, voxel size = 1.15 × 1.18 × 2 mm, FOV = 220 mm, and acquired matrix = 192 × 186.MR images were acquired for 72 preterm infants  28 + 4 weeks, mean postmenstrual age (PMA) at scan 40 + 3 weeks). Imaging was performed on a Philips 3‐Tesla system  using an eight‐channel phased array head coil. Single‐shot echo‐planar diffusion tensor imaging was acquired in the transverse plane in 15 noncollinear directions using the following parameters: repetition time (TR): 8000 msec; echo time (TE): 49 msec; slice thickness: 2 mm; field of view: 224 mm; matrix: 128 × 128 (voxel size: 1.7531 × 1.753 × 2 mmwww.fmrib.ox.ac.uk/fsl) (Smith and Nichols b = 0) image and corrected for differences in spatial distortion owing to eddy currents. Images were brain extracted using Brain Extraction Tool (BET v2.1), diffusion tensors calculated voxelwise, and FA maps generated.Diffusion tensor imaging (DTI) analysis was performed by using FMRIB's Diffusion Toolbox  as implemented in FMRIB's Software Library   distance as implemented in PLINK 1.9  was collected by asking mothers (and fathers when present) to define themselves according to a list of options. The terms were drawn from Ethnic Category National Codes as in Department of Health Guidance at the time. Parental self‐reported ethnicity was summarized into broader categories for the purposes of data visualization by aggregating all “White” subcategories into a single group “White”, all “Black” subcategories into “Black”, and all “Asian” subcategories into “Asian”. In cases where either one parent self‐reported as “Mixed” or if there was a discrepancy between maternal and paternal ethnicities, the term “Mixed” was applied. Where parents were both from an ASEAN member state (two cases) the individual was classified by the authors as “SE Asian”. These aggregated ethnic categories were used to label the datapoints of the PCA plot of the first two principal components of the IBS variance‐standardized relationship matrix. This illustrates the correspondence between the first two components of genetic ancestry and ethnicity, and provides an overview of the cohort population mixture.λ 0.99, 100 subsamples, 20 iterations with 2000 × 10 plus 4000 × 10 model fits per iteration. Model adjusted for GA and PMA; λ 0.99, 100 subsamples, 20 iterations with 2000 × 10 plus 4000 × 10 model fits per iteration.Pathways sparse reduced rank regression estimates the regression coefficients in the linear model with multivariate responses, subjected to constraints. The model accounts for potential biasing factors such as pathway linkage disequilibrium and size by using an adaptive, weight‐tuning procedure. Pathway weightings in the regression model are adjusted according to the empirical bias in pathway selection frequencies, obtained by fitting the PsRRR model with a null response. Depending on the degree of penalization, some coefficients are driven to zero, thus performing variable selection so that only SNPs within associated pathways are retained in the model. Pathways are ranked in order of importance using a resampling strategy, with highest ranked pathways having highest selection frequency and highest correlation with the phenotype. SNPs were mapped to genes (NCBI GRCh37) and KEGG pathways (Kyoto Encyclopedia of Genes and Elements), after excluding cancer‐related pathways due to high redundancy in gene membership as previously described  and brain imaging features, applying an additional penalty function on the regression coefficients that incorporates network structure information. The model selects functionally related genes that are associated with the trait and identifies important SNPs within the selected genes. GGGL‐1 further imposes that SNPs in functionally related genes have similar effects on the phenotypic trait.μ controls the weight of the prior knowledge added to the squared loss function. When μ is large the model relies more on prior knowledge from the network. Using GGGL‐1 with μ = 0.1, 1, and 10 resulted in the same set of genes with selection probability > 0.4 (a threshold representing a step change in the probability distribution). Using GGGL‐2 with μ = 0.1, 1, 10 also yielded very stable results, with 27 of the top 30 SNPs appearing in all three lists for different values of μ, and the same genes were selected in the three instances. Both versions of the method (GGGL‐1 and GGGL‐2) produced comparable results, with the same genes and SNPs being ranked highly by both approaches (Tables S8 and S9). Genes with more than one occurrence and a selection probability threshold >0.4 were retained, at a threshold representing a step change in the probability distribution.The regularization parameter A group white matter skeleton was constructed and used to extract voxel‐wise fractional anisotropy (FA) values for each of the 72 individuals. These values were serially adjusted for important clinical variables: effect of gestational age (GA) at birth, effect of postmenstrual age (PMA) at scan and genetic ancestry, resulting in three separate phenotypes. In all instances the phenotypes underwent dimension reduction with principal component analysis (PCA). The elbow of the scree plot was selected in each case to determine the number of components used for the subsequent analysis .The statistic images Fig.  indicateWe use the term population stratification to refer here to allele frequency differences between subpopulations within a population. Genetic ancestry is a cause of population stratification, and the term here refers to the use of empirical methods to assign an ancestry classification  as summarized in Table S4.The PsRRR method was used to identify and rank biological pathways predictive of white matter integrity. Pathways were ranked by stability selection in order of selection frequency, with each adjusted phenotype analyzed in turn Table . These eP ≤ 0.005) (Table Lipid pathways were significantly over‐represented in the top ranking pathways adjusted for PMA (5) Table , bold an5) Table .The GGGL method was applied to genes in the most highly ranked KEGG pathway  . This alP‐value 7.7 × 10−4.Analysis of transcriptional regulation using the PASTAA algorithm  , four of which are under common transcriptional control by EGR‐4.The focus of this study is on white matter development among preterm infants, and in this population TBSS detects variability in white matter features that are related both to adverse perinatal events and later neurodevelopment  at birth, which can be viewed as degree of prematurity and length of early separation from the placenta, and postmenstrual age at scan (PMA), which captures ongoing development ex utero. It is known that FA increases in white matter as part of ongoing development in term and preterm infants  and this is linked to degree of prematurity (GA). On adjustment for GA the main remaining effects are likely to be genetic influence and environmental variables. In this latter scenario, key lipid metabolic pathways remain an important determinant of white matter integrity in the context of a greater variety of biological processes. It is tenable that the genes highlighted here are involved in processes contributing to normal white matter myelination, and that the disruption of these physiological processes in a subset of preterms contributes to the observed variability in FA within the preterm group.Multiple sources of evidence indicate that a well‐balanced and carefully timed fatty acid supply during the neonatal period is a determinant of growth, visual development, and cognitive development (Fleith and Clandinin The peroxisome proliferator‐activated receptors (PPARs) are ligand‐activated transcription factors belonging to the superfamily of nuclear hormone receptors, with an important role in nutrient homeostasis. They are involved in cell membrane structure, signaling, inflammation and biotransformation  in modifying the phenotypic consequences of an allele and its association with disease risk. The impact of alleles may be modified by the local genetic architecture  and GA plus PMA (right).Figure S2. Frequency density distributions of residuals in TBSS phenotypes adjusted for PMA, GA and PMA, or GA, PMA, and ethnicity.Figure S3. First two components from principal component analysis of population stratification based on pairwise identity by state (IBS).Table S4. Mapping of SNPs to genes and pathways.Figure S5. Null selection frequencies for all KEGG pathways in the PsRRR model.Table S6. PsRRR pathway rankings with null and empirical selection frequencies adjusted for PMA.Table S7. PsRRR pathway rankings with null and empirical selection frequencies adjusted for GA and PMA.Table S8. GGGL‐1 gene selection frequencies >0.4.Table S9. GGGL‐2 top thirty SNPs.Click here for additional data file.


Every known link between a genetic variant and blood pressure improves the understanding and potentially the risk assessment of related diseases such as hypertension. Genetic data have become increasingly comprehensive and available for an increasing number of samples. The availability of whole-genome sequencing data means that statistical genetic models must evolve to meet the challenge of using both rare variants (RVs) and common variants (CVs) to link previously unidentified genome loci to disease-related traits. Penalized regression has two features, variable selection and proportional coefficient shrinkage, that allow researchers to build models tailored to hypothesized characteristics of the genotype-phenotype map. The following work uses the Genetic Analysis Workshop 18 data to investigate the performance of a spectrum of penalized regressions using at first only CVs or only RVs to predict systolic blood pressure (SBP). Next, combinations of CVs and RVs are used to model SBP, and the impact on prediction is quantified. The study demonstrates that penalized regression improves blood pressure prediction for any combination of CVs and RVs compared with maximum likelihood estimation. More significantly, models using both types of variants provide better predictions of SBP than those using only CVs or only RVs. The predictive mean squared error was reduced by up to 11.5% when RVs were added to CV-only penalized regression models. Elastic net regression with equally weighted LASSO and ridge components, in particular, can use large numbers of single-nucleotide polymorphisms to improve prediction. The potential number of lives affected by successful early identification of patients at high risk for hypertension has motivated researchers across a spectrum of fields. On the frontier of risk prediction is the identification of genetic variants linked to traits such as high blood pressure. Advancements in sequencing have fostered the identification of a growing number of loci related to blood pressure. One such study performed by the International Consortium for Blood Pressure Genome-Wide Association Studies identified 29 single-nucleotide polymorphisms (SNPs) related to systolic blood pressure (SBP) . A seconOne promising tool that may be able to leverage risk information simultaneously in both CVs and RVs is penalized regression. The range of available penalties allows researchers to estimate models with a mixture of two desirable properties: variable selection and proportional shrinkage of regression coefficients. The following work systematically measured the advantages of the different types of penalized regression methods in the prediction of SBP using only CVs, only RV, or combinations of the two.The primary source for genotypic, phenotypic, and covariate data was Genetic Analysis Workshop 18 (GAW18) data files. GAW18 data is provided for approximately 1000 Mexican American individuals consisting of 20 pedigrees enriched for type 2 diabetes. The pedigrees contained between 21 and 76 individuals. The phenotype of interest was the SBP measure from the first time point. Genotype data for more than 8,000,000 genome locations was derived from sequencing data for all odd-numbered chromosomes, representing all sequencing data made available by GAW18. Approximately one-third of the variants were common. The analysis accounted for the covariates age, gender, smoking status, and antihypertensive medication.The pairwise correlation structure resulting from either a family structure or a cryptic population structure was removed using an estimate of the variance-covariance matrix. We estimated the variance-covariance structure as a function of the identity-by-state (IBS) matrix calculated from all available genome-wide association study data. EMMAX software  was usedi Ybe the SBP value at the first examination for subject i's minor allele count  for SNP n samples; ε is a β. More specifically,Let Y and X by Y*, as a function of the new genotype and covariate matrix, X*. To be clear, the model used in the current study is Y*=X*β+ε*, where where S matrix . Kang etS matrix . For Kan mapping . The prep as p. One possible remedy is to introduce regularization or penalization on regression coefficients. We obtained predictions of Y * by first obtaining We first consider the unpenalized regression model. MLE is asymptotically unbiased with fixed kelihood : lβ-λPβ.L1-penalty (TLP) [α, which is restricted to 0[Candidate penalties that perform variable selection are LASSO , SCAD 77, and thty (TLP) . LASSO rgression . Ridge rgression  is a hybcted to 0 is Pβ=1-n = 253 for each), and a sequence of models was then fit on the training set. The sequence was defined by incremental increases in both the penalty and penalty-specific parameters . The sequence of penalty (and tuning parameter when applicable) values used to fit the models spanned a range comprehensive enough to allow identification of the values which optimized performance for SCAD, LASSO, elastic net, and ridge regression. The additional tuning parameter, We restricted our study to the top 1000 CV SNPs and top 1000 RV SNPs as identified by the marginal significance of a Kruskal-Wallis test of the minor allele counts and SBP values for the 759 samples. The real-data observations were randomly divided into equally sized training, tuning, and testing sets . Define Models were fit in a directed way based on the number and type of variants. First, we examined only the top 10, 100, and 1000 most significant CVs. Then we repeated the examination using only the top 10, 100, and 1000 RVs. Next, we added 1, 10, 100, and 1000 of the complementary type of variant to the model. For example, after fitting a model with only the top 10 CVs, four models were fit using these same 10 CV SNPs s glmnet  and ncvrs glmnet . TLP ests glmnet .Y* from the 100 randomly created testing data sets are presented in Figure Y* for the best model using the fixed number of CVs and 1, 10, 100, or 1000 RVs. For example, the column labeled CV = 10,RV>0 gives the smallest PSME from the four models using exactly the top 10 CVs and the top 1, 10, 100, or 1000 RVs. Similarly, the fourth column describes the model with the smallest PMSE using the fixed number of RVs and 1, 10, 100, or 1000 CVs. Figures Descriptions of the PMSE of It is evident from Table Comparisons across models based on the top 10, top 100, and top 1000 SNPs revealed an interesting pattern. As the number of candidate SNPs increased, the sparse SCAD, LASSO, and TLP penalties were generally superior to the nonsparse ridge penalty. Differences were small, at most 0.1555 mm Hg, and need confirmation on different SBP real data sets. The conclusion should also be corroborated with simulated SBP data sets generated from genetic models reflecting a comprehensive range of possible SBP genetic architectures. Furthermore, although reductions in PMSE occurred within the same variant composition across the three top SNP groupings , the gains were often less than those made just by adding the complementary type of variant to the model. Combined, these two results suggest that the true number of strong causative variants is at most moderate and includes both RVs and CVs. Ridge regression was the best or nearly identical to the best penalty choice when only the top 10 CVs or RVs were used, indicating that all of these top variants are integral in understanding the association between genotypes and SBP. TLP was a top performer with models using only the top 10 or top 100 RVs. As more SNPs of any type were included, the elastic net equally weighted to LASSO and ridge was generally superior. That is, there was a need for a selection element to distinguish noise from true effect, and there was a need for a nonsparse penalty feature to still incorporate larger numbers of SNPs in the regression model. This perhaps indicates that beyond a small set of strong causative SNPs, there are many SNPs that are truly associated with the outcome, but the majority of them have small marginal effects sizes. This could prove important when considering that previous research has found at least 29 causative SNPs; thus, undiscovered variants associated with SBP may have at most moderate effect sizes.The strongest conclusion can be drawn about the effect of including RVs in addition to CVs when predicting SBP. The PMSE was reduced by up to 11.5%, and generally reduced between 4% and 9%, when RVs were added to CV-only penalized regression models. This was true when any of 10, 100, or 1000 top SNPs were used. PMSE comparisons of single-variant type models to combined-variant type models revealed that both RVs and CVs explain variance in SBP. Every penalty considered in the study improved SBP prediction over OLS. This was true whether estimation used only CVs, used only RVs, or used both types of variants. The elastic net penalized regression was best at leveraging the information in the additional SNPs (RVs or CVs) and produced the best overall models.  Caution is needed when making conclusions about the TLP because of the limited number of combinations of λ and The authors declare that they have no competing interests.WP designed the study framework, helped develop methodology, and helped rewrite the manuscript; EA performed the statistical analysis, drafted the manuscript, and helped rewrite the manuscript; and XS helped develop methodology and helped develop software used to conduct the analysis. All authors read and approved the final manuscript.


The Hovawart is a working and companion dog breed of German origin. A few hundred Hovawart dogs are registered annually in Finland. The most common disease with a proposed genetic background in Hovawarts is hypothyroidism. The disease is usually caused by lymphocytic thyroiditis, an autoimmune disorder which destroys the thyroid gland. Hypothyroidism can be treated medically with hormone replacement. Its overall incidence could also be reduced through selection, provided that the trait shows an adequate genetic basis. The aim of this study was to estimate the heritability of hypothyroidism in the Finnish Hovawart population.The pedigree data for the study were provided by the Finnish Kennel Club and the hypothyroidism data by the Finnish Hovawart Club. The data included 4953 dogs born between 1990 and 2010, of which 107 had hypothyroidism and 4846 were unaffected. Prior to the estimation of heritability, we studied the effects of gender, birth year, birth month, and inbreeding on susceptibility to hypothyroidism. Heritability was estimated with the probit model both via restricted maximum likelihood (REML) and Gibbs sampling, using litter and sire of the dog as random effects. None of the studied systematic effects or level of inbreeding had a significant effect on susceptibility to hypothyroidism. The estimated heritability of hypothyroidism varied from 0.47 (SE = 0.18) using REML to 0.62 (SD = 0.21) using Gibbs sampling.Based on our analysis, the heritability of hypothyroidism is moderate to high, suggesting that its prevalence could be decreased through selection. Thus, breeders should notify the breed association of any affected dogs, and their use for breeding should be avoided. Hypothyroidism is one of the most common hereditary diseases in Finnish Hovawart dogs, and actually the most common endocrine disease in all dogs . The preThe list of dogs diagnosed with hypothyroidism was provided by the Finnish Hovawart Club. Only dogs born between 1990 and 2010 were used in this study resulting to 107 affected dogs in the data. The prevalence of hypothyroidism in Finnish Hovawart dogs by birth year is presented in Fig. Prior to genetic analysis, we studied the influence of gender, birth year, birth month, and level of inbreeding on hypothyroidism susceptibility using the generalized linear model in IBM SPSS Statistics v.21. Birth months were classified into four seasons: December to February, March to May, June to August, and September to November. The effect of inbreeding on susceptibility to hypothyroidism was tested by logistic regression, only including individuals born between 1990 and 2010 and having a pedigree completeness value [A is the additive relationship matrix between the sires and maternal grandsires and I is the identity matrix with diagonal elements of Variance components were estimated via the restricted maximum likelihood (REML) method  with a pAdditionally, we estimated the variance components using a Bayesian analysis via Gibbs sampling. The theoretical basis of the Gibbs sampling approach for mixed linear models is presented by Wang et al.  and is nectively . A probi package .The prevalence of hypothyroidism in the Finnish Hovawart population was similar than observed previously for other breeds –5. HowevP value = 0.78); the incidence of hypothyroidism in males and females was 2.1 and 2.2 %, respectively. This is contrary to the findings of Panciera [P value = 0.08) and was included in the final mixed model, whereas birth season was not significantly associated with hypothyroidism .With our data, the gender of the dog had no effect on its susceptibility to hypothyroidism . The corresponding posterior means for the sire and litter variances were 0.23 (SD = 0.09) and 0.26 (SD = 0.14), respectively. The difference in heritability estimates between the two methods (REML and Gibbs sampling) is relatively large but not significant, considering the standard error and standard deviation of the estimates. Also the generalized linear mixed model approach tend to over shrink random effects and their corresponding variances [The REML method with the probit model gave an estimated variance of 0.07 (SE = 0.07) for the litter effect and of 0.14 (SE = 0.06) for the sire effect. The estimated heritability of the underlying liability, thus, was 0.47 (SE = 0.18) [Eq. The diagnosis of hypothyroidism is challenging, because all affected dogs do not show clinical signs and not all may have been tested in a laboratory. Moreover, notification to the breed club about the disorder is on a voluntary basis, which means that the list of hypothyroid dogs maintained by the Finnish Hovawart Club may be incomplete. Our study data can therefore include dogs given a healthy status although they were, in fact, affected. Also, the data only contained information that is routinely recorded for registration purposes: parents, gender, birth date, color, and breeder, most of which had no influence on susceptibility to hypothyroidism. On the other hand, the data did not enable us to study the effects of other, perhaps more important factors that might increase the risk to acquire hypothyroidism, such as the dogs’ nutritional status or exercise level.Despite the limitations of the data, our results offer clear evidence that hypothyroidism involves a genetic component. The analysis did not allow to determine whether the genetic component is due to only one or just a few risk genes, or whether the genetic background is more polygenic in nature. Based on the immune-mediated character of hypothyroidism it can be postulated that the variation in the DLA area probably explains at least some of the obtained heritability , 12–14. The prevalence of hypothyroidism in the Finnish Hovawart population is relatively high. Our pedigree analysis gave moderate to high heritability estimates for this trait, depending on the method applied. The high heritability of hypothyroidism suggests that selection against it should be effective by using only healthy dogs for breeding.


Following the publication of the ENCODE project results, there has been increasing interest in investigating different areas of the chromosome and evaluating the relative contribution of each area to expressed phenotypes. This study aims to evaluate the contribution of variants, classified by minor allele frequency and gene annotation, to the observed interindividual differences. In this study, we fitted Bayesian linear regression models to data from Genetic Analysis Workshop 18 (n = 395) to estimate the variance of standardized and log-transformed systolic blood pressure that can be explained by subsets of genetic markers. Rare and very rare variants explained an overall higher proportion of the variance, as did markers located within a gene rather than flanking regions. The proportion of variance explained by rare and very rare variants decreased when we controlled for the number of markers, suggesting that the number of contributing rare alleles plays an important role in the genetic architecture of chronic disease traits. Our findings lend support to the "common disease, rare variant" hypothesis for systolic blood pressure and highlight allele frequency and functional annotation of a polymorphism as potentially crucial considerations in whole genome study designs. The proportion of phenotypic variance explained by genetic factors is influenced by multiple variant attributes. First, an analysis of several complex traits by Yang et al showed that genic regions explain more variation than intergenic because causal variants are more likely to be located in or near the genes, particularly the protein-coding regions . HoweverTo quantify the relative contribution of subsets of variants  to genetic variance of systolic blood pressure (SBP) we fitted Bayesian linear regression models to the Genetic Analysis Workshop 18 (GAW18) data. To control for the effects of the number of variants in each of the subsets, we performed sensitivity analyses with a fixed number of markers in each category to determine the subset with the highest variance contribution per variant.SBP was first log transformed (logSBP) to control skewness and kurtosis. Subsequently, for ease of interpretation, logSBP was standardized to unit variance. To use the available longitudinal data on SBP and covariates and to maximize sample size, a cross-sectional data set was formed by selecting the first available data of 4 visits. Covariates recorded at the time of the visit used for SBP included age, gender, use of tobacco products, and use of medication for blood pressure.ith individual μ is an intercept, The whole genome regression models  used forur group . The sofWith the goals of maximizing the biological plausibility of the analysis and reducing the requirements on computational power, we selected a set of 54,309 variants in regions of the genome enriched for blood pressure. The set of variants was identified as follows: We initially selected 84 genes implicated in pathways regulating blood pressure that were included on the SA Biosciences Human Hypertension PCR Array . Of the 84 genes reported, only 31 were located on the odd chromosomes and thus available through the GAW18 data release; for our analysis, we selected all variants in or near these 31 genes . Table n = 4470). Variants were further classified on the basis of functional annotation  and allele frequency in the GAW18 data set . To determine functional annotation, we used VCFtools [p = 49,839) categorized by MAF and functionality. Based on these classifications, we identified 12 sets of markers and fitted whole genome regression models as described above to each set.Subsequently, we excluded variants that did not pass quality control or had more than 2 alleles n = 440. Varianm. We reported the average of the Because of variation in the number of variants across the categories in Table Tables p = 500). Overall, estimates of explained variance slightly decreased compared to those obtained from models including all variants in each category, but the relative contributions of each class of markers remained similar, with rare and very rare variants located in genic regions explaining slightly more outcome variability than common flanking variants.Table Using sequence data from a set of 31 biologically relevant genomic regions, we established that the proportion of logSBP variance explained by genetic markers in hypertension-related pathways is roughly 11% of the phenotypic variation that remains after accounting for systematic, nongenetic effects. The proportion of variance explained by regression on markers decreases with the MAF, regardless of whether the variant is located within or near known genes. However, the observed trends across MAF categories were not statistically significant.To interpret our findings, it is important to distinguish between estimating the proportion of variance explained by the markers, which is the focus of our article, and estimating heritability. Indeed, our estimates of proportion of variance explained by genetic factors (roughly 11% after accounting for differences as a result of nongenetic effects) are likely to be smaller than the true heritability of the trait because, as a result of imperfect linkage disequilibrium between alleles at markers and those at causal loci, some proportion of genetic variance is likely to have remained unaccounted. Our findings are consistent with evidence from several recent studies, which suggest that the genetic architecture of blood pressure is likely to involve a large number of modestly associated, and many yet undiscovered, variants -12.The level of linkage disequilibrium between alleles at markers and those at causal loci depends on linkage disequilibrium decay and on marker density. Consequently, the proportion of variance captured by a maker set is related not only to the nature of the marker set and the genetic architecture of the trait, but also to the number of genetic markers in the set . To accoAlthough our variants were included in the models based on biological relevance, the family structure of the GAW18 data implicates identity by descent as an important contributor to shared genetic variance. To explore other contributions to SBP variance, we conducted sensitivity analyses using only unrelated individuals (data not shown) and found that the basic pattern held true, with rare variants explaining the highest proportion of outcome variance regardless of functional annotation.As whole genome sequence data become widely available and computational software evolves in its ability to handle a large number of genetic variants, future studies may consider repeating our analyses with a finer classification of markers. Specifically, genic regions could be further subdivided into coding or noncoding regions, or into exonic, intronic, 5′ and 3′ untranslated region, downstream, upstream, and splicing variants. For both genic and flanking regions, another approach would distinguish between variants that encode microRNAs and/or other regulatory molecules, or consider variance explained by differential DNA methylation. Additionally, future studies may evaluate whether the tradeoff between the proportion of outcome variance explained and the multiple testing burden is more favorable for randomly selected  variants across the genome or for variants located within a priori defined biological pathways. Finally, in our implementation we used shrinkage estimation procedures; however our proposed hypothesis could also be tested using statistical methods that perform variable selection and shrinkage simultaneously. In future, our findings may be extended beyond estimating the proportion of genetic variation to whole genome prediction, fully realizing the potential for clinical and public health applications of deep sequence data.We have investigated the tradeoff between proportion of blood pressure variance explained using subsets of the whole genome sequence, and found that rare and very rare variants contribute more outcome variance regardless of their functional annotation.The authors declare that they have no competing interests.SA designed the overall study and drafted the manuscript. HWW, DZ, SS, GC, and AIV developed the analytic strategy and contributed to the interpretation of the data. HWW, GW, DZ, and AIV conducted statistical analyses. GC developed the software used to fit the model and wrote sections of the manuscript. All authors read and approved the final manuscript.