Genetic association studies have been proved to be an efficient tool to reveal the aetiology of many human complex diseases and traits. When the phenotype is binary, the logistic regression model is commonly employed to evaluate the association strength of the genetic variants predispose to human diseases because the maximum likelihood estimator of the odds ratio based on case-control data is equivalent to that from the same model by taking the data as being arisen prospectively. This equivalence does not hold for the proportional odds model and using it to analyze the case-control data directly often results in a substantial bias. Through putting a parameter of the minor allele frequency in the modified likelihood function under the condition that the Hardy-Weinberg equilibrium law holds within controls, a consistent estimator is obtained. On the basis of it, we construct a score test statistic to test whether the genetic variant is associated with the diseases. Simulation studies show that the proposed estimator has smaller mean squared error than the existing methods when the genetic effect size is away from zero and the proposed test statistic has a good control of type I error rate and is more powerful than the existing procedures. Application to 45 single nucleotide polymorphisms located in the region of TRAF1-C5 genes for the association with four-level anticyclic citrullinated protein antibody from Genetic Analysis Workshop 16 further demonstrates its performance. There are two types of phenotypes: continuous and discrete. The majority of the discrete phenotypes are binary and ordinal. The logistic regression model is a major tool to analyze the binary phenotypes because the odds ratio estimator from the logistic regression model based on case-control data is equivalent to that from the same model by taking the data as being sampled from a prospective study45A retrospective study is highly popular in genetic epidemiology study because of its economic cost and substantially reduced study duration compared with a prospective design. The data in a retrospective design are not drawn from the general population and they are randomly sampled from each subpopulation and the numbers of subjects chosen from each individual subpopulation are usually matched. In the last decade, the retrospective case-control genetic association studies, especially genome-wide association studies, have been considered as a big success in searching for the deleterious genetic susceptibilities12Several procedures were proposed to analyze the retrospective data with ordinal responses in the literatures. An ad hoc approach is to use the proportional odds model10−4 for the HWE test to ensure that there is no possible systematic genotyping error in the sampled individuals. On the other hand, checking whether the HWE law holds in case population has been used as an association test for fine-mapping of the disease loci15et al.A and a. Denote the allele frequency of A by p. Under the HWE principal, the genotype frequencies of AA, Aa and aa are p2, 2p(1 − p) and (1 − p)2, respectively.The Hardy-Weinberg equilibrium (HWE) law is a very important principal in population genetics. It is a routine to check whether the observed genotypes satisfy the HWE law in control population before conducting an association test, because deviations from HWE can indicate many problems such as population stratification, genotyping error and so on1213−7.To the best of our knowledge, most of the exiting methods in the literatures focused on the estimation of the parameters. Although the test statistic such as the score test or the Wald test derived from the proportional odds model is still valid and has been used in practice, such as the CHD studyJ = 4, we consider J = 4 in the simulations with Pr(Y = 1) = 0.98, Pr(Y = 2) = 0.01, Pr(Y = 3) = 0.006, and Pr(Y = 4) = 0.004, which results in θ1 = 3.89, θ2 = 4.59 and θ3 = 5.51 under β = 0. We choose β ∈ {ln1.2, ln1.4, ln1.6, ln1.8} and the minor allele frequency (MAF) p ∈ {0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5}. Let n1 ∈ {1, 200, 500, 300, 200} and n2, n3 and n4 be drawn from a multinomial distribution Mul, where n1 = n2 + n3 + n4 and q = τ is the probability vector which is proportional to the corresponding prevalence rates of the case statuses with (P(Y = 2), P(Y = 3), P(Y = 4))τ. The reason why we choose several values of n1 is to make the power comparable for different genetic effect sizes. We consider two significance levels: 0.05 and 0.001. 1,000 and 50,000 replicates are conducted to calculate the empirical type I error rates and powers for the significance levels of 0.05 and 0.001, respectively.We compare the performances of three estimators: proMLE (the MLE derived from the likelihood function by taking the data as being arisen prospectively), modMLE (the MLE derived from the modified likelihood function) and hweMLE (the proposed method). What needs illustration is that the parameters used in this section are defined in the following “Notation” section. Since in the real application analyzed later, β = ln 1.2, β = ln 1.4, β = ln 1.6, and β = ln 1.8, respectively, where we set the same value of n1 (= 500). 1,000 replicates are conducted. As expected, the proMLE is biased and the proposed hweMLE are unbiased. Interestingly, the proMLE underestimates β in most cases with the median values being smaller than the true values, while the modMLE overestimates β a little bit with the median values being greater than the true values. The absolute value of bias of the proMLE increases as β increases. For example, when MAF = 0.25, the bias of the proMLE for β = ln 1.2, ln 1.4, ln1.6, and ln 1.8 are −0.023, −0.051, −0.068, and −0.091, respectively. From these boxplots, when β is away from zero, the proposed hweMLE performs the best, followed by modMLE, then proMLE based on the bias of the median value. For instance, when β = ln1.2 (= 0.182) and MAF = 0.10, the median values of proMLE, modMLE and hweMLE are 0.163, 0.202 and 0.192, respectively, while for β = ln 1.4 (= 0.336) and the same MAF, the median values of proMLE, modMLE and hweMLE are 0.276, 0.340 and 0.338, respectively. β is relatively large.js, j = 2, 3,…, J based on the reciprocal of case-control design. So we compare two test statistics. One is the score test derived from the proportional odds model by taking the data as being arisen prospectively. For convenience, we denote it by proT; Another is the proposed hweT. n = 1,000. 1,000 and 50,000 replicates are conducted to calculate the empirical type I error rates. The results indicate that both proT and hweT can control the type I error rates correctly with the empirical values being close to the nominal level. For example, when the MAF is 0.20 and the nominal level is 0.05, the empirical type I error rates of proT and hweT are 0.052 and 0.044, respectively, and when the MAF is 0.15 and the nominal level is 0.001, those of proT and hweT are, respectively, 0.00086 and 0.00078.As is shown in the “Methods” section, the observed Fisher information matrix of the modMLE is close to singular since there is an equality among Δβ. In details, we set n = 1,000, 500, 300, and 200 for β = ln 1.2, ln 1.4, ln 1.6 and ln 1.8, respectively, under the nominal significance level of 0.05, and n = 2,400, 1,000, 600, and 400 for β = ln 1.2, ln 1.4, ln 1.6 and ln 1.8, respectively, under the nominal significance level of 0.001. We conduct 1,000 and 50,000 replicates for the significance level of 0.05 and 0.001. n = 1,000, MAF = 0.35, and β = ln 1.4, the power of hweT is 0.582, which is much larger than that 0.522 of proT under the significance level of 0.001.In this part, we explore the power performances of proT and hweT. For the convenience, we assume D, to test whether the HWE law holds in the controls. When the HWE law holds in the controls, D = 0. The HWE test is given by nc0, nc1 and nc2 are the counts of the subjects possessing the genotype 0, 1 and 2, respectively, and nc = nc0 + nc1 + nc2. Under that D = 0, the HWE test follows the standard normal distribution. The results in −7 using the proT or the hweT. However, we can identify another five SNPs including rs10760130, rs10985073, rs2900180, rs7037673, and rs1468673 with p-values being less than 0.0001 using the hweT, while only three SNPs rs1953126, rs881375 and rs2900180 can be identified using the proT. In addition, we use the Fisher-combined method to combine the p-values over these 45 SNPs as Tcom for the proT and hweT are 408.9 and 512.2. Based on 1,000 bootstrap replicates, we calculate the p-values of Tcom for the proT and hweT. Both are less than 0.001. This indicates that the gene TRAF1-C5 is associated with the anti-CCP measure and that the hweT can detect association signals easily than the proT.The region of TRAF1-C5 in human genome has been shown to be associated with rheumatoid arthritis (RA) based on both genome-wide association study20When using the logistic regression to handle the binary response outcome in genetic association studies, it has been shown that the odds ratio estimate based on the MLE is equivalent to that from the same model by taking the data as being arisen prospectively45G = 0|Y = 1) = p0, Pr(G = 1|Y = 1) = p1, Pr(G = 2|Y = 1) = p2. Thus there is one additional parameter that needs to be estimated in the proposed modified likelihood function. At this point, the number of parameters is larger than that under the assumption of HWE. Hence, the biases of the estimators tend to bigger than those under the assumption that the HWE law holds.HWE principal is very important in genetic association studies. It is often considered to be a cornerstone for further statistical inference. Departure from HWE often result from inbreeding, population migration and genotyping errors. Researchers have suggested that the deviation of HWE among cases can provide additional evidence for the associations between genetic variants and human diseases17O(n−1) order for the quasi-likelihood estimates from GEE, where n is the total sample size, because it is derived based on the first-order approximation of the Taylor expansion about the estimating equation. Thus in our case, if we use the sandwich variance estimator to construct the test statistic, this may result in inflated type I error rate. Hence, we use the summation of the first derivatives of the likelihood function on the individual observation to estimate the variance of the MLE. It should be noted that the used variance estimate is a consistent estimate based on the law of large numbers.The sandwich variance estimate is a common tool used to estimate the variance of quasi-likelihood estimates from generalized estimating equations (GEE)Y be a J ordered status response variable and G be a random variable taking the genotype values of the subjects at a SNP locus. Without loss of generality, let Y = 1 denote the status of a healthy individual, and Y = j denote the status of a diseased subject, j = 2, 3,…, J. Then the standard proportional odds modelConsider a biallelic SNP and the genotype at a marker locus is coded as 0, 1 or 2, with the value corresponding to the copy number of a certain candidate allele. Let β is the parameter of interest, which is called log-odds ratio when J = 2, and θj, j = 1, 2,…, J − 1 are the intercepts. Denote ϕ(x) = 1/(1 + exp(−x)) for x ∈ where nj subjects who are randomly sampled from the jth subpopulation for j = 1, 2,…, J. Denote Let If we take the data as being collected from a prospective study, the prospective likelihood function isθ = τ and τ denotes the transpose of a vector or a matrix. The corresponding log-likelihood function iswhere β when β ≠ 0 and the bias increases as β increases. So, Cosslettβ:As shown in Cosslett2, Δ3,…, ΔJ)τ and Δ1 = n1/n.where Δ =  because the observed Fisher information matrix of τ is close to be singular. So, in the following part, we will derive a MLE through incorporating this equality and the HWE law. Suppose that the HWE principle holds in the control population with the minor allele frequency p. Thus Pr(G = 0|Y = 1) = (1 − p)2, Pr(G = 1|Y = 1) = 2p(1 − p), Pr(G = 2|Y = 1) = p2. From the Supplemental Material, we set w = [(1 − p)2/ϕ(θ1) + 2p(1 − p)/ϕ(θ1 − β) + p2/ϕ(θ1 − 2β)]. Then, the modified likelihood function is rewritten asBased on the reciprocal of case-control design where all case groups are randomly sampled from the case population, the structure among different case groups in the sample is the same as that in the general case population. So each case group should have the same degree of importance which yields Δand the log-likelihood function ism0, m1 and m2 are the numbers of the subjects with genotypes 0, 1 and 2, respectively, in the sample. We adopt two steps to estimate the parameters. We first estimate the parameter p using the observations in controls and denote the estimator by p almost surely. Then we optimize lh according to β and θ under β and θ through. Denote the estimator of τ by β and th element of the matrix I−1,where lh,ij = ln(Δj) + ln[ϕ(θj − gijβ) − ϕ(θj−1 − gijβ)] − ln{Δ1ϕ(θ1 − gijβ) + Δ2[1 − ϕ(θ1 − gijβ)]} for i = 1, 2,…, nj and j = 1, 2,…, J.and Aβ = n1Aβ1/Aβ2, where θ under β = 0 by In genetic association studies, the most concern of investigators is whether the genetic variant is associated with the disease. One can construct the Wald test statistic based on the asymptotic normality of and the score test statistic (denote it by hweT) iswhere How to cite this article: Zhang, W. et al. Fitting Proportional Odds Model to Case-Control data with Incorporating Hardy-Weinberg Equilibrium. Sci. Rep.5, 17286; doi: 10.1038/srep17286 (2015).


PLINK was used in the single marker GWAS, and INTERSNP was used to perform the two-marker GWIS, focusing only on SNPs with p ≤ 0.01 for the GWAS analysis. Age, sex, and diagnosis were used as covariates in both analyses. Corrected p values using the Bonferroni method were reported. The GWAS analysis revealed significant hits within or proximal to APOE, APOC1, and TOMM40 genes, which were previously implicated in AD. The GWIS analysis yielded 8 novel SNP-SNP interaction findings that warrant replication and further investigation.Alzheimer's disease (AD) is the most common neurodegenerative disorder. Using discrete disease status as the phenotype and computing statistics at the single marker level may not be able to address the underlying biological interactions that contribute to disease mechanism and may contribute to the issue of “missing heritability.” We performed a genome-wide association study (GWAS) and a genome-wide interaction study (GWIS) of an amyloid imaging phenotype, using the data from Alzheimer's Disease Neuroimaging Initiative. We investigated the genetic main effects and interaction effects on cingulate amyloid-beta (A The pathologic cascade in AD involves two primary hallmarks: amyloid- tangles . Genetic tangles . The lasnd EPHA1 . Howevernd EPHA1 . More rend EPHA1 .Traditional GWAS analyses used discrete disease status as the phenotypic trait of interest despite the fact that LOAD is a clinically heterogeneous disorder. Recently, researchers started to explore intermediate quantitative traits (QTs), such as clinical or cognitive features, biochemical assays, or neuroimaging biomarkers, in genetic association testing. This may have the potential to address the issue of clinical heterogeneity in LOAD. These QTs are often measured as continuous variables and thus exhibit a higher genetic signal-to-noise ratio. Further, most intermediate QTs are more proximal to their genetic bases than disease status. Thus, the incorporation of intermediate QTs can potentially increase statistical power to detect disease-related genetic associations , 12. An β peptides in cerebrospinal fluid (CSF), selective measures of brain atrophy using magnetic resonance imaging (MRI), and imaging of glucose hypometabolism and amyloid using positron emission tomography (PET) [18F-AV-45 or AV-45) or/and Pittsburgh compound-B -6-hydroxybenzothiazole). These amyloid measures have been studied as biomarkers for classifying AD [Over the past 10–15 years, studies have identified robust and predictive biomarkers for AD including levels of tau and amyloid-hy (PET) . PET imafying AD –17. All  CR1 and APOE using quantified Aβ PET as the outcome variable [In addition, most genetic association studies compute statistics at the single marker level and ignore the possible underlying biological interactions that contribute to the development of disease  and coulvariable  and betwvariable , 22 and variable  in case-β) load in an effort to better understand the genetic etiology of cingulate cortical Aβ deposition (a LOAD biomarker).With these observations, in the present work, we conducted a quantitative genetics study of an AD-associated amyloid imaging phenotype and examined both single marker main effects and two-marker interaction effects at the genome-wide level. Specifically, we investigated the main and interaction effects of genome-wide markers on cingulate amyloid-beta (Ahttp://adni.loni.usc.edu/). The ADNI was launched in 2003 by the National Institute on Aging (NIA), the National Institute of Biomedical Imaging and Bioengineering (NIBIB), the Food and Drug Administration (FDA), private pharmaceutical companies, and nonprofit organizations, as a $60 million, 5-year public-private partnership. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early Alzheimer's disease (AD). Determination of sensitive and specific markers of very early AD progression is intended to aid researchers and clinicians to develop new treatments and monitor their effectiveness, as well as lessen the time and cost of clinical trials.Data used in the preparation of this paper were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database  to conduct genetic association and interaction analyses.We applied for and were granted permission to use data from the ADNI cohort (N = 602) included 190 healthy control (HC), 215 early MCI (EMCI), 152 late MCI (LMCI), and 45 AD subjects. For the present work, analyses were restricted to subjects with both genotyping data and AV-45 PET data available. The study sample (The genotyping data of the participants were collected using either the Illumina 2.5 M array (a byproduct of the ADNI whole genome sequencing sample) or the Illumina OmniQuad array , 25, 26.n = 117, 175 SNPs were excluded based on criteria 1 and 2); and (3) Hardy-Weinberg equilibrium test of p ≥ 10−6 (n = 997 SNPs were excluded) using control subjects only. Participants were excluded from the analysis if any of the following criteria were not satisfied: (1) call rate per participant ≥ 90% (3 participants were excluded); (2) sex check (1 participant was excluded); and (3) identity check for related pairs .Quality control (QC) was performed using the PLINK software (version 1.07) . SNPs noPopulation stratification analysis was performed using EIGENSTRAT  and confhttp://adni.loni.usc.edu/) for 987 ADNI-GO/2 participants. We also downloaded the cerebellum SUVR measure and used it to normalize the cingulate SUVR measure. The normalized SUVR was used as the quantitative trait (QT) in our analyses. After excluding 383 participants due to the lack of genotyping data, 602 individuals remained in the further analysis.A previous AV-45 PET study  showed tβ 1-42 peptide (Aβ-42), total tau (t-tau), and tau phosphorylated at the threonine 181 (p-tau181p), measured in CSF samples, are potential diagnostic biomarkers for AD [In addition, amyloid-s for AD –33. Amons for AD , QC was http://www.broad.mit.edu/mpg/haploview/) and R (http://www.r-project.org/), respectively.For GWAS examining the main effects, linear regression was performed using PLINK to determine the association of each SNP to the AV-45 measure. An additive genetic model was tested with covariates including age, gender, and diagnosis . Manhattan plots and Q-Q plots were generated using Haploview  was performed on all possible SNP pairs among the top 10,000 SNPs, using two-marker analysis. The computation was conducted in a linear regression framework. We examined the association between SNP-SNP interactions and the AV-45 measure while controlling for relevant covariates at the baseline scan, including age, sex, and clinical diagnosis. This resulted in a total of approximately 50 million unique SNP pairs to be tested from the ADNI dataset. Interactions were considered significant if their Bonferroni corrected p value < 0.05.For GWIS examining the interaction effects, the INTERSNP software  was applR2) in the AV-45 measure accounted for by these interaction terms. We first included the same set of covariates  in the linear model. After that, we included APOE status, the closest SNP to the BCHE SNP identified in a prior amyloid GWAS study [R2 for the significant models was calculated in SPSS as ΔR2 = R2 (full model with interaction term) − R2 (reduced model without interaction term). Significant effects were plotted in SPSS as well.For identified significant interactions, we applied hierarchical linear regression using IBM SPSS 20 to estimate the amount of variance , which codes for the APOC1 [ APOE region, including adjacent APOC1 and TOMM40, were significantly associated with the AV-45 level in cingulate.he APOC1 . A few op value < 0.05) (CLSTN2)-rs7644138 (FHIT), rs7916162 (TACC2)-rs2326536 (PRNP∗), rs2295873 (TACC2)-rs7794838 (IGFBP3∗), rs2295874 (TACC2)-rs2326536 (PRNP∗), rs13056151 (BCR)-rs17594541 (MAGI2), rs13426621 (LOC388942)-rs7037332 (TYRP1∗), rs16936424 (LOC387761)-rs10504164 (N/A), and rs16939265 (HNF4G∗)-rs6854047 (RWDD4∗).The INTERSNP model we tested included age, sex, and diagnosis as covariates. Eight SNP pairs showed significant interaction effects on the cingulate AV45 measure (corrected  < 0.05) : rs21949 APOE status was then accounted for an additional 16.1% of variance, followed by the closest SNP to the BCHE SNP identified in [CLSTN2)-rs7644138 (FHIT), the SNP main effects accounted for 3.4% of variance, and the interaction term accounted for 4.9% of variance (8.3% combined). For rs7916162 (TACC2)-rs2326536 (PRNP∗), the main effects accounted for 2% of variance, and the interaction accounted for 4.9% of variance (6.9% combined). For rs2295873 (TACC2)-rs7794838 (IGFBP3∗), the main effects accounted for 3.7% of variance, and the interaction term accounted for 4.1% of variance (7.8% combined). For rs2295874 (TACC2)-rs2326536 (PRNP∗), the SNP main effects accounted for 3.7% of variance, and the interaction term accounted for 4.1% of variance (7.8% combined). For rs13056151 (BCR)-rs17594541 (MAGI2), the main effects accounted for 3.5% of variance, and the interaction term accounted for 2.6% of variance (6.1% combined). For rs13426621 (LOC388942)-rs7037332 (TYRP1∗), the main effects accounted for 4.2% of variance, and the interaction accounted for 2.3% of variance (6.5% combined). For rs16936424 (LOC387761)-rs10504164 (N/A), the main effects accounted for 3.7% of variance, and the interaction term accounted for 1.7% of variance (5.4% combined). For rs16939265 (HNF4G∗)-rs6854047 (RWDD4∗), the main effects accounted for 2.7% of variance, and the interaction term accounted for 1.3% of variance (4.0% combined).ified in  accounteN = 499) with CSF biomarker data available, all 8 identified interactions remained statistically significant when performing hierarchical linear regression using the CSF phenotypes  instead of the AV-45 measure as outlined earlier  with cingulate amyloid burden. The risk variants at these pairs had low main effects but explained a relatively high-level variance of the amyloid deposition in cingulate . Given amyloid and tau phosphorylation as major AD hallmarks, it is not surprising to observe the genetic interaction effects on both the amyloid load and relevant CSF biomarkers  regulates cell survival and apoptosis [ MAGI2 SNP associated with hippocampal atrophy using the ADNI data [In line with our hypothesis, we identified multiple significant genetic interactions associated with cingulate amyloid deposition. Several genes found in this study have already been implicated in AD, thus lending confidence to the analytic procedure and results. These genes include PRNP , 39, IGF[ IGFBP3 , 41, and [ MAGI2 , 43. For disease . Ikonen poptosis . Potkin DNI data . Perhaps APOE and BCHE and provided more accurate estimate of the interaction effects on amyloid burden. (5) CSF data were used in this study to cross-check the identified interactions, which had the potential to serve as an indirect validation strategy or provide complemental information.This study had several methodological and technical advantages over other imaging genetics studies in addition to the above interesting findings. (1) To our knowledge this is the first genome-wide study to explore how SNP-SNP interactions influence cingulate amyloid burden, measured using florbetapir PET scan information. (2) Using continuous quantitative traits as phenotypes confers higher statistical power than using conventional clinical status. (3) The sample in this study included HC, EMCI, LMCI, and AD, thus providing a continuous and wide spectrum of the disease progression in the dataset. (4) Our approach embraced, rather than ignored, the confounding factors including age, sex, diagnosis, and previously identified risk genesOur study has several limitations. (1) We used single marker main effect value to select SNPs for interaction analysis, which could miss significant interactions between SNPs with insignificant main effects. (2) The small cell size in the interaction analyses might introduce false positives. (3) Our approach is mostly data-driven, without utilizing any existing biological knowledge , which may reduce the statistical power and result interpretability.β) load. The single marker analyses revealed significant hits within or proximal to APOE, APOC1, and TOMM40 genes, which were previously implicated in AD. The interaction analyses yielded a few novel interaction findings associated with cingulate amyloid burden, such as those between CLSTN2 and FHIT, between TACC2 and PRNP, between TACC2 and IGFBP3, and between BCR and MAGI2. Each of these SNP pairs demonstrated significant interaction effects while their individual main effects were not prominent. This suggests that searching for interaction effects may help solve the problem of missing heritability to some extent. Future studies should attempt to replicate these results in independent datasets with neuroimaging and genetic data, as they become available. Additional pathway analysis and gene sets enrichment analysis could be performed to help understand the genetic interactions between SNPs on amyloid imaging phenotypes and potentially provide critical functional evidence in support of the statistical association findings.We performed GWAS and GWIS using amyloid imaging as the quantitative phenotype and investigated the genetic interaction effects on cingulate amyloid-beta (A


Populus trichocarpa (black cottonwood) genotypes originating across western North America. Variation in 74 field-assessed traits  was investigated for signatures of selection (comparing QST -FST) using clustering of individuals by climate of origin (temperature and precipitation). 29,354 SNPs were investigated employing three different outlier detection methods and marker-inferred relatedness was estimated to obtain the narrow-sense estimate of population differentiation in wild populations. In addition, we compared our results with previously assessed selection of candidate SNPs using the 25 topographical units (drainages) across the P. trichocarpa sampling range as population groupings. Narrow-sense QST for 53% of distinct field traits was significantly divergent from expectations of neutrality (indicating adaptive trait variation); 2,855 SNPs showed signals of diversifying selection and of these, 118 SNPs (within 81 genes) were associated with adaptive traits (based on significant QST). Many SNPs were putatively pleiotropic for functionally uncorrelated adaptive traits, such as autumn phenology, height, and disease resistance. Evolutionary quantitative genomics in P. trichocarpa provides an enhanced understanding regarding the molecular basis of climate-driven selection in forest trees and we highlight that important loci underlying adaptive trait variation also show relationship to climate of origin. We consider our approach the most comprehensive, as it uncovers the molecular mechanisms of adaptation using multiple methods and tests. We also provide a detailed outline of the required analyses for studying adaptation to the environment in a population genomics context to better understand the species’ potential adaptive capacity to future climatic scenarios.Forest trees generally show high levels of local adaptation and efforts focusing on understanding adaptation to climate will be crucial for species survival and management. Here, we address fundamental questions regarding the molecular basis of adaptation in undomesticated forest tree populations to past climatic environments by employing an integrative quantitative genetics and landscape genomics approach. Using this comprehensive approach, we studied the molecular basis of climate adaptation in 433  In addstudies ,56), yet, yetFSTstudies  .i.e. critical day length in autumn phenology [2 up to 0.94; Picea [Evidence is emerging that for perennial trees to effectively sense short day signals, henology , a tempehenology . Allele 4; Picea . For aut4; Picea ,60), butATNRT2:4 and ATNRT2:1 were strongly aligned with temperature variation (R2~90%) in P. trichocarpa. Moreover, these SNPs were pleiotropic for multiple autumn phenology traits, height, and leaf rust resistance (ATNRT2.1/2.2/2.3/2.4/2.6). This implies that a deletion event occurred in this clade whose functional significance remains elusive to date [SNP allelic frequencies within both nitrate transporter genes sistance . Nitrate to date . Phyloge to date . This ac to date .Arabidopsis, loss of function of ATNRT2.1 primes salicylic acid signaling and PR1 up-regulation [NRT2.4 and PTNRT2.1 are strongly down-regulated in incompatible interactions, while no expression change is apparent in compatible interactions . The identified nitrogen transporters might be important in nitrogen storage and nitrogen remobilization to recycle nutrients during the progression of leaf senescence [In addition, our results revealed that adaptive genetic variants within both poplar nitrate transporters were also associated with leaf rust resistance ; S3 Tabl. In Arabgulation . In poplnescence . They manescence ,67) and/nescence . The effnescence  and the Pectin esterase gene Potri.012G014500 (SNP scaffold_12_1811250) represents another example for which significant associations with climate (here: temperature) and several adaptive traits were found  Tables. P. trichocarpa). The key findings provided SNPs whose allelic frequencies were most diverged among populations from different climate clusters and these SNPs tended to be associated with mapped genes underlying phenotypic variation. This phenotypic variation itself diverged among the different climate clusters. Our study dissected the influence of climate , yet much of the variation in phenology is also attributed to photoperiod , soil/root microbial diversity, groundwater, and other ecological interactions also of potential importance. Reciprocal transplants will be necessary to elucidate the effects of gene × environment plasticity on the expression of traits with spatially heterogeneous selection [The high adaptive potential of tree populations is considered the result of positive effects of long-distance gene flow based on its interactions with divergent selection across the contrasting environments , while loperiod ,72,45). ,45. P. toperiod ,74,59) i,59 iP. toperiod ,56). Whi. WhiP. telection , but canelection . These felection .P. trichocarpa Torr. & A. Gray genotypes growing in a common garden. These genotypes came from 140 unique geographic locations spanning two thirds of the species’ range   , Fig 1)..P. trich–138°W)  were cl–138°W) .P. trichocarpa [i.e. candidate genes) or genes based on expression analyses. Because of the rate of linkage disequilibrium (LD) decay in P. trichocarpa, between 67–134k SNPs would be required to include all common variants throughout the genome at LD = 0.2 (assuming a 403 Mb assembled genome length and an average of 3–6 kb for r2 between common variants to drop to 0.2). Therefore, some SNPs were selected as representative SNPs to “tag” genes and genetic regions with high LD, and thus represent a group of SNPs (the haplotype). For this study, we further filtered array SNPs for: i) minor allele frequency (MAF) <0.05, ii) >10% missing data, and iii) Illumina’s GenTrain score <0.5, thereby reducing SNP numbers to 29,354. This filtering is not biased towards higher frequency SNPs (i.e. older variants established at much higher frequencies within the population over time) as a wide distribution of allele frequencies (MAF>0.05) was considered for the analysis.Trees were genotyped using an Illumina iSelect array with 34,131 SNPs from 3,543 candidate genes designed for chocarpa . The chachocarpa ). BrieflMelampsora xcolumbiana) resistance traits [Phenotyping of genotype accessions within the common gardens and climate of origin data were obtained from previously published work . In e traits  were repi.e. trees collected at the same latitude/longitude), we randomly selected a single genotype representing a geographical region . Eigenvalues for principal components from sPCA provided a cumulative picture about contributing factors, including the genetic variance and the spatial autocorrelation . Large positive eigenvalues reflect the importance of the proportion of the genetic variance along with a strong positive autocorrelation in the global pattern (i.e. IBD), while large negative eigenvalues indicate the importance of the proportion of the genetic variance along with negative autocorrelation indicating the existence of discrete local genetic clusters.Since forest tree species usually have extensive geographic ranges, exhibit extensive gene flow and have low levels of population stratification , we inven is the number of populations , wij is weight set at 0 or 1 depending on whether populations are considered neighbours in each 200 km lag test, xi is the allele frequency in the ith population, and We used the "global.test" and "local.test" functions in the "adegenet" package to infer the statistical significance of each type of genetic structure. These functions are based on a spectral decomposition of the connection matrix into Moran's eigenvector map and test for association of those eigenvectors from Moran's eigenvector map with Moran's I . To invei.e. independently of the genetic data). Climate variables were obtained using ClimateWNA [Since our initial investigation of population structure with sPCA indicated the presence of only one global structure consisting of IBD and lack of local discrete clusters, any marker-based inference about genetic clusters might be highly unreliable . TherefoimateWNA  and inclimateWNA . Using KimateWNA , we spliP. trichocarpa for their adaptive potential , p and a are vectors of random climate cluster and individual tree additive genetic effects, X and Z are incidence matrices assigning fixed and random effects to measurements in vector y, the cluster effects are following p~N enable estimating broad-sense QST directly without the use of any relationship matrix, while narrow-sense QST estimation was based on variance components estimated in the mixed linear model considering the realized relationship matrix [broad-sense QST were estimated in the model considering a as the vector of clonal genotypic values following a~N strictly based on neutral markers (see below) allowing inferences about trait evolution based on selection or genetic drift , [The measurements of all ecology and disease traits using clonal ramets (p matrix  as in Eq trait), .Narrow-sense heritability (h2) was based on variance components estimated in the mixed model as follows:β is the vector of fixed effects (intercept and cluster) and a is the random vector of additive genetic effects following the description of narrow-sense heritability was estimated as follows:QST (i.e. PST) ([β is the vector of fixed effects (intercept) and p is the vector of random cluster effects, the effect of individuals within cluster is found within the error variance.. PST) ,24) was  was Narrrding to . The varrding to  using thFST outlier analysis (using Fdist2) employing the same climate clusters as for QST analysis, 2) unsupervised spatial ancestral analysis (SPA), and 3) SPA with climate as a covariate. Additionally, we compared our results with FST outlier analysis (using Fdist2 and BayeScan) that were reported in [To identify SNPs putatively under selection and also associated with adaptive traits ,43,39), ,39, 43,3orted in  using 25FST values for SNPs were calculated among the four climate clusters . We implemented the Fdist2 program within the LOSITAN project [FST outlier detection. Fdist2 compares the distribution of FST values of sampled loci to the modeled neutral expectation of FST distribution using coalescent simulations [FST values conditioned on heterozygosity and outside the 99% confidence interval were considered candidate outliers. project  for SNP ulations . We emplP. trichocarpa populations have known structure related to IBD . Only those correlations that remained significant after Bonferroni correction for each of the three test statistics  were retained.We investigated correlations between the outlier SNPs (based on climate clusters) and the environmental variables that defined the established climatic clusters . SubpopuQST values with the simulated distribution of QST-FST values for a neutral trait using previously provided R scripts [QST values based on mean FST for neutral markers and mean QST for neutral traits  and i.e., expected between-population variance) given in FST values of 29,233 SNPs present  within the simulated neutral envelope of FST values  with QST replaced by the FST in P-values were obtained by testing whether the null hypothesis that the estimated narrow-sense QST for each tested trait is statistically equal to the expected QST for a neutral trait [Finally, we compared observed  scripts . In brie traits ,98). For. ForQSTal trait .P. trichocarpa, confounding effects of population stratification were adjusted using principal component analysis . rrection ,45). Furrrection  and the S1 FigFST outlier loci detection and distribution of empirical FST estimates conditioned on expected heterozygosity (HE). The envelope of values corresponding to neutral expectations at 99% CI level (with mean FST = 0.0078), solid line, was constructed with the infinite allele model according to [FST estimates along the 19 poplar chromosomes and additional scaffolds (abbrev: scaff); the 121 identified outlier loci are indicated by red circles above their FST value bars. A goodness-of-fit test assuming a uniform distribution was performed to test whether the observed frequencies of ‘outlier loci’ along the 19 poplar chromosomes differed significantly from the expected value. Following the rejection of the null hypothesis , we declared ‘outlier loci hotspots’ if the number of outliers at a given chromosome was equal or above the maximum value  for assessed outlier clusters from a randomly generated data set using the 118 outliers found across the 19 chromosomes, and running 1,000 replicates, which identified significant clustering of outliers on chromosome 15.(a) (PDF)Click here for additional data file.S2 Fig2) of allelic frequencies (following arcsine transformation) on temperature and precipitation, respectively  calculated among the four distinct climate clusters  and PTNRT2.4B  with 97% DNA sequence similarity [Simple linear regression Click here for additional data file.S1 Table(XLS)Click here for additional data file.S2 TableIn red and dark blue are 1% cutoffs (spa = 2.78025 and spa = 1.50795), in orange and light blue are 5% cutoffs (spa = 2.12467 and spa = 1.08868) in unsupervised SPA and climate SPA analyses, respectively.(XLSX)Click here for additional data file.S3 TableComprehensive results are provided in (XLS)Click here for additional data file.


Equine recurrent laryngeal neuropathy (RLN) is a bilateral mononeuropathywith an unknown pathogenesis that significantly affects performance inThoroughbreds. A genetic contribution to the pathogenesis of RLN issuggested by the higher prevalence of the condition in offspring ofRLN-affected than unaffected stallions. To better understand RLNpathogenesis and its genetic basis, we performed a genome-wide association(GWAS) of 282 RLN-affected and 268 control Thoroughbreds.LCORL/NCAPG locus on ECA3 previously shown to affectbody size in horses. Using height at the withers of 505 of these horses, weconfirmed the strong association of this locus with body size, anddemonstrated a significant phenotypic and genetic correlation between heightand RLN grade in this cohort. Secondary genetic associations for RLN onECA18 and X did not correlate with withers height in our cohort, but didcontain candidate genes likely influencing muscle physiology and growth:myostatin (MSTN) and integral membrane protein 2A(ITM2A).We found a significant association of RLN with theThis linkage between body size and RLN suggests that selective breeding toreduce RLN prevalence would likely reduce adult size in this population.However, our results do not preclude the possibility of modifier loci thatattenuate RLN risk without reducing size or performance, or that the RLNrisk allele is distinct but tightly linked to the body size locus on ECA3.This study is both the largest body size GWAS and the largest RLN GWASwithin Thoroughbred horses to date, and suggests that improved understandingof the relationship between genetics, equine growth rate, and RLN prevalencemay significantly advance our understanding and management of thisdisease. Musculus cricoarytenoideusdorsalis (CAD), that prevents maximal abduction of the arytenoid cartilageand contributes to the development of airway obstruction. The degree of CADparesis/paralysis is graded endoscopically from 1 to 4 [Equine recurrent laryngeal neuropathy is a common upper respiratory tract disorder of horses and a significant cause oftheir reduced athletic performance . RLN repm 1 to 4  and a sim 1 to 4 .Recurrent laryngeal neuropathy is an important axonopathy for horses in competitiveevents, because the resultant paresis of the intrinsic laryngeal muscles leads toobstruction of air flow during intense exercise, an abnormal inspiratory noise knownas “roaring,” and, most importantly, reduced performance. Despite itsimportance, however, the etiology of the axonopathy is unknown. RLN has particulareconomic importance in Thoroughbred racehorses because these animals enter racetraining as 2-year-olds. Even though all horses going through sales areendoscopically examined for evidence of RLN, the condition may not yet be clinicallyevident in some animals  so a diath and 18thcentury that were crossbred to local mares [The prevalence of RLN varies, ranging from between 2-11% in Thoroughbreds ,9 and 35al mares . As a coal mares , making P = 1.62 × 10-6) and 31(P = 1.69 × 10-5) associatedwith RLN. Neither of these chromosome regions contained regions associated withneuropathies in other mammals or candidate genes that could be immediately linked toperipheral neuropathy. The present study reports on use of a newer Illumina EquineSNP70 BeadChip to conduct a GWAS on a large cohort of Thoroughbred horses that werephenotypically characterized for RLN and height. Specifically, we sought to identifyrisk loci associated with RLN and in doing so, gain additional insight into theetiopathogenesis of this disease. Because of the clinically reported connectionbetween height and RLN, we also examined their genetic association.The presence of genetic factors contributing to the pathogenesis of RLN is suggestedby the observation that offspring of RLN-affected stallions are more likely to beaffected with the disorder than are offspring from unaffected stallions ,16. ElucThe protocol for phenotypic characterization and obtaining blood samples and theinformed consent forms were approved by the Animal Use and Care Committee ofMichigan State University (MSU) and Cornell University. Information on the risksand benefits of the investigation was provided to the horse trainers/owners andinformed consent was signed before at the time the sample was taken. Phenotypicdata and venous blood samples (collected in 10 ml EDTA tubes) were obtainedfrom 573 Thoroughbred horses evaluated by large-animal specialists from twoveterinary teaching hospitals (Michigan State University and Cornell University)and by participating board-certified veterinary surgeons at private practicesthroughout the US (n = 10). A total of 505 horses at least2 years old had recorded heights. In each unsedated horse, avideoendoscopic examination of the larynx was performed and a laryngeal grade of1 to 4 [Samples submitted by practitioners were shipped on dry ice to the PulmonaryLaboratory, MSU and then stored at -20°C. All samples were then processedby personnel in the Cornell Veterinary Biobank . DNA wasThe 573 Thoroughbred samples were genotyped using Illumina Equine SNP70 arrayscontaining 65,157 markers. Genotype calls were made in GenomeStudio using aGenCall threshold of 0.25. Twenty samples with high missingness were re-run andan additional 29 samples passing QC were run multiple times as technicalreplicates. A total of 1,470 SNPs were excluded for high missingness (>10%)and 7,751 monomorphic or rare SNPs  were excluded. Duplicate concordance at theremaining sites was >99.99%.P < 1 × 10-4based on Hardy-Weinberg equilibrium) and 24 were excluded for low heterozygosity(P < 1 × 10-32);filtering was more relaxed for low heterozygosity because population structurewould be expected to depress heterozygosity somewhat from Hardy-Weinbergexpectation. The final unphased dataset contained 566 horses genotyped at 56,679markers with a mean call rate of 99.77%. Principal component analysis wascomputed using Eigenstrat [For the final dataset, only one array per sample was retained, and an additionalseven samples were excluded because their imputed genotypic sex did not matchthe phenotypic records, indicating a possible misidentification of thesesamples. Two SNPs (BIEC2_403636 and BIEC2_424368) were excluded because theyshowed evidence of probe hybridization to multiple genomic locations, and twoother SNPs (BIEC2_346860 and BIEC2_840760) were excluded due to low duplicateconcordance rates. Additionally, 197 markers were excluded for highheterozygosity  and afamily-wise error rate of α = 0.05.Genome-wide association was performed using GEMMA  after imTo assess association to haplotypes in the ECA3 candidate region a set of SNPscentered on BIEC2_808543 and encompassing approximately 2 Mb(ECA3:104737609–106517004) was selected. We further filtered this set toinclude only markers with a genotyping rate > 99% and aMAF > 0.01, leaving 45 remaining markers. Linkage disequilibriumbetween markers was examined using Haploview v4.2 . HaplotyLCORL) and non-SMC condensing I complex,subunit G (NCAPG) genes and correcting for sex and gelding, the strongest association with RLNwas centered on a SNP, BIEC2_808543, adjacent to the ligand-dependent nuclearreceptor corepressor-like (P = 1.3 × 10-22) centeredon the same SNP, BIEC2_808543, in LCORL/NCAPG locus that was thestrongest association for RLN. All 12 SNPs passing the Bonferroni significancethreshold were in this region . We estimated thatthe entire SNP70 panel explains 0.24 ± 0.11 of the variation inwithers height after accounting for sex and gelding [We performed GWAS on height using sex  as acategorical covariate for the 505 horses of at least two years of age withrecorded heights Figure B. The st gelding .LCORL/NCAPG minor (G) allele count at BIEC2_808543 hadan effect of increasing height 3.7 cm (S.E. = 0.4) and explained 16.6% ofthe variation in height in the dataset. This effect was roughly additive andconsistent in sires, mares, and geldings (Table P = 0.08). GWASfor height including BIEC2_808543 genotype as a covariate did not reveal anyadditional associated regions  analysis  to estimP = 8.18 × 10-23). Thisgenomic region includes both the LCORL and NCPG genes , acompact muscle mass close to the trunk, and high rates of maximal oxygenconsumption. Although selection for height apparently has been accompanied by anincreased prevalence of RLN, the genetic basis for this is currently unknown.The current study, which used the largest cohort of RLN-affected and unaffectedhorses ever assembled from a single breed revealed a very strong genetic correlationbetween RLN and height. This association was previously well known across breeds, buthas never been demonstrated over the small range of heights (152 to 178 cm)spanned in the present study. This was evident not only in the clinical observations, but also in the GWAS results. The haplotypeassociations for height and RLN were coincident and the genetic correlation betweenthe traits was >90%.The phenotype of RLN was defined based on the presence (cases) or absence (controls)of endoscopically visible failure of abduction of the left arytenoid and associatedvocal fold. This clinical and functional definition was used because veterinariansin the field use it and because it has important implications for athleticperformance. There were two reasons for selection of grade 3 as the threshold scorefor a case: first, horses with this grade are unable to maintain vocal foldabduction and therefore require surgical correction to restore their performance;and second, failure of abduction is easily recognized endoscopically by experiencedveterinarians . However, histopathological examination ofthe intrinsic muscles of the larynx can reveal muscle fiber-type regrouping(7evidence of axon death and reinnervation from a neighboring axon) in some horsesthat do not have clinical evidence of laryngeal paresis  and therLCORL/NCAPG region and body size in the horse [LCORL and NCAPG in growth is supported by well-validatedassociations to human height [Recently, several independent studies have documented strong associations between thehe horse ,31, but n height ,33, and n height .HMGA2, ZFAT, and LASP1) as well as aThoroughbred-specific QTL on ECA28, but no signal for these loci was detected inthis dataset [HMGA2 is poor on the IlluminaSNP70 product, the same markers for ZFAT are included in this analysis andsegregate at ~ 20% in our cohort, but do not appear to impact withers height. Theassociation of ZFAT with height is replicated in a study of 1,077Franches-Montagnes horses [ZFAT may bedependent on genetic background. The previous estimate that 59% of the variation inheight in Thoroughbreds was explained by LCORL/NCAPG and the QTLon ECA28 is significantly higher than our estimate of 24% based on the entire SNPdataset. This may reflect an overestimation in their analysis due to small samplesize (N = 48) or an underestimation in ours due to the small range ofThoroughbred heights, environmental variability, and/or measurement error.Within the Thoroughbred breed, the ECA3 locus appears to have the single largesteffect on withers height Figure B. Previo dataset . Althougs horses , suggestLCORL and NCAPG are genes encoding transcriptionfactors and could have diverse targets with varying physiological functions. Thus,the same causative mutation may be implicated for both body size and RLN phenotypes.Lastly, given extensive LD across the ECA3 locus, RLN and body size may beinfluenced by two independent genetic changes associated by linkage. Further workwill be needed to distinguish which of these three hypotheses is the most likelyexplanation for the association between this ECA3 locus and RLN.Our study confirms anecdotal evidence and previous reports in draft breedsidentifying an association between RLN and large size in horses ,12. HoweLCORL/NCAPG locus, the chromosome 18 and X candidate loci arenear to genes that are likely to affect muscle physiology and growth in horses(MSTN and ITM2A). Interestingly, the primary associationsignal in [MYO9B), a candidate gene thatmay be very relevant to the pathology of a distal axonopathy like RLN. Yet, ourstudy suggests that QTLs on chromosome 21 and 31 identified by Dupuis etal.[Besides the ignal in  is also is etal. are not is etal.; Dupuis The results of this study add further confirmation to the association between heightand the clinical occurrence of RLN and indicate the close relationship between thegenetic loci determining these two characteristics. However, despite beingstatistically significant, LCORL/NCAPG accounted for only 6% of the variation inRLN. However, in these Thoroughbreds, the proportion of variation of RLN explainedby the entire SNP70 set (“chip heritability”) was 45%, which, althoughmuch higher than in most complex diseases , suggestSome of the information in this paper has been used in a patent application.ARB and SAB participated heavily in experimental design, data analysis, andmanuscript preparation. ALB was instrumental in establishing the cohort of samples.EC prepared DNA samples, established the DNA bank, and participated in dataanalysis. KCO, RJT and ZZ participated in experimental design and data analysis. MCprepared samples and was involved in data analysis. DMA contributed samples to thecohort, was involved in data analysis and manuscript preparation. JES wrote thefirst plan to establish the cohort and conducted the genetic analysis (she is now inprivate industry). NER (the corresponding author) assisted JES with the initialexperimental design, oversaw and participated in collecting the majority of thesamples for the cohort, participated in discussions of data, and prepared the finalmanuscript submission. All authors reviewed the final manuscript.Principal component analysis (PCA) of genotype data shows nostratification according to RLN status or evidence for significantpopulation substructure within the cohort. Red = RLNaffected, black = control.Click here for fileSummary of horses used for height and RLN association mapping.Click here for fileLCORL/NCAPG. The RLN association (A) is shown with sex asthe covariate in GEMMA  and height (B). There is little genomic inflation ofP-values in the data in uncorrected PLINK (gray) or population structurecorrected GEMMA (black) association scans for height (B) in our dataset.The top dozen SNPs are all linked to the ECA3 locus nearA Figure B. GenomiClick here for fileA: GWAS of RLN using sex, gelding, and height as covariates yieldsadditional suggestive associations on ECA18 and ECA23 (489 horses: 274cases and 215 controls). B: GWAS of RLN without covariates yields anadditional signal on chromosome X.Click here for fileLCORL/NCAPG) allele count as covariates shows noother significant genetic associations with these traits.Genome-wide association of RLN (A) and height (B) using sex, gelding, andBIEC2_808543 (Click here for filePartitioning the variance explained by each chromosome for height(x-axis) and RLN grade (y-axis) in our sample (550 horses) afteraccounting for sex and gelding as covariates. The variance explained forRLN case/control was not estimable by the REML method with sex andgelding covariates; therefore, RLN grade (1-4) was used instead. REMLestimated total variance explained genome-wide for height and RLN gradeis 59% and 61%, respectively, somewhat higher than the estimatesobtained by the LMM method or by bivariate REML (see Methods).Click here for fileP-values resulting from a haplotype association test for withersheight.Click here for fileP-values resulting from a haplotype association test for RLN status(case/control).Click here for file


Large-scale genetic studies are often composed of related participants, and utilizing familial relationships can be cumbersome and computationally challenging. We present an approach to efficiently handle sequencing data from complex pedigrees that incorporates information from rare variants as well as common variants. Our method employs a 2-step procedure that sequentially regresses out correlation from familial relatedness and then uses the resulting phenotypic residuals in a penalized regression framework to test for associations with variants within genetic units. The operating characteristics of this approach are detailed using simulation data based on a large, multigenerational cohort. As biological techniques for assaying genetic variation have advanced, so have methods for candidate gene, genome-wide, exome, and whole genome association studies. In addition to methodological advancements purely statistical in nature, each progression has resulted in new analytical complexities, including those relating to problems of assuring reliable data quality and handling massive multiple testing and computational challenges. Increasingly, the ability to analyze the complex data resulting from genetic studies relies on having specialized software and vast computational resources. Methods are needed that are able to appropriately respect data complexity and are also accessible to investigators who wish to prevent an expensive computing investment.Family-based association methodologies in the specific context of next-generation sequencing have been proposed . Like anThe transition from designing genome-wide association studies (GWAS) that rest upon the common disease and common variant hypothesis to exome and whole genome studies that are better equipped to ascertain effects from rare variations has resulted in much interest for methods to group or aggregate variants in order to test for the multiple rare variant hypothesis that rarer variants of larger effect underlie common disease variation. In what follows, we present a method that handles complex pedigrees in a computationally accessible manner while also incorporating information over a genetic functional unit.Whole genome sequencing for a subset of the San Antonio Family Studies (SAFS) participants was conducted through the Type 2 Diabetes Genetic Exploration by Next-generation sequencing in Ethnic Samples (T2D-GENES) Consortium. We examined the 1,215,399 variants on chromosome 3 that were genotyped on the 955 fully phenotyped subjects.The simple and fast way to test the effect of a genetic marker on a trait is to use a contrast of the frequencies or means among genotype groups in a linear model. However, this method does not explicitly take into account relationships among family members when available, which can lead to both false-positive and false-negative associations. The use of a linear mixed model is a potential solution to this problem when family-based data is available. These models adjust for familial relationships by modeling the polygenic component between individuals as a random effect but are computationally intensive. Recently, Aulchenko et al  developeStep 1: adjustment for family structure. First, we estimated the kinship matrix, Φ, in the SAFS pedigrees using a linkage disequilibrium-pruned subset of all common single-nucleotide polymorphisms (SNPs) genotyped in 955 participants [icipants . For theicipants . These 2icipants , we usedi yis the phenotype of the ith individual, μ is a grand mean, i Gand i eare random additive polygenic and residual effects, respectively, for individual i. The vectors G and e, consisting of all polygenic and residual effects, follow zero-mean multivariate normal distributions with variance-covariance matrices n × n) identity matrix, n is the number of subjects, and where analysis .Step 2: penalized regression including a gene-based group penalty. Instead of using the linear model in the original approach proposed by Aulchenko et al, we applied a lasso-type group penalized regression to better incorporate rare variants. This type of penalized regression has been applied to many forms of genetic analysis, including microarray data [jth SNP. The factor 2 makes the value of the weight range between 0 and 1. In our study, we adopted this method for the whole genome sequencing data with the residuals generated via step 1 from the family data. Briefly, we have an objective function to be minimized that comprises the underlying sums of squares from ordinary regression (first term) modified by a conventional lasso penalty and, finally, the group penalty, as below:ray data  and GWASray data . Friedmaray data  introducray data . In this2 norm, m SNPs, β is an Gth group, often used to adjust for gene size. Here, jth SNP, and Gth group, where for our purposes G is defined by gene. The use of the 2 Lnorm on the effects within a group encourages the incorporation of rare causal variants inside the same gene. To test the power of this approach on rare variants in particular, we further excluded common variants  and repeated the analysis. These analyses were performed using the statistical software Mendel (Version 12.0) [where on 12.0) ,10. Thiscausal genes. Table PAK2 contains 4 SNPs used to model SBP and DBP, but, although none of these map within PAK2 using GRCh37/hg19, PAK2 can still be found in the fitted models.We used GRCh37/hg19 build annotations to map chromosome 3 variants to genes; 521,355 of the 1,215,399 variants were located in a total of 1,165 genes. There are 188 causal variants, ie, variants with nonzero effect sizes for either diastolic blood pressure (DBP) or systolic blood pressure (SBP) at exam 1; these map to 31 unique genes. One hundred sixty-eight of these variants are causal for DBP and 134 for SBP. Genes with at least 1 causal variant are here referred to as Table ou et al , and usiMAP4 is consistently discovered using our approach, which is not surprising because the variants within MAP4 confer more than 5% heritability for both DBP and SBP. No other gene encompasses that amount of heritability for either trait. In fact, the next highest heritability is that of FLNB for SBP (0.28%), and the other genes are correspondingly much less reliably detected. A few genes  are discovered with at least 10% probability in some models, although it is difficult to infer with confidence any pattern of characteristics for detection when these are so low. It appears that detection probability increases with more weight placed toward the pure lasso penalty, which is consistent with Figure FLNB, MUC13, SCAP, and ZBTB38) the improvement is considerable. Some interesting patterns in results from Table FLNB shows a markedly higher detection probability when testing SBP with all variants using a 90% lasso proportion; this discrepancy is because a common causal variant unique to SBP is removed from consideration when examining only common variants and is more reliably detected with pure lasso.or SBP 0.8%, and tPer-gene false-positive rates are reasonably maintained , partially because of the upper bound constraint that the penalty tuning parameter imposes. That is, because only approximately 134 variants can enter the final regression model, there is a limit to the number of times each noncausal gene can be falsely discovered. False discovery rates  are large, often above 90%, which is a known problem with penalized regression strategies.Methods for handling complex pedigrees are varied and are often computationally intensive. The advent of next-generation sequencing exacerbates these complexities. We have proposed a novel method for approaching this problem that is easily implemented in freely available software packages and eases computational burden by regressing out correlations caused by familial relatedness in an initial step, preventing the estimation of often complex mixed models for each variant under consideration.We chose to use a kinship matrix estimated based on genomic data to adjust for family structure because this confers several advantages over pedigree kinship ,4,6. FirMethods that group variants within genes and treat the gene as a functional unit, as in the group lasso portion of our approach, can efficiently borrow information across the gene without necessarily testing the disease-influencing variants. We find that, in the context of the GAW18 simulated data, our method can successfully and consistently discover disease genes with sufficient heritability, but is largely underpowered when heritability is below 0.03%. More exploration of the proposed approach through simulation is warranted to examine cross-validation strategies for choosing lasso parameters, the related effects of linkage disequilibrium structure, and private variants. Notably, private variants  will not likely be detected using our proposed methodology. The incorporation of tracking genetic transmissions will be required to do so, and any method that treats familial relatedness as a nuisance, as we do, in order to utilize test statistics for independent subjects will be severely underpowered for these variants.The authors declare that they have no competing interests.SS, XW, and DWF designed the overall study. XD, SS, KN, and DWF conducted the statistical analyses and created the tables and figure. SS and DWF drafted the manuscript, which was revised by XD, KN, and XW. All authors discussed the project throughout, read, and approved the final manuscript.


High-throughput experimental methods have fostered the systematic detection of millions of genetic variants from any human genome. To help explore the potential biological implications of these genetic variants, software tools have been previously developed for integrating various types of information about these genomic regions from multiple data sources. Most of these tools were designed either for studying a small number of variants at a time, or for local execution on powerful machines.https://yiplab.cse.cuhk.edu.hk/vas/). It provides a large variety of information useful for studying both coding and non-coding variants, including whole-genome transcription factor binding, open chromatin and transcription data from the ENCODE consortium. By means of data compression, millions of variants can be uploaded from a client machine to the server in less than 50 megabytes of data. On the server side, our customized data integration algorithms can efficiently link millions of variants with tens of whole-genome datasets. These two enabling technologies make VAS a practical tool for annotating genetic variants from large genomic studies. We demonstrate the use of VAS in annotating genetic variants obtained from a migraine meta-analysis study and multiple data sets from the Personal Genomes Project. We also compare the running time of annotating 6.4 million SNPs of the CEU trio by VAS and another tool, showing that VAS is efficient in handling new variant lists without requiring any pre-computations.To make exploration of whole lists of genetic variants simple and accessible, we have developed a new Web-based system called VAS (Variant Annotation System, available atVAS is specially designed to handle annotation tasks with long lists of genetic variants and large numbers of annotating features efficiently. It is complementary to other existing tools with more specific aims such as evaluating the potential impacts of genetic variants in terms of disease risk. We recommend using VAS for a quick first-pass identification of potentially interesting genetic variants, to minimize the time required for other more in-depth downstream analyses. High-density microarrays and massively parallel sequencing have made genome-wide detection of genetic variants from human DNA samples systematic, efficient and inexpensive. In these experiments, it is common to observe hundreds of thousands or even millions of loci in the DNA of a studied sample that differ from the reference genome. To explore possible links between these variants and the phenotypes of the sample, it is necessary to first analyze the potential biological significance of each variant.Early-days analysis methods have focused on the potential impacts of genetic variants in coding regions, the functional consequences of which are usually related to alterations to the corresponding proteins. There have been many successful software tools for classifying coding variants into those that are synonymous, missense and nonsense, whether they may affect splicing or cause frameshift, and the level of disruption to protein functions and structures–6.On the other hand, it is now well-recognized that many functionally important genetic variants do not change the coding sequences directly but rather perturb gene regulation, 8. For A large amount of data containing cell/tissue-specific features have been produced for various human cell types in large-scale studies such as ENCODE and RoadConcerning the data uploading issue, some tools restrict the maximum number of genetic variants per job to a small value, while others do not set an explicit limit but practically cannot handle full lists of millions of variants–13, 17. Regarding the data integration issue, most tools use a relational database to store the collected data. As a result, a table join between a stored feature and the uploaded genetic variants is often performed by time-inefficient algorithms that make use of standard tree-based indices. Although more efficient linear-time sort-merge join algorithms are available, it could be difficult to instruct the query optimizer to use them. Some tools attempted to solve this problem by pre-computing the results of a large amount of table joins, 21, whiTo overcome these two issues, we have developed VAS (Variant Annotation System), a tool for efficient genomic data integration.The overall workflow of VAS is shown in FigureVAS provides a large variety of genomic features collected from different data sources Table. To helpA user uploads a list of genetic variants and selects the features to be integrated through a user-friendly Web interface. Multiple data formats are supported for the input list of genetic variants, including VCF and white-space-delimited lists. In our test, uploading 3 million genetic variants involved less than 50 megabytes of data transfer Figure. The enaUpon submitting the input variants and the selected genomic features, the data integration job is added to a queue on the server side. The back-end system adopts a scalable design that allows executing multiple jobs on different computing nodes in parallel. The user is redirected to a waiting page that provides the latest status of the job. Optionally, if an email address is entered, an email notification will be sent to the user when the job is finished.We store data in a customized file format without relying on a relational database, which facilitated our design of linear-time integration algorithms that can efficiently identify overlapping genomic regions in different data files. Specifically, for each feature, the genomic regions containing feature values are sorted according to their genomic locations. Special pointers are added to particular locations (such as the start of each chromosome) in the genome to allow direct access of these locations without a sequential scan of all regions from the beginning of the file.We provide two types of data integration. The first one is identifying genomic features overlapping exactly the locations of the input genetic variants . The second one is identifying genomic features overlapping a flanking window of each input genetic variant. Both types of integration are performed by sort-merge algorithms.For the first type of data integration, we first sort the input variants according to their locations. We then use a pointer to scan through all the genetic variants and all the genomic feature regions sequentially. Whenever a region of the genomic feature is encountered, we add it to a feature queue. Any genetic variant that is then encountered before the end of the region will be annotated with the region and the result is stored in the variant map . We randomly downloaded 5 lists of genetic variants with at least one variant reported to have high clinical importance according to the report on the Web site Project home page:https://yiplab.cse.cuhk.edu.hk/vas/Operating system: VAS can be accessed from any platform by using one of the listed Web browsersProgramming languages: PHP, PythonOther requirements: We recommend accessing VAS by using Google Chrome (version 35 or higher), Microsoft Internet Explorer (version 10 or higher), or Mozilla Firefox (version 24 or higher), with JavaScript enabled and a minimum screen resolution of 1024 pixels x 768 pixelsAny restrictions to use by non-academics: Nil


In this paper, we demonstrate how these robust tests can be applied to the replication study of GWAS and how the overall statistical significance can be evaluated using the combined test formed by p values of the discovery and replication studies.In genome-wide association study (GWAS), robust genetic association tests such as maximum of three CATTs (MAX3), each corresponding to recessive, additive, and dominant genetic models, the minimum  With the advance of biotechnology and substantial reduction of genotyping costs, a genome-wide association study (GWAS) using hundred thousand markers in several thousand individuals is now increasingly utilized and has been successful in detecting genetic associations across the entire genome with complex human traits –6. Amongp value of Pearson's Chi-square test and additive CATT, and the asymptotic properties of this approach were studied in detail by Joo et al. [Two most frequently used association tests are the allele-based test (ABT) and the genotype-based test (GBT). ABT compares the allele frequencies between cases and controls, while GBT compares the genotype distributions of cases and controls. The Cochran-Armitage trend test (CATT) , 8 is a o et al. . These mp values of the discovery and replication studies. The importance of replication or validation in GWAS has been well recognized [In this paper, we illustrate how these robust tests can be applied to a replication study of GWAS and how overall statistical significance can be evaluated using the combined test formed by cognized , 17, andcognized , 19.p value for the replication data set, given the significant result of the discovery stage, using robust tests. In the next section, a combined test of the p values of the discovery and replication data sets is proposed, together with the way to evaluate the statistical significance for the combined test. Simulation studies are conducted to compare the type I error rates and powers of various analytical strategies. For illustration purposes, the summarized methods are applied to a non-small-cell lung cancer data set and at the end there is a discussion.The paper is organized as follows. We first describe the data structures and notation and review existing robust association tests for a single data set. Then we describe how to obtain the A and B, let the frequencies of B in cases and controls be p = P(B∣case) and q = P(B∣control). Denote three genotypes by G0 = AA, G1 = AB, and G2 = BB. In case-control association studies, r cases and s controls are independently sampled from each population. The observed genotype counts for  are  in the cases and  in the controls. Disease prevalence is denoted by k = P(disease) and penetrance by fi = P(disease∣Gi) for i = 0,1, 2. Two genotype relative risks (GRRs) are denoted by λ1 = f1/f0 and λ2 = f2/f0 using f0 > 0 as baseline penetrance. Under the null hypothesis of no association H0 : f0 = f1 = f2 = k or alternatively H0 : λ2 = λ1 = 1. Genetic model is recessive (REC), additive (ADD), multiplicative (MUL), and dominant (DOM) when λ1 = 1, λ1 = (1 + λ2)/2, λ1 = λ21/2, and λ2 = λ1, respectively.For a marker with two alleles H0. The test is given by ni = ri + si for i = 0,1, 2 and n = r + s. Under Hardy-Weinberg equilibrium (HWE), an allele-based test (ABT) and CATT with scores  for , where 0 ≤ x ≤ 1, are given by x0, x1, x2) =  [x for the recessive (REC), additive/multiplicative (ADD/MUL), and dominant (DOM) models are x = 0,1/2 and 1, respectively [Zx and ZABT asymptotically follow a standard normal distribution under H0. Zx can be used even when HWE does not hold. However, without the HWE assumption, ZABT does not follow a standard normal distribution due to the correlation between two alleles.The association in case-control studies can be tested using various methods which have been extensively studied. The general association between the disease status and the SNP can be tested using Pearson's Chi-square test which has an asymptotic Chi-square distribution with 2 degrees of freedom under 0, x, 1) . The optectively , 20. BotZ0 | , |Z1/2 | , |Z1|). Parametric bootstrap or permutation methods can be used to find the p value of MAX3 [A robust test, MAX3 proposed by Friedlin et al. , can be  of MAX3 .p values of Pearson's Chi-square test and CATT under the additive genetic model Z1/2 be Pchi2 and P1/2, respectively. WTCCC [Pchi2, P1/2). Joo et al. [p value of MIN2 can be obtained as H1 and H2 are the cumulative distributions of Chi-square distributions with 1 and 2 degrees of freedom.Let the y. WTCCC  proposedo et al.  derived p and Δq, where Pr(BB)−{Pr(AB)/2 + Pr(BB)}2 and Δp and Δq denote the HWD coefficient in cases and controls, respectively.On the other hand, Song and Elston  considerp, Δq) to determine the genetic models in their two-phase method. Their two-phase statistic ZGMS is given by ZGMS = Z0 if ZH > c, Z1 if ZH < −c, and Z1/2 otherwise, where c = Φ−1(1 − αH) for αH = 0.05. The asymptotic correlations between ZH and three CATTs under HWE were derived and the significance level was adjusted accordingly to control the desired type I error. Based on the observation that this method assumes B is the risk allele, Joo et al. [ZGMS when either one of the alleles can be a risk allele. They chose the risk allele based on the sign of Z1/2; that is, if Z1/2 > 0, B is the risk allele, and Z0, Z1/2, and Z1 are chosen for REC, ADD, and DOM models, respectively. If Z1/2 < 0, the respective test statistics are chosen to be −Z1, −Z1/2, and −Z0. They incorporate this property in defining the test statistic for genetic model selection (ZGMS) and calculating the p value. Let Θ0(z) = {z : z > c}, Θ1/2(z) = {z : |z | <c}, and Θ1(z) = {z : z < −c}. Then, the p value of this method can be obtained byρx = Corr in   are replaced by their consistent estimates. Here, t = zGMS and (−t∧0) = min. Moreover, zGMS and z1/2 are the observed values of ZGMS and Z1/2, respectively.Zheng and Ng  used theo et al.  studied ZH is very low especially for low to moderate GRRs, but the unlikely genetic model can be successfully excluded. This led to genetic model exclusion method ZGME which is the same as the ZGMS described above except Zx for x = 0,1/2,1 is replaced by Zx* where p value of GME can be obtained ast = zGME.While studying the properties of GMS, Joo et al.  noticed p value of robust association tests, including MAX3, MIN2, ZGMS, and ZGME, can be obtained as described in p value of replication data using the robust method, we use the same analytic method that was used for discovery and the risk allele identified by it [In the discovery stage, the ed by it . This meZxi). Also, denote x = 0,1/2,1. Then, for CATT with a preselected genetic model, Px(2) = 1 − Φ(sign(Zx(1) · Zx(2))·|Zx(2)|) using a one-sided p value given the direction of association from the discovery stage, and Px(2)∗ = 1 − Φ(sign(Zx(1)∗ · Zx(2)∗)·|Zx(2)∗|). Moreover, denote the test statistics and p values using Pearson's Chi-square test from the ith stage as Tchi2i)( and Pchi2i), PMIN2(2), PGMS(2), and PGME(2), can be obtained as follows: Suppose that, for simplicity of notation, our interest is in GWAS with two stages, one for discovery and the other for replication, although the methodology described below can be extended to multistages for replication. Let p values, the p values from the two stages are independent under the null hypothesis. This is because, under the null hypothesis, the probability of Z1/2 being positive or negative is simply 1/2, and the probability of the selection of a certain genetic model is also a constant (αH for the recessive and dominant models and 1 − 2αH for the additive model).It is important to note that even though the direction of the test statistics and the selected genetic models are used to obtain the second stage p values from the discovery and replication stages of GWAS. We consider using p values rather than the test statistics because test statistics can have complex forms and obtaining the distribution of the joint test can be difficult. On the other hand, calculating a p value for each data set might be relatively simple, and the distribution of p values under the null hypothesis of no association is easy to handle.For a given robust test, we can consider the joint analysis by combining p values after −2log transformation; that is, ZFC = −2w1log(P(1)) − 2w2log(P(2)), where Pi) where πs = ND/(ND + NR), and ND and NR are sample sizes of the discovery and replication data sets. A linear combination of two P values after taking the inverse of the standard normal cumulative distribution is given by αD, which means that markers with P(1) < αD are selected and replicated in the replication stage. The p value of combined test can then be obtained as pFC = PH0(P(1) < αD, ZFC > zFC) where the observed value of ZFC is zFC. The pFC are calculated as ezFC/2−(1 + zFC/2 + logαD) for equal weights where zFC > −2logαD and (w1/(w1 − w2))ezFC/2w1− − (w2/(w1 − w2))ezFC/2w2−αdw1−w2)/w2− < αD, ZFC > CFC) ≤ α. Similarly, for the ZLC, the p value is calculated as pLC = PH0(P(1) < αD, ZLC > zLC) = zLC > zαD/21− where the observed value of ZLC = zLC.There are several methods for combining test statistics from two stages , and twoπs) of 0.5 and 0.6 were considered for the minor allele frequency (MAF) of 0.3 and 0.4. We considered M = 10 markers to control the genome-wide false positive rate at α = 0.05 with the Bonferroni correction. We did not consider a larger M such as 300,000 or 500,000 because this will require more than 10 million simulations to show a stable estimate of the type I error rate. With M = 10, we performed 20,000 simulations which result in less than 10% of a coefficient of variation for a significance level 0.05/M = 0.005 for each marker [Z1/2, Pearson's Chi-square test, MIN2, MAX3, GMS, and GME. For the second stage analysis, we considered a replication-based analysis, ZFC, and ZLC as proposed above. The results are based on the situation under HWE (HWE coefficient F = 0). As expected, all tests control the type I error reasonablly well, and similar results were obtained when a slight deviation from HWE is present with F = 0.05 (results not shown).h marker . The tesM = 10 markers, a disease prevalence of 10%, the same genotype relative risk for two stages (r1 = 1.4 and r2 = 1.4), and 1,000 cases and 1,000 controls. 2,000 simulations were performed under HWE (F = 0) to control the genome-wide false positive rate at α = 0.05. The recessive, additive, and dominant models were assumed for the first, second, and third rows. Both joint analyses showed better power performances compared to the replication-based analysis .We examined the empirical powers of different tests considered above. In Z1/2. In the replication stage, 168 SNPs with p value less than 1 × 10−4 in the first stage based on Z1/2 were tested using 804 patients and 1470 control samples. We identified additional 234 SNPs using MIN2 in the first stage which could be studied in the replication stage if MIN2 was used instead of Z1/2 since MIN2 produces stronger evidence for the additional SNPs than Z1/2 does. The Manhattan plots of using MIN2 and Z1/2 are presented in rs385272 located in chromosome 2, which had a p value of 1.37 × 10−7 which reached significance level at Bonferroni correction in discovery samples alone, whereas Z1/2 yielded a p value greater than 1 × 10−4. Even though there is possibility of false positive findings, these SNPs could have been selected for replication if robust methods were used.The GWAS on non-small-cell lung cancer (NSCLC) by Yoon et al.  studied Z1/2 in Yoon et al. [rs2131877 that was reported by Yoon et al. [αD = 5 × 10−5 so that all these exemplary SNPs can be selected in the discovery stage; the p value of combined test based on four robust methods  as well as Z1/2 and Pearson's Chi-square test is presented in rs2131877 was found to be significant with Bonferroni correction  by all except MAX3 method.Since we do not have replication data for these additional SNPs selected using MIN2 because the first stage selection was based on n et al. , just fon et al. . When thp values of the discovery and replication studies.In genetic association studies, efficiency robust tests whose performance does not depend on the underlying genetic model have been extensively studied, and their power benefit over a wide range of genetic models has been well recognized. In this paper, we described how the idea of these robust association tests can be applied to the replication studies and further how overall statistical significance can be evaluated using the combined test formed by p value of each stage might be relatively simple. Because the asymptotic distribution of the p value under the null hypothesis of no association is easy to handle, the combined test using p values rather than the test statistics themselves can provide computational convenience.When the robust tests are used, the test statistic of each stage can have a complex form and thus dealing with the distribution of the joint test can be difficult, whereas calculating the p values of these two methods. In our limited experience, the linear combination and Fisher's combination are fairly comparable. Fisher's combination seems to perform slightly better than the linear combination when there exists some heterogeneity between stages in terms of the genotype relative risk, while the linear combination seems to perform slightly better in most of other situations. However, the difference is extremely minor. Further research is required for the thorough comparison of various methods of combining p values in the application of efficiency robust tests to the replication of genetic association studies.There are several methods for combining test statistics from two stages and Won et al.  comparedIn a genetic study where the purpose of considering a replication stage is to validate or replicate the genetic findings from the discovery stage, which is the case considered in this paper, the analysis in the replication stage utilized the test statistic or genetic model that is selected as being the best in the discovery stage and also the direction of the risk allele, following guidelines for exact replication in genetic association studies. If the purpose is to simply combine the evidence from different data sources such as in meta-analysis, other strategies may be devised. Further research, again, is required to provide fully detailed properties of such methods.p values. However, even though it was minor in some situations, the joint anlysis presented better power performance than the replication-based analysis in our study. This type of joint analysis raised concerns about the exact meaning of replication [Power gain of a joint analysis over the conventional replication-based analysis was thoroughly studied by Skol et al. , 19. In lication . Howeverlication  mentionep values of the SNPs under a specific genetic model, usually an additive model, which were further genotyped in the replication stage and were finally defined to be significantly associated with a phenotype of interest. As emphasized in this paper, one may have a better chance of finding many missing SNPs by applying more powerful and robust methods that consider different genetic models simultaneously. Therefore, we urge the community to share test results under not only an additive model but also other genetic models, although they were not significant at a stringent significance level, so that future research may have enriched data resources, to which robust tests can be applied in association studies.We illustrated how the proposed methods can be applied in the real data that studied the association of SNPs with non-small-cell lung cancer (NSCLC) in discovery and replication stages. In the original study reported by Yoon et al. , SNPs we