A number of studies have been conducted to investigate the predictive value of common genetic variants for complex diseases. To date, these studies have generally shown that common variants have no appreciable added predictive value over classical risk factors. New sequencing technology has enhanced the ability to identify rare variants that may have larger functional effects than common variants. One would expect rare variants to improve the discrimination power for disease risk by permitting more detailed quantification of genetic risk. Using the Genetic Analysis Workshop 17 simulated data sets for unrelated individuals, we evaluate the predictive value of rare variants by comparing prediction models built using the support vector machine algorithm with or without rare variants. Empirical results suggest that rare variants have appreciable effects on disease risk prediction. The potential of common genetic variants detected from genome-wide association studies to predict the risk of complex diseases has been investigated in a steadily increasing number of empirical studies. So far, these studies generally show limited predictive value of genetic factors -4. This Recently, substantial advances have taken place in developing new sequencing technologies  . SequencThe aim of this study is to determine whether rare variants provide valuable predictive information beyond that provided by common variants and environmental covariates alone. To this end, we examine the effect of adding collapsed rare SNPs to prediction models that include both environmental covariates and common genetic variants in the Genetic Analysis Workshop 17 (GAW17) simulation data sets. The prediction models are built using an SVM algorithm ,9,10, whThe first data set of GAW17 consists of a collection of 697 unrelated individuals from the 1000 Genomes Project. There are 200 replicates of simulated trait information and a number of nongenetic covariates such as age, sex, and smoking status. SNP genotypes were obtained from the sequence alignment files provided by the 1000 Genomes Project for their pilot3 study . Includep-values less than a prespecified threshold from association analysis, with adjustment for covariates. Here, rare variants are defined as SNPs with minor allele frequency (MAF) less than 5% [To assess the effect of rare variants on global disease risk prediction, we consider prediction models built using an SVM algorithm. The SVM is one of the popular classifiers in the field of machine learning and delivers state-of-the-art performance in a wide variety of biological applications . In esse than 5% ,13.p-values less than a prespecified threshold  are used for disease risk assessment in the next step. For the analysis of rare variants (MAF < 5%), SNPs are first collapsed by the presence or absence of minor alleles within each gene in each individual [p-value threshold used to select variants ranged from 1.0 × 10−5 to 0.01.The association between disease and common SNPs (MAF ≥ 5%) is evaluated using Fisher’s exact test by comparing allele counts between case subjects and control subjects. SNPs with dividual -17. For The SVM training algorithm is applied to these variants and to the covariates Age, Sex, and Smoking status. The genotype data for common SNPs are coded 0, 1, or 2, reflecting the number of minor alleles. Rare variants are coded 1 or 0, corresponding to the presence or absence, respectively, of minor alleles within each gene. Prediction models are built to discriminate between case subjects and control subjects. The risk prediction model is built using the SVM algorithm in the training data set, and the prediction error of the model is assessed in the validation data sets.To evaluate the predictive value of rare variants, we conducted two experimental studies. In the first experiment, the set of SNPs included in the risk prediction model was selected from the first trait replicate, and the prediction model was built on the same data set. Prediction error was assessed on the remaining 199 trait replicates. In the second experiment, for each trait replicate we randomly divided the data into a training set and a validation set. SNP selection and risk prediction models were performed on the training set, and prediction error was estimated from the validation set. We repeated this procedure in each of the 200 trait data sets. In this second experiment, the size of the training set took values from 300 to 600, with an increment of 100.http://www.csie.ntu.edu.tw/~cjlin/libsvm). We trained the soft-margin linear SVM classifiers [C = 1, the default value of the R package.We used the R package e1071 to build the risk prediction models. This package is an interface to the LIBSVM implementation of the SVM algorithm  curve analysis to the validation data sets. The ROC is a widely used tool to evaluate the discrimination ability of a binary classifier. In ROC analysis, the discriminatory power of the prediction model is usually measured as the area under the ROC curve . This is the probability that a randomly chosen positive sample will have higher predicted risk than that of a randomly chosen negative sample. We compared the AUC values of prediction models combining both common and rare variants with the AUC values of models incorporating only common variants.p-value cut points were selected from 1.0 × 10−5 to 0.01. All SNPs reaching the predefined threshold were included in the prediction models. We evaluated the prediction potential of rare variants by comparing models including rare variants with models without rare variants.Seven p-value thresholds, despite the fact that a higher threshold allows more genetic factors in the prediction model. This suggests that adding less significant SNPs does not provide sufficient improvement in discrimination power. In comparing the AUC values between models with and without rare variants, we found that incorporation of rare variants slightly improved the prediction. The improvement was statistically significant if we compared the two AUC values using a paired T-test. However, the magnitude of the improvement was relatively small (<0.4%).Table In the first experiment, the prediction model was built on the first trait replicate. We also evaluated the performance of prediction models using other trait replicates as a training set. Models built on different replicates could choose quite different sets of genetic variants, but the results of these models were similar to the results shown in Table p-value threshold increases. Therefore prediction models perform better with more variants included in the model. However, this relationship is not apparent when the size of the training set is 300. With such a small training set, the estimation of association between disease and variants is not accurate. A larger p-value threshold may result in a higher proportion of false-positive variants included in the prediction model, thus leading to smaller AUC values.Table In the second experiment, the differences between the AUC values of models with rare variants and models without rare variants were significant at the 0.05 level, and the differences were much larger than the differences in the first experiment. The largest AUC difference between two models was more than 6%, suggesting a great potential for the improvement of prediction models through incorporation of rare variants.By using prediction models built on the GAW17 simulated data sets and using the SVM algorithm, we conducted two experiments to assess the value of rare variants in complex disease risk prediction. In our studies, including rare variants marginally improved the classification of risk prediction models in the first experiment and substantially improved the classification in the second experiment. In both experiments, rare variants had an appreciable effect on disease risk prediction.In the SVM literature, two kernel functions are commonly used: the linear kernel:k = x′y (1)and the radial kernel:We applied both the radial kernel and the linear kernel in our two experiments. The predictive values contributed by rare variants were similar between the two kernels. The results presented in this paper are limited to the linear kernel for its good interpretability .C = 1 are shown in this paper.In addition, we used different penalty parameters, ranging from 0.001 to 1000, to build the prediction models. Although the SVM algorithm tends to assign different weights for risk factors under different penalty parameters, the performance of the prediction models are similar for different penalty parameters. Results for the penalty parameter The AUC value is one of the popularly used statistics for model comparison. We also computed other measurements of discriminatory power for prediction models, such as accuracy, true-positive rate, false-positive rate, positive predictive value, and negative predictive value. The difference between the two types of prediction models have similar patterns to the results for the AUC value (data not shown)..The authors declare that there is no competing interestCW and ZW designed the study, and CW performed the statistical analysis and drafted the manuscript. ZW participated in the design of the study and provided critical revisions of the manuscript. KMW and ATD participated in designing the study, reviewing and editing the manuscript. JH directed the study and revised the manuscript. All authors read and approved the final manuscript.


Triad families are routinely used to test association between genetic variants and complex diseases. Triad studies are important and popular since they are robust in terms of being less prone to false positives due to population structure. In practice, one may collect not only complete triads, but also incomplete families such as dyads (affected child with one parent) and singleton monads (affected child without parents). Since there is a lack of convenient algorithms and software to analyze the incomplete data, dyads and monads are usually discarded. This may lead to loss of power and insufficient utilization of genetic information in a study.We develop likelihood-based statistical models and likelihood ratio tests to test for association between complex diseases and genetic markers by using combinations of full triads, parent-child dyads, and affected singleton monads for a unified analysis. A likelihood is calculated directly to facilitate the data analysis without imputation and to avoid computational complexity. This makes it easy to implement the models and to explain the results.TGFA) gene and cleft palate in an Irish study.By simulation studies, we show that the proposed models and tests are very robust in terms of accurately controlling type I error evaluations, and are powerful by empirical power evaluations. The methods are applied to test for association between transforming growth factor alpha ( In family-based studies, one might collect triads, sib-ships, parent-child dyads, general pedigrees or some combinations. In modern times, large multi-generation pedigrees are not common, and small nuclear families are more practical to collect. In our birth defects studies, almost all families contain only a single affected child with or without parents. They are basically triad families allowing for missing parents . In famiIn practice, one may collect not only complete triads, but also incomplete families such as dyads (affected child with one parent) and singleton monads (affected child without parents). Here the terminology of dyads and monads are taken from Weinberg . Since tTriad studies are popular and important because the triad families are relatively easy to collect. More importantly, the results of triad studies are robust in terms of being less prone to false positives due to population stratification. To analyze triad data, TDT analysis is usually performed . To use In this paper, we develop likelihood-based statistical methods to test for association between complex diseases and genetic markers by using combinations of full triads, parent-child dyads, and affected singleton monads for a unified analysis. Our research interest is stimulated by our oral clefts study . We assuTGFA) gene and cleft palate in the Irish study -TGFA gene of an Irish study to show the association found previously . Le. LeF,M, R=p2ψ2+2pqψ1+q2, C=AA,Aa,aa, and i=1,2,⋯,6. There are 10 combinations . The results are presented in Table where ni are sub-sample sizes of the ten entries in Table where G and C, where G=M or G=F is the genotype of the parent and C is the genotype of the affected child. Given the sampling scheme of using the affected child as a proband, the conditional probabilities of parent-child pair genotypes P can be derived. For instance, we may calculateFor parent-child dyads, denote the genotypes at the candidate gene locus as Table AA, one of its parents has genotype Aa, and the genotype of the other parent is missing” which does not specify which parents, father or mother, having Aa or missing genotypes. However, the present paper takes an ordered example of mother having Aa genotype and father having missing genotype, which is different from the unordered case in Schaid and Sommer [Our derivation above in (2) is different from that of Schaid and Sommer , p1119, d Sommer . To maked Sommer  calculatIn practice it is easy to make mistakes by applying the unordered result like (4) in Schaid and Sommer  directlys affected singleton monads, assume s2 of them have genotype AA, s1 of them have genotype Aa, and s0 of them have genotype aa (s2+s1+s0=s). Then, P(AA|D)=p2ψ2/R, P(Aa|D)=2pqψ1/R, and P(aa|D)=q2/R. Let us denote the likelihood of the affected singleton monads as Laffected−Monads. Then, we have the following log-likelihoodFor the Based on the three log-likelihoods (1), (3), and (5), we may calculate the log-likelihood of full data as follows H0:ψ1=ψ2=1. There is only one parameter p to estimate under the null hypothesis H0 and the log-likelihood is equal to p.Under the null hypothesis of no association between the disease and the marker locus, we have HUnr:ψ1≥0,ψ2≥0. Let ψ1,ψ2 and p under HUnr. The likelihood ratio test statistic of association isWithout any restrictive condition on the parameters, one gets an unrestricted alternative hypothesis Unr is approximately chi-square distributed with 2 degrees of freedom (DF) by large sample theory, when the sample size is sufficiently large.HDom:ψ1=ψ2. Let ψ1 and p under HDom, respectively. The LRT of association isUnder a dominant model, one imposes a restriction of an alternative hypothesis HRec:ψ1=1,ψ2≥0. Let ψ2 and p under HRec, respectively. The LRT of association isIf a recessive disease model is desired, one has an alternative hypothesis ψ1 and p under HMult, respectively. The LRT of association isUnder a multiplicative model, an alternative hypothesis is HAdd:ψ2=2ψ1−1. Let ψ1 and p under HAdd, respectively. The LRT of association isIf an additive model is used, one has an alternative hypothesis Dom, Rec, Mult, and Add are approximately chi-square distributed with 1 DF by large sample theory, when the sample size is sufficiently large. In Appendix-A of Additional file TDT=(b−c)2/(b+c), where b=n2+2n5+n6+n8 and c=n3+n6+2n7+n9[W=b+b1,Acom=(b+c)/2+(b1+c1)/2,Vcom=(b+c)/4+(b1+c1)/4,b1=m2+m5, and c1=m3+m6[TDT is approximately chi-square distributed with 1 DF and zcom is approximately normally distributed when the sample size is sufficiently large.Using the notations in Table  c1=m3+m6. By largModels. For instance, p is the allele frequency of allele A,n is the number of triad families, m is the number of dyad families, and s is the number of monads. Hence, n,m, and s are sample sizes for triads, dyads, and monads, respectively.In our simulation, we use the same notations as those in the section of n=100 of triad families, a sample size m=25 of dyads and no monads s=0, a given allele frequency p=0.5, and parameters ψ1=ψ2=2.6. By using the 10 probabilities of Table p=0.5 and parameters ψ1=ψ2=2.6, we generate the triad counts Unr using unrestricted model. The process is repeated 100,000 times. The number 0.84541 is the the proportion of the Unr test values calculated for the 100,000 samples, that exceed the 95-th percentiles of the ψ1 and ψ2 are taken to be 1 under the null hypothesis of no association using the multinomial distribution.For power calculations, the data are simulated under disease models using the multinomial distribution. For instance, let us look at the upper left corner cell 0.84541 of empirical power in Table The authors declare that they have no competing interests.RF, AL, JFT, and JLM contributed to the conception and design of the study, analysis and interpretation of data, and AL and ZL wrote the programs, and performed the simulation analysis. All authors have been involved in writing the manuscript and approved this final version. All authors read and approved the final manuscript.Supplementary material.Click here for file


Questions remain regarding the utility of self-reported ethnicity (SRE) in genetic and epidemiologic research. It is not clear whether conditioning on SRE provides adequate protection from inflated type I error rates due to population stratification and admixture. We address this question using data obtained from the Multi-Ethnic Study of Atherosclerosis (MESA), which enrolled individuals from 4 self-reported ethnic groups. We compare the agreement between SRE and genetic based measures of ancestry (GBMA), and conduct simulation studies based on observed MESA data to evaluate the performance of each measure under various conditions.Four clusters are identified using 96 ancestry informative markers. Three of these clusters are well delineated, but 30% of the self-reported Hispanic-Americans are misclassified. We also found that MESA SRE provides type I error rates that are consistent with the nominal levels. More extensive simulations revealed that this finding is likely due to the multi-ethnic nature of the MESA. Finally, we describe situations where SRE may perform as well as a GBMA in controlling the effect of population stratification and admixture in association tests.The performance of SRE as a control variable in genetic association tests is more nuanced than previously thought, and may have more value than it is currently credited with, especially when smaller replication studies are being considered in multi-ethnic samples. The use of self-reported race and ethnicity (SRE) in genetic and epidemiologic studies has been much discussed in the literature -8. Some Some studies have found SRE to be closely related to an individual's genetically estimated ancestry proportions ,10 and hWe used ancestry informative markers (AIMs) and phenotypic data on left-ventricular hypertrophy (LVH) collected in the context of the Multi-Ethnic Study of Atherosclerosis (MESA) to address two related questions: (1) what agreement is there between SRE and clusters created based on the genotyped AIMs? (2) In multi-ethnic genetic association studies, does SRE provide comparable type I error control to that provided by genetic ancestry background measures, such as individual ancestry proportions or genetic background scores? To address these two questions we compared three sets of measures; SRE, individual ancestry proportion (IAP) estimates obtained using STRUCTURE ,12, and LVH is a condition where the ventricular mass increases as existing cells of the LV enlarge or hypertrophy . LVH is κ between the two classification procedures is about 0.83 when all four self-reported ethnicities are analyzed jointly. However, the agreement between these two methods is almost perfect (κ = 0.98) when the self-reported Hispanics were not considered in the analysis. The marginal homogeneity test, which in this case is testing the null hypothesis that 2 classification methods are consistently assigning the same individuals to the same cluster, is rejected at the 0.05 significance level when the self-reported Hispanics are included in the analysis Ward's minimum variance and the K-means clustering algorithm applied on the 96 AIMS yielded similar clusters. Here we present only the results from the K-means algorithm. Figure Q =  denote the ancestry proportion of an individual in the dataset where i qrepresents the proportion of alleles that this individual has inherited from the th iancestral population. Q therefore estimates, for each individual, the proportions of African, Chinese, European and American ancestry estimated based on the genotyped AIMs. In general, for individuals who have not experienced a recent admixture event in their lineage, it is expected that the Q vector will have 3 of its components very close to 0 and the fourth component close to 1. All significant deviations from these numbers can be seen as either evidence of recent admixture events, a sign of discrepancy between self-reported ancestry and estimated ancestry given that each MESA participant self-identified with only one ethnic group, or weak resolution due to limited ancestry informativeness content of the marker panel. The average ancestry proportion estimate is given for each self-reported ethnic group in Figure We also ran the program STRUCTURE to obtain individual ancestry proportion estimates assuming 4 ancestral populations. Let Q = q1,q,q3,q4 de-14, which shows strong evidence that the mean LV mass is different between the 4 self reported ethnic groups. The association of SRE with the distribution of the LV ejection fraction is more pronounced compared to what is observed with adjusted LV mass. Although the observed mean and standard deviation of the LV ejection fraction in each ethnicity appear to be very close, its overall distribution varies in the 4 ethnic groups. The Kruskal-Wallis test comparing this distribution in the 4 ethnic groups has a p-value of 2 × 10-24, which strongly suggests that the distributions are different in the 4 self-reported ethnic groups. These two results confirm the presence of the previously identified ethnic specific differences in the distribution of both BSA adjusted LV mass and the LV ejection fraction. We also note that after standardizing both phenotypic variables, SRE explains only 3% of the variation explained in LV mass while it explains 19.5% of the variations in LV ejection fraction. Therefore, it is expected that variations in ethnicity will play a more important role in determining the level of LV ejection fraction compared to adjusted LV mass.The summary statistics for the distribution of BSA LV mass and the LV ejection fraction by self-reported ethnicity is given in Table It is known that population stratification can lead to confounding issues in genetic association studies . To asseAs mentioned above, the second set of simulation studies was designed to better understand when SRE might perform well as a control variable in genetic association tests.These simulations showed that when the confounder is univariate - that is, when there are exactly 2 ancestral populations, some type I error inflation may still occur when SRE is used as a control variable. We observed this inflation even in the absence of misclassification error. If there is a discontinuity point, as would most likely be the case when a sample of African Americans and European Americans is collected, the bigger the gap in the observed ancestry proportion the smaller the type I error inflation. Therefore, it is safe to conclude that the inflation rate depends on the composition of the sample. For example, when the study sample is comprised of admixed individuals derived from intermating between exactly 2 ancestral populations, the type I error inflation can be very small when the study sample is composed of individuals whose ancestry proportions are near the extreme values (near 0 or 1). This type I error inflation becomes substantial when the sample is composed of individuals whose ancestry proportion is near 50%. Figure The type I error rates observed for simulation 3 are displayed in Figure 2 from  to  to make it independent but identically distributed with G1 then the observed type I error rate dropped from 0.165 to 0.07. We repeated the same experiment by keeping the frequency of G1 fixed and changing the frequency of G2 to . The observed type I error was 0.18; as demonstrated above, by changing the order of the first 2 components of the allele frequency vector, this type I error is reduced to 0.065. The type I error associated with the scenario described in figure We ran additional simulations studies in order to better understand the effect of misclassification errors on the type I error rate. We considered two scenarios that are described in Figure We also considered the effect of misclassification error between self-reported ethnicity and true ethnicity when the study sample is made of 4 ancestral groups according to their representation in the MESA sample. We looked at the effect of discrepancy between true and self-reported in each ethnic group separately. We observed a small, almost negligible, type I error inflation when a misclassification rate varying between 5% and 15% occurred in exactly one ethnic group for individuals whose self-reported ethnicity is simulated to be European-Americans, African-Americans or Hispanic-Americans. However, significantly higher type I error inflation follows when there are discrepancies between SRE and true ancestry for individuals whose initial ancestry was Asian. This result makes sense intuitively because, as can be seen in Figure We focused on the utility of self-reported ethnicity as a control variable in genetic and epidemiologic studies. We used data collected in the MESA for LVH traits, specifically, LV mass and ejection fraction, to illustrate our points. LVH is one of the strongest determinants of cardiovascular outcomes. Ethnic differences in the distribution of both LV mass and the LV ejection fraction have been reported in many studies, and we found significant evidence of an ethnicity related effect on these phenotypes in the MESA sample.We observed a high degree of agreement between self-reported ethnicity and two GBMAs computed using genotyped ancestry informative markers. The self-reported Hispanic-Americans were by far the most heterogeneous group represented in this dataset. This result is not surprising given the current definition of the term "Hispanic" which refers to a group of individuals who are culturally and genetically quite diverse. It is now well accepted that the ancestry distribution of self-reported Hispanics reflects, at different degrees, the genetic contribution of the three ancestral populations Africans, Europeans and Native American .Another factor that may explain the genetic heterogeneity detected among the self-reported Hispanics may be the lack of individuals from the Native American ancestral groups represented in the sample. The initial panel of ancestry informative markers used in the MESA study was chosen based on their capacity to distinguish between individuals of Chinese, African and European ancestry. This panel might not be adequate to detect subtle variation between individuals who self identified as Hispanics. Following this analysis, a new panel of markers known to be particularly informative for Hispanics was typed in an effort to better understand the observed variation in the estimation of ancestry in this ethnic group. However, judging by the observed type I error rates, it appears that ancestry proportions estimated with the current marker panel work well as control variables in all the association tests that we have considered.K = 2 is discouraged. Our analyses simply suggest that there may be cases where SRE might provide adequate type I error control in the presence of population stratification. This MESA sample, given its composition, seems to be one of these cases. SRE would not necessarily perform as well in other cases. Investigators, in general, will be best served by allocating part of the resources available for their study to genotype the appropriate set of ancestry informative markers. However, SRE seems to be a valuable alternative in multi-ethnic samples when the misclassification rate is likely to be small. This observation is particularly true for smaller studies like candidate gene and other replication studies where a relatively small number of markers are being considered. It is also not unexpected; for example, Walcholder et al.  denotes the new order after sorting the residuals.Compute a new pseudo-phenotype m = 1 to K {(for each marker)For m ≠ i) {If  = 0.6. We also looked at the effect of various misclassification rates in these association tests.N = 2000 be the total number of individualsLet For gap = 0.05 to 0.5 by 0.05{x from Draw individual ancestry proportion x ∈ and 2 otherwise.Set true ethnicity to 1 if m represent the misclassification rate.Let m = 0.025 to 0.15 by 0.025{For s from a Bernoulli (m).Draw the random variable s = 0) then SRE = true ethnicityIf .Compute α1 and σ such that the effect size is equal to 0.5 in Figure ## Note that we set Y is also independent of 2G.## Note that conditional on the individual ancestry, the random variable Fit the following 3 linear regressions:y = β0 + β1x + β1G2 + e1. y = β0 + β1ethn + β2G2 + e2. y = β0 + β1sre + β2G2 + e3. β2 is statistically significant than 0 at the 0.05 level in each case.Test whether Y and 2 Gwill constitute a type I error.## A statistically significant association observed between }β2 is statistically significant for each control variable. These proportions for the ancestry proportion and SRE regressions are shown in Figure Repeat the experiment 10,000 times for each configuration of the gap value and the misclassification rate and count the proportion of times that the parameter We wanted to determine how SRE, when used as a control variable against population stratification, would perform in a multivariate setting. That is, when the number of number of ancestral populations is greater than 2. This simulation procedure resembles the previous one, except that the individual ancestry proportions are drawn from a Dirichlet distribution. We used a different set of parameters for each ethnic group in order to create the conditions needed for confounding to occur. The parameter used to generate the Dirichlet distribution can be represented by a 4 × 4 matrix, where the rows represent the expected individual ancestry proportions in each ethnic group, and for a fixed row, the columns represents the expected individual ancestry proportion from each ancestral population.N = 2000 individuals divided equally into 4 subsets such that k n= 500 for k = 1,2,3,4 is the sample size in each ethnic group.1) Let th ksubgroup be k.2) Let the SRE for an individual in the 3) We considered 4 cases; the parameters used for the Dirichlet distribution in each case are chosen as follows:(a) Proportions that are very close to ancestry proportion estimates obtained with STRUCTURE in the MESA sample;(b) Values near a 4 × 4 identity matrix with diagonal elements equal to 0.97 and off diagonal elements equal to 0.01;(c) Values very close to what would be observed in equally admixed individuals, that is all proportions are set 0.25;(d) Different ancestry proportions where the contribution of one specific ancestral population is clearly greater in each admixed population. That is the diagonal elements are set to 0.55 and off diagonal elements at 0.15 and made sure that the row and column sums are equal to 1.P1 =  and P2 =  be the frequency of the reference allele of the markers G1 and G2 respectively in each ancestral population. These frequencies were chosen arbitrarily with the only constraint being that they vary greatly between the 4 ancestral populations. Confounding will occur if the distribution of the trait is also different in the 4 ancestral populations.4) Let G1 and G2 in the admixed population (5) The allele frequencies of G1 and G2 according to these averages.6) Draw Y is simulated from a normal distribution where the parameters are based on the distribution of the LV ejection fraction observed each ethnic group observed in the MESA study. That is, we used the mean in each group as the intercept in the model described in the next step. We then compute the pooled variance and set σ2 represents the pooled variance.7) The outcome variable 8) The remaining steps are similar to those taken in simulation 2. That is, we fitted the following linear regressions:y = β0 + β1x1 + β2x2 + β3x3 + β4G2 + e. The variables 1x, 2 xand 3 xare the first 3 components of the individual ancestry proportion vector drawn from the Dirichlet distribution.1. y = α0 + α1SRE + α2G2 + e. The variable SRE has 4 levels, which are defined in step 2. Therefore, 1 αis a vector with 4 components.2. β4 and α2 are statistically different than 0 at the 0.05 level in each model, and repeated each experiment 10,000 times. The results of this simulation procedure can be seen in Figure We then tested whether adx1, adx2 and adx3 such that adx1 + adx2 ≤ 1, define 3 or 4 specific regions. Figure adx1, adx2, adx3), it is not exactly clear what the correct ethnicity assignment should be for the individuals whose ancestry proportions fall in region IV. There is no such ambiguity in Figure To better understand when the use of SRE as a control variable may fail, we devised a situation where it may be unclear which ethnicity to assign to individuals whose ancestry proportions take specific values. To facilitate the graphical representation of each scenario, we focused on the case where there are exactly 3 ancestral populations. In this case, an individual's ancestry proportion can be represented by a vector with 3 components, adx1 and one for adx2. The ancestry proportions are each drawn independently from a uniform distribution, which has been rescaled such that the proportions add up to 1. We then excluded the ancestry proportions that fell in the regions defined by the gaps. In Figure adx1 and adx2 that fall between 0.1 and 0.3. The range of excluded values went from 0.1 to 0.5 in 5b. As in all previous cases, we considered 2 markers G1 and G2. We used G1 to simulate the trait, and G2 to test for association with the simulated trait. All significant association is seen as type I error. We use the vector  as the allele frequency in the 3 ancestral populations for G1 and  for G2. We also considered various effect sizes for evaluating the contribution of admixture in the confounding pathway. We also changed the allele frequencies vector to account for the fact that k, the number of ancestral populations, is now 3 instead of 4. The remaining simulation steps are again similar to those described in simulation 3. The results of this simulation analysis can be seen on Figure a,b) represents the coefficient associated with the variables adx1 and adx2 in the model. The error term in all models is drawn from a normal distribution with 0 and variance 1 such that the effect size associated with adx1 and adx2 are equal to a and b respectively.The simulation steps are similar to those described in simulation 2, except that there are now two gap values: one for As can be seen in Figure k n= 500 for k = 1,2,3,4 be the sample size in each ethnic group.1) Let th ksubgroup be k.2) Let the true ethnicity of any individual in the k xfrom Dir(kα) where k αis based on the ancestry proportions observed in the MESA study. These proportions are displayed in Figure 3) Draw 0.09, 0.02, 0.84, 0.05) for the African-Americans,  for the European-Americans,  for the Hispanic-Americans and  for the Chinese-Americans.That is,  We let 5) The remaining simulations steps are similar to those taken in simulation 3, and are not repeated here.MESA: Multi-Ethnic Study of Atherosclerosis; SRE: self-reported ethnicity; GBMA: genetic based measures of ancestry; AIMs: ancestry informative markers; IAP: individual ancestry proportion; GBS: genetic background score; AA: African-Americans; CA: Chinese-Americans; EA: European-Americans; HA: Hispanic-Americans; LVH: Left ventricular hypertrophy; LV: Left ventricular; MRI: magnetic resonance imaging; BSA: body surface-adjusted; SAT structured association test.JD conceived the manuscript, conducted the analyses, interpreted the results and drafted the manuscript. DTR and DBA helped conceive the simulation experiments, interpret the results and edit the manuscript. KMR, LKV and MAP helped during the analyses and reviewed the manuscript. DAB, JHY and DKA made the data available, helped interpret the results and reviewed the manuscript. All authors have read and approved the manuscript.


Considerations in applying association mapping (AM) to plant breeding are population structure and size: not accounting for structure and/or using small populations can lead to elevated false-positive rates. The principal determinants of population structure in cultivated barley are growth habit and inflorescence type. Both are under complex genetic control: growth habit is controlled by the epistatic interactions of several genes. For inflorescence type, multiple loss-of-function alleles in one gene lead to the same phenotype. We used these two traits as models for assessing the effectiveness of AM. This research was initiated using the CAP Core germplasm array (n = 102) assembled at the start of the Barley Coordinated Agricultural Project (CAP). This array was genotyped with 4,608 SNPs and we re-sequenced genes involved in morphology, growth and development. Larger arrays of breeding germplasm were subsequently genotyped and phenotyped under the auspices of the CAP project. This provided sets of 247 accessions phenotyped for growth habit and 2,473 accessions phenotyped for inflorescence type. Each of the larger populations was genotyped with 3,072 SNPs derived from the original set of 4,608.a priori knowledge of the candidate genes, based on re-sequencing. The re-sequencing data were used to define allele types of the determinant genes based on functional polymorphisms. In a second round of association mapping, these synthetic markers based on allele types gave the most significant associations. When the synthetic markers were used as anchor points for analysis of interactions, we detected other known-function genes and candidate loci involved in the control of growth habit and inflorescence type. We then conducted association analyses - with SNP data only - in the larger germplasm arrays. For both vernalization sensitivity and inflorescence type, the most significant associations in the larger data sets were found with SNPs coincident with the synthetic markers used in the CAP Core and with SNPs detected via interaction analysis in the CAP Core.Significant associations with SNPs located in the vicinity of the loci involved in growth habit and inflorescence type were found in the CAP Core. Differentiation of true and spurious associations was not possible without Small and highly structured collections of germplasm, such as the CAP Core, are cost-effectively phenotyped and genotyped with high-throughput markers. They are also useful for characterizing allelic diversity at loci in germplasm of interest. Our results suggest that discovery-oriented exercises in AM in such small arrays may generate a large number of false-positives. However, if haplotypes in candidate genes are available, they may be used as anchors in an analysis of interactions to identify other candidate regions harboring genes determining target traits. Using larger germplasm arrays, genome regions where the principal genes determining vernalization sensitivity and row type are located were identified. Hordeum vulgare subsp. vulgare) (reviewed in 13]). In VRN-H1, VRN-H2 and VRN-H3) interact in an epistatic fashion to determine vernalization sensitivity. Vernalization sensitivity occurs only in one of the possible allele combinations at these loci: Vrn-H2_vrn-H1vrn-H1vrn-H3vrnH3 VRN-H1 anrding to  and convribed in . We clasVRS1 in POPA3 . SNP markers with unknown position and/or more than 10% missing data and/or Gentrain scores below 0.6 were excluded from all the analyses.Genome-wide, mostly EST-based, SNP markers were developed as described in detail in Close et al. 2009). Three 1536-plex pilot Oligonucleotide Pool Assays  were designed and used to genotype the 102 accessions using the Illumina GoldenGate BeadArray SNP detection platform . Based o. Three 1http://www.harvest-web.org/). For the calculation of population structure, markers with Minor Allele Frequency (MAF) below 0.05 were excluded. The program was run five independent times for each value of the maximum population number (K = 1 to K = 7) with 25,000 burn in and 25,000 MCMC iterations. For each run, an estimated logarithm of the probability (ln Pr(X|K) was calculated. After determining the most probable number of subpopulations, the program was run again with 100,000 burn in and 100,000 MCMC iterations in order to obtain the population structure matrix (Q matrix). This matrix defines the estimated membership of each individual to each population, expressed in proportion, and was used to correct for population structure in further analysis. Principal Components Analyses (PCA) were also used to determine population structure using SNP data and SAS v9.1 PROC PRINCOMP .Population structure in the CAP Core was determined with STRUCTURE 2.2  using thThe Loiselle coefficient  was usedhttp://www.maizegenetics.net[r2 and corresponding p-values (two-sided Fisher's exact test). For the calculation of LD, markers with Minor Allele Frequency (MAF) below 0.05 were excluded. The extent of LD was calculated for each chromosome for the whole set of 102 accessions, and also for the different groups of growth habit and inflorescence type: spring two-rowed (54 accessions), spring six-rowed (33 accessions) and winter six-rowed (17 accessions). The calculation was based on the approach described by [r2 values between all unlinked loci and, separately, r2 values for the loci in the same chromosome. For each chromosome, r2 with p-values below 0.001 were plotted against genetic distance (cM), and a second-degree smoothed loess curve with 90% confidence limits was fit using SAS PROC LOESS (SAS v9.1). The SAS macro bctrans [r2 values in order to approximate a normal distribution. The parametric 95th percentile of the distribution was used as a threshold to consider that LD was likely caused by genetic linkage. The intersection of the Loess curve and its confidence bands with a baseline drawn at the threshold value was considered to estimate the extent of LD, with confidence intervals, for each chromosome.TASSEL 2.1 etics.net was usedribed by . We calc bctrans  was used bctrans  of the up-value below 0.2 in the M+Q+K model were selected for an analysis of interactions. To find the interactions that best explained the phenotypic data, we performed a likelihood ratio test (LRT) subtracting the -2 log likelihood for the model that included the two markers being tested, their interaction and the population structure from the model that included only the population structure. Since these tests are for fixed effects parameters, -2 log likelihood for the models was estimated using the maximum likelihood method (ML) of SAS PROC MIXED (SAS v9.1). The asymptotic null distribution of the test statistic is a χ2 with degrees of freedom equal to the difference in the number of fixed-effect parameters between the two models. Validation of the results of AM analysis of row type in the CAP I, II and III sets was done using TASSEL 2.1 and a M+Q+K model. Population structure was determined by means of PCA using SAS PROC PRINCOMP and relatedness of individuals was corrected using a the K matrix calculated with TASSEL 2.1.Genome-wide association analysis was performed using TASSEL 2.1. Four models were tested: (i) only markers as a fixed factor (M), (ii) markers as a fixed factor and the population structure defined by the Q matrix as a co-variate (M+Q), (iii) markers as a fixed factor and the genetic background as a random polygenic effect with a variance covariance structure defined by the K matrix (M+K), and (iv) markers as a fixed factor, the Q matrix as covariate and genetic background as a random polygenic effect (M'+Q+K). A False Discovery Rate multi-test adjustment  was perfAM: association mapping; BOPA barley oligonucleotide pool assay; CAP: coordinated agricultural project; EST: expressed sequence tag; InDels: insertions/deletions; LD: linkage disequilibrium; MAF: minor allele frequency; OWB: Oregon Wolfe Barley; PCA: principal components analysis; POPA: pilot oligonucleotide pool assay; QTL: quantitative trait locus; SNPs: single nucleotide polymorphisms.ACM, PS, PMH, designed the experiment, interpreted the results and wrote the manuscript; PS scored the phenotypes; PS and TF performed re-sequencing; ACM and PS performed statistical analyses; TJC analyzed the SNP genotype data; GM and KS provided insightful interpretations of key results. All authors read and approved the final manuscript.Table S1. Passport information and phenotypic characterization of the CAP Core for flowering time and inflorescence type and genotypic characterization for the main loci controlling these traits.Click here for fileTable S2. Accession numbers of the sequences deposited with GenBank.Click here for fileTable S3. Primers used for genotyping and/or re-sequencing VRN-H, PPD-H, FR-H, and VRS1 specific genes in the CAP Core.Click here for fileFigure S1. Neighbor-Joining phylogenetic cluster analyses of several re-sequenced genes of the barley CAP Core set. Confidence values on the branches are based on 1000 bootstraps. For each gene, sequence length and number of lines used is indicated in the heading of each cluster. Number of genotypes per haplotype is indicated in brackets.Click here for fileText S1. Detailed results of re-sequencing.Click here for fileFigure S2. Schematic representation of VRN-H1 intron 1 found in the barley CAP Core set (102 genotypes). Deletion types are named based on the first cultivar whose sequence was deposited in GenBank. Edges of exons 1 and 2 are indicated by flanking black boxes. Gaps represent deletions (>50 bp) relative to the full-length Strider allele. The 2.8 kb barley-wheat conserved region (dashed line) and the 436 bp vernalization critical region (dotted box) are indicated. Number of genotypes per deletion type is denoted.Click here for fileFigure S3. Sequence alignment of HvHOX1 from seven accessions with different VRS1 haplotypes. Similarity is shown to four levels.Click here for fileTable S4. Detailed information about SNP markers highly significant in the analysis of interactions in the CAP Core.Click here for fileTable S5. POPA SNP markers developed from re-sequencing of loci related to vernalization sensitivity and inflorescence type.Click here for file


F-statistic-based robust joint analysismethod for quantitative traits using the combined raw data from both stages in the framework oftwo-staged GWASs. Explicit expressions are obtained to calculate the statistical significance andpower. We show using simulations that the proposed method is substantially more robust than theF-test based on the additive model when the underlying genetic model is unknown. An example forrheumatic arthritis (RA) is used for illustration.Genome-wide association studies (GWASs) in identifying the disease-associated genetic variantshave been proved to be a great pioneering work. Two-stage design and analysis are often adopted inGWASs. Considering the genetic model uncertainty, many robust procedures have been proposed andapplied in GWASs. However, the existing approaches mostly focused on binary traits, and few workhas been done on continuous (quantitative) traits, since the statistical significance of these robust testsis difficult to calculate. In this paper, we develop a powerful  F-test statistics. Note that these tests just focus on single-marker analysis in one-stage analysis.Genome-wide association studies (GWASs) have identified a large number of genomic regions ) with a wide variety of complex traits/diseases. In a GWAS, two most common types of data, qualitative (or binary) and quantitative (or continuous) traits, are analyzed and two contentious points are often faced; one is how to construct the test statistic considering the genetic model uncertainty and the other is how to evaluate the statistical significance for controlling the false positive rates efficiently e.g., , 2). Con. Con2]).P-values  based on some efficient tests are further screened on the remaining subjects, are often adopted in practice ..P-valuesF-statistic, since F-test is commonly employed from the linear regression model for quantitative trait, and Li et al. [F-statistics is more powerful than So and Sham's method by extensively numerical simulation.In genetic association studies, especially GWASs, genetic markers are routinely tested under the assumption of additive effects. Although convenient to use, those tests are optimal only when the true underlying genetic model is additive so that they are not robust against the genetic model misspecification. To our best knowledge, few work has been done on the two-stage joint analysis for quantitative trait GWASs allowing for genetic model uncertainty. Here, we attempt to develop a joint analysis method with data fusion in the two-stage design using i et al.  show thaF-test from the numerical results of power comparison when the real genetic model is unknown. After that, an illustrative example for rheumatic arthritis (RA) is presented. Finally, we give some discussion of this paper in The content of this paper is organized as follows. In n individuals are randomly selected to be genotyped in a two-staged GWAS for a certain quantitative trait and that π is the sampling proportion in Stage 1. Let n1 = nπ and n2 = n(1 − π) be the sample sizes for Stages 1 and 2, respectively. Consider a biallelic marker with two alleles G and g. Without loss of generality, we assume that G is the minor or high-risk allele. We suppose that the total m SNPs are genotyped on the samples of Stage 1, and SNPs with P-values less than γ in Stage 1 will be further genotyped and tested in Stage 2. Let the significance level be α, and then the genome-wide significance level per SNP is α/m with the Bonferroni adjustments. Let Y1 = ′ and Y2 = ′ be the observed quantitative outcome vectors for Stage 1 and Stage 2, respectively. Without loss of generality, we assume that the first n10 individuals in Stage 1 have the genotype gg, the second n11 individuals in Stage 1 have the genotype Gg, and the last n12 subjects in Stage 1 possess the genotype GG. Similarly, the first n20 subjects in Stage 2 have the genotype gg, the second n21 individuals in Stage 2 have the genotype Gg, and the last n22 subjects in Stage 2 possess the genotype GG. Let 0k = k×1′ and 1k = k×1′, and let Ok×j be the k × j matrix with all its entries being zero and In be the n × n identity matrix. Assume that F-statistic-based MAX3 by Li et al. [β0 is the nuisance parameter for the intercept, β1 is the parameter of interest for genetic effect, and gi is the genotype value, which takes 0, 1, or 2 corresponding to the count of G at a marker locus for the ith subject, i = 1,2,…, n1. The hypotheses of interest aregi in the previously stated equation is coded differently for the three common genetic models. Let XR1 = , XA1 = , and XD1 =  be the design matrices under three commonly used genetic models, where GR1 = ′ corresponds to the recessive model, GA1 = ′ corresponds to the additive model, and GD1 = ′ is for the dominant model. Denote X1 = , where x11 = ′ and x12 = ′. The modified F-test statistics under the recessive, additive, and dominant models for Stage 1 are given by We firstly briefly introduce i et al.  just usiP-values less than γ will be genotyped on the remaining n2 subjects in Stage 2. Following the previous notation for Stage 1, corresponding to the recessive, additive, and dominant models, the genotype data in Stage 2 are denoted by GR2 = ′, GA2 = ′, and GD2 = ′, respectively, and the design matrices are XR2 = , XA2 = , and XD2 = , respectively. Denote X2 = , where x21 = ′ and x22 = ′. Then, we can obtain three modified F-test statistics under the recessive, additive, and dominant models for Stage 2 similarly, and denote them by F2R, F2A, and F2D. Let Y = ′, GR = ′, GA = ′, and GD = ′. Denote N0 = n10 + n20, N1 = n11 + n21, and N2 = n12 + n22 for the combined sample sizes from two stages, corresponding to three genotypes. Then the proposed F-test statistics under three genetic models on the basis of the combined data are as follows:XR = , XA = , XD = , and We now give the proposed robust joint analysis. In the framework of two-stage design GWAS of quantitative traits, the SNPs with u1, which is the solution ofα/m, in order to control the false positive rate, we haveuJ is the cut-off point for the joint statistic. Once we have u1 and uJ, the power is calculated byIn order to calculate the power of the proposed joint analysis, we have to get the thresholds, which is determined by the significance level. Denote the threshold for choosing the promising SNPs in Stage 1 by Z1R,Z1A,Z1D,ZJR,ZJA,ZJD,RSS1,RRSJ)′ under both H0 and H1.We now give the detail to calculate the cut-off point and power above. The left side of  can be fH0 or H1 holds, RSS1 and RSSJ and ′ are mutually independent ′ by V1 = (vkl)3×3, whose entries are v11 = v22 = v33 = 1, VJ = (vkl*)3×3 be the correlation matrix of ′ with v11* = v22* = v33* = 1, 1/σ2 ~ χn1−32, RSSJ/σ2 ~ χn−62, andρ = (ρkl)3×3 is the correlation matrix between ′ and ′, withNote that whether H1, for a given odds ratio OR = exp(β1) for subjects with two copies of risk allele corresponding to recessive model or one copy of risk allele corresponding to additive or dominant models, we have the following:(i) when the true genetic model is recessive, μR = ′ withwhere (ii) when the true genetic model is additive, μA = ′ withwhere (iii) when the true genetic model is dominant, μD = ′ withwhere Under H0(F1MAX ≤ u1) and PrH0(FJMAX ≤ uJ) and PrH0. The details are included in H1(F1MAX ≤ u1) and PrH1(FJMAX ≤ uJ) and PrH1 are essentially similar.We develop a method for simplifying the calculations of PrF-test statistics F1A and FJA. For convenience, we refer to the proposed method as MAXFJ and AFJ for the other one. We choose the sample size n = 2000, and m = 5 × 105. The proportion of subjects genotyped in Stage 1 has three levels π = 0.3,0.4,0.5. We set the genome-wide significance level as α = 0.05 and that the significance level per SNP as α/m = 1 × 10−7. In Stage 1, the P-value threshold for SNPs selected for followup is set to be 1 × 10−4 and 2 × 10−4. We assume that the Hardy-Weinberg equilibrium holds in the general sample population, and then there are on average n × (1 − MAF)2, 2n × MAF × (1 − MAF), and n × MAF2 individuals with genotype gg, Gg, and GG, respectively, where the minor allele frequency is set to be 0.15, 0.30 and 0.45. To make the power comparison more distinctly, we specify different genetic effect parameters β1 under three genetic models as follows: β1 = 0.5 for the recessive model, β1 = 0.3 for the additive model, and β1 = 0.4 for the dominant model.We conduct simulation studies to evaluate the performance of the proposed method under three commonly used genetic models . We mainly compare the power of two approaches; one is the proposed method in this paper, and the other is the joint analysis based on the γ = 1 × 10−4 and γ = 2 × 10−4, respectively. They indicate that MAXFJ is more efficiency robust than AFJ across various inheritance models. As expected, AFJ is more powerful than MAXFJ under the additive model. However, MAFJ performs much more powerful than AFJ when the true genetic model is recessive. For instance, in π = 0.4 and MAF = 0.3, the powers of AFJ and MAXFJ are 0.101 and 0.529, respectively. In summary, MAXFJ is substantially more powerful than AFJ in two-staged GWAS of quantitative traits, when the model for AFJ is misspecified.The power results are displayed in Tables 5 SNPs [π = 0.3,0.4,0.5 three simulation circumstances. For π = 0.3, thirty percent of individuals were randomly sampled from all cases and controls and were used as the data from Stage 1, and the rest of individuals were treated as the data of Stage 2. The P-values of AFJ and MAXFJ were calculated, respectively. We repeated the above procedure 1,000 times and saved the corresponding P-values. A base-10 logarithm transformation and an opposite transformation were successively applied to these P-values, and the histogram and density of these transformed data were obtained  is an autoimmune disease  which mainly attacks synovial joints. About 1% of the common adult population worldwide is affected by RA . It has 5 SNPs –20. It i5 SNPs  pointed 5 SNPs  showed tobtained . SimilarF-statistics over three different genetic models. The denominator of the used F-statistic, which is constructed without assuming any genetic model, is different from the commonly used one. This adoption can reduce the computation intensity. Taking advantage of an ingenious design matrix, we successfully construct the common denominator of three F-test statistics for the joint analysis with combined raw data from both stages. The statistical significance  for the proposed joint analysis method can be calculated with the derived analytic expressions on the basis of the asymptotic distributions, which greatly reduce the complexity and computational intensity compared with the resampling-type permutation and bootstrap procedures. Our numerical results demonstrate that this novel approach has the greater efficiency robustness for genetic model uncertainty than the F-statistic-based joint analysis which assumes the additive genetic model.We have developed a feasible two-stage design and the corresponding robust joint analysis approach for quantitative trait GWASs. The method is based on the In this work, we did not investigate the power of joint analysis based on other existing robust association methods for quantitative traits such as So and Sham's method. We find that it is very difficult to extend So and Sham's method (score test-based MAX3) to two-staged GWASs with quantitative outcomes, since it is almost impossible to derive the joint distribution of score tests from two stages.For simplicity, here we do not take into account the effects of covariates in the considered two-stage design. However, in real application, the proposed method can be easily applied to the situation including one or more covariates as shown by the original MAXF by Li et al. . It is i


Logistic regression (LR) classifiers have been used successfully in the single-trial analysis of EEG data, especially in tasks of perceptual decision-making ,2, but hTo establish a baseline for comparison, we utilize EEG data from a previous study using LR to classify neural response to a two-choice forced-decision face vs. car visual task . In thisz (area under receiver-operating characteristic) value while decreasing its variability (determined from bootstrapping), which increases with a low number of epochs. Once converging to subject-specific values of δ* and N*, we then test the classifier solution for statistical significance using the false discovery rate across all windows [Recognizing the dependence of classifier performance on these discrete parameters, we use a genetic algorithm to explore the δ vs. N design space. In doing so, we track an objective function whose value depends on maximizing an epoch window's leave-one-out A windows , as therz from that of the earlier study. Second, we vary δ  and N  simultaneously and converge using a genetic algorithm  mutation probability, [max and that δ* is a subject-specific parameter that differs from the heuristics offered by previous work. Finally, on a group level, we find that the components of our objective function exhibit distinct variation with respect to δ and N, with an epoch's maximum Az optimizing for low N and low δ, while its Az variability minimizes for high N and maximizes for low N, nearly irrespective of δ.First, minimizing our objective function with N held constant at its maximum, we find that δ* can be tuned in a subject-specific way and we find on average a 3.7 ± 1.1% improvement in maximum A


Multiple rare variants either within or across genes have been hypothesised to collectively influence complex human traits. The increasing availability of high throughput sequencing technologies offers the opportunity to study the effect of rare variants on these traits. However, appropriate and computationally efficient analytical methods are required to account for collections of rare variants that display a combination of protective, deleterious and null effects on the trait. We have developed a novel method for the analysis of rare genetic variation in a gene, region or pathway that, by simply aggregating summary statistics at each variant, can: (i) test for the presence of a mixture of effects on a trait; (ii) be applied to both binary and quantitative traits in population-based and family-based data; (iii) adjust for covariates to allow for non-genetic risk factors and; (iv) incorporate imputed genetic variation. In addition, for preliminary identification of promising genes, the method can be applied to association summary statistics, available from meta-analysis of published data, for example, without the need for individual level genotype data. Through simulation, we show that our method is immune to the presence of bi-directional effects, with no apparent loss in power across a range of different mixtures, and can achieve greater power than existing approaches as long as summary statistics at each variant are robust. We apply our method to investigate association of type-1 diabetes with imputed rare variants within genes in the major histocompatibility complex using genotype data from the Wellcome Trust Case Control Consortium. Rapid advances in sequencing technology mean that it is now possible to directly assay rare genetic variation. In addition, the availability of almost fully sequenced human genomes by the 1000 Genomes Project allows genotyping at rare variants that are not present on arrays commonly used in genome-wide association studies. Rare variants within a gene or region may act to collectively influence a complex trait. Methods for testing these rare variants should be able to account for a combination of those that serve to either increase, decrease or have no effect on the trait of interest. Here, we introduce a method for the analysis of a collection of rare genetic variants, within a gene or region, which assesses evidence for a mixture of effects. Our method simply aggregates summary statistics at each variant and, as such, can be applied to both population and family-based data, to binary or quantitative traits and to either directly genotyped or imputed data. In addition, it does not require individual level genotype or phenotype data, and can be adjusted for non-genetic risk factors. We illustrate our approach by examining imputed rare variants in the major histocompatibility complex for association with type-1 diabetes using genotype data from the Wellcome Trust case Control Consortium. Despite the recent successes of genome-wide association studies (GWAS), which can be well powered under the common disease, common variant hypothesis, the majority of the genetic component of many complex traits remains unexplained. For example, hundreds of common genetic variants, in at least 180 loci, have been associated with height in studies of up to more than 180,000 individuals. However, the individual effects of these variants are modest and their cumulative effect explains just over 10% of the phenotypic variation in height Recently published methods for the analysis of multiple rare variants illustrate that power can be greatly increased by combining information in a joint analysis in comparison to studying individual variants one at a time Generalised C-alpha test. We report the results of simulations to investigate the power of our test to detect rare variant association with a quantitative trait, and compare performance with existing approaches.Here, we introduce a C-alpha test for the analysis of rare genetic variation for association with both binary and quantitative traits based on normally distributed measures of effect at each site. Measures of effect at each site can be calculated from re-sequencing, array genotyping or imputed data or taken directly from summary measures of effect available, for example, from meta-analysis or published data. Our test assesses the evidence for a mixture of increaser, decreaser and neutral effects in a gene, region or pathway and can be applied to both population and family-based association studies and can adjust for covariates to allow for non-genetic risk factors, such as indicators of population stratification. We refer to our test as the The HLA class II genes in the major histocompatibility locus (MHC) play a major role in susceptibility to type-1 diabetes (T1D) K variants, each with a minor allele frequency (MAF) less than a pre-defined threshold and assayed in a sample of individuals measured for a binary or a quantitative trait. Suppose that at each variant a normally distributed estimate of the effect of the minor allele on the trait of interest can be obtained. For example, in a case-control association study such an estimate may be the log allelic odds ratio obtained as a coefficient in a logistic regression; or in a quantitative trait association study, the estimate may be the per-allele increase in phenotypic value obtained as a coefficient in a linear regression. For each variant alone, there is unlikely to be enough information to make inference about association, unless the sample size is unfeasibly large. However, if the gene is not associated with the trait, then the distribution of estimates across all variants will be Gaussian with mean zero. Conversely, if variants in the gene are associated with the trait, there will be a mixture of Gaussian distributions with different means, manifested as “overdispersion”, which can be detected by a C-alpha test.Consider a gene, region or pathway containing k, k = 1,…,K. We assume that S is simply the sum of the differences between the variance of the observed measures of association and the expected variance under the null hypothesis. To standardise S, we require the estimated normalizing varianceNORM significantly larger than that expected using a one-tailed test of size α. The quantities S and c are easily derived using methods detailed in Zelterman and Chen More formally, let NORM is Gaussian depends on: (i) the degree of sparseness in the data, as summarised by the relationship between sample size and MAF at each variant; (ii) the number of variants that are considered and (iii) the independence of variants. When the data are too sparse, because the sample size is too small and/or the MAF too low, the maximum likelihood estimates of effect size computed at each site are typically unstable. Furthermore, the discrepancy between the empirical variance of the estimates, and their variance under the reference asymptotic distribution can be large, resulting in inaccurate type I error NORM to Gaussian by the central limit theorem. To estimate significance accurately for low MAF, where small numbers of variants are considered or where variants are correlated, standard permutation testing is required. See For genetic association studies, the expected effect of a minor allele is zero, so that We conduct simulations to investigate the performance of the Generalised C-alpha test for the identification of rare variants associated with a binary or quantitative trait. We compare the performance of the Generalised C-alpha test to three existing approaches: (i) the optimized score-based variance component test  traits. The test models the number of minor alleles, ky, at variant k out of a total of kn observations by a binomial  distribution, where k = 1,…,K. Under the null hypothesis, kp = 0p, the proportion of cases present in the sample. Under the alternative hypothesis, kp can take on a mixture of values across the K variants, with some variants deleterious , some protective (i.e. with greater frequency in the controls than the cases kp<0p), and some neutral . It can then be shown that the Binomial C-alpha test statistic is simply:BIN has a standard Gaussian distribution under the null hypothesis of no association, which is rejected for values of ZBIN significantly larger than that expected for a one-tailed test of size α. The Binomial C-alpha test cannot adjust for covariates and cannot be directly applied to imputed data.The GRANVIL models the trait value of an individual as a function of the proportion of rare variants at which they carry at least one minor allele in a generalised linear regression framework. GRANVIL can thus be applied to binary and quantitative traits, can incorporate imputed genotypes, and can allow adjustment for covariates. However, GRANVIL is a burden test, and thus assumes the direction of effect of all rare variants is the same, within the same gene or pathway.Our simulations make use of a simple model of population genetics to generate high-density haplotype data in 30–200 kb genomic regions, designed to represent a gene. Haplotypes are then randomly paired together to form individuals for analysis, and quantitative trait values are generated according to their genotypes at rare causal variants, selected at random according to the underlying trait association model. In the trait association model that we consider here, we assume that the expected phenotypic value of an individual is determined by the net effect of a combination of increaser causal variants, which serve to elevate the mean trait value in the population, and decreaser causal variants, which serve to reduce it. The trait association model is parameterised in terms of: (i) the maximum MAF of each individual causal variant; (ii) the total MAF of all causal variants in the gene; (iii) the relative proportion of increaser and decreaser causal variants; and (iv) the joint contribution of the causal variants in the gene to the trait variance. Full details of the simulation process are described in −5 significance levels, respectively. Simulations are repeated 10,000 times for each set of parameter values.The Generalised C-alpha test, SKAT-O and GRANVIL are applied directly to the simulated quantitative trait. However, to apply tests designed for binary traits, we dichotomise the quantitative distribution by assigning individuals as “cases” if they belong to the upper 50% of the trait distribution, or “controls” otherwise. The Generalised C-alpha test, as well as the Binomial C-alpha test, is then applied to the dichotomised trait. The significance of the Generalised C-alpha and Binomial C-alpha test statistics are evaluated empirically by standard permutation testing  signals of association with T1D in genes on chromosome 6 using the Generalised C-alpha test applied to rare variants using genotype data from the WTCCC −6 is required to ensure genome-wide significance. Hence for each analysis, we performed 600,000 permutations and declared genome-wide significance for a given gene if less than 1 of 600,000 (<1.7×10−6) permutations resulted in a C-alpha test statistic larger than the original.For each gene, the Generalised C-alpha test was applied to SNPs in two MAF ranges: 0.1%<MAF<0.5% (very rare) and 0.5%<MAF<1% (rare). Measures of effect at each SNP used in the Generalised C-alpha test were the log odds ratios estimated from single SNP additive tests of association using simple logistic regression. The Generalised C-alpha test was applied to the original data and then, in order to determine a permuted p-value, to repeated permutations of the case/control status and covariate data , the Generalised C-alpha performed better than all the other tests compared. In the examples we selected, it performed equally as well or better than SKAT-O for variants with more than ∼15–25 copies of the minor allele  for any combination of risk or protective variants . However, the SKAT-O was optimal for variants with fewer copies of the minor allele. In our qualitative analyses of a binary trait, the Binomial C-alpha test and the Generalised C-alpha test were comparable for variants with MAF>∼0.5% but the power of the Generalised C-alpha test declined for variants with fewer than ∼15–20 copies of the minor allele .The power of the Generalised C-alpha test applied to the quantitative and the dichotomised traits decreases rapidly as the number of copies of the minor allele for included rare variants falls below ∼10 in the models we have considered . Rapid decreases in power with decreasing MAF are likely to be a consequence of increasing sparseness leading to violation of the assumptions of asymptotic normality in the Generalised C-alpha test. Of course, in a given region, the total number of variants considered increases as the minimum MAF decreases – in simulations for 10,000 individuals, the number of variants in our simulated 100 kb region when minimum MAF is 0.5% is 28 increasing to 94 for a minimum MAF of 0.1% - and losses in power are also a consequence of an increased number of non-causal variants being included in this total, but this factor affects the power of all the tests similarly .Computation time for the Generalised C-alpha depends on the sample size, the number of markers and the method used to estimate the normally distributed measures of effect at each variant. To analyse all ∼160 markers sequenced on 5,000 or 10,000 individuals in a 100 kb region and obtain permuted p-values with 1,000 permutations in a Generalised C-alpha test of association required ∼5.0 s and ∼10 s, respectively, for a quantitative trait (using estimates of effect derived from linear regression) and ∼20% longer for a binary trait (using estimates of effect derived from logistic regression). Increasing the number of permutations to 100,000 increased the run times ∼20-fold. Halving the number of markers analysed only marginally reduced run times. These estimates were based on simple code programmed in R and run on a Unix operating system. Coding in a language that allows faster numerical computation times is expected to reduce run times.p<1.7×10−6) evidence of association with T1D were observed with rare variants in 17 genes throughout the 7.5 Mb extended Major Histocompatibility Complex (MHC) region (ranging from the GNL1 gene to the COL11A2 gene). The strongest signal of association was observed at C6orf10 , which contains rare variants previously implicated in susceptibility to T1D After QC and imputation, the WTCCC data comprised 2,938 T1D cases and 1,963 controls with directly or imputed genotypes available at 490,888 SNPs with 0<MAF<1%, located in 1,611 distinct genes on chromosome 6; gene boundaries were identified from the UCSC human genome database (build 37). Common SNPs in the MHC have been previously associated with T1D We have developed the Generalised C-alpha test for the analysis of multiple rare variants that display a mixture of increaser and decreaser effects on a binary or quantitative trait. The Generalised C-alpha test is a score test combining routinely calculated Gaussian distributed measures of effect at multiple variants in order to increase the power to detect an effect at the gene, region or pathway level. The Binomial C-alpha test for binary traits, The Generalised C-alpha test performs better than SKAT-O when the data is not too sparse: in our examples we showed the Generalised C-alpha was optimal as long as there were at least 15–25 copies of a minor allele at each rare variant. When data is sparse, so that either the sample size is too small and/or the MAF is too low, estimates of allelic effects at each SNP are not robust, and the asymptotic assumptions on which the Generalised C-alpha test are based are inappropriate. Similarly, for testing rare variant association with a binary trait, we have shown that the Generalised C-alpha test has lower power that the Binomial C-alpha test in the presence of variants with very low minor allele counts: a minimum MAF>∼0.5% is recommended in order to achieve comparable power in these tests.In any application, the Generalised C-alpha test works on the assumptions that there are (i) a sufficiently large set of variants; (ii) that estimates of effect based on these variants are robust and independent and; (iii) normally distributed. These assumptions are often unrealistic: they are violated for example, in the presence of linkage disequilibrium, small sample size, low MAF or few variants. Hence, it is imperative that permutation testing is employed for accurate estimation of significance. For analysis of the whole genome, 1,000 permutations, for which a simply coded version of the test can be run in a matter of seconds, is recommended as a first approach; regions where the test is significant with a p-value<0.001 can then be rerun with 100,000 or more permutations for an accurate estimate of genome-wide significance.Unlike the Binomial C-alpha test, the Generalised C-alpha test can naturally adjust for additional covariates and can easily incorporate imputed variation. Unlike SKAT-O, the Generalised C-alpha test can be applied to summary statistics, without requirement of the individual level genotype data. For example, the Generalised C-alpha test can be quickly and easily applied to published data. However, this is recommended only for discovery as permutation testing cannot be implemented in this case and test statistics are likely to be inflated leading to increased type I errors: In this case, any regions identified would require further investigation for any confirmation of association.Evaluation of rare variants extracted from existing GWAS data via imputation up to re-sequencing reference panels, such as those made available by the 1000 Genomes Project, has been demonstrated to be feasible http://atgu.mgh.harvard.edu/plinkseq/. R package is available from http://www.well.ox.ac.uk/~rivas/calphanorm.tar.gz.In summary, the Generalised C-alpha test is a novel, flexible and powerful method for the analysis of rare genetic variation. There is no single alternative test, amongst those we have considered, that is uniformly most powerful over all models and genetic architectures. Our test, however, has the unique advantage that it can be applied to summary statistics from published literature, without the need for individual level genetic data. The fact that the Generalised C-alpha test simply aggregates data from summary statistics allows for great flexibility in general allowing direct application to both binary and quantitative traits, to population  and family based data , and to imputed genotype data whilst simultaneously allowing for the adjustment of additional covariates. We are already using the method in our analyses and it is currently implemented using the R-PLINK/SEQ library available from: Figure S1Power Comparisons. Power to detect association in a region is shown for the Generalised C-alpha test, SKAT-O and the GRANVIL test applied directly to the quantitative trait and for the Generalised C-alpha and the Binomial C-alpha tests applied to the dichotomised quantitative trait. (A) Power is shown as a function of the percentage of causal variants in a region of size 100 kb that are risk as opposed to protective when the minimum MAF of variants considered is fixed at 0.5% for a sample size of 5,000. Results show that as the proportion of risk causal variants approaches 50%, the C-alpha and SKAT-O tests maintain power and that the Generalised C-alpha applied directly to the quantitative trait has optimal power. (B) Power is also shown as a function of the minimum MAF of variants considered when the percentage of risk causal variants in a region of size 100 kb is fixed at 50% for a sample 10,000 individuals. Results show that the power of the Generalised C-alpha test is optimal for variants with MAF>∼0.3% but the SKAT-O is optimal for lower MAF. For quantitative traits, the power of the Generalised C-alpha test remains better than the Binomial C-alpha applied to a dichotomized version of the trait as long as variants have MAF>∼0.12%. For binary traits, the Binomial C-alpha test has greater or equivalent power than the Generalised C-alpha test.(TIF)Click here for additional data file.Figure S2Power By Region Size. Power is shown as a function of region size when the percentage of risk causal variants is fixed at 50%, the minimum MAF of variants considered is fixed at 0.5% for a sample size of 10,000 individuals. Here, the region size is a proxy for the number of variants considered and results show that power decreases for all methods as the number of non-causal variants included increases. Results are presented for a model assuming a total MAF of 5% for all causal variants in the region, a maximum MAF of any individual causal variant of 1% and where causal variants account for 0.6% of the phenotypic variance. The trait mean is determined by the net effect of the risk causal variants, which serve to increase the mean trait value, and the protective causal variants, which serve to decrease the trait mean. Power is estimated at a 5% significance level over 10,000 replicates of data. Significance in each replicate of data is assessed empirically by random permutation of the trait value and recalculation of the test statistic: permutation occurs 1000 times to ensure accurate assessment at a significance level of 5%.(TIF)Click here for additional data file.Text S1A flexible approach for the analysis of rare variants allowing for a mixture of effects on binary or quantitative traits: Supplementary Methods(DOCX)Click here for additional data file.