Motivation: For the past few decades, many statistical methods in genome-wide association studies (GWAS) have been developed to identify SNP–SNP interactions for case-control studies. However, there has been less work for prospective cohort studies, involving the survival time. Recently, Results: Through simulation studies, we compared the power of Cox-MDR with those of Surv-MDR and Cox regression model for various heritability and minor allele frequency combinations without and with adjusting for covariate. We found that Cox-MDR and Cox regression model perform better than Surv-MDR for low minor allele frequency of 0.2, but Surv-MDR has high power for minor allele frequency of 0.4. However, when the effect of covariate is adjusted for, Cox-MDR and Cox regression model perform much better than Surv-MDR. We also compared the performance of Cox-MDR and Surv-MDR for a real data of leukemia patients to detect the gene–gene interactions with the survival time.Contact:leesy@sejong.ac.kr; tspark@snu.ac.kr P < 5 × 10−8. However, it is noted that the effective sizes of the loci identified via GWAS are relatively small, and these individual loci may not be useful in assessing risk in personal genetics, as mentioned by Recently, massive amounts of information for single-nucleotide polymorphisms (SNPs) across the whole genome have become available from high-throughput technology, which allows genome-wide association studies (GWAS) to be performed. As recently reviewed by In early GWAS, statistical methods for identifying susceptibility have considered a single SNP at a time and have selected a subset of the top few SNPs from a ranked list of SNPs. Then, replication studies have been implemented to determine whether these associations held for other samples. Some of the replication studies, however, show that significant associations are not found from the top ranked list. Recently, this single-locus approach has been moved into a multiple-loci approach because most complex diseases are associated with multiple genes and their interactions. However, the traditional parametric approach, such as the logistic regression model, has limited power in detecting non-linear patterns of interaction and needs a large amount of study samples when multiple SNPs and gene–gene interactions are considered.As a dimensional reduction strategy, et al. (2010) proposed a novel approach for identifying gene–gene interactions with survival times using SNP information in the frame of MDR, called Surv-MDR. The Surv-MDR method modifies MDR's constructive induction algorithm to classify multi-level genotypes as high- and low-risk groups using a log-rank test instead of case control ratios. In addition, balanced accuracy is replaced by log-rank test statistics and is used as a score to determine the best model. Surv-MDR was shown to have better performance than that of traditional Cox regression models through simulation experiments and was successfully applied to the identification of SNP–SNP interactions associated with survival time in bladder cancer data  be the observed time and let iδ = I(iT* ⩽ iC*) be an indicator for uncensored observation. The observed data consists of  pairs as well as covariates that may be vector-valued and/or time-varying, notated as iX(t). Here we consider only time-fixed covariates, notated as iX. The counting process formulation replaces the pair of variables  with the pair of functions (iN(t),iY(t)), where iN(t) = I and iY(t) = I(iT⩾t). Here, iN(t) is a counting process that represents the number of observed events by time t for the ith individual and iY(t) is a predictable process that represents the risk set at time t.Let 0(t) is an baseline hazard function, iX is the predictor variable vector coding gene–gene and gene–environment interaction of interest, iZ is the vector coding for the covariates and β and γ are the corresponding parameter vectors to iX and iZ, respectively. Then, we call β the target effects and γ the covariate effects.The Cox model assumes a hazard function of the following formith individual to classify each cell of multi-locus genotype combinations into either high- or low-risk groups. The martingale residual for the ith individual is obtained from the null model of no target effects (i.e. β = 0) and is specified as follows:We propose a new method, called Cox-MDR, using a martingale residual value of the M SNPs in the dataset. For a given dataset, fit a reduced Cox model with no SNPs effect, adjusting for covariates. Obtain martingale residuals from the fitted Cox model.Assume that there are a total of k-way interaction, select k SNPs among M SNPs and construct all possible k-way contingency tables using these k SNPs.For a given k SNPs, calculate the sum of martingale residuals over those individuals who have the corresponding genotype.For each multi-locus genotype combination defined by the If the sum of martingale residuals is positive, classify the cell corresponding to this genotype as a high-risk group. Otherwise, classify the cell as a low-risk group.The process of classifying each cell into high- and low-risk groups is summarized as follows:k-way interactions and to identify the best model. The balanced accuracy has been proposed by TP denotes the true positives, TN true negatives, FP false positives and FN false negatives, respectively. Then, Cox-MDR uses 10-fold cross-validation to determine the best model similarly as implemented in MDR.The Cox-MDR method shares the same reduction strategy as MDR except for replacing the case-control ratios by the sum of the martingale residuals in each cell. For each dataset, balanced accuracy is used to evaluate all possible As mentioned in et al. (2010).Through simulation studies, we compare the power of Cox-MDR with those of Surv-MDR and Cox regression model without and with adjusting for covariates. For comparison, the simulation setting is constructed similar to that of Gui We consider 2 disease-causal SNPs among 10 unlinked diallelic loci with the assumption of Hardy–Weinberg equilibrium and linkage equilibrium. For the covariate adjustment, we consider only one covariate which is associated with the survival time but has no interactions with any SNPs. We generate simulation datasets from different penetrance functions, which define a probabilistic relationship between a status of high- or low-risk groups and SNPs. We consider eight different combinations of two different minor allele frequencies of  and the four different heritability of . For each of the 8 heritability-allele frequency combinations, a total of 5 models were generated, which yields 40 epistatic models with various penetrance functions, as described by ikf be an element from the ith row and the kth column of a penetrance function. Assuming that SNP1 and SNP2 are the two disease-causal SNPs, we have the following penetrance function:x is an indicator variable with value 1 for the high-risk group and 0 for the low-risk group, and we set β = 1, γ = 0.0, 1.0, 2.0 and z is an adjusting covariate generated from N. In addition, the baseline hazard function follows a Weibull distribution with the shape parameter of 5 and the scale parameter of 2, and the censoring time is generated from a uniform distribution, U.Let P-value < 0.05. γ= 0.0, 1.0, 2.0, respectively. In For the power comparison, we ran Surv-MDR and Cox-MDR on 100 simulated datasets for each of 40 models, including 2 disease-causal SNPs, and we selected the best model over all possible two-way interaction models without and with adjustment of covariates, respectively. The power of Cox-MDR is defined in the same manner as that of Surv-MDR in For type I error, we also evaluate the performance of Cox-MDR when there is no SNP effect on the survival time. In other words, we check whether the type I error is well preserved under the null hypothesis. To do this, we select randomly 20 datasets from each of 40 models and remove the 2 disease-causal SNPs, which create a total of 800 null datasets. We ran Cox-MDR on 800 datasets and estimated the percentages of times that Cox-MDR included the 2 disease-causal SNPs in the chosen model out of 800 datasets. From the simulation result, the type I error is estimated as 0.02, which is smaller than the nominal level of 0.05. When we tried to select randomly 100 datasets from each of 40 models, which is a total of 4000 null datasets, the type I error varied from 0.014 and 0.040, which is also consistently smaller than 0.05.As shown in γ=2.0) whereas the corresponding powers are 0.746, 0.544 and 0.446 in γ=1.0), respectively. This implies that the power of all methods decreases substantially when the effect of covariate is large and it is not adjusted for. It is noted that there is a similar trend of power when the effect of covariate increases from 2.0 to 3.0 (data not shown). However, the deteriorating power is recovered by adjusting for covariate as shown in On the other hand, as shown in In summary, the simulation results show that Surv-MDR has a good power only when the minor allele frequency is 0.4 and heritability is more than 0.3. Since Surv-MDR cannot adjust for covariate, there is no gains in power by adjusting for covariate and has worse power when the effect size of covariate is larger. On the other hand, Cox-MDR has reasonable power across all combinations of minor allele frequency and heritability and gains substantial power by adjusting for covariate. Cox regression model also has comparable power with Cox-MDR though it is more sensitive to low heritability. This implies that it is very important to adjust for covariates when some of confounding factors are associated with the survival time in detecting significant gene–gene interactions. The availability of covariate adjustment is an important advantage of Cox-MDR and Cox regression model over Surv-MDR.We analyze a real example of leukemia patient data to illustrate the procedure of Cox-MDR and compare it with Surv-MDR. The data consist of 97 acute myeloid leukemia (AML) patients with demographic and clinical variables and 139 SNPs information. Among those, 40 patients were dead and 57 patients were alive until the termination of study. We consider two variables of age and sex as adjusting covariates in comparing the power of Cox-MDR and Surv-MDR. Likewise the simulation results, we implement Cox-MDR with and without adjusting for covariates up to two-way interactions whereas Surv-MDR is implemented without adjustment of covariate.P-value < 0.05 in et al. (2010). The top three one- and two-way models from these results were listed in the right hand side in P-value obtained from the Cox regression model and the permutation P-value.First, we ran Surv-MDR and Cox-MDR with all of 139 SNPs for one- and two-way models and the effects of age and sex were adjusted for Cox-MDR. Since the available sample size is only 97 and censoring is heavy, 10-fold cross validation provides too small test set to evaluate the best model. Instead, we implemented 2-fold cross validation with a replication of 100 times and listed the top 3 one- and two-way models in the first column of P-values in detecting the two-way interactions when all 139 SNPs were considered. However, when 118 SNPs were considered, NT5C3 rs12155477 was selected as the top one-way model and also appeared in detecting two-way models.As can be seen in On the other hand as shown in P-values of the log-rank test for the equality of two survival curves. The Surv-MDR attribute in P = 0.001. However, the Surv-MDR attribute of NT5C3 rs12155477 and NT5C3 rs7776847 do not separate two survival curves clearly as shown in P = 0.339. The other attributes also separate two survival curves significantly (data not shown).In order to compare the performance of Surv-MDR and Cox-MDR, we plot the survival curves for the high- versus low-risk groups by the attribute of SNPs pairs selected as two-way models in P-values for the log-rank test.Similarly, we plot the survival curves for high- and low-risk groups by the attributes of two-way models defined by Cox-MDR in P-values of NC5C3 rs12155477, TYMS rs1004474 and TYMS rs2847153 are 0.844, 0.487 and 0.324, respectively. However, Surv-MDR and Cox-MDR select these SNPs in one-way and two-way models and the combinations of these SNPs make subjects separate high- and low-risk groups significantly. This implies that there may be gene–gene interactions associated with the survival time which cannot be detected by ordinary approach using a Cox model.It is noted that the SNPs defined as attributes of Surv-MDR and Cox-MDR are not included in the list of To identify the complexity of gene–gene and/or gene–environment interactions on common diseases, many plausible approaches have been developed by extending existing methods into a more general framework. In this article, we propose the Cox-MDR method by extending the main idea of GMDR to the survival phenotype. Cox-MDR uses the martingale residual of the Cox regression model as a score to classify multi-loci genotype combinations into high-and low-risk groups. Since the martingale residual reflects the unexplained part beyond what is explained by the adjusted covariates excluding the genetic factors, we can evaluate whether genetic factors have an independent association with the survival time using the martingale residual.Through the simulation study, we compared the performance of Cox-MDR to those of a Cox regression model and Surv-MDR. All of the three methods showed the common trend that the power increases steadily as heritability increases from 0.1 to 0.4 although the degree of increase varies depending on the minor allele frequency and the effect size of covariate. When the minor allele frequency is 0.2, Cox-MDR and Cox regression model have higher power than Surv-MDR but Surv-MDR has higher power than Cox-MDR and Cox regression model when the minor allele frequency is 0.4 regardless of the effect size of covariate.It is noted that the power of three methods decreases substantially as the effect size of covariate increases from 1.0 to 2.0 as shown in From the result of real data analysis, it is noted that the top ranked SNP pairs identified by Surv-MDR and Cox-MDR separate the survival curves for the high- and low-risk groups significantly except for one case. Each of these SNPs has no significant main effects in the Cox model but the gene–gene interaction effect defined by their pairs has substantial impact on the separation of two survival cures between high- and low-risk groups.a and SNP2 with B and b, in which there are nine possible genotypes from the combinations of SNP1 and SNP2. For the cell of genotype , Surv-MDR computes a log-rank test statistic comparing the survival time between samples with the genotype  and without this genotype. However, for the cell of genotype , those patients who have the genotype of  are also used for computing the log-rank test statistics as the alternative group for comparing the survival time between samples with a genotype of  and without this genotype combination. In this way, each individual contributes several times to compute the log-rank test statistic as either one of the samples with or without the genotype of interest. This overlapping usage may cause distinctions between high- and low-risk groups to become contaminated, which yield rather low power. However, the martingale residual of each individual is taken into account once to discriminate each cell into high- or low-risk groups, such as the case-control ratios in MDR.Comparing Cox-MDR and Surv-MDR, Surv-MDR is a non-parametric approach based on the log-rank test statistic whereas Cox-MDR is a semi-parametric approach based on the martingale residual of a Cox model. When Surv-MDR calculates a log-rank test statistic to pool multiple combinations of genotypes into two-level attributes, each log-rank test statistic is comparing the survival time between samples with and without the genotype combination. For example, we consider two-way interactions between SNP1 with allele A and 9 combinations (e.g. pairwise interactions between 100k SNPs) with 1000 samples and 10 cross validation in ~6.4 days on a workstation with Intel Xeon 2.4GHz CPU and 12G RAM. If Cox-MDR is extended to parallel computing system, our Cox-MDR can be feasible to analyze pairwise interactions for GWAS.The execution time of our Cox-MDR has linear relationship with sample size and combinations of SNPs. We expect that Cox-MDR can evaluate 5 × 10In conclusion, Surv-MDR needs more intensive computations for a large number of SNPs and has a big weakness with which there is no way to adjust for covariates whereas, Cox-MDR requires less intensive computations by using the martingale residual score and has a great advantage of being able to adjust for covariate. Moreover, the Cox-MDR method could be extended to other types of high-dimensional data such as copy number variation (CNV) and next generation sequencing (NGS) data.For the next research topic, we plan to work on using the standardized residual for the parametric regression models in the frame of GMDR. Our key idea can be extended to the parametric regression models with various error distributions such as Weibull, log-normal and logistic distributions.Funding: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Education, Science and Technology of Korea (MEST) (20110507 and 2012R1A3A2026438).Conflict of Interest: none declared.


Clinical binary end-point traits are often governed by quantitative precursors. Hence it may be a prudent strategy to analyze a clinical end-point trait by considering a multivariate phenotype vector, possibly including both quantitative and qualitative phenotypes. A major statistical challenge lies in integrating the constituent phenotypes into a reduced univariate phenotype for association analyses. We assess the performances of certain reduced phenotypes using analysis of variance and a model-free quantile-based approach. We find that analysis of variance is more powerful than the quantile-based approach in detecting association, particularly for rare variants. We also find that using a principal component of the quantitative phenotypes and the residual of a logistic regression of the binary phenotype on the quantitative phenotypes may be an optimal method for integrating a binary phenotype with quantitative phenotypes to define a reduced univariate phenotype. Clinical end-point traits are usually binary (affected/unaffected) in nature. However, these end points are often governed by quantitative precursors. On the other hand, a single quantitative trait may not be a sufficiently good surrogate for the end-point trait, and it may be more optimal to analyze a genetically relevant multivariate phenotype vector that includes both quantitative and qualitative phenotypes. Association analyses of multivariate phenotypes involve multiple statistical challenges, the primary one being the construction of the phenotype, particularly in the presence of both quantitative and binary traits in the multivariate phenotype vector. We assess the performances of some genetic association methods for certain choices of the multivariate phenotype vector using the data on the simulated phenotypes in the framework of the 1000 Genomes Project provided in Genetic Analysis Workshop 17 (GAW17).For our analyses, we used the GAW17 data on the three quantitative traits  and the single binary trait for 697 individuals along with their genotypes at all the available 24,487 single-nucleotide polymorphisms (SNPs) distributed over the 22 autosomal chromosomes. We used data on age, sex, and smoking status (defined as a binary variable) as covariates because these factors could be potential confounders in the association analyses. We did not remove any SNP based on its minor allele frequency (MAF) because one of our goals was to identify rare variants involved in the etiology of the phenotypes. We performed our analyses on all 200 available replicates in the GAW17 data set.Likelihood-based methods, such as variance components ,2, have Z): (1) the first principal component of Q1, Q2, Q4, and Z (denoted T1); (2) the first principal component of Q1, Q2, and Q4 only (denoted T2); (3) the first principal component of Q1, Q2, and Q4 with Z as a covariate (denoted T3); (4) a risk score of Z using Q1, Q2, and Q4 as predictors (denoted T4); and (5) the first principal component of Q1, Q2, Q4, and the proportion of the risk score of the binary trait unexplained by Q1, Q2, and Q4 (denoted T5). The principal components are computed on the basis of the variance-covariance matrix of the phenotypes included in the multivariate phenotype vector. The pairwise correlations (averaged over the replicates) between the different traits are as follows: : 0.23; : −0.31; : 0.55; : 0.01; : 0.4; and : −0.53. The risk score is defined as the conditional probability of an individual being affected with respect to the binary trait (Z = 1) given the trait values  corresponding to the three quantitative phenotypes Q1, Q2, and Q4, respectively, and is computed using a binary logistic model:We define five univariate phenotypes based on the three quantitative traits Q1, Q2, and Q4 and the single binary trait  and a modification of a novel quantile-based approach developed by our group . The ANOociation ,8.The quantile-based regression approach is a model-free alternative that tests for equality of marker allele frequencies within different quantile intervals of the quantitative trait; it is based on the Armitage trend test . We noteIn our modified approach, we compute the ratio of the between-quantile variance to the within-quantile variance of the proportion of allele frequencies along the lines of the ANOVA statistic for different numbers of quantile intervals ranging from 2 to 10. The optimal number of quantile intervals is determined by the maximum of this ratio, and the Armitage trend test is performed based on this number. Unlike the original regression test statistic, which depends on the number of fixed quantiles, the Armitage trend test statistic is distributed as a chi-square with 1 degree of freedom, irrespective of the optimal number of quantiles.To correct for multiple testing, we use the false discovery rate (FDR) procedure  with an The GAW17 simulation model was available to us. To assess the power of detecting association, we considered the causative SNPs that modulated any of the component phenotypes in the multivariate phenotype vector. The empirical power at each such SNP was obtained as the proportion of replications in which the SNP was significantly associated with the relevant phenotype definition. Both the ANOVA and the quantile-based regression were adjusted for age, sex, and smoking status. Although the ANOVA approach incorporated these variables as covariates at the individual level, the quantile-based regression incorporated the mean values of these variables within each quantile interval as covariates. In Table T5 yielded the maximum power and T4 the minimum power among the five phenotype definitions for both the ANOVA and the quantile-based method. We also found that T2 (which does not involve the binary trait Z) performed better than T3 (which uses Z as a covariate) with respect to the causative SNPs modulating Q1 or Q2. The maximum empirical power was obtained for the SNP C13S523 in the FLT1 gene on chromosome 13 (modulating Q1) for all five phenotypes . The two flanking SNPs C13S522 and C13S524 (modulating Q1) also exhibited significant evidence of association based on T1 and T5 in a large proportion of replications.We found that ANOVA identified more causative SNPs than the quantile-based method did. VNN1 gene on chromosome 6 (modulating Q2) were greater than 0.5 based on T1 and T5 and greater than 0.4 based on T2 and T3. Similarly, the SNPs C4S1877, C4S1878, and C4S1889 in the gene KDR on chromosome 4 (modulating Q1) were significantly associated based on T2 and T5 in more than 50% of the replications using ANOVA and in more than 40% of the replications based on T1 using ANOVA and based on T1, T2, T3, and T5 using the quantile-based method. All the significant SNPs mentioned have common variants (MAF > 0.01), except for C4S1877 and C4S1889, which have rare variants (MAF = 0.0007).The empirical powers corresponding to SNP C6S5380 in the P1K3C3 on chromosome 18 with T1 and T5, and C17S4578 in the gene PRKCA on chromosome 17 with T3 (both modulating the latent liability trait and hence the binary trait Z).We also obtained significant evidence of association in more than 30% of the replications at C18S2492 in the gene T1–T5) we found that both the model-based ANOVA and the model-free quantile-based regression yielded the maximum power for the phenotype defined by the principal components of the quantitative traits and the proportion of the risk score unexplained by the quantitative traits (T5) and the minimum power for the phenotype defined by the risk score of the binary trait as a function of the quantitative traits (T4). This finding can be explained by the fact that the risk score defined by T4 does not capture the complete information on variability of Q1 and Q2 and contains only that part of the variability that explains Z. Hence there is a loss of information and power when using T4 as an association phenotype. On the other hand, T1, which involves computation of principal components using the binary trait Z as a variable, is expected to capture less information on covariability of Q1, Q2, and Q4 compared to T5, which uses the continuous residuals of Z on Q1, Q2, and Q4 in the computation of the principal components. Because the binary trait Z is a function of Q1 and Q2, removing the effect of Z from a principal component involving Q1 and Q2 leads to a reduction in the information on both Q1 and Q2, and hence T3, which uses Z as a covariate, is less powerful than T2 (which does not involve Z) in identifying SNPs associated with Q1 or Q2.We have developed a method that integrates quantitative phenotypes with binary traits to construct a reduced univariate phenotype for association analysis of a multivariate phenotype vector. Among the five approaches to defining the reduced phenotype (labeled The ANOVA approach provides consistently higher power than the quantile-based method. The ANOVA is a genotype-based method that models the distribution of quantitative phenotypes conditioned on genotypes. On the other hand, the quantile-based method is allele based and models the distribution of allele frequencies conditioned on phenotypic quantiles. If the sample size is reasonably large, then the asymptotic properties of the ANOVA statistic hold and the ANOVA is expected to be more powerful than the model-free quantile-based method. Although the modification of the original quantile-based method , which aThe authors declare that there are no competing interests..SG and IM developed the proposed method. SS wrote the computer codes and performed the data analyses. IM participated in the compilation and interpretation of the results. SG drafted the manuscript. All authors read and approved the final manuscript


Genome-wide Complex Trait Analysis, GCTA) for the first time to estimate genetic influence on CU. We also report the first genome-wide association (GWA) study of CU as a quantitative trait. We compare these DNA results to those from twin analyses using the same measure and the same community sample of 2,930 children rated by their teachers at ages 7, 9 and 12. GCTA estimates of heritability were near zero, even though twin analysis of CU in this sample confirmed the high heritability of CU reported in the literature, and even though GCTA estimates of heritability were substantial for cognitive and anthropological traits in this sample. No significant associations were found in GWA analysis, which, like GCTA, only detects additive effects of common DNA variants. The phrase ‘missing heritability’ was coined to refer to the gap between variance associated with DNA variants identified in GWA studies versus twin study heritability. However, GCTA heritability, not twin study heritability, is the ceiling for GWA studies because both GCTA and GWA are limited to the overall additive effects of common DNA variants, whereas twin studies are not. This GCTA ceiling is very low for CU in our study, despite its high twin study heritability estimate. The gap between GCTA and twin study heritabilities will make it challenging to identify genes responsible for the heritability of CU.Callous-unemotional behavior (CU) is currently under consideration as a subtyping index for conduct disorder diagnosis. Twin studies routinely estimate the heritability of CU as greater than 50%. It is now possible to estimate genetic influence using DNA alone from samples of unrelated individuals, not relying on the assumptions of the twin method. Here we use this new DNA method (implemented in a software package called  Callous-unemotional behavior (CU) – defined by low levels of empathy, absence of guilt and emotional unresponsiveness – is currently under consideration as a subtyping index for conduct disorder in DSM-V Individual differences in CU are estimated to be moderately to strongly heritable using the twin design that compares resemblance in monozygotic (MZ) twins and dizygotic (DZ) twins in community samples of children and adolescents  studies for psychiatric phenotypes have shown that genome-wide “hits” are often in genes that were not previously hypothesized to influence the phenotype or not in traditional genes at all After more than a thousand published GWA studies across the life sciences Another reason for the missing heritability gap could be that twin studies have overestimated heritability. A new method, implemented in a software package called Genome-wide Complex Trait Analysis (GCTA), uses DNA alone to estimate genetic influence from samples of unrelated individuals, not relying on the assumptions of the twin method GCTA genetic similarity is not limited to the genotyped SNPs themselves, but also includes unknown causal variants to the extent that they are correlated with the SNPs. Mendel’s second law of inheritance is that genes  are inherited independently , but Mendel did not know that genes can be on the same chromosome, in which case they are not inherited independently (linkage disequilibrium). This violation of Mendel’s second law is complicated by the fact that during meiosis, on average each pair of chromosomes – one from the mother and one from the father – crosses over (recombines) once; in the population, genes on the same chromosome are separated by this process of recombination to the extent that they are not close together on the chromosome. GCTA provides a lower-limit estimate of heritability because it misses genetic influence due to causal variants that are not highly correlated with the common SNPs on genotyping arrays.A difference between GCTA estimates and twin-study estimates of heritability is that GCTA only estimates additive genetic effects, whereas the twin method captures nonadditive as well as additive genetic effects. Additive genetic effects are caused by the independent effects of alleles, which add up in their effect on a trait; nonadditive genetic effects are those that interact. Because GCTA adds up the effect of each SNP, it does not include gene-gene interaction effects; the twin method captures nonadditive as well as additive genetic effects because the DNA sequence of identical twins is virtually identical and thus they share all genetic effects, including nonadditive ones . Parents were given a letter describing the general purpose of the study and written parental consent was required. It was made clear that participation was voluntary and participants could withdraw from the study whenever they wished.The sample was drawn from the Twins Early Development Study (TEDS), a multivariate longitudinal study which recruited over 11,000 twin pairs born in England and Wales in 1994, 1995 and 1996 In order to make our twin sample as comparable as possible to our GCTA and GWA samples, we selected those twin pairs for whom one member of the twin pair was chosen for the GCTA and GWA analyses. For GCTA and the discovery sample of the GWA analysis, we included unrelated individuals by selecting only one member of each twin pair for whom GWA genotyping and CU data were available. For the GCTA analysis, we verified that the unrelated individuals were less genetically related than fourth-degree relatives (genetic relatedness >.025), the standard GCTA exclusion criterion.Based on these selection criteria, our twin analyses included 1099 MZ pairs and 1787 DZ pairs. Our GCTA and GWA discovery sample included 2,930 children; the slightly smaller number of twin pairs was caused by twin pairs for whom the co-twin did not have CU data.https://www.wtccc.org.uk/ccc2/). A total of 3,747 DNA samples from unrelated children in TEDS were sent for genome-wide DNA array genotyping used in our GCTA and GWA analyses. In total, 3,665 samples were successfully hybridized to Affymetrix GeneChip 6.0 SNP genotyping arrays  using experimental protocols recommended by the manufacturer . The raw image data from the arrays were normalized and pre-processed according to the manufacturer’s guidelines .DNA was extracted from buccal cheek swabs and sent to the Wellcome Trust Sanger Institute, Hinxton, UK for genotyping as part of the Wellcome Trust Case Control Consortium 2 (https://mathgen.stats.ox.ac.uk/genetics_software/chiamo/chiamo.html). Where there was a sufficient quantity of DNA, samples were also re-genotyped on a panel of 30 SNPs  using the Sequenom iPlex Gold assay .Genotypes for the Affymetrix arrays were called using CHIAMO  to infer identify by descent along the genome between pairs of individuals. Among pairs of closely related individuals, we excluded the member of the pair with the lowest call rate, iteratively repeating this procedure to obtain a set of individuals with pairwise identity by descent less than 5% p-value was lower than 10−20. Association between the SNP and the plate on which samples were genotyped was calculated and SNPs with a plate effect p-value less than 10−6 were also excluded. In addition, SNPs were manually filtered for call quality by visual inspection of the hybridization intensity plots using EVOKER software (http://sourceforge.net/projects/evoker/). The above filters removed 22.7% of the SNPs, leaving 699,388 autosomal SNPs for further analysis.An index of information (Fisher) for the allele frequency at each of 932,533 called SNPs was calculated using SNPTEST version 2.1.1 In order to increase the number of SNPs used in our GCTA and GWA analyses, imputation was carried out using the IMPUTE version 2 software CU at 7, 9, and 12 years of age was assessed by each child’s school teacher using a paper  questionnaire. Teacher ratings were obtained towards the end of the academic year when the class teacher had known the child for most of the academic year. In the U.K. there are no systematic differences with regard to placing mono- versus dizygotic twins to same or different classes. The percentage of twins rated by the same teacher is 65% at age 7, 58% at age 9, and 33% at age 12.Teachers are familiar with a broad range of children and have expertise regarding normative child development. Teacher ratings have been found to show higher internal consistency and stability than parent ratings broad heritability, in contrast to narrow heritability, which is limited to additive genetic effects. In twin analysis, the additive genetic model assumes that DZ twins are half as similar as MZ twins because the genetic relatedness of DZ twins is 50% for additive genetic effects, whereas the relatedness of MZ twins is 100%. This twofold greater genetic resemblance of MZ as compared to DZ twins is the reason why heritability is often estimated by doubling the difference between MZ and DZ correlations. .MZ and DZ twin intraclass correlations were calculated and standard twin model-fitting was used to estimate additive genetic (A), common or shared environment (C), and residual or non-shared environment (E) ADE, in which ‘D’ refers to dominance . In the ADE model, dominance discounts DZ resemblance from 50% for the A parameter to 25% for the D parameter. However, this adjustment does not cover the extreme epistatic case in which the DZ correlation could be zero despite a high MZ correlation. However, even in this extreme case – for example an MZ correlation of 0.80 and a DZ correlation of 0.00 – twin model-fitting would detect genetic influence and would cap the heritability estimate at 80%, as suggested by the MZ correlation of 0.80.In contrast, the hallmark of non-additive genetic effects is that the DZ correlation is less than half the MZ correlation because epistatic (inter-locus) gene-gene interactions scarcely contribute to DZ similarity but are shared entirely by MZ twins. If non-additive genetic effects are important, the twin method will detect these effects, although its ability to estimate these effects is limited. For example, if MZ and DZ twin correlations are 0.80 vs. 0.20, respectively, simply doubling the difference between MZ and DZ correlations – an additive genetic model which would be inappropriate given the non-additive pattern of twin correlations – would yield a heritability estimate of 120%. However, heritability cannot exceed the MZ twin correlation, so that the heritability estimate in this example would be constrained to be 80%. Model-fitting would show that the ACE model does not fit the data in this example. An allowance is made for non-additivity in a twin model called In summary, the twin design can detect the presence of non-additive genetic effects, although it is limited in its ability to distinguish additive and non-additive genetic effects. Greater detail about distinguishing additive and non-additive genetic variance in twin designs is available e.g. . As is uWe used the software package Genome-wide Complex Trait Analysis GCTA;  to estimGenome-wide association (GWA) analysis was conducted using a linear regression approach implemented in SNPTEST v2.0 r2>0.2) and to exclude high linkage disequilibrium genomic regions so as to ensure that only genome-wide effects were detected p<0.05. We caution that the inclusion of principal components as covariates may not be sufficient to remove biases in estimation due to population structure Following SNP quality control and SNP imputations, described earlier, we performed several preliminary analyses prior to GWA analysis. First, principal component analysis (PCA) was used to attenuate GWA biases due to population structure. PCA was conducted on a subset of 105,556 autosomal SNPs post QC, selected after pruning to remove SNPs in high linkage disequilibrium (http://www.cran.r-project.org).For the GWA analysis, each autosomal SNP was tested for association with CU at 7, 9, and 12 using a multivariate method that was similar in its essentials to the test of a longitudinal composite but required slightly less restrictive statistical assumptions. Using a linear regression framework, we calculated score statistics to test a hypothesis that the SNP had an equal effect on CU at each age. The test evaluated a single parameter and hence had 1 degree of freedom. Sex, birth year/school year cohort, and the first eight principal components of the genotype data were included as covariates in the regression model. The statistical framework was able to account for missingness in the outcome variables as a function of these covariates, assuming that data for these covariates were missing at random Probability values were adjusted by genomic control The GCTA estimate of genetic variance was 0.07, which was not significant given our sample size. The standard error of 0.12 suggests that the proportion of variance explained by the common SNPs tagged by our genome-wide genotypes is highly likely to be less than 20%, which suggests a wide gap with our twin study heritability estimate of 64%. Unlike twin analysis, GCTA does not discriminate C and E because each individual is from a different family. In GCTA, E is a residual term that refers to all variance (including error of measurement) that cannot be attributed to additive genetic effects of the common SNPs included on the DNA array.We tested 699,388 genotyped autosomal SNPs and 1,024,929 imputed autosomal SNPs that passed quality control thresholds. The analysis included 2,930 children with admissible data for both genotype and CU. The quantile-quantile (QQ) plot presented in p<5×10−8p<5×10−6 are summarized in No single SNP achieved genomewide significance using a conventional significance threshold of The sample size of 2,930 would, in a univariate analysis, be sufficient to detect a quantitative trait locus (QTL) explaining 1.0% of the variance with 49% power, or a QTL explaining 1.3% of the variance with 77% power Our twin study heritability estimate of 64% for CU is consistent with previously published results Although our sample size of nearly 3000 children entails a large standard error (0.12) for our GCTA heritability estimate (0.07), the 95% confidence interval suggests that the true estimate is GCTA heritability is less than 20%. In other words, even if the true GCTA heritability estimate for CU were at the top of this confidence interval, it would imply a wide gap between the GCTA and twin heritability estimates. This gap between GCTA heritability and twin study heritability could be called ‘missing GCTA heritability’. The missing GCTA heritability gap for CU, if true, would greatly increase the difficulty of identifying GWA associations because GWA, like GCTA, is limited to the additive effects tagged by the common SNPs on our DNA array.Why is missing GCTA heritability so much greater for CU than for some other traits in the same study? Although it is possible that the twin study overestimated heritability for CU, converging evidence from several sources suggests that twin estimates are valid What we do know for certain is that GCTA is limited to additive effects of common DNA variants that are tagged by commercially available DNA arrays, whereas the twin design is not. Because identical twins are identical in terms of all inherited DNA sequence variants, they share all genetic effects – small or large, nonadditive or additive, rare or common. For this reason, as compared to twin studies, GCTA underestimates heritability to the extent that heritability is caused by nonadditive genetic effects and the effects of rare variants The first hypothesis – that nonadditive genetic effects led to the low GCTA estimate of heritability for CU – is not supported by our twin results. As mentioned in the Methods section, the hallmark of nonadditive gene-gene (epistatic) interactions is that the DZ twin correlation is less than half the MZ twin correlation. However, in our twin analysis of CU, the DZ correlation (0.31) is almost exactly half the MZ correlation (0.63), providing no support for the hypothesis of nonadditive genetic influence.Our twin model-fitting also found that the additive AE model best fit the data. Indirect support for suggesting that nonadditive genetic variance may not be a major factor causing missing GCTA heritability for CU comes from the general conclusion from quantitative genetic research that most genetic variance is additive The second hypothesis –less common DNA variants contribute to the GCTA heritability gap for CU – seems certain to be part of the general explanation for missing heritability Because CU seems unlikely to be under strong negative selection, rare variants of large effect seem an unlikely hypothesis as to why the GCTA heritability gap is so much greater for CU than for other traits.p<5×10−8). Three SNPs achieved suggestive significance at a less stringent threshold (p<5×10−6), none of which are close to the coding regions of well-documented candidate genes implicated in previous genetic association studies of CU, or any of the suggestive hits identified by the previous pooling study of extreme CU that co-occurred with conduct problems Whatever its cause, the low GCTA heritability estimate, if true, implies that identifying DNA variants responsible for the heritability of CU will be even more difficult than it is for most complex traits. Because GCTA heritability, not twin study heritability, creates a ceiling for GWA analysis, the low GCTA estimate for CU doomed our GWA from the outset, even beyond the relatively small sample size that was limited to detecting genome-wide significant additive effects tagged by common variants that yield associations accounting for more than 1% of the total variance. This first genome-wide association study of childhood CU in a community sample of 2,930 individuals found no associations that met stringent genomewide correction for multiple testing (These results suggest that, for CU in particular, not only will much larger samples be needed to detect associations that account for very small effect sizes, but that methods to identify gene-gene interactions and whole-genome sequencing to detect rare variants may be needed in order to detect DNA variants that are responsible for the heritability of CU.


Aortic root diameter is a clinically relevant trait due to its known relationship with the pathogenesis of aortic regurgitation and risk for aortic dissection. African Americans are an understudied population despite a particularly high burden of cardiovascular diseases. We report a genome-wide association study on aortic root diameter among African Americans enrolled in the HyperGEN study. We invoked a two-stage, mixed model procedure to jointly identify SNP allele and copy number variation effects.CRCP and KCTD7 genes on chromosome 7 (p = 4.26 × 10-7); and the SIRPA and PDYN genes on chromosome 20 (p = 3.28 × 10-8).Results suggest novel genetic contributors along a large region between the The regions we discovered are candidates for future studies on cardiovascular outcomes, particularly in African Americans. The methods we employed can also provide an outline for genetic researchers interested in jointly testing SNP and CNV effects and/or applying mixed model procedures on a genome-wide scale. African Americans are known to be at increased risk for adverse cardiovascular outcomes including arterial stiffness and hypertension ,2. This The success of GWAS relies on the assumption that the heritability in common diseases can be captured by relatively few common genetic variants in the form of single nucleotide polymorphisms (SNPs) - sometimes referred to as the common disease, common variant hypothesis ,19. HoweCRCP and KCTD7 genes on chromosome 7 and the other between the SIRPA and PDYN genes on chromosome 20.Two recent studies have investigated the genomic architecture of CNVs in African Americans ,22. ThesPhenotypic data was obtained on 1,086 self-reported African American family members and individuals enrolled in the HyperGEN study . HyperGEet al. [The present study cohort consisted of 421 families. Families varied in size from single individuals to larger families, up to 10 . Genotype-based quality control was assessed by checking markers for Mendelian inconsistencies, potential patterns of missing data, and allele frequency measurements  were conducted on all autosomal SNP marker loci genotyped on the Affymetrix 6.0 array that passed quality control thresholds . The residuals were regressed on allelic and copy number state via lm using R version 2.10.1 [lmer function in R [-7 were defined as genome-wide significant [Residuals for use in the GRAMMAR approach were obtained from the PROC MIXED procedure in SASn 2.10.1 . Genome-ion in R . The mosnificant ,17.-3. In general, these markers appeared to be uniform across the genome, with the exception of a few regions. Chromosome 7, in particular, had numerous markers making it into the top 1,000. Many of these markers were found in a 500 kb region spanning the CRCP and KCTD7 genes. It should be noted that neither loci within genes previously found to be associated with thoracic aortic aneurysm , nor regions previously identified as being associated with aortic root size reached inclusion into the top 1,000 [Genome-wide results were generated via a two-stage procedure (see Methods). The GRAMMAR procedure (first stage), alone, does not provide accurate parameter estimates and significance values. The systematic depression of the test statistic, which can be seen in the quantile-quantile (QQ) plot . The copy number estimate at this locus was nearly three times its standard error. Also, a 40 kb copy number variable region on chromosome 2 within a gene gap between LPIN1 and TRIB2 genes showed relatively large copy number effect estimates - about four times its standard error. At this location, a rare deletion was correlated with higher aortic root diameter (pCNV = 2.97 × 10-5).While the loci that reached or approached genome-wide significance predominantly achieved that threshold based upon SNP effects, the second largest signal, found within the African Americans have been disproportionately represented in genetic association studies on cardiovascular traits despite a well-documented high burden of cardiovascular diseases. We implemented a non-traditional statistical approach to test for genetic associations with aortic root diameter in African American family members enrolled in the HyperGEN study. This method allowed us to jointly test for SNP and CNV effect while controlling for family structure.SIRPA and PDYN genes. However, a clear relationship with either gene to cardiovascular traits beyond the results of this study has not been discovered. SIRPA is involved in negative regulation with numerous growth factor signaling receptors; and PDYN has been found to be associated with nonlesional temporal lobe epilepsy [CRCP. The product of the CRCP gene is known to interact with CRLR to facilitate adrenomedullin (ADM) mediated signaling [CRLR has been found to be significantly decreased in the umbilical artery and uterus of women with pregnancy-induced hypertension [We identified two novel regions of genetic association on chromosomes 7 and 20. The significant SNPs in each region  have different allele frequencies in YRI and CEU populations in HapMap  - providepilepsy . Meanwhiignaling . CRLR hartension . Also, prtension .-3 in the second stage; and among the most significant findings, nine were in the top ten most significant findings in both stages. Second, the joint analyses we performed frequently reverted back to a conventional SNP analysis due to the majority of the genome being copy number invariant. However, we were cautious about using a SNP-only approach throughout the genome as it would require us to either treat non-two-copy genotypes as missing, or force them into a two-copy state. And finally, the HyperGEN sib-pair ascertainment scheme restricted our ability to distinguish inherited copy number variation from variants arisen through de novo mutation as the direct transmission of CNVs from both parents to offspring was unobservable.The two-stage procedure, joint analysis, and ascertainment scheme offered some limitations to the present study. First, it is possible that we missed a true association that did not pass the first stage criteria; but after comparing the results between the loci that were included in both steps, we are fairly comfortable as the results were consistent across stages. Only three of the top 1,000 did not have p-values less than 3 × 10We found no findings that proved copy number variation was playing a role in the pathogenesis of aortic root diameter, but we do not believe CNVs should be ruled out altogether. In a few cases we found large estimates with promising significance values. Unfortunately, the low frequency of copy number events restricted our ability to separate their effects from more frequent SNPs. Also, copy number variants potentially cover numerous loci. In terms of copy number alone, we expect perfect or near-perfect linkage disequilibrium among those loci. Therefore, perhaps the same criteria imposed for genome-wide significance on SNPs should not hold for CNVs when looking at a "per locus" level.CRCP perhaps shows the most promise. However, the large region encompassing this gene shows strong patterns of linkage disequilibrium  procedure.Click here for file


SPA has three advantages. First, it accounts for the interaction information and gains considerable power in the presence of unknown and complicated G×G or G×E interactions. Secondly, it does not sacrifice the marginal detection power; in the situation when rare variants only have marginal effects it is comparable with the most competitive method in current literature. Thirdly, it is easy to extend and can incorporate more complex interactions; other practitioners and scientists can tailor the procedure to fit their own study friendly. Our simulation studies show that SPA is considerably more powerful than many existing methods in the presence of G×G and G×E interactions.Recently more and more evidence suggest that rare variants with much lower minor allele frequencies play significant roles in disease etiology. Advances in next-generation sequencing technologies will lead to many more rare variants association studies. Several statistical methods have been proposed to assess the effect of rare variants by aggregating information from multiple loci across a genetic region and testing the association between the phenotype and aggregated genotype. One limitation of existing methods is that they only look into the marginal effects of rare variants but do not systematically take into account effects due to interactions among rare variants and between rare variants and environmental factors. In this article, we propose the  Despite of the success of large scale biological studies such as GWAS in discovering many disease variants, most of which are common variants with minor allele frequency (MAF) greater than 0.05, for diabetes, heart disease, Alzheimer disease, etc., the variants identified thus far confer relatively small risk, explain a small fraction of familial clustering, and add little practical value in disease prediction. The issue of so-called “missing heritability” has been a serious concern that has attracted considerable attention and discussion recently. In genetic association studies, the ‘common-disease common-variants’ (CDCV) model states that common diseases are caused by common variants with MAFs greater than 5% or 1%. K is the number of markers and Due to the low frequencies and large number of rare variants, traditional single-marker association tests that have worked well for common variants will in general lack power for rare variants. summation of partition approach (SPA), a robust model-free method that not only tests the marginal effects of rare SNPs but also naturally incorporates G×G and G×E interactions. As with existing methods, SPA is based on aggregating information across rare variants in a region of interest. We shall demonstrate the power of SPA and compare with existing methods for both dichotomous and quantitative phenotypes. Simulation studies show that in disease models without interactions, the performance of SPA is comparable to or even better than the most competitive existing method in current literature, and in the presence of G×G interactions, SPA substantially outperforms all the other methods. Another advantage of our procedure is its simplicity and extensibility. We also demonstrate in this article how to incorporate an environmental factor in the proposed framework and show that the augmented test score is powerful in detecting G×E interactions. Similar approaches can be taken to account for interactions with common variants or other covariates. In addition, we compare the proposed method with several existing tests on the dataset provide by Genetic Analysis Workshop 17 (GAW17) and find that SPA is robust for detecting different genes. When large volumes of datasets with rare variants become available in the near future, the proposed procedure will become a powerful tool to detect complicated interaction effects in various genetic regions and it will help us to better understand the mechanisms of complex human diseases.The aforesaid methods have been shown to work well in different simulated models . However, all these tests only consider marginal effects from rare variants and they do not systemically address the issue of interactions among rare variants (G×G), or between rare variants and covariates, such as environmental factors (G×E). Therefore, additional statistical methods are needed to generate scientific knowledge on the etiology of complex diseases where interactions among genetic, biological and environmental variables work together to produce a phenotype. In this article, we propose the I derived from what we now know as the Partition Retention (PR) method. To better understand the motivation and rational behind SPA, we briefly review a general framework that has been adopted for detecting common variants with interactions. A core element in this framework is the influence score n subjects with a response variable Y and K discrete explanatory variables {1,…, XKX}. If each iX can take three discrete values, we generate a partition Π with K3 non-overlapping partition elements. Let in be the number of subjects in partition i, i, and We demonstrate a basic tool adopted by our method. Suppose there are Y, the normalized I, Y) is asymptotically distributed as a weighted sum of K discrete variables with K3 partition elements each containing non-overlapping observations. This influence measure captures any discrepancy between the conditional mean and the grand mean of Y and thus is able to detect X-Y association regardless of the structure of dependence. It can be easily generalized to any discrete random variables with finite number of outcomes.It has been shown that under the null hypothesis that none of the predictors has influence on AN is the number of affected individuals, UN is the number of unaffected individuals, and i. Several variations of this partition-based method have been successful at identifying influential common variants and their interactions in human diseases, such as Rheumatoid Arthritis summation of partition approach (SPA). We introduce below several test statistics of SPA, including the marginal test score I1, G×G interaction score I2, and G×E interaction scores In case-control studies, the influence measure can be rewritten as: The general framework mentioned above can be extended to rare variants association analysis for both dichotomous and continuous phenotypes.N unrelated individuals, among which AN are cases and U = N−NAN are controls. The region of interest G contains K rare variants and the genotype of the I1 that accounts for all marginal information contributed by these K rare SNPs is defined as: in is the total number of In population-based case-control studies, suppose there are I1 is defined as: in is defined as above. Different from the original influence measure, I1 recognizes the partition elements formed by individual SNP and hence the partitions from different SNPs are not non-overlapping any more; therefore, I1 does not suffer from the sparseness of rare variants. Under the null hypothesis of no influence, the differences between i are small, so a large I1 value indicates that some rare variants in the region might be associated with the disease phenotype. Additionally, since each term of I1 is the squared difference between the conditional average and the grand average, it can detect both directions of departure from the expected difference zero, which renders I1 the ability to capture the association even in a region with both risk and protective rare variants. Unlike PR's influence measure I, the statistical property of I1 is more complicated to obtain since the dependence between partition cells created by different SNPs will not asymptotically disappear even under the null hypothesis of no influence. Therefore, in our analyses we will rely on the method of permutation to assess its statistical significance.For continuous traits, I1 is that it considers little interactions among rare SNPs. From the general framework, we propose a second SPA test score I2 that evaluates G×G interactions among rare variants.In order to increase the power of detecting the genotype-phenotype associations as well as to elucidate the biological pathways that underpin disease, more and more attentions have been given to the identification of interactions between SNP loci. K3 partition elements for all levels of interactions. However, due to the low frequencies of rare variants, the higher order (>2) interaction information among rare SNPs in current sample size will be small. For example, if the sample size is 1,000 and the SNP frequency is 0.01, the expected number of observing one specific rare variants triplet is 1,000×0.013 = 10−3. If a region contains 20 independent rare SNPs, the expected total number of rare variants triplets would be i and j, we consider three aggregated cells: individuals with rare variants only on SNP i (denoted mM), individuals with rare variants only on SNP j (denoted Mm) and individuals with rare variants on both SNPs (denoted mm). Note that we do not consider the cell MM where individuals have no rare variant at either position. For dichotomous trait, the SPA test score I2 for G×G interaction is defined as: i or j), mM, Mm, and mm. For quantitative trait, 2I is defined as: mM, Mm, and mm. If two rare variants have interactions, the difference between the conditional average and the unconditional average will be large, leading to a large I2 value. Again, permutation is used to evaluate the significance of the test statistic I2. Even though m-tuples (m≥3) of rare SNPs.As the genotype at each SNP position can take 3 values, in theory we are facing a maximum of p* that is a compromise between I1 and I2. We first evaluate the significance of I1 and I2. Then the adaptive test score is defined as: p(I1) and p(I2) are the p-values of I1 and I2 separately. We evaluate the significance of p* by permutation.When we are unclear whether G×G interaction is involved in the onset of disease, we propose an adaptive score E has J levels. The SPA test score for detecting the effect of the environmental factor is expressed as: i on level j that are from cases, and j. I1 by building additional overlapping rare variants partition cells to J non-overlapping partitions created by the environmental factor. The significance of Increasing evidence have shown that gene and environmental (G×E) interactions are widely involved in the etiology of complex diseases, including diabetes, cancer and psychiatric disorders I1, I2, p* and We simulated several scenarios for the purpose of evaluation and comparison of our test scores with several existing rare variants association methods. The genotype consists of 20 independent rare variants in each scenario. Scenario ‘Null-1’ is a ‘null model’ where none of the 20 variants affects the phenotype. For dichotomous traits, the phenotypes are determined by the baseline penetrance only. This is the null setting for For empirical power comparisons, we generate three different sets of simulations. The first set of simulations are marginal effect models, in which the MAF of all SNPs are uniformly distributed between 0.0001 and 0.01. In scenario 1, 5 out of the 20 rare SNPs are risk SNPs and the effect size is constant. Scenario 2 is similar to scenario 1 except that the risk effect is negatively correlated with MAF. Scenario 3 has 5 protective variants and 5 deleterious variants with effect size negatively correlated with MAF. The second set of simulations contains 2-way G×G interaction between rare variants, with MAF 0.01 for all 20 SNPs. In scenario 4, 50% of the SNPs (10 out of 20 SNPs) have interaction effects. Scenario 5 is similar to scenario 4 but 75% of the SNPs are involved in G×G interactions. Both main effect and G×G interaction effect exist in scenario 6. The third set of simulation models involves G×E interaction effects with a binary environmental factor. Scenario 7 has positive G×E interaction effects and environmental marginal effect; scenario 8 has both positive and negative G×E interaction effects. Logistic regressions or linear regression was used to generate dichotomous or quantitative phenotypes. 1,000 repetitions were simulated for each scenario with four different sample sizes, each having equal number of cases and controls. Detailed simulation models are provided in Table S1 in 1I, 2I and p* with existing methods: CMC, WS, VT, RB SKAT (with the weighted linear kernel) and SKATint (a modified SKAT score with the weighted quadratic kernel) in a series of simulation scenarios, including marginal effect models and G×G interaction effect models for both dichotomous traits and continuous traits. RB only deals with binary outcomes, so it is not included in our analysis for continuous traits. We also evaluated the power of Material and Methods for details of simulation models; numerical results from our simulation studies are presented in Table S2−S7 in We compared the power of SPA test scores I1, I2 and p* are presented in I1, I2 and p* are well controlled at both significance levels for either dichotomous or continuous trait, even when the sample size is small, indicating that the proposed tests are valid methods. Additional results of type I error for competing methods are presented in Fig. S1 in The empirical type I error rates for I1, I2 and p* with competing methods in three marginal effect models when (1) only risk variants exist and the effect size is constant, (2) only risk variants exist and the effect size is negatively correlated with MAF, or (3) a mixture of risk and protective rare variants exists.We compare the power of I1 and SKAT are comparable and they are both superior to the other tests  or the second most powerful (for quantitative traits) method even in scenarios where the effect size is negatively correlated with MAF, showing that its good performance is intrinsic and is not driven by a specific weighting scheme. The test score I2 does not show a high power in these marginal effect models as it is designed to detect G×G interaction effects but not the marginal effect.In all three marginal effect scenarios, the performance of er tests , 2 and 3 effects . It is wI2 over all the other methods is apparent for both dichotomous and continuous traits . The advantage of the G×G interaction association score s traits . For dicI2 is still consistently the most powerful test and p* is the second best, followed by SKATint . Here the marginal effect is set to be small so that it will not mask the interaction effect.  SKATint . For conMaterials and Methods), denoted by Global and Local respectively. As Global is appropriate for testing the null hypothesis of no association at all , and Local is appropriate for testing the null hypothesis of no E marginal effect.For the G×E interaction score Global and Local in these two null settings. In null-1, both Global and Local are correctly controlled at levels α = 0.05 and 0.01. In null-2, Local still hits the target level while Global has significant higher values. This is because Global is able to test any effect from genetic or environmental factors, including the E marginal effect; hence the results of Global in null-2 are indeed the power of Global in the presence of E marginal effect. On the other hand, Local removes the E marginal effect, so it shows the correct type-I error in both null-1 and null-2.The type I error of Global, Local and competing methods when (1) the phenotypes are affected by E marginal effect and positive G×E effect, (2) the phenotypes are affected by E marginal effect and both positive and negative G×E effects. In computation, SKAT and SKATint regress the phenotype on the environmental factor when calculating the test statistic Global and Local use the environmental factor as in their definition. All the other methods work on the phenotype and the genotype directly. The results show that Global has much higher power than all the other tests because it takes into account both E marginal and G×E interaction effects, and Local outperforms all the remaining methods that do not consider G×E interaction effects  provided genotypes of 3,205 autosomal genes on 697 individuals from the 1000 Genome Project. A dichotomous phenotype was simulated from a linear model using SNPs from 34 genes and most causal SNPs were rare variants. A total of 200 simulation replicates were carried out and the genotype was held fixed for all replicates. See I1, I2 and p* depends on the sample size, the number of variants and the number of permutations. On a 2.66 GHz laptop with 4 GB memory, to reach a significance level of 10-4, the computation times to analyze a region with 20 SNPs for 600, 1000, 1500 and 2000 individuals are 3, 5, 7, 10 sec for I1, and are about 1000, 1400, 1900, 2500 sec for I2.The computation time of I1 is designed to detect marginal effects of rare variants and I2 is designed to capture the G×G interaction effects among rare variants. In various marginal effect models, I1 is more powerful than most approaches examined in our study. Its performance is comparable to SKAT, which is regarded as the most competitive existing method. In G×G interaction models or in the scenario with both marginal and G×G interaction effects, I2 is superior to all the other methods in terms of detection power. The adaptive score p* is a compromise between I1 and I2 and has the advantage of both test scores. Its performance is just a little shy of the better of the two scores I1 and I2, for both marginal effect models and interaction effect models. Therefore, p* is a self-tuning adaptive score that is able to gain power automatically regardless of the simulation scenario. In practice when we have no clue of how the genotype affects the phenotype, we suggest to use the adaptive score p*. A significant p-value of p* indicates a potential true signal from either marginal or interaction effects of rare variants. In our study, we focus on the situation with 20 rare SNPs. If the SNP number changes to 30, the simulation results , a flexible robust model-free framework for rare variants association analysis that incorporates both G×G and G×E interactions. The proposed SPA test scores create partitions from individual SNP and combine the information across all rare variants in a region of interest. I1 that incorporates covariates. It can be used to test the hypothesis of no association at all  using ‘global permutation’ or to test the hypothesis of no E marginal effect using ‘local permutation’. By ‘local permutation’, I2 and the resulting augmented score could be used to detect E×G×G 3-way interactions between an environmental factor and two rare variants.The insight of SPA is similar to the partition retention (PR) influence measure as in I1 can be viewed as a special case of The information measure Population stratification has been shown to be an important problem for common variant association analysis. For rare variants, this problem is more likely to occur due to their low frequency and possible uneven distribution among populations. It is straightforward to control population stratification in our approach as we can consider population as an environmental factor and apply it to A major advantage of SPA is that it is highly extensible. The building blocks of SPA are the partitions formed by individual rare variant and it is easy to incorporate complex interactions. As demonstrated in the article, we are able to take into account interactions with environmental factors. Similar approaches can be applied when considering interactions with common variants or other covariates. It can also be generalized to other research areas to benefit the practitioners and scientists in various fields. We believe that the proposed framework of SPA will offer substantial opportunities in detecting potential complicated interactions. Once interaction effects indeed exist, our approach is capable of identifying these interactions and thus adding to the detection power.http://www.columbia.edu/rf2283/Software.htmlThis paper presents a simple novel (and easily implemented) tool SPA as an alternative to existing statistical methods for rare variants association studies, with a unique additional feature that SPA can easily incorporate various forms of interaction effects. This addition may add considerable power to disease-related detection in the future. From our studies, if the underlying model is a simple linear additive model with only marginal effects, the powers of SPA are comparable to several existing methods. However, if the model is more complex with interaction effects, the proposed approach provides a more powerful alternative in rare variants association analysis so that there is a better chance to find disease-associated factors. With the development of next-generation sequencing techniques, more and more data with a large amount of rare variants will be generated. It is highly unlikely that the disease phenotype is associated with genetic factors through a simple linear main effect model, so the proposed approach is going to be a powerful and rewarding tool to explore the complicated interaction effects revealed by larger datasets. It is worth noting that any interaction pattern, whether it is linear or nonlinear, can be detected by SPA, since it is model-free and is not subject to any distribution assumptions. Therefore, it is very robust and effective regardless of how the genotype affects the phenotype. The R code of the proposed test scores is available to download at File S1The supporting information file for article “A Robust Model-free Approach for Rare Variants Association Studies Incorporating Gene-Gene and Gene-Environmental Interactions”. It contains the files: Table S1. Models to generate simulated phenotypes; Table S2. Power of different methods for dichotomous traits in scenarios 1∼6 (α = 0.05); Table S3. Power of different methods for dichotomous traits in scenarios 1∼6 (α = 0.01); Table S4. Power of different methods for continuous traits in scenarios 1∼6 (α = 0.05); Table S5. Power of different methods for continuous traits in scenarios 1∼6 (α = 0.01); Table S6. Power of different methods for dichotomous traits in G×E interaction effect models (α = 0.05); Figure S1: Type I error for different methods in various sample sizes with nominal α levels 0.05 (left) and 0.01(right); Figure S2: Power comparison in scenarios 1∼6 for dichotomous traits with 500 cases and 500 controls when the SNP number is 30.(DOC)Click here for additional data file.


The aim of this study was to estimate the components of variance and genetic parameters for the visual scores which constitute the Morphological Evaluation System (MES), such as body structure (S), precocity (P) and musculature (M) in Nellore beef-cattle at the weaning and yearling stages, by using threshold Bayesian models. The information used for this was gleaned from visual scores of 5,407 animals evaluated at the weaning and 2,649 at the yearling stages. The genetic parameters for visual score traits were estimated through two-trait analysis, using the threshold animal model, with Bayesian statistics methodology and MTGSAM  threshold software. Heritability estimates for S, P and M were 0.68, 0.65 and 0.62 (at weaning) and 0.44, 0.38 and 0.32 (at the yearling stage), respectively. Heritability estimates for S, P and M were found to be high, and so it is expected that these traits should respond favorably to direct selection. The visual scores evaluated at the weaning and yearling stages might be used in the composition of new selection indexes, as they presented sufficient genetic variability to promote genetic progress in such morphological traits. In Brazilian genetic improvement programs, consideration has been given to growth traits involving the evaluated of weight at different standard ages its gain as criteria for selection in Zebu beef-cattle. The use of these criteria has effectively contributed to the increase in productive performance indices, especially for the Nellore breed. It is noticeable that present day Nellore herds, when it comes to production, are in no way similar to those introduced into Brazil through importation in 1962.Although, the Nellore population has excelled in body weight, there has been little improvement in finishing precocity and carcass quality. According to et al. 2004; Although the carcass quantitative traits, evaluated through ultrasound, allow direct selection for such traits, other indirect measurements can also lead to genetic progress in meat quality. With this in mind, use has been made in various selection programs of information on morphological traits as evaluated by visual scores, with a view to obtaining genetic progress in finishing precocity and carcass maturity and quality in Zebu beef-cattle. Nonetheless, there are only a few studies on estimating genetic and environmental parameters for these visual evaluation traits , the market trend is to seek animals which are more precocious and economically viable, and that remain less time in pastures or feedlots, thereby shortening the production cycle. With this in mind, the National Association of Breeders and Researchers  and the company Brasilcomz developed the Morphological Evaluation System , which applies modern procedures to collecting data on visual scores for the traits body structure (S), precocity (P), musculature (M) and navel (N), aiming at improving genetic evaluation, and through this, the generation of new selection indexes.According to Lôbo Care must be taken in the genetic evaluation of morphological traits, as data do not present a normal distribution. Therefore, the aim of this work was to estimate variance components and genetic parameters for those visual scores which constitute the Morphological Evaluation System such as body structure (S), precocity (P) and musculature (M) in Nellore beef-cattle, evaluated at the weaning and yearling stages, through the use of threshold Bayesian models.Information was resorted to from visual scores of 5,407 animals evaluated at weaning and 2,649 at the yearling stage, respectively born to 135 and 224 bulls and 2,274 and 2,314 cows. The animals were of the Nellore breed and belonged to herds which took part in the Brazilian Nellore Program of the National Association of Breeders and Researchers (ANCP). The visual scores of MES evaluated in this paper were: body structure (S), precocity (P) and musculature (M). The animals were visually evaluated by means of MES methodology , in whicScores from one to six points could be attributed to the tested animals for each morphological variable. When compared to its contemporary group, an animal which is considered intermediate (three or four points) for a specific trait, represents the reference for the classification of those below (one or two points) and above (five or six points) the mean. Data collection for the MES was done by qualified technicians from Brasilcomz.On applying MES methodology to visual evaluation, the animals from the same management lot were individually evaluated individually by the same technician. Each lot was composed of animals of the same sex and time birth abiding under the same food and sanitary conditions. The entire management lot was taken into consideration, in an attempt to visualize the average profile for each morphological trait evaluated. In this way, visual evaluation was comparative, the score being given to each individual in relation to the others. The distribution of scores as a percentage for each morphological trait is presented Structuring of data files was done by means of the Statistical Analysis System  program. In order to guarantee data consistency of morphological traits, the interval in the age of the animals followed a variation from 150 to 270 days in relation to 210 days (age of 7 months) and from 490 to 610 days in relation to 550 days (age of 18 months). GLM and REG procedures from SAS  software were used to check environmental effects that influence morphological traits. Weights at 120 days (W120) were included in analyses in order to minimize the effects of selection on visual scores. The individual weight (W120) of 6,169 animals was used, mean and standard deviation being 130 kg and 19 kg, respectively.Contemporary groups for categorical morphological traits were defined, taking into account the farm, year, season of birth, management lot and the technician responsible for rating lot. As to weight at 120 days, contemporary groups included animals from the same farm, year and season of birth, sex and management lot, on reaching the stipulated age. The effect of season of birth was divided into four classes: animals born in the months of January to March, April to June, July to September and October to December.Genetic parameters for visual score traits were estimated through two-trait analyses by means of the threshold animal model, using Bayesian statistic methodology with MTGSAM  threshold software, developed by For visual score traits, sex and age of the dam (six classes) were considered as fixed effects, contemporary groups as random effects and the age of animal at the moment of data collection as covariate (linear effect). The complete model can be represented in matricial notation as follows:y is the vector of observations , β is the vector of fixed effects , a is the vector of random effects which represent additive genetic values from each animal, p is the vector of non-correlated random effects of permanent maternal environmental effects, c is the vector of random effects not correlated to contemporary groups, e is the vector of residual random effects, and X, Z1, Z2 and Z3 are the incidence matrices which link the observations to fixed and random effects: additive genetic direct and non-correlated , respectively. The relationship matrix included 26,893 animals from the Nellore breed.in which Threshold models usually present certain some problems in estimating variance components and predicting genetic values, when there are many levels of fixed effects  is the vector of the location parameters of order s with β , and order s with a, p and c , W is the known incidence matrix of order r for s, I is the identity matrix of order r for r; and yi falls into category j , given the vectors β, a, p, c and t  is given by:in which t1 < t2...< tj-1, with t0 = -∞ and tj = ∞, where j is the number of categories. Observable data are dependent on the underlying variable which is limited between two unobservable thresholds  for each animal i are defined by Uiin the underlying scale:Categorical traits are determined by unobservable continuous variables, in underlying scale, this having fixed threshold initial values, in which resholds .  Therefn is the number of observations. After specifying the thresholds t0 to t6, it is necessary that one of the thresholds (from t1 to t5) be adjusted to an arbitrary constant. In the genetic analyses for morphological traits evaluated at the yearling stage, it was assumed t1 = 0, in such a way that the vector of estimable thresholds was defined as:in which t1 = 0 and t2 = 1, in such a way that the vector of estimable thresholds was defined as:For the two-trait analyses of visual scores evaluated at weaning, another parameterization was adopted variances was assumed as Inverted Wishart for genetic random effects, and non-correlated and residual in the traits studied, this including mutual covariance. Uniform initial distribution was defined as much for fixed as well as for threshold effects. Parameter A 1,000,000 total cycle sampler chain, a 300,000 cycle burn-in and a 1,000 cycle thinning interval to obtain 700 initial samples, were used implementation of Gibbs Sampling. Thereafter, the analysis continued, each time 100,000 cycles being added until estimates obtained in the last analysis were equal to the previous one. This criterion was adopted in order to verify whether convergence had really been achieved. Sample analysis, serial correlation and convergence of the Gibbs chain were undertaken with support from the GIBANAL  program.The Monte Carlo error was estimated by calculating the variance in samples of genetic parameters of traits and dividing this variance by the number of samples  function, which resulted in EPDs (expected progeny differences) expressed as a percentage (%). The PROC CORR procedure from SAS  software was applied to verify rank or Spearman rank-order correlations.Descriptive statistics of variance components and genetic parameters of morphological traits, evaluated by MES methodology at weaning, are presented in The estimates of heritability for S (0.68), P (0.65) and M (0.62) indicate high genetic variability for visual scores, which is explained by the differences in biotypes or morphological types existent in Nellore cattle. It is important to point out that the application of MES methodology to morphological evaluation is novel, and thus selection for the visual scores S, P and M is also relatively recent in Brazil.A similar result for P heritability was encountered by et al., 2004) reported low estimates for heritability coefficients calculated for the maternal genetic effect, when considering the traits conformation (C), precocity (P) and musculature (M) in Nellore beef-cattle. According to It is important to emphasize that the maternal additive genetic random effect of visual scores evaluated at weaning was not included in the model due to the difficulty in attaining Gibbs chain convergence, when applying threshold models to an animal model. Other authors  and visual scores evaluated at weaning were high for all morphological traits . Based oIn Rank or rather Spearman rank-order correlations of EPDs (expected progeny differences) in morphological traits evaluated through MES methodology at weaning and the yearling stage, are presented in Rank correlations among visual scores evaluated at different ages were 0.78 for S, 0.61 for P and 0.66 for M. These results show that classification or ranking of the animals changed when a certain morphological trait was evaluated at different ages (weaning and yearling). It was noted that for S, the higher rank correlation (0.78) reflects the intense genetic association of this trait with body weight, and that, no matter the age at visual evaluation, genetic expression become evident. The same was not the case with P and M, in which genetic differences were better presented at the yearling stage.In conclusion, visual scores from the Morphological Evaluation System, evaluated at weaning and the yearling stage, could be used in composing new selection indexes, as they presented sufficient genetic variability to promote genetic progress in such morphological traits.


Univariate genome-wide association analysis of quantitative and qualitative traits has been investigated extensively in the literature. In the presence of correlated phenotypes, it is more intuitive to analyze all phenotypes simultaneously. We describe an efficient likelihood-based approach for the joint association analysis of quantitative and qualitative traits in unrelated individuals. We assume a probit model for the qualitative trait, under which an unobserved latent variable and a prespecified threshold determine the value of the qualitative trait. To jointly model the quantitative and qualitative traits, we assume that the quantitative trait and the latent variable follow a bivariate normal distribution. The latent variable is allowed to be correlated with the quantitative phenotype. Simultaneous modeling of the quantitative and qualitative traits allows us to make more precise inference on the pleiotropic genetic effects. We derive likelihood ratio tests for the testing of genetic effects. An application to the Genetic Analysis Workshop 17 data is provided. The new method yields reasonable power and meaningful results for the joint association analysis of the quantitative trait Q1 and the qualitative trait disease status at SNPs with not too small MAF. Statistical methods for the univariate association analysis of quantitative and qualitative traits have been well developed in the literature. Complex human diseases are often characterized by multiple traits. These traits tend to be correlated with each other because of common environmental and genetic factors. In the genetic analysis of complex diseases, it is natural to account for the correlations among multiple traits and to model them simultaneously. Joint genetic linkage analysis of multiple correlated phenotypes has been studied by Jiang and Zeng , Mangin Recently, Liu et al.  developeIn this paper we extend the joint linkage analysis of multivariate qualitative and quantitative traits described by Williams et al. ,8 to assn independent individuals. Let iY1 be the qualitative trait and iY2 be the quantitative trait for the ith individual. Let Xi denote a vector of covariates, including the intercept and environmental variables, and let Zi denote a vector of genotype score(s) at the major single-nucleotide polymorphism (SNP) locus. We may also include gene by environment interaction terms. We assume that iY1 is determined by a latent continuous variable Suppose that the data contain We consider the following model:β are regression coefficients for the environmental effects, the γ are regression coefficients for the genetic effects, and the subscripts 1 and 2 denote the qualitative and quantitative traits, respectively, and the where the 12 is the variance of εi1, σ22 is the variance of εi2, and ρ is the correlation between εi1 and εi2.where σσ1 = 1 and α = 0.To ensure the identifiability of the model, we fix iY2, it can be shown that Given and variance:θ ≡  takes the form:Therefore the likelihood function for the unknown parameters where:and Φ is the cumulative distribution function of the standard normal distribution.θ, denoted To obtain the maximum-likelihood estimator (MLE) of s et al. . We can θ under the null hypothesis. The LRT follows a chi-square distribution asymptotically with the degrees of freedom being the difference of the number of free parameters under the null hypothesis and the number under the alternative hypothesis.where x-axis denotes the correlation between the two Wald test statistics for the two traits. Figure We first conducted a small simulation study to evaluate the type I error rates of the association test based on the univariate analysis at significance levels of 0.01 and 0.05. Figure We next performed a joint association analysis of the quantitative trait Q1 and the qualitative trait disease status from the GAW17 data. Before conducting the analyses, we had knowledge of the answers. Sex, Age, and Smoking were included as covariates in the model. We evaluated the additive effect of each of the 11 true SNPs on chromosome 13 on each individual trait as well as the pleiotropic effect on both traits. The genotype score was coded as the number of minor alleles at each SNP locus. For comparison, we also performed a univariate association analysis and used the sum of the LRT statistics for testing genetic effect on an individual trait as the test statistic for testing the pleiotropic effect.Table In this paper, we described a likelihood-based approach for the joint association analysis of quantitative and qualitative traits in unrelated individuals. As expected from the true data generation model, our association analysis of the GAW17 data reveals that there is strong correlation between the quantitative trait Q1 and the qualitative trait disease status. In this case, the joint association test of pleiotropic effects is more appropriate than the test based on univariate analysis, which ignores the trait correlation.Two factors might have contributed to the small difference between the joint analysis and the univariate analysis using the GAW17 data set. First, both methods had low power with rare variants, for example, when MAF < 0.0014. On the other hand, when the MAF was moderate and the effect size was relatively large, both methods yielded power close to or equal to 100%; therefore we are unlikely to see the difference between the two methods under such situations. In fact, it may not be fair to compare the powers between the two methods because the univariate analysis tends to inflate type I error rates in the presence of correlation between the qualitative trait and the quantitative trait.In its current form, the single-SNP joint analysis is more ideal for common variants and less ideal for rare variants. To improve the power of the association tests with rare variants, we could apply the approaches of Morris and Zeggini  based onThe authors declare that there are no competing interests.GD was responsible for the study design and helped to draft the manuscript. MY performed the statistical analysis and drafted the manuscript. Both authors read and approved the final manuscript.