Phenotypic variation in human intellectual functioning shows substantial heritability, as demonstrated by a long history of behavior genetic studies. Many recent molecular genetic studies have attempted to uncover specific genetic variations responsible for this heritability, but identified effects capture little variance and have proven difficult to replicate. The present study, motivated an interest in “mutation load” emerging from evolutionary perspectives, examined the importance of the number of rare (or infrequent) copy number variations (CNVs), and the total number of base pairs included in such deletions, for psychometric intelligence. Genetic data was collected using the Illumina 1MDuoBeadChip Array from a sample of 202 adult individuals with alcohol dependence, and a subset of these (N = 77) had been administered the Wechsler Abbreviated Scale of Intelligence (WASI). After removing CNV outliers, the impact of rare genetic deletions on psychometric intelligence was investigated in 74 individuals. The total length of the rare deletions significantly and negatively predicted intelligence . As prior studies have indicated greater heritability in individuals with relatively higher parental socioeconomic status (SES), we also examined the impact of ethnicity (Anglo/White vs. Other), as a proxy measure of SES; these groups did not differ on any genetic variable. This categorical variable significantly moderated the effect of length of deletions on intelligence, with larger effects being noted in the Anglo/White group. Overall, these results suggest that rare deletions (between 5% and 1% population frequency or less) adversely affect intellectual functioning, and that pleotropic effects might partly account for the association of intelligence with health and mental health status. Significant limitations of this research, including issues of generalizability and CNV measurement, are discussed. Despite a good number of candidate gene and SNP (single nucleotide polymorphism) association studies, very little progress has been made—so little, in fact, that a recent review led the authors to conclude, “it is not possible confidently yet to name one genetic locus unequivocally associated with the quantitative trait of intelligence”  show that many loci contribute effects, with no one locus accounting for more than 1% of the variation If no one locus accounts for much variation in g, a likely possibility is that g is just such a phenotypic feature g is associated with developmental instability, as assessed by composite measures of random variations in bodily symmetry. Meta-analyses show human developmental instability is associated with reduced psychometric intelligence and neurodevelopmental disorders g independent of developmental instability too; a great many genes are expressed in the brain Researchers have proposed that g (or any other phenotypic trait) is that, technologically, it is not possible to measure genome-wide mutations at this time. But recent advances in genomic studies have led to the discovery of one potentially important class of rare variations that can be assessed with current technology, copy number variations (CNVs). It was once thought that the “normal” human genome could be defined by a shared reference genomic structure, one specifying all single nucleotide sites. In its extreme form, this view implies that all genetic variation between any two  individuals consist merely of the aggregate of base differences across all 3 billion or so single nucleotide sites. Geneticists have long recognized the existence of exceptions—insertions, deletions, or inversion of long chromosomal segments in individual genomes. Recent discoveries, however, show that “exceptions” are anything but unusual One problem with testing these ideas by examining the association between the genome-wide mutation load and Ultimately, CNVs originate as mutational events Recent studies have revealed an elevated incidence of rare deletions in schizophrenia, autism, and mental retardation g.Not all diseases are associated with specific CNVs g across its normal range. The definition of “rare” is somewhat arbitrary and different researchers have used different criteria. In studies of schizophrenia, 1% frequency (percentage of a sample with the variant) has often been used as a cutoff for rare g, prompting our decision to use the 5% figure. Hence, variants we aggregated include those that qualify as both rare and infrequent . Inclusion criteria were (1) age between 21 and 55; (2) within 21 days of their last drink; (3) drinking levels of more than 14 drinks per week  or 21 drinks per week  during four consecutive weeks within three months of beginning the study; (4) negative drug screen for opiates, cocaine, or amphetamine; (5) must meet DSM IV criteria for alcohol dependence; and (6) must have a Clinical Institute Withdrawal Assessment (CIWA) All subjects provided written consent. The study was approved by the University of New Mexico Human Research and Review Committee according to principles expressed in the Declaration of Helsinki.Participants provided at least 5 ml of saliva into a sterile 50 ml conical centrifuge tube. DNA was then extracted from the sample, purified, and hybridized. Detection of 1,199,187 SNP and CNV markers across the entire genome was performed using the Illumina Human 1 M Duo BeadChip Array according to the manufacturer's directions. We did not analyze DNA from the X and Y chromosomes, reducing the number of SNPs to 1,147,842. The data were further scanned and 10,272 loci with missing measurements were removed. The median distance between adjacent markers was approximately 2.5 kbp.Details of the series of procedures used to quantify CNVs are described in Chen, Liu, and Calhoun All participants were administered the vocabulary and matrix reasoning subtests of the Wechsler abbreviated scale of intelligence WASI; , from whGenetic data was available for 202 participants, while WASI data were available for only 77. Initial evaluation of CNV numbers was performed on the larger sample, so as to maximize accurate evaluation of the shape of the frequency distributions. Six samples were discarded using Need's criteria, leaving a sample of 196 participants. At total of 13,557 CNVs were detected, 7249 deletions, 6308 insertions . The observed frequency distribution of CNV number was markedly skewed, as a few participants had extremely high numbers of CNVs. As oversensitivity of CNV detection for these individuals likely led to unrealistically high values Pearson correlation coefficients between rare CNV variables and intelligence test performance are presented in p values >0.17). Hence, these findings are not driven by an association between FSIQ and severity of alcohol dependence. One might also wonder, however, whether they are driven by duration of alcohol abuse. If this were the case, one would anticipate correlations between age, as a proxy measure for duration of alcohol abuse, and the three cognitive measures. None of these correlations was significant .Adding total ADS score as a covariate did not alter these findings, and total ADS did not correlate with any CNV variable , for which ethnicity might serve as a proxy variable. Prior research has found that parental SES affects heritability estimates for intelligence. Data on parental SES were not available for this sample. In New Mexico, where the sample was collected, individuals of other ethnicity , on average, come from lower SES backgrounds than those of Anglo/White ethnicity.In these regression analyses (run on PASW [formerly SPSS] 17.0 GLM univariate), FSIQ served as the criterion variable. Predictors entered were (1) sex, (2) ethnicity, (3) rare deletion length, (4) rare insertion length, and (5) interactions of each of the CNV variables with sex and ethnicity. Results revealed a main effect of deletions, partial eta  = 0.38, F  = 11.00, p = 0.002. This effect was significantly moderated by ethnicity, partial eta  = 0.39, F  = 11.48, p = 0.001. Ethnicity also had a main effect independent of deletions and insertions, β = 0.53 F = 24.36, p<.001. No other effect was statistically robust, all p>.28.To evaluate the robustness of the effect across varying definitions of infrequent or rare deletion, we repeated this regression analysis twice, each time substituting a measure of CNV length based on a different, more stringent definition of rare CNVs,. Specifically, we calculated length using criteria for rare CNVs of “≤3%” and “≤1%”. In each analysis, the effect of deletion length was statistically significant, with the ≤3% definition yielding slightly stronger effects than our original <5% definition, and the ≤1% definition yielding slightly weaker effects: partial eta  = .41, p<.001, and. 31, p = .013, respectively. Both analyses also yielded the significant interaction between deletion length and ethnicity. In sum, then, the effect of infrequent deletions in this sample is not peculiar to a criterion of a relative frequency of 5% or less.To examine the nature of the deletion x ethnicity interactions, we separately computed correlations for the Anglo/White and Other groups. As can be seen in We computed the residual variance in FSIQ with variation associated with deletion length in each group was removed. There was no difference in residual variance across groups, p = 0.41. Hence, the Anglo/White group had significantly more variance in FSIQ associated with deletions, but there was no significant difference in the amount of variance in IQ not associated with deletions.No significant differences related to ethnicity category were found in independent samples t-tests  for any of the four genetic variables (number and length of rare insertions and deletions). We emphasize, then, that, while rare CNV deletion lengths covary differently for the two ethnic categories , main effects of CNVs do not contribute to differences in cognitive performance of the two groups.The greater the size of rare and infrequent deletions, as represented by the number of base pairs lost, the lower an individual's psychometric intelligence. In contrast to SNP effects investigated in prior studies, the current effect size is substantial, accounting for 9% of the phenotypic variance in the full sample and 45% in the Anglo/White sub-sample.There are several important implications of this finding. Before discussing them, however, we consider the generalizability of the current results. Our sample consisted of individuals with alcohol dependence. Do associations between intelligence and rare deletions exist in other populations? Only replication in other samples of healthy controls and clinical groups will provide a clear answer to this critical question. At present, however, no specific observations suggest the relationship is unique to the current sample. First, overall WASI performance is solidly in the average range, only very slightly below the mean level of the general population. Second, controlling for a measure of alcohol dependence (the ADS) did not diminish the association. Third, the greater genetic effect in the Anglo/White sample is consistent with prior studies in healthy children and adolescents showing greater heritability in higher SES groups The measure of CNV deletions we used aggregates many different deletions. The genetic effect demonstrated here is different in kind than that revealed by studies of individual SNPs or large aggregations of SNPs e.g., . The rarnone of these deletions. And for any two individuals possessing even a handful of them, the probably that they share precisely the same ones is vanishingly small.These facts have two important implications. First, different people have different deletions. Indeed, given the distribution of infrequent deletions, it stands to reason that the large majority of random pairs of individuals share particular deletion may do no better than any individual SNP). Our sample is of insufficient size to examine the effects of many individual CNV deletions. But with 3364 individual rare variations possible, it seems unlikely that only a small subset carries the effect we observe. The reason is simple: Each CNV deletion can possibly account for only a tiny amount of total variation in the aggregate measure of infrequent deletions. Were it the case that only a small subset  had reliable associations, their variation accounted for would be overwhelmed by the variation in the remaining non-relating deletions (say >90%), leaving a very small amount of variation accounted for. Hence, the only plausible view is that a sizeable number of infrequent deletions covary with IQ.Second and relatedly, it must then be the case that many different individual CNVs have common effects on intellectual functioning, with no one CNV deletion possessing any more than a very small effect. Though the total number of possible deletions is too many for us to examine all, we We did not screen CNVs to have any particular function or be located in any particular place in the genome. Why, then, would deletions in many regions of the genome affect intellectual functioning? We discuss two important considerations.First, there may exist massive pleiotropy The association of rare deletions with intelligence may, in this context, shed light on why lower intelligence predicts more comorbid health and mental health problems: holes in the genome may have widespread consequences, especially for brain function. For example, two disorders commonly comorbid with alcohol problems are the “externalizing” disorders of ADHD and antisocial behavior. Each of these disorders is associated with reduced intelligence Second, it is possible that genetic material prone to deletion is more associated with intellectual functioning than randomly selected genetic material in the human genome. One process generating both deletions and insertions, nonallelic homologous recombination (NAHR), tends to produce CNVs near “hot-spots” every rare CNV deletion affects intellectual functioning. Almost certainly, many do not. But again, were it the case that only very small proportion of deletions did so, there would exist only a very weak association between FSIQ and a composite of deletions. As this is not what we observed, it seems very likely that a meaningful, substantial proportion of CNVs are associated with intellectual functioning.Naturally, we do not suggest that As emphasized in the We found that rare deletion length predicted intelligence in our Anglo/White sample, but not in our Other Ethnicity sample. These results are consistent with previous findings that the heritability of g is greater in children and adolescents coming from high SES backgrounds There are two sets of important limitations of the current study. The first relates to our sample. Though our results have clear implications for the origins of comorbidity among externalizing disorders, and the potential to help account for the genetic vulnerability associated with alcohol abuse, these issues can be best pursued in future studies that provide diagnoses of all possible comorbid disorders and include a healthy control group. The lack of a significant association between alcohol dependence and intelligence and the aggregate CNV measures may be due to the fact that the sample was limited to only individuals with alcohol dependence. In light of the numerous failures of replication in history of molecular genetic studies of intelligence The second set of important limitations concerns our assessment of CNVs. Though we believe we have achieved valid and reliable estimates, it is undoubtedly the case that we did not capture all rare CNVs in our participants' genomes. Some smaller CNVs and those in regions not as well mapped by the reference genome may have gone undetected. Whether inclusion of such CNVs would strengthen or weaken our findings is unknown. We have made two major assumptions in the quantification of CNVs, which should be systematically evaluated in larger studies. First, we arbitrarily defined rare CNVs as those occurring in 5% or fewer of our participants. However, use of 1% and 3% cutoffs led to similar effect sizes. An important question for future research is how different phenotypes relate to CNVs occurring at different population frequencies. Possibly, variations in intelligence within the normal range are subject to less selection pressure than phenotypic variation associated with debilitating disorders such as schizophrenia or autism, and hence, associated with relatively more common variants.As more common CNVs might be less deleterious


P = 0.01 in males) and 5%  of genetic variance in an independent sample of multiplex families. Overall, our work demonstrates that prioritization of GWAS data based on functional genomics identified common variants associated with autism and provided additional support for a common polygenic background in autism.Autism spectrum disorders (ASD) are highly heritable complex neurodevelopmental disorders with a 4:1 male: female ratio. Common genetic variation could explain 40–60% of the variance in liability to autism. Because of their small effect, genome-wide association studies (GWASs) have only identified a small number of individual single-nucleotide polymorphisms (SNPs). To increase the power of GWASs in complex disorders, methods like convergent functional genomics (CFG) have emerged to extract true association signals from noise and to identify and prioritize genes from SNPs using a scoring strategy combining statistics and functional genomics. We adapted and applied this approach to analyze data from a GWAS performed on families with multiple children affected with autism from Autism Speaks Autism Genetic Resource Exchange (AGRE). We identified a set of 133 candidate markers that were localized in or close to genes with functional relevance in ASD from a discovery population (545 multiplex families); a gender specific genetic score (GS) based on these common variants explained 1% ( Autism Spectrum Disorders (ASDs) are characterized by impairments in social interaction and communication, restricted interests, and repetitive behaviors with a 4:1 male to female ratio  or chromosomal abnormalities but several common variants including CNVs and single nucleotide polymorphisms (SNPs) have also been strongly linked to autism  but have little value for risk prediction  have been performed to decipher the genetic etiology of autism that is attributable to common variants  with only a few variants having shown significant associations and replicated in an independent population or in endophenotypes  . The GS was tested for association in this validation sample and the proportion of genetic variance explained evaluated.http://www.agre.org/), were selected to improve the power of the GWAS by homogenizing the phenotype  or that had a call rate less than 90% or a minor allele frequency less than 5% were removed. Mendelian transmission of alleles was checked for every SNP; genotypes that were inconsistent with Mendelian inheritance in one or several families were considered as unknown in all members of the families showing the error. SNPs identified in the discovery population were genotyped in the University of Pennsylvania collection, as previously described  were estimated using a generalized estimating equation (GEE) model with an independence correlation matrix to account for the non-independence of individuals from the same family  software  using a GEE in males and females separately. Inflation effect on GRRs because of SNPs selection according to consistent direction of allele effect in the validation sample was evaluated based on 1,000 bootstraps resampling  under a recessive model with P = 8.8 × 10−9. Allantoicase participates in the uric acid degradation pathway, but its activity is absent in mammals  and in females (P = 8.7 × 10−7).One SNP was discarded from subsequent analyses because of genotyping failure in the validation population. From the subset of 132 SNPs selected by the scoring strategy  of a link with autism, some had been previously linked to genetically-related neuropsychiatric disorders such as schizophrenia and bipolar disorder and may be potentially implicated in autism, as has been demonstrated for genes such as HTR2A, GRIN2A, and RBFOX1  Ma, . A subseWe acknowledge that different ways of scoring SNP-gene pairs could have given slightly different results. Nevertheless, the simple scoring system that was developed in this study provides a good prioritization of SNPs with regard to our focus of mining real statistical signals from “noise” in the GWAS. The major limitation of the proposed scoring approach is its dependence on available information about genes; however, its power will increase with the evolution of knowledge.p-value selection only identified a limited number of SNPs. Common variants displaying liberal association p-value  explain a proportion of genetic variance in autism (Anney et al., p-value from GWAS could identify additional genes and enhance our understanding of the common genetic basis of autism.In conclusion, adding function evidence of genes to GWA results is a powerful alternative to allele score approach in order to identify common variants associated to autism. In a sample of moderate size, the scoring strategy like the CFG method allow to extract and prioritize biologically relevant signals when classical GWA studies based on Jérôme Carayol and Thomas Rio Frio participated in the design of the study, the analysis of data, the interpretation of data, and to draft the manuscript. Beth Dombroski, Bérengère Génin, Karine Fontaine, Francis Rousseau, and Céline Vazart participated to the genotyping of the samples and the interpretation of data. Claire Amiet, David Cohen, Thomas W. Frazier, Antonio Y. Hardan, Gerard D. Schellenberg, and Geraldine Dawson participated to the interpretation of data, and to review the manuscript. All authors read and approved the final manuscript.Jérôme Carayol, Claire Amiet, Karine Fontaine, Bérengère Génin, Francis Rousseau, Céline Vazart, and Thomas Rio Frio are currently salaried employees of IntegraGen. Jérôme Carayol, Francis Rousseau, and Thomas Rio Frio have patent applications with IntegraGen. Gerard D. Schellenberg and Beth Dombroski declare that they have no competing interests; Antonio Y. Hardan, David Cohen, and Geraldine Dawson are compensated consultants for IntegraGen. Thomas W. Frazier has received a clinical research grant from IntegraGen.


P) and family (K) levels. We also compared type-I error rates among models in analyses of publicly available human and dog datasets. The models corrected for none, one, or both structure levels. Correction for K was performed with linear mixed models incorporating familial relationships estimated from pedigrees or genetic markers. Linear models that ignored K were also tested. Correction for P was performed using principal component or structured association analysis. In analyses of simulated and real data, linear mixed models that corrected for K were able to control for type-I error, regardless of whether they also corrected for P. In contrast, correction for P alone in linear models was insufficient. The power and precision of linear mixed models with and without correction for P were similar. Furthermore, power, precision, and type-I error rate were comparable in linear mixed models incorporating pedigree and genomic relationships. In summary, in association studies using samples with both P and K, ancestries estimated using principal components or structured assignment were not sufficient to correct type-I errors. In such cases type-I errors may be controlled by use of linear mixed models with relationships derived from either pedigree or from genetic markers.Population structure is known to cause false-positive detection in association studies. We compared the power, precision, and type-I error rates of various association models in analyses of a simulated dataset with structure at the population (admixture from two populations;  The power of an association study depends on the phenotypic variance explained by the causal variant, the extent of linkage disequilibrium (LD) between the causal variant and the markers, and, not least, the size of the study sample. Several recent association studies have thus focused on collecting large samples to obtain higher powers of detection Population stratification refers to the inclusion of individuals from isolated subpopulations in the population of interest. In such a population, individuals from a subpopulation are, on average, more closely related to each other than to other individuals in the population as a whole. Population structure is common in nature. It manifests in the form of herds, colonies, and ethnic groups, and as a consequence of geographic isolation and natural or artificial selection Genetic association studies aim to correlate differences in trait level with differences in genotypes at a tested marker. This correlation is assumed to arise from the co-occurrence of the marker with the quantitative trait locus (QTL), i.e., LD. The presence of LD is usually equated with physical linkage. However, in a stratified population with differences at the trait level, markers with different frequencies across populations will also correlate with trait levels. Thus the use of a structured population in an association study may yield false associations due to differences in allele frequencies among subpopulations Examination of an unstratified sample is thus optimal in an association study Several methods, such as structured analysis (SA) The study uses a simulated dataset and 2 publicly available datasets, so no ethical approval was required.−3/locus/generation) and drift. The HP consisted of 5,000 individuals that were randomly mated for 1,000 generations. The genome was assumed to have 20 chromosomes of 100 cM each with 40,000 evenly distributed bi-allelic SNP markers with equal frequencies (0.5) for the two alleles in the base population. Only two QTLs with two equally frequent alleles were simulated, both located on chromosome 1 at 20 cM and 80 cM. The QTL mutation rate was 2.5×10−5/generation. The number of recombination per Morgan (per chromosome) was sampled from a Poisson distribution of mean = 1 and the cross-overs were randomly placed on the chromosomes. QTL effects were introduced in the last generation of the HP for a trait  with a heritability of 0.3. The allelic effects were sampled from a gamma distribution with a shape parameter of 0.4 as implemented in QMSim An admixed population  was simuNext, two populations were generated from the HP by mating two separate parent groups in isolation. The parent groups  were randomly selected from the last generation of the HP. Each sire was mated to 20 dams, with each mating producing one offspring. This procedure yielded a total of 1,000 offspring with an equal sex ratio  from each parent group. Fifty randomly selected male offspring from each group were used as sires for the next generation, whereas all female offspring along with 500 randomly selected dams from the previous generation were used as dams for the next generation. This mating scheme was continued for 30 generations.stF from all 40,000 SNP markers, was 0.025 [standard deviation (SD) = 0.0006]. The allele frequencies of the markers , following the mating scheme described above. After admixture, QTLs in the last generation explained 0–13% of the total phenotypic variance. The individuals' ancestries, calculated as average genetic contributions from one parent population to individuals in the admixed population over four generations, are presented in Chromosome 1, which contained two simulated QTLs, was used to assess the power and precision of the models and five chromosomes (chromosomes 2–6) without simulated QTLs (null chromosomes) were used to assess the false-positive rate (type-I error). A total of 100 replications were used to compare the models.In addition to the simulated datasets, the models for GWA studies were applied to the following two publicly available datasets to assess their ability to control type-I error.This human stature dataset was collected from 1,315 European American individuals genotyped for 637 genetic markers as part of the Genetics of Lipid Lowering Drugs and Diet Network (GOLDN) study 1, F2, and backcrosses). The samples were genotyped for 23,500 SNPs and hip dysplasia was measured using the Norberg angle This dataset contains hip dysplasia data from 292 dogs of two breeds (Labrador retriever and greyhound) and their crosses (Fb) of an SNP was estimated by successively fitting the marker in the following models. The simulated data contained two confounding factors: admixture (P) and familial relatedness (K). The models compared corrected for none, one, or both factors.The allele substitution effect (b was then tested against the null hypothesis (b = 0) using a t-test with Bonferroni correction for multiple testing, at a significance level of α = 0.05. All analyses were carried out with the DMU software package The significance of estimated Linear Model (LM). A linear model (LM) ignoring K and P was used to estimate the allele substitution effect:Linear Mixed Model Including Pedigree-Based Relationship (LMMped). An LMM accounting only for K was used to test the associations of single SNPs with the phenotype:Linear Mixed Model Including Genomic Relationship (LMMgmat). This LMM was similar to the LMMped accounting for K, differing only in the estimation of the variance-covariance matrix of Structured Association. A model based on the clustering method of the STRUCTURE software package P. One thousand markers evenly distributed over the genome were selected and individuals were assigned to two clusters (k = 2) in a structure linkage model P.4a) Linear Model with STRUCTURE (LMstr). This model was similar to the LM, with additional fixed regression (4b) Linear Mixed Model with STRUCTURE (LMMstr). This model was similar to LMMped, with the addition of fixed regression of the phenotype on Principal Component Analysis (PCA). PCA was performed to correct for P in the sample. Principal components (PCs) for the covariance matrix, estimated from all 40,000 SNP markers, were calculated. SNP genotypes and phenotypes were corrected for the first two PCs, as described elsewhere P. Although PCs did not separate the human samples, two PCs were used to correct the genotypes and phenotypes for comparison. The corrected genotypes  Principal Component Analysis in a Linear Model (LMpca). Here, P while ignoring K:5b) Principal Component Analysis in a Linear Mixed Model (LMMpca). Here, K and P:Simulated Data. SNPs on five null chromosomes (with no simulated QTL) were used to assess the type-I error rates of all described models. A total of 1,000,000 SNPs over 100 replicates were fitted into the models to estimate b. The distribution of −log10p-values of the t-test was plotted against the null (uniform) distribution in a quantile-quantile (QQ) plot. Deviation from the null distribution was used to assess the type-I error rate. In addition, the number of observed significant SNPs on null chromosomes was compared with the expected number at different significance levels.k = 2), with the inclusion of sex, age, and squared age as covariates 10p-values of the t-test from the expected uniform distribution in a QQ plot was used to assess the type-I error rate.The human data were analyzed with LM, LMMped, LMMpca (first two PCs), and LMMstr (k = 2), and LMMpedB, a model similar to LMMped but with the addition of breed (as proportion of Labrador retriever) as a fixed regression. The 1,000 randomly chosen markers were assumed to be unassociated with the causal variants for hip dysplasia. The distribution of −log10p-values of the test of association was compared with the expected distribution under the assumption of no association in a QQ plot. Deviation from the expected distribution was used to assess the type-I error rate.The data on hip dysplasia in dogs were fitted to the LM, LMMped, LMMpca (first two PCs), LMMstr . A QTL was considered detected when an SNP within 1 cM of the simulated QTL showed significance after Bonferroni correction  at α = 0.05. The absolute distance between the detected and simulated QTL positions was calculated to ascertain the precision of the methods in locating the QTL. Power and precision were estimated separately for small effect . Among LMMs, LMMpca closely followed the expected type-I error rate, whereas LMMgmat was the most conservative, showing a lower than expected type-I error rate.The distributions of −logibutions . In geneK  showed significantly higher false-positive results than expected (p<0.01), whereas LMMs that corrected for K  showed better control of false-positive results.The average number of significant SNPs on the null chromosomes is compared with expected numbers under the null hypothesis at different significance levels in α = 0.05. In contrast, all models tested showed 53–60% power to detect large-effect (explaining >5% of n = 99) = 0.9925; p = 0.8031]. Overall, LMMped had the highest power, detecting 32% of simulated QTLs, followed by LMMstr (31.7%), LMMpca (30.7%), and LMMgmat (28.1%); however, this difference was also not significant .The power of the models to detect two simulated QTLs on chromosome 1 using 100 replicates with 199 QTLs is summarized in Out of the total 199 simulated QTLs, 53 were detected by all the four models, the majority (50) of which being large effect QTL . On the other hand 133 QTLs were not detected by any of the models, majority (95) of them being small effect QTL .The precision of the models, measured as the absolute distance (cM) between the positions of simulated and detected QTLs, is described in K and P showed the highest false-positive rate. The three LMMs , showed similar distributions of −log10 p-values, which were slightly higher than expected.For the Human GOLDN dataset, type-I error was compared among the LM, LMMped, LMMstr, and LMMpca. The distribution of test statistic is given in a QQ plot . The LM The dog data were analyzed using the LM, LMMped, LMMpedB, LMMstr, and LMMpca. The type-I error rates for these methods are compared in a QQ plot in K  showed higher rates of false-positive associations than did LMMs accounting for K  and LMpca (two PCs). Thus our findings indicate that correction for structure at the population level alone is not sufficient in samples with structure at the population and family levels. In such samples, LMMs can be effectively used to control for false-positive results.In contrast, correction for Among LMMs, LMMgmat was the most conservative and consequently suffered a slight loss of power. In this model, the relationship matrix (G) was estimated from the whole genome, including markers linked to the simulated QTL. This probably led to correction of part of the QTL effects in the data, thereby reducing the model's power to detect QTLs.r2 = 0.07) between adjacent markers and a similar LD between markers and the QTL. Increasing the marker density and sample size might increase the power to detect small-effect QTLs in such a dataset.The compared models had powers of ∼60% in detecting large-effect QTLs. In our simulation, the same set of QTLs were segregating in the two ancestral populations, which might not accurately reflect the situation in real populations, where subpopulations may have different causal variants affecting the same trait, thereby reducing the power to detect such QTLs. Thus, our power results likely reflect the upper boundary of detection power in structured populations for QTLs with similar effects using the same sample size. In our study, none of the tested methods had the power to detect small-effect QTLs in the simulated dataset. This finding might be explained by a lower average LD  used in the simulation. The mutation rate for markers assumed in this study was higher than what is observed in real life  the first principal component captures the genetic variation due to population stratification .Replicates i and iii show separation along the first axis whereas replicates ii and iv show separation along the second axis. Over 100 replicates, in 86 cases the first PC had the highest correlation (mean of 0.77) with the ancestries, in 9 cases with the second PC (0.67), in 3 cases with third PC (0.58) and in 2 cases with the fourth PC (0.52).(EPS)Click here for additional data file.Figure S2Top two axes of variation in the Dog hip dysplasia datashows separation of Dog breeds, Labrador (L), Greyhound (G), and crosses along the top 2 PCs. The first PC had a correlation of 0.91 with the breed proportions.(EPS)Click here for additional data file.Figure S3Correlation between the top 20 PCs and the known ancestries in the simulated dataset (100 replicates).(EPS)Click here for additional data file.


Although the efficacy of genomic predictors based on within-breed training looks promising, it is necessary to develop and evaluate across-breed predictors for the technology to be fully applied in the beef industry. The efficacies of genomic predictors trained in one breed and utilized to predict genetic merit in differing breeds based on simulation studies have been reported, as have the efficacies of predictors trained using data from multiple breeds to predict the genetic merit of purebreds. However, comparable studies using beef cattle field data have not been reported.Molecular breeding values for weaning and yearling weight were derived and evaluated using a database containing BovineSNP50 genotypes for 7294 animals from 13 breeds in the training set and 2277 animals from seven breeds  in the evaluation set. Six single-breed and four across-breed genomic predictors were trained using pooled data from purebred animals. Molecular breeding values were evaluated using field data, including genotypes for 2227 animals and phenotypic records of animals born in 2008 or later. Accuracies of molecular breeding values were estimated based on the genetic correlation between the molecular breeding value and trait phenotype.With one exception, the estimated genetic correlations of within-breed molecular breeding values with trait phenotype were greater than 0.28 when evaluated in the breed used for training. Most estimated genetic correlations for the across-breed trained molecular breeding values were moderate (> 0.30). When molecular breeding values were evaluated in breeds that were not in the training set, estimated genetic correlations clustered around zero.Even for closely related breeds, within- or across-breed trained molecular breeding values have limited prediction accuracy for breeds that were not in the training set. For breeds in the training set, across- and within-breed trained molecular breeding values had similar accuracies. The benefit of adding data from other breeds to a within-breed training population is the ability to produce molecular breeding values that are more robust across breeds and these can be utilized until enough training data has been accumulated to allow for a within-breed training set. One key advantage of genomic predictors is that they can be estimated early in the life of the animal and thus allow for increased accuracy of estimated breeding values (EBV), particularly for young animals, which have not yet produced progeny. However, the benefit of the inclusion of genomic predictions into EBV estimates is proportional to the amount of genetic variation that is explained by the genomic predictor . To dateA total of three within-breed and two across-breed training populations were used:1. A multiple-breed training population, which will be referred to as the MB population, that included five breeds  from a database containing 50K genotypes assembled from purebred beef cattle breeds. Animals in the database were primarily artificial insemination (AI) sires that had a substantial influence on their respective breeds and EBV with reasonably high accuracies. The only exception was the Limousin breed, for which a large number of DNA samples originated from previous DNA testing, such that only about 58% of the genotyped animals were AI sires. The MBV were trained on de-regressed EBV  (includi2. Single-breed Angus (AN), Hereford (HH), or Limousin (LM) subsets that contributed to the MB MBV, collectively referred to as the single-breed training population (SB). The difference between MB and SB is that training was performed separately for each breed, as opposed to simultaneously for all five. The MBV trained on these single-breed training sets were computed both for subsets of the evaluation population that included the same breeds as the training set and subsets that included different breeds.3. A multiple-breed training population consisting of AI sires from 13 breeds with high accuracy EBV and referred to as MB_2K. De-regressed EBV (adjusted for base differences and including weights to account for variable accuracy) were treated as phenotypes for training, following the methods of . Due to http://bigs.ansci.iastate.edu/). No pre-analysis filtering of SNPs (single nucleotide polymorphisms) based on minor allele frequency (MAF) was performed. The MB_2K predictions used a BayesCπ model [2e/W, with W weights from [2g with probability 1-π. The SNP effect variance and the residual variance had scaled inverse Chi-squared prior distributions. Parameter π had a uniform  prior distribution in the BayesCπ model, but was assumed known in the BayesC model.The numbers of animals per breed in the SB, MB and MB_2K sets are presented in Table Cπ model . The MB hts from . The SNPThe evaluation population consisted of genotyped animals from seven breeds in the herds of 24 seedstock producers from the Northern Plains region of the US plus any genotyped animals in their 4-generation pedigrees. The numbers of genotyped animals in the evaluation populations are summarized in Table The accuracies of the various MBV trained as described above, were evaluated based on the estimated genetic correlations between each of those MBV and the corresponding phenotypes in the evaluation population. Correlations were estimated using bivariate mixed linear models in which the traits were the MBV and the corresponding phenotypic trait. The genetic correlations between the trait and MBV reflect the accuracies of MBV since the square of these correlations represents the proportion of genetic variance explained by the genomic information ,8,16,19.Bos indicus, British x Continental, and Bos indicus x Continental breed composition. The model for MBV included a fixed effect for the intercept and a random direct genetic effect with the variance-covariance matrix proportional to the numerator relationship matrix. The environmental variance for MBV was fixed at 0.01% of the environmental variance of the phenotypic trait, and environmental covariances between the MBV and trait phenotype were assumed to be zero. The restricted maximum likelihood (REML) estimates of the variance components were obtained using ASReml [The model for the phenotypic trait in the bivariate model included fixed effects for contemporary group, breed composition, and heterosis, and random direct and maternal additive genetic effects. An effect of heterosis was expected in some breeds if the breed association previously had an open herd book or currently registers composite animals . The heterosis effect was modeled as a direct effect using three covariates for the proportions of the animal’s British x g ASReml . VariancGenetic parameters for weaning and yearling weights estimated using single-trait analyses of the evaluation population field data from 24 herds are presented in Table g2) in weaning and yearling weights , and thus a reduction in the variability of MBV among animals, when SB MBV were applied to animals from a different breed (Table There was a reduction in the proportion of genetic variance explained (red Table , illustrThe average model incorporation frequencies, i.e. the proportion of iterations of the MCMC chain with which an individual SNP enters the model, were within 0.0012 of the prior model frequency of 0.01 = 1 - π for all SNPs and for each MBV. The numbers of SNPs with model frequencies and effects that exceeded various thresholds are presented in Table As suggested by , one benThe accuracy of within- or across-breed trained MBV are substantially lower for breeds that are not included in the training set, since the estimated genetic correlations between trait MBV and their corresponding phenotypes cluster around zero. This is true even for breeds that are closely related, such as Angus and Red Angus. The addition of training data from other breeds produces an MBV where the SNP effects are concentrated onto a smaller set of informative SNPs. However, the accuracy of the across-breed trained MBV is similar to the within-breed trained MBV.The authors received no funding from The American Angus Association, Red Angus Association of America, American International Charolais Association, American Gelbvieh Association, American Hereford Association, North American Limousin Foundation, or American Simmental Association to support this study. The authors declare that they have no competing interests.SDK, MLS, and KJH conceived and designed the study and assisted in the interpretation of the results. SDK analyzed the data. GLB, LAK, WMS, RMT, and EJP developed the MB_2K MBV, provided genotypes, and assisted in the interpretation of the results. MS and DJG developed and provided the genotype database and estimated the MB and SB MBV. RDS and JFT provided Angus and Limousin genotypes. SDK and MLS wrote the draft of the paper. SDK, MLS, GLB, KJH, LAK, WMS, RMT, MS, DJG, JFT, and EJP contributed to the preparation of the final manuscript. EJP directed the overall project. All authors read and approved the final manuscript.


In designing genome-wide association (GWA) studies it is important to calculate statistical power. General statistical power calculation procedures for quantitative measures often require information concerning summary statistics of distributions such as mean and variance. However, with genetic studies, the effect size of quantitative traits is traditionally expressed as heritability, a quantity defined as the amount of phenotypic variation in the population that can be ascribed to the genetic variants among individuals. Heritability is hard to transform into summary statistics. Therefore, general power calculation procedures cannot be used directly in GWA studies. The development of appropriate statistical methods and a user-friendly software package to address this problem would be welcomed.This paper presents GWAPower, a statistical software package of power calculation designed for GWA studies with quantitative traits, where genetic effect is defined as heritability. Based on several popular one-degree-of-freedom genetic models, this method avoids the need to specify the non-centrality parameter of the F-distribution under the alternative hypothesis. Therefore, it can use heritability information directly without approximation. In GWAPower, the power calculation can be easily adjusted for adding covariates and linkage disequilibrium information. An example is provided to illustrate GWAPower, followed by discussions.http://dl.dropbox.com/u/10502931/GWAPower.zipGWAPower is a user-friendly free software package for calculating statistical power based on heritability in GWA studies with quantitative traits. The software is freely available at:  Statistical power and sample size calculation is an important step during experiment design in genome-wide association (GWA) studies. It estimates the probability that a true genetic effect can be detected under the experimental constraints and other reasonable assumptions. To date, more than 230 GWA studies have been conducted and reported. Among these, case control design (>100 studies) and cohort studies with unrelated subjects (>70 studies) are the two most popular designs. In the majority of cohort studies, one or more phenotypic variables are quantitative traits. A number of statistical power calculation softwares have been developed for case-control GWA studies . This reGeneral statistical power calculation procedures for quantitative measures often assume normality and require information of the first two distribution moments (means and variances) of the quantitative measure. These power calculation procedures cannot be applied directly in genetic studies. One challenge is that a key parameter, i.e. the size of the hypothesized genetic effect, is represented in a genetic term, "heritability". In its broad sense, heritability is defined as the amount of phenotypic variation in the population that can be ascribed to the genetic variation among individuals. If the total variability of the phenotype is V(t) and the variations that can be explained by genetic variants and error terms are V(g) and V(e), respectively, where V(t) = V(g) + V(e), then the heritability (H) is estimated as: H = V(g)/V(t). Although H is intrinsically related to the distribution of the quantitative measure, the exact mathematical transformation formula is not clear; given the value of H, the first two moments of the quantitative measure cannot be determined directly.However, a power calculation procedure based on heritability is possible for GWA studies, since most applied genetic models in GWA analysis are simple and have only one degree of freedom. Using straightforward algebra, a sound statistical procedure and convenient graphical user interface software have been developed to estimate the statistical power for quantitative traits in GWA studies.2. In general, the ANOVA F-test can be used as the base of power calculations. However, this approach requires the non-centrality parameter λ of the F-distribution, which is defined as:The heritability (H) can be estimated by the Analysis of Variance (ANOVA) approach. For independent random samples drawn from natural populations, H is the coefficient of determination, or Ri is the phenotype mean of genotype group i; σ2 is the error variation; and ri is the number of individuals in each genotype group i. Therefore, distribution parameters such as means and variances are still necessary for this approach.where τd degrees of freedom, then X2 has F distribution . This is true for central and non-central distributions [We provide a solution that avoids assuming the non-centrality parameter in power calculations. In GWA data analysis, some popular genetic models have one degree of freedom including the dominant model, the recessive model and the additive model. When these genetic models are assumed, the one-way ANOVA model (dominant and recessive models) and the simple linear regression model (additive models) are used in the majority of GWA data analyses for quantitative traits. In these studies, the F-statistic is related to Student's t statistic, i.e. if a random variable X has a t distribution with ibutions . This suFor dominant and recessive models, the one-way ANOVA table and the terminologies are described in Table It should be noted that (1) the absolute values of SST and SSM are unknown, as are the absolute values of numerator and denominator of the t-statistic. However, in this approach, only the relative values are necessary to calculate power; (2) the t-test is associated with the ANOVA table. In this study, an ANOVA (and the associated F-test) is used owing to its ability to accommodate the concept of heritability and to adjust for covariate effects. Therefore, if a dominant or a recessive genetic model is considered, the t-test compares two sample means within an ANOVA framework, although an ANOVA t-test is different from a "two sample t-test".GWAPower calculates statistical power using the following input parameters: 1) The heritability (a number in the range 0~1); (2) Total sample size; (3) The number of SNPs in the GWA study; (4) The type 1 error rate: The default value is calculated using a Bonferroni correction. A snapshot of the main page of the software is presented in Figure  The heriGWAPower considers other design factors so that users have the options to explore the power of the study under various situations. The factors include:(1) Covariates: users can input the percentage of the variation explained by the covariates, in addition to the genetic effects and the number of total covariates. The information is included in the ANOVA table and the statistical power is adjusted accordingly, i.e. the degree of freedom of the t-test is N-2-number of covariates;2) between the SNP marker and the hypothesized causal variant. When the LD between the genotyped and "causative" marker is r2, the sample size is increased by 1/r2, because in order to have the same power to detect a causative variant when using a marker that is r2 away, an increase in sample size of 1/r2 is required [(2) Linkage Disequilibrium (LD): users can input the two-locus LD  the statistical power; 2) a family of power curves outputted with different heritability and sample size combinations, as demonstrated in Figure  a familyThe GWAPower package was developed using Microsoft Visual C++ 2008 and consists a main dialog graphical user interface (GUI). Users can download and install the compiled versions of the GUI that run as stand-alone applications under Windows operating systems. A demonstration of the GUI for power calculations with the corresponding parameters setting is presented in Figure The beta-version of GWAPower has been used to support the designs of several GWA studies -11. A GWThe GWAPower package provides a simple and useful statistical power calculation procedure for GWAS with quantitative traits. It is designed specifically to allow genetic researchers to use the genetic term, heritability, instead of the general statistical term, 'phenotype means of each genotype', in power calculations. Current users have reported that the tool is very user-friendly and easy to operate.Although GWAPower is developed for GWA studies, it can be used for general genetic studies if researchers wish to use heritability as the parameter for genetic effect size. However, this approach should be used with caution. Some complicating factors are not considered such as population stratification, various interactions among genetic variants, and environmental factors. Furthermore, Bonferroni correction for multiple tests is known to be over-conservative. Therefore, for complicated designs with multiple confounding factors, researchers are encouraged to consult a statistical geneticist.One reviewer has reported that another software "genetic power calculator" (GPC)  can be uSF and LL derived the method. SF and SW developed the software. SF, SW, CCC and LL wrote the manuscript. All authors read and approved the final manuscript.


Significance of genetic association to a marker has been traditionally evaluated through statistics that are standardized such that their null distributions conform to some known ones. Distributional assumptions are often required in this standardization procedure. Based on the observation that the phenotype remains the same regardless of the marker being investigated, we propose a simple statistic that does not need such standardization. We propose a resampling procedure to assess this statistic’s genome-wide significance. This method has been applied to replicate 2 of the Genetic Analysis Workshop 17 simulated data on unrelated individuals in an attempt to map phenotype Q2. However, none of the selected SNPs are in genes that are disease-causing. This may be due to the weak effect that each genetic factor has on Q2. The traditional approach to hypothesis testing is to construct a test statistic whose exact or approximate distribution under the null hypothesis is known. Alternative methods for assessing significance of a test statistic include resampling techniques, such as the bootstrap and permutation. These methods dominate modern genetic association studies.Statistics used in these methods typically make use of a standardization factor, for instance, the standard error of the observed genetic effect size. Different model assumptions result in different standardization factors. For instance, in a case-control genetic association study, we are interested in testing whether the difference in the frequency of a reference allele between case subjects and control subjects is 0. The popular allelic test and the Cochran-Armitage test for trend are based on two different estimates of the variance of this difference. The estimate for the allelic test requires Hardy-Weinberg equilibrium in the combined case-control population, whereas the estimate for the Cochran-Armitage test does not. The properties of these two tests are rather different .It is a brilliant idea to construct a test statistic through standardization. It makes it possible to assess the significance of the observed effect size with respect to a reference distribution. However, in genome-wide association studies there are typically tens of thousands of markers. Presumably most of these markers are under the null hypothesis . The presence of these null markers provides a natural reference distribution. Standardizing the effect size in this setting appears to be unnecessary and awkward.We propose a novel resampling approach that aims to directly assess the significance of genetic effect size. The null distribution generated from this method is for a randomly selected single-nucleotide polymorphism (SNP). Hence it naturally addresses the genome-wide significance of genetic effect size at a SNP and obviates the issue of multiple testing. To define the genetic effect size, we observe that the phenotype remains the same in a genome scan. Only the testing locations are different. This observation leads to a simple representation of the genetic effect size. This approach is demonstrated by means of a genome-wide association study on phenotype Q2 in replicate 2 of the Genetic Analysis Workshop 17 (GAW17) simulated data on unrelated subjects.n denote the number of individuals in a study. Let iy denote the measure of a quantitative phenotype y on individual i. The score of the genotype at a SNP is denoted by g. Assuming that the two alleles of the SNP are denoted by A and B, we set g = 0 for genotype AA, g = 1 for genotype AB, and g = 2 for genotype BB. The genotype score on individual i is denoted by ig.Let y, we consider a regression in which the response variable is the genotype score g and the independent variable is the phenotype y. The  regression sum of squares for g equals:To identify SNPs associated with phenotype ig and iy, respectively. Because the denominator in expression (1) remains constant in a genome scan, a natural measure of association would be:where p covariates jx, j = 1, …, p, then phenotype iy is replaced by the residual Note that in a case-control study ijx is the value of jx for individual i. Similarly, genotype g is replaced by the residual g* of the regression of g over jx, j = 1, …, p.where We use the statistic:n* is the number of subjects whose genotype is nonmissing at the SNP being investigated. The sample mean of S. So is the sample mean of n* is to make the statistic S on the same scale because n* is expected to vary across the genome.to measure the strength of association, where In comparison, the usual method for detecting association would consider the following regression:β1, denoted The least-squares estimate of The statistic for testing for association is:g is 0 in a multiple linear regression that includes ix, …, px as covariates and y as the response. Because the total sum of squares for T.where S: (1) Compute residuals p-value [S for the randomly selected marker. (5) Repeat steps 2–4 the desired number of times, say K. (6) Let S(k) denote the value of statistic S in the kth iteration. Suppose that the value of statistic S is s for the observed data. Its p-value is computed as:We propose the following resampling procedure to evaluate the genome-wide significance of ion (3).  RandomlyI(·) is the indicator function satisfying I(True) = 1 and I = 0.where Because the SNP selected in each iteration is random, the significance obtained in this way is genome-wide.p-values from the t-distribution with 1 degree of freedom for statistic T are presented in Figure p-value less than 5 × 10−4  shows no apparent deviation from uniform distribution (data not shown). Thus all SNPs are used to determine the reference distribution for statistic S. P-values obtained from the resampling procedure are presented in Figure p-value that is less than 5 × 10−4. None of these SNPs are close to the 19 SNPs that have p-values less than 5 × 10−4 for statistic T, except for C20S2223 on chromosome 20 . C20S2223 is located in gene PRIC285 identified by statistic S.The reference distribution for statistic p-value is assessed through a t-distribution or a standard normal distribution. This strategy works beautifully in general but is awkward in the context of genome-wide association studies. The presence of abundant unassociated SNPs provides a natural reference distribution that is pertinent to the data. We have proposed a simple statistic S and a resampling method for generating its reference distribution. Although a distributional assumption is necessary to compute the standard error of Significance of genetic association at a SNP has been traditionally evaluated through a standardized test statistic such that its significance can be evaluated through a known distribution. For instance, in Eq. (7) the standard error of The essence of the proposed method is the existence of a large number of similar features—SNPs in the current context. The basic principle is applicable to other situations that involve a large amount of similar features, for instance, gene expression levels in gene expression studies.S can be defined as the difference in frequency of the reference allele between the two phenotype groups. More generally, one can use a proportional odds model with genotype as the response and phenotype as the predictor. A proportional odds model is appealing for dealing with covariates and both continuous and dichotomous traits. We are actively investigating this possibility.We have focused on a continuous trait. As alluded to earlier, for a dichotomous trait, if there are no covariates, then the statistic K = 10,000,000 resampling iterations took less than two days on a computer running the Windows Vista operating system. The iteration coverage is about 408 per SNP . We have also used K = 100,000; the simulated p-value remains almost the same. The resampling procedure needs to be done only once, regardless of the number of SNPs.The proposed method is resampling based and requires intensive resampling. However, computation for each sample is fast. The overall computation time for We are investigating the utility of our approach in other association study settings, such as gene-gene interaction and population stratification.We had no knowledge of the GAW17 disease-generating model when this analysis was conducted. It turns out that none of the genes in Table The authors declare that there are no competing interests.KW conceived of the study and participated in its design and coordination and helped to draft the manuscript. JH helped to draft the manuscript. All authors read and approved the final manuscript.


Analysis of expression quantitative trait loci (eQTL) aims to identify the genetic loci associated with the expression level of genes. Penalized regression with a proper penalty is suitable for the high-dimensional biological data. Its performance should be enhanced when we incorporate biological knowledge of gene expression network and linkage disequilibrium (LD) structure between loci in high-noise background.We propose a network-based group variable selection (NGVS) method for QTL detection. Our method simultaneously maps highly correlated expression traits sharing the same biological function to marker sets formed by LD. By grouping markers, complex joint activity of multiple SNPs can be considered and the dimensionality of eQTL problem is reduced dramatically. In order to demonstrate the power and flexibility of our method, we used it to analyze two simulations and a mouse obesity and diabetes dataset. We considered the gene co-expression network, grouped markers into marker sets and treated the additive and dominant effect of each locus as a group: as a consequence, we were able to replicate results previously obtained on the mouse linkage dataset. Furthermore, we observed several possible sex-dependent loci and interactions of multiple SNPs.The proposed NGVS method is appropriate for problems with high-dimensional data and high-noise background. On eQTL problem it outperforms the classical Lasso method, which does not consider biological knowledge. Introduction of proper gene expression and loci correlation information makes detecting causal markers more accurate. With reasonable model settings, NGVS can lead to novel biological findings. Genetic loci that affect the expression levels of mRNA are called expression quantitative trait loci (eQTL). Considering mRNA transcript abundance as a quantitative trait, the aim is to detect the associated genetic loci, which is the key to understanding the regulation network and disease phenotype. Thanks to the high-throughput and advanced sequencing technology, genome-wide linkage and association studies ,2 have s1 penalty produces interpretable models with some coefficients exactly 0. Two of its extensions are appealing. The Group Lasso or GLasso [2 penalty to each group of variables. The elastic net method [1 and L2 penalties together, becomes ideal for " large p small n" problems with highly correlated data. However these excellent Lasso-based methods [The simplest mapping ideas are regression-based methods, but traditional methods have some disadvantages. Single QTL regression and the interval mapping method  tend to r GLasso  on the ot method , on the  methods ,11 are nPrevious studies have demonstrated that incorporating biological information on genes with the same function would increase the accuracy of detection of hot spots -14. SincBesides the gene expression network, correlations between markers or linkage disequilibrium (LD) structure are very informative. The true causal SNPs are rarely genotyped but may be in LD with near markers. In addition the epistatic effects among different SNPs can be very large, with each individual SNP' s effect very low. In these cases we need complex models rather than linear ones to describe the LD structure. Wu et al. proposed to group SNPs into SNP sets based on LD structure in the association study, and then test the joint effect of each SNP set . We applY1, Y2,..., YG, where Yg = T, g = 1,..., G and we combine them to form the entire gene expression vector p markers can be divided into J blocks describing the J marker sets, where the jth marker set for the ith individual is xj, i = , j = 1,2,..., J, p = p1+p2+ · · · +pJ being the total marker number. Then the marker data matrix is XM = , where Xj = T. We combine them to get the entire marker data matrix X = diag . Note that all the G traits come from the same genotype data and the marker data matrix is the same for all traits. We then regress all gene traits Y on the marker data X. After the location and scale transformation, we can assume that the regressors are standardized and each response is centered, obviating the need to consider the intercepts.Suppose that the dataset has n samples and p markers. We have G quantitative gene traits E edges and G vertices. Each Vertex represents a trait, an edge u~v indicates that gene trait u and v are linked on the network. Let' s define the degree u dof the vertex u as the total number of edges linked to u; and suppose u > 0 dfor each u. To describe the structure of the network, we use matrix L similar to [p by p block element L of L is defined as:Let' s consider a network that is represented by a graph with milar to . The p bIp is the identity matrix of order p. Since L is always non-negative definite, it can be decomposed as L = SST, where SGp × Ep is the matrix in which, taking every p by p matrix as one block, the block rows are indexed by the vertices and block columns are indexed by the edges of the graph such that each block column corresponding to an edge u~v has an entry u, an entry v and zero elsewhere.where 1 λand 2λ, we define our network-based group variable selection (NGVS) criterion:For any pair of fixed non-negative tuning parameters β = T, βg = T, βj, g = T and the norm is L2 norm. The first term is the sum of squared errors. The second term is identical to the GLasso penalty only with an additional sum over gene traits and is used for selecting marker sets. The weights where u~v denotes the sum over all unordered pairs  for which u and v are linked on the network. The third term actually reveals the assumption that genes which are highly correlated and truly regulated by the same QTLs tend to have the same effect. The NGVS estimator where ΣThe following lemma shows that minimizing our NGVS criterion is equivalent to solving a GLasso-type optimization problem, thus can be computed by some efficient existing algorithms.LEMMA 1. Given dataset  and two fixed tuning parameters , define an artificial dataset  bywhere S is the decomposition of L. Let and Then the NGVS criterion can be written asLet be the solution to the above GLasso minimization problem; then the solution to (2) is 2 due to the possible bias of double shrinkage. From Lemma 1, the NGVS problem can be reformulated as an equivalent GLasso problem by augmenting the dataset from Gn to Gn+Ep. Therefore, when doing variable selection, this model can select all Gp variables if Gn+Ep > Gp. GLasso can only select at most Gn variables before it saturates. By choosing a network with the total number of edges bigger than G(p-n)/p, even when n is much smaller than p, we can overcome the limitation. This can be easily accomplished by using a smaller correlation threshold or making the network sufficiently big.Following Zou and Hastie , the NGVLEMMA 2. is determined by Equation (2). Assume that gene u and v are only linked with each other on the network and the corresponding response vectors are equal, i.e. Yu = Yv, then for any λ2 >0 where is the estimated coefficients for gene g.2 λ> 0. This lemma shows the grouping effect of NGVS, which means that coefficients corresponding to highly correlated gene traits on the network tend to be the same. Therefore, our method can borrow information from traits with the same underlying function.Lemma 2 is true since the penalty is a strictly convex function with Some algorithms are available for solving the GLasso problem. Yuan and Lin provided an iterative algorithm , but theThe key idea of BCGD method is to combine a quadratic approximation of the log-likelihood with an additional line search. We first pick a zero vector as the initial coefficient vector, denoting no groups have been selected. Then by approximating the nonlinear log-likelihood by a second-order Taylor expansion at β of the last iteration and replacing the Hessian of the log-likelihood by a proper matrix, the minimization direction is found and β is updated by a point of that direction. Thus, either a new group will be selected, or the coefficients of previously selected groups will be changed slightly. The algorithm is fast in computing a whole range of solutions given sufficiently small grid on penalization parameters and then generating the selection order. The algorithm is available in the R-package grplasso.Biological information incorporated by our proposed NGVS method mainly include gene expression network and loci correlation, that is, the way to form marker sets. Proper grouping of markers based on the prior knowledge can increase the power to detect causal SNPs, while bad division of marker sets may probably harm the results since the unlinked loci may dilute the effect of causal loci. Basically, all grouping ideas can be divided into three categories: LD-based, knowledge-based, and convenience-based. In GWAS, grouping SNPs in or near a gene is an ideal method; while in linkage analysis, because of the limited number of markers, grouping highly correlated markers produces good results. Genes that are located within a gene pathway often share biological functions and could be considered as a group. A more detailed analysis about how marker sets are formed can be found in .The gene expression network, if properly set, could come from any source such as GO or KEGG pathways ,16, clus2, we are able to generate a selection order of the marker sets for a wide range of choices of λ1. We call this the big scale selection order as it describes the ranking of importance for the groups of markers. Once the relative importance of marker sets is established, further selection order of markers within each marker set can be produced by various existing methods. We call this the small scale selection order. To generate the final selection order of individual QTLs, we try to combine the two different scales together. Hence we need to go over a three-stage procedure: firstly getting the big scale selection order with NGVS; then finding the small scale selection order by any method suitable for low-dimensional selection; finally, combining the two selection orders together according to the three criteria discussed below.With our NGVS method, for each fixed λp or GCV criterion, it is time-consuming for two-dimensional tuning parameters. Based on our experiments, when considering big scale ranking, the results are quite stable against different 2λ's. So we use 2 λ= 10 in our analysis and for this given 2λ, we let 1 λvary over a wide range of grid to give the big scale selection order. The step size of 1 λshould be small enough to guarantee that at most one new marker set is selected at a time. In the second stage, we have reduced the problem to be a " large p small n" variable selection problem. So all methods designed for low-dimensional ranking should be suitable, though we prefer to use the GLasso, which can select additive and dominant effects as one group. The small scale selection orders within each marker set are obtained without considering the loci structure and the co-expression network. This is because small scale differences of each gene trait are allowed. Furthermore, for a low-dimensional problem, simple selection methods are accurate enough to detect QTLs and considering the network may lead to bias (see the first simulation). Once the selection orders in two scales are ready, we apply three criteria to combine them in the final stage. Firstly, the most significant loci in each marker set are ranked according to the big scale selection order with NGVS; secondly, loci within each marker set are ranked according to the small scale selection order with GLasso; thirdly, when several loci satisfy the first two criteria, the locus with the smallest p-value for single QTL regression should be selected ahead of the others. Here, single QTL regression means assessing the significance of each individual SNP using the likelihood ratio test. The final selection order of all the markers will be determined uniquely by these three criteria. The final order is a combination of macro-order based on prior biological information, micro-order within each small group and single QTL p-values ranking. If we want to detect the causal QTLs of a certain trait, we can identify as significant the first desired number of loci in the final selection order.In the first stage, we face a " large p small n" variable selection problem. Though we can select the optimal parameters by Cross-validation or some kind of CHowever, if we care more about general findings for a class of gene traits, we should pick out, according to the big scale order, marker sets which are identified as significant in most traits, then form the final selection orders and make conclusions only using markers in these identified marker sets. The whole process is shown in Figure Four assumptions of our method are listed here: (1) the distribution of the error term is normal; (2) markers that are in high LD regions together reflect more information than one single marker; (3) highly correlated traits tend to be determined by the same loci; (4) covariate, additive, dominant and epistatic effects should be assumed based on some prior knowledge. Covariates such as age and sex sometimes are quite influential for gene expressions, hence cannot be ignored. If one covariate is significant, typically, we add it into the regression model as one additional group, together with the interactions of the covariate with markers. As for the additive and dominant effect, we assume that one SNP has only additive effect or both effects. Models containing two effects can provide more accurate QTL detection than the additive models as is shown by our results.th ggene trait and its corresponding th jmarker set, then by representer theorem [j, g hof markers within the marker set can be written as:Epistatic effect refers to the interaction of multiple genetic variants. However, how this joint activity really happens is hard to detect. We propose two possible ways to describe epistasis - kernel model and near interaction model. If we focus on only the  theorem , the relK is some kernel function defining the epistatic relationship. Thus we extend the original regression model for the gth trait Yg = 0βXg+εg to be:where Yg = 0βXg+εg, but with an increased degree of freedom when j<np. Note that kernel function is applied to each marker set. Thus with the same division of marker sets, we can treat kernels as J×n correlated new variables and get the big scale selection order as before. Essentially, kernel function projects nonlinear relationship into a higher dimensional space and the regression is then modelled linearly in the new space. The kernel can also be intuitively interpreted as the measure of similarity between two individuals. After this representation, we need only to introduce different kernels to specify the epistatic model and here we present five kernels we will use in our analysis.For example, the linear kernel(1) Linear Kernel: (2) Polynomial Kernel: (3) Gaussian Kernel: (4) Identical-by-state (IBS) Kernel: (5) Weighted IBS (WIBS) Kernel: jk qis the minor allele frequency (MAF) for the kth marker in the jth group. The first kernel is linear while the second adds the interactions of SNPs to the qth order. The next two kernels basically give various ways to measure the distance between two individuals. And the WIBS kernel is a correction of IBS for the rare alleles because they are usually more informative than common alleles. [where alleles.  offers mDespite the many choices of kernels, it can be advantageous to switch to traditional two-locus interactions because, if showen to be significant, the biological interpretation is easier. We can extend each marker set to contain all the interactions between SNP pairs in the set and treat each of them as one new variable. However, when the number of SNPs in one marker set is large, we tend to only add near interactions. In high LD regions, we may consider interactions of two loci a little farther apart, while in low LD regions or with SNPs not densely genotyped, interactions of adjacent markers are enough. We will consider the adjacent-locus interaction model together with different kernel models in our analysis.To evaluate the performance of our proposed NGVS method, we simulated two models: the first to illustrate the advantages and disadvantages of the method in a low-dimensional setting; the second to mimic the high dimensional real problem.Z1,..., Z7 denoting genotypes according to a centered multivariate normal distribution whose covariances were 1, Z 2) = 0.8 Cov = 0.4|i-j| Cov≠. Then i Zwas trichotomized as -1, 1, 0 if it is smaller than Φ-1(1/4), larger than Φ-1(3/4) or in between respectively, representing genotype aa, AA, Aa. And we considered the linear model:First, we generated seven latent variables 1, i, Z2, i, Z3, i Zwere one realization of 1, Z2, Z3 Zfor individual i; i, g's Ydenoting the gene expressions determined by additive and dominant effects of loci 1, 2, 3, which were modelled as two dummy variables - one for genotype AA and one for Aa; the expression network of two linked genes was also considered, i.e. g = 1, 2; i, j, g~U, i = 1,2,3, j = 1,2 rwas a scaling factor used to perturb the effect size of the marker on trait g; finally εi=  ~ N where Σij = 0.5σiσj for i ≠ j, Σii= σi2 and Σ was determined by our choice of the signal-to-noise ratio (SNR), which is defined as the expected value of the mean square over the variance of expressions. Two different SNRs of 1 and 5 and two different ways to form the marker sets were tried. For each case, 50 simulated datasets were generated independently to calculate sensitivity and specificity.where 1 Zand 2 Zor four corresponding dummy variables are grouped into one marker set; and the two dummy variables for each of the other 5 loci build up the other 5 marker sets. This division for marker sets assumes that we have some prior knowledge about the higher correlation between 1 Zand 2Z.We call this marker set formation with SNR = 1 and SNR = 5 model 1 and 2 respectively. The second way to construct sets is to group 1, Z2 Zand 4 Zor six corresponding dummy variables into the first marker set; 3 Zand 5 Zare grouped into the second one and 6Z, 7 Zin the third one. This division represents a bad set formation because every significant locus is tangled with some insignificant one. We call this division with SNR = 1 and SNR = 5 model 3 and 4 respectively.The first way to construct marker sets is that We compared three methods: 1) our proposed NGVS which combines gene expression network and loci structure; 2) GLasso which scans that information and merely selects additive and dominant effects simultaneously trait by trait; (3) the traditional Lasso method which only considers additive model. In NGVS, the selection order was constructed as illustrated above. And in the other two methods, selection orders were obtained by applying a wide range of different tuning parameters. Sensitivity and specificity of identifying the first three loci in the final selection order as significant loci are reported in Table  our prop GLasso wFrom Table Z1,..., Z200 according to a centered multivariate normal distribution. The covariances are set to decrease as the distance between markers increases and 0 when the markers are more than 10 markers apart. Like before, i's Zwere trichotomized as -1, 0, 1. We considered the following model:In the second simulation, we considered a simulated dataset including 60 samples, each with the data of 200 markers, or 400 dummy variables, and 5 gene traits, to mimic a real linkage analysis. We first generated genotypes denoted by 3, i, Z4, i, Z13, i, Z27, i Zbelong to three different marker sets 1,2,3; i, j, g, εi, g rare defined as before and SNR = 5. Twenty simulated datasets were generated independently to calculate True Positives (TPs) and False Positives (FPs). Our main aim here is to find out the causal markers 3, Z4, Z13 Zand 27Z.where E>G(p-n)/p. For the marker set formation, r markers rj-r+1,..., rj were grouped to form the jth marker set, j = 1,...,(200/r) and r = 1, 2, 5, 10. Note that r = 1 means we do not actually have a marker set and select the QTLs individually. In brief, we can either choose to use the gene expression network or not and choose among 4 different marker set formations - a total of 8 possibilities. When the network was utilized, we applied our proposed NGVS method under the 4 different marker set formations. If the network was not taken into account, we considered each individual gene trait respectively, but still maintained the 4 different marker set structures. We call this single trait selection, which means loci structure was taken into account but the QTLs were selected for each trait individually. In single trait selection, if r = 1, it is just GLasso used to select two dummy variables of one locus as a group for each trait. Besides the 8 possibilities, we also compared the selection orders of the single QTL regression and the Lasso. Single QTL regression assumed linear simple regression and tested whether the slope was significantly different from zero by likelihood ratio test. The selection order came from the ranking of p-values. Note that our method is a combination of the big scale NGVS, the small scale GLasso and the p-values coming from single QTL regression as the adhesive tool of the two scales.We applied our three-stage selection procedure. We first selected significant marker sets using our NGVS method based on the additive-and-dominant-effect model; then GLasso was used to find the causal markers within each marker set; finally we decided the final selection order for all the loci. The full network of the 5 gene traits considered here obviously satisfies our requirement r = 2 performed the best. The effect of the scale of marker sets is shown through the first 4 columns of Figure r = 10 and r = 5, the 5 gene network protected NGVS from suffering the power decrease generated by containing too many unlinked loci into marker sets. Borrowing information among correlated gene traits reduces the risk of using improper division of marker sets. Comparing NGVS with GLasso, we conclude that adding a proper loci grouping and gene expression network indeed can improve the performance. Single QTL regression and the GLasso are probably useful for coarse ranking, but not good enough. In addition, the comparison between NGVS with Lasso illustrated the importance of considering the additive and dominant effects together. Methods lacking the description of the latent two-effect mechanism can only select markers of strong effect. In summary, when the sample size is smaller than the marker size, our method considering two-effect model, marker sets and correlated traits together can discover more casual QTLs of moderate effect and give more true positives.The total TPs and FPs of the 5 gene traits of the 20 simulated datasets if we identified the first k loci in the final selection orders as significant are shown in Figure 2 mice generated by crossing strain C57BL/6J (B6) with BTBR. Lan et al. found that B6-ob/ob mice, when made obese, are resistant to diabetes while BEBR-ob/ob mice are not. Then the 60 animals were genotyped at 192 MIT microsatellite markers, an average of approximately 10 cM apart across the entire genome. The liver mRNA was quantified by Affymetrix M430A and B arrays. The dataset was processed using the robust multi-array average (RMA) normalization method [We analyzed a published mouse linkage dataset deposited at the gene expression omnibus (GEO) by Lan et al. . This dan method . Previoun method ,14. Lan In our analysis, we considered two ways to form marker sets: loci within one chromosome as a marker set and loci within the boundaries where significant correlation decrease happens as a marker set. The smaller marker set formation was shown by the black squares in Figure D2Mit297, D2Mit241, D2Mit9) on Chr 2, or loci at around 30 cM, were significantly linked with GPCR genes. This region was identified as the most significant by 9 of the 16 traits in the first co-expression network under chromosome marker set scale and by all except one under smaller marker set scale. There were also weak signals that loci on Chr 10 at 40 cM have effect on the expression levels of some genes. The second co-expression network did not generate very consistent results among traits, because the 17 genes used were not highly correlated. However, we can still identify the 30 cM region on Chr 2 by 13 of the 17 genes in the top 4 selected loci with the smaller marker set division. The marker sets constructed by correlations can be treated as a more detailed division of the marker sets formed by chromosomes. It is clear that marker set formation by correlations generate more consistent and convincing results than naively setting markers within each chromosome as a group. However, in this real data case, due to the high correlations, even marker set formation by chromosomes may improve the results. Additive model and single trait selection gave specious results (not shown). The successful reproduction of existing results proved the effectiveness of NGVS.The results were consistent with Lan et al. . AccordiAnkrd24, Sfrp1, Ergic1 each connected with the other two. The following six models were formulated: (1) the additive linear model; (2) the two-effect linear model; (3) the additive linear model including adjacent interactions between markers; (4) the two-effect linear model including adjacent interactions between markers; (5) the two-effect sex-dependent model treating sex as one additional group; (6) the additive model with 5 different kernels - linear, polynomial (q = 2), Gaussian (d = 1), IBS and WIBS.In the above analysis, we only considered linear models. Covariate and epistatic effects based on different assumptions are discussed below. To simplify the analysis, we only considered a co-expression network of trait All the big scale selection orders are shown in Table D2Mit241, D2Mit9) on Chr 2 and marker 107 (D10Mit20) on Chr 10. The region marked by D2Mit241 and D2Mit9 was obviously hot spot for those GPCR genes. Under two-effect adjacent interaction model 4 with loci on Chr 2 and 10, we went further to discover the significant epistatic effects by treating each interaction term as one variable and applying GLasso to genes in the first co-expression network. We found that 9 of the 16 gene traits exhibited the epistatic effect between marker 15 and 16 (D2Mit297 and D2Mit241) on Chr 2; 15 of the 16 gene traits showed the effect between marker 106 and 107 (D10Mit148 and D10Mit20) on Chr 10. It is interesting that the most significant epistatic effects occurred together with their additive and dominant effects. Under two-effect sex-dependent model 5 with loci on Chr 10 and 11, we tried to detect the sexual distinction. For Chr 10, there was no significant evidence for difference between males and females. However for Chr 11, interaction of sex with marker 118 (D11Mit99) was identified by 13 of the 16 gene traits. So we believe that D11Mit99 denotes a region which has a regulation mechanism related to sex. The results above still need further biological study.In order to get general findings, we picked out marker sets Chr 2 and Chr 10, as they were identified as significant by all the 3 traits under most of the model assumptions. Then we made final selection orders for the 3 traits only using loci on Chr 2 and Chr 10. We identified marker 16, 17 (We have proposed a penalized regression method called the network-based group variable selection. The basic idea of our method is along the ongoing efforts to incorporate prior biological knowledge into data analysis. In eQTL, we hope to combine information from both the correlated gene expression traits and the loci structure ,17,18. B2 and make λ1 vary with a sufficiently small step to generate the selection order. When the network is complex and the number of SNPs is large, our method is quite expensive. The storage of high-dimensional matrix is also a problem for eQTL.However, the method has some limitations. First of all, the method is designed for high-dimensional biological data such as linkage analysis or genome-wide association study, thus it is not very effective for low-dimensional selection problems. Our method is especially powerful for high-dimensional and very noisy data. In addition, combining more information means longer computation time and larger storage space. Though our method is powerful for detecting causal SNPs with moderate or weak effect, we need to try different tuning parameters λD2Mit297 &D2Mit241 and between D10Mit148 &D10Mit20 are significant; and sex may have some effect on the expressions through marker D11Mit99 on Chr 11. Although all these conclusions need to be tested by additional research, it is clear the NGVS has the power and flexibility to handle high-dimensional problems with high-noise data successfully.We applied our method to two simulations and one real linkage dataset to demonstrate the capability of the NGVS. Simulation one compared three methods for a low-dimensional model setting and we concluded that our method is suitable for problems with high-noise background. Simulation two mimicked the real linkage data. It showed that considering the proper loci grouping, the co-expression network and the additive and dominant effects simultaneously is essential for obtaining convincing results. Under the framework of our method, we also considered many different models including kernels, interactions, and covariates in the real data analysis. All the results led to the co-regulatory regions on Chr 2, 10 for GPCR genes, which replicated the findings of Lan et al. . FurtherThe authors declare that they have no competing interests.WW initiated the project, invented the NGVS method, completed the simulation experiments and the linkage data analysis, and drafted the manuscript. XZ provided advice for important intellectual content and revised the manuscript. All authors read and approved the final manuscript.


In experiments with many statistical tests there is need to balance type I and type II error rates while taking multiplicity into account. In the traditional approach, the nominal  Development of novel statistical methods to accompany technological advances in biological sciences is essential in order to further scientific progress. Human genetics is a prime example of a field where data is currently being produced at an accelerating rate. The sheer volume of data and its complexity create new challenges for statistical science. How to design studies appropriately, and how not to drown in the data while extracting useful signals? Studies now routinely involve multiple testing of statistical hypotheses on a massive scale. Thus, both design and analysis of large-scale experiments must entail considerations of multiplicity.Design of studies often involves sample size and power calculations. In exploratory studies where many statistical tests are anticipated, it is reasonable to assume that most test statistics and their respective P-values are generated under the condition where the null hypothesis holds. One would also assume that there are multiple real signals. In reality, effect sizes (e.g. odds ratios) for these true signals follow some distribution, although it might be tempting to assume, for simplicity, a single “typical” effect size. In multiple testing contexts, statistical power can be defined as the probability of correctly rejecting at least one  of true signals among, for example, the first ten top results?We will seek a simple and practical way with a minimum number of assumptions to allow for such computations.In the simulation experiment to construct For the next motivating example, we considered a genome-wide association study (GWAS) with Power to detect any particular associated variant at the Power to detect at least one of the 75 true signals is 0.19, at the More revealing are calculations based on the ranks of true signals:Probability that the best hit  given byFor continuously distributed effect sizes, Gamma the true (continuous) effect size distribution; (2) the distribution reduced to three bins  allows to detect less than one half of the actual number of true signals. Doubling the sample size makes the number of discovered signals to rise considerably more sharply, as the number of true signals increases. The combined sample (15000 cases and 15000 controls) allows to detect nearly all true signals.Classical counterparts to ranking and posterior probabilities are statistical power and P-values. Power is calculated for a fixed significance threshold while in our ranking approach it is the number of top hits that is fixed. “Fixed” in this context simply means not-random: it is possible to vary the number of top hits by incrementing Ranking probabilities, rFDR, as well as standard power calculations when multiple tests are anticipated, all depend on the effect size distribution and the number of true signals. Specification of these parameters can be challenging. Here, we investigated whether the usage of the mean of the effect size distribution would provide acceptable precision. Wacholder et al. P-value distribution functions, https://sites.google.com/site/entrust2013/. The programs can also be used to choose a sample size such that a desired number of true signals would rank among the top R programs to compute the expected number of true (truly associated) signals among top